**Information Capacity of Quantum Transfer Channels and Thermodynamic Analogies**

Bohdan Hejna

36 Will-be-set-by-IN-TECH

[19] Kauzmann, W. (1959) Some factors in the interpretation of protein denaturation. *Adv.*

[20] Lazaridis, T., and Karplus, M. (2003) Thermodynamics of protein folding: a microscopic

[21] Lee, B., and Richards, F. M. (1971) The interpretation of protein structures: estimation of

[22] Novotny, J., Bruccoleri, R., and Karplus, M. (1984) An analysis of incorrectly folded protein models. Implications for structure predictions. *J. Mol. Biol.* 177, 787-818.

[24] Reynolds, J. A., Gilbert, D. B., and Tanford, C. (1974) Empirical correlation between hydrophobic free energy and aqueous cavity surface area. *Proc. Natl. Acad. Sci. USA*

[25] Richards, F. M. (1977) Areas, volumes, packing, and protein structure. *Ann. Rev. Biophys.*

[26] Rose, G. D., Fleming, P. J., Banavar, J. R., and Maritan, A. (2006) A backbone based theory

[27] Tuñón, I., Silla, E., and Pascual-Ahuir, J. L. (1992) Molecular surface area and

[28] Wang, Y., Zhang, H., Li, W., and Scott, R. A. (1995) Discriminating compact nonnative structures from the native structure of globular proteins. *PNAS*, 92, 709-713. [29] Wu, H. (1931) Studies on denaturation of proteins XIII. A theory of denaturation. *Chinese Journal nof Physiology* 4, 321-344 (1931). A preliminary report was read before the XIIIth International Congress of Physiology at Boston, Aug. 19-24, 1929 and published in the *Am. J Physiol.* for Oct. 1929. Reprinted in *Advances in Protein Chemicsty* 46, 6-26 (1995).

[23] Popelier, P. (2000) *Atoms in Molecules: An Introduction*. (Prentice Hall).

hydrophobic effect. Protien Engineering 5(8), 715-716.

*Protein Chem.* 14, 1-63 (1959).

71(8), 2925-2927.

*Bioeng.* 6, 151-176.

of protein folding.

*PNAS* 103(45), 16623-16633.

view. *Biophysical Chemistry*, 100, 367-395.

static accessibility. *J. Mol. Biol.* 55:379-400.

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50466

## **1. Introduction**

We will begin with a simple type of *stationary stochastic*<sup>1</sup> systems of quantum physics using them within a frame of the **Shannon** Information Theory and Thermodynamics but starting with their *algebraic representation*. Based on this algebraic description a model of information transmission in those systems by defining the Shannon information will be stated in terms of variable about the system state. Measuring on these system is then defined as a spectral decomposition of measured quantities - *operators*. The information capacity formulas, now of the *narrow-band* nature, are derived consequently, for the simple system governed by the **Bose–Einstein** (B–E) Law [bosonic (photonic) channel] and that one governed by the **Fermi-Dirac** (F–D) Law [fermionic (electron) channel]. *The not-zero value for the average input energy needed for information transmission existence in F–D systems* is stated [11, 12].

Further the *wide–band* information capacity formulas for B–E and F–D case are stated. Also the original *thermodynamic* capacity derivation for the wide–band photonic channel as it was stated by **Lebedev–Levitin** in 1966 is revised. This revision is motivated by apparent relationship between the B–E (photonic) wide–band information capacity and the *heat efficiency* for a certain *heat cycle*, being further considered as the demonstrating model for processes of information transfer in the original wide–band photonic channel. The information characteristics of a *model reverse* heat cycle and, by this model are analyzed, the information arrangement of which is set up to be most analogous to the structure of the photonic channel considered, we see the necessity of returning the transfer medium (the channel itself) to its initial state as a condition for a *sustain, repeatable* transfer. It is not regarded in [12, 30] where a single information transfer act only is considered. Or the return is

*t*=*t*0+1 about a stochastic system. If these probabilities do not depend on the beginning *t*0, a stationary stochastic system is spoken about.

©2012 Hejna, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

<sup>1</sup> We deal with such a system which is taking on at time *t* = 0, 1, . . . states *θ<sup>t</sup>* from a state space **Θ**. If for any *t*<sup>0</sup> the relative frequencies *IB* of events *<sup>B</sup>* <sup>⊂</sup> **<sup>Θ</sup>** is valid that <sup>1</sup> *T t*0+*T* ∑ *IB*(*θt*) tends for *T* → ∞ to probabilities *pt*<sup>0</sup> (*B*) we speak

regarded, but by opening the whole transfer chain for covering these *return energy needs from its environment* not counting them in - *not within the transfer chain only* as we do now. *The result is the corrected capacity formula for wide–band photonic (B–E) channels being used for an information transfer organized cyclicaly*.

### **2. Information transfer channel**

An *information*, *transfer* channel K is defined as an arranged *tri–partite* structure [5]

$$\mathcal{K} \stackrel{\text{Def}}{=} [X, \mathfrak{e}, Y] \text{ where } X \stackrel{\text{Def}}{=} [A, \ p\_X(\cdot)], \text{ } Y \stackrel{\text{Def}}{=} [\mathcal{B}, \ p\_Y(\cdot)] \text{ and} \tag{1}$$

The *information capacity* of the channel K (both discrete an continuous) is defined by the

over all possible probability distributions *q*(·), *p*(·).It is the maximum (supremum) of the medium value of the usable amount of information about the input message *x* within the

**Remark:** For continuous distributions (densities) *<sup>p</sup>*[·](·), *<sup>p</sup>*[·|·](·|·) on intervals <sup>X</sup> , Y ∈ **<sup>R</sup>**,

Equations (4), (5 are valid for both the quantities *H*(·) and *H*(·|·), as well as for their respective

The most simple way of description of stationary physical systems is an *eucleidian* space Ψ of

Physical quantities *α*, associated with a physical system **Ψ** represented by the Eucleidian space Ψ are expressed by *symmetric* operators from the linear space *L*(Ψ) of operators on Ψ, *α* ∈ **A** ⊂ *L*(Ψ) [7]. The supposition is that any physical quantity can achieve only those *real* values *α* which are the *eigenvalues* of the associated *symmetric operator α* (symmetric *matrix* [*αi*,*j*]*n*,*n*, *n* = dim Ψ). They are elements of the *spectrum* **S**(*α*) of the *operator α*. The eigenvalues *α* ∈ **S**(*α*) ⊂ **R** of the quantity *α* being measured on the system **Ψ** ∼= Ψ depend on the (inner) states *θ* of

• The *pure states* of the system **Ψ** ∼= Ψ are represented by *eigenvectors ψ* ∈ Ψ. It is valid that the *scalar project* (*ψ*, *ψ*) = 1; in *quantum physics* they are called normalized wave functions. • The *mixed states* are nonnegative quantities *θ* ∈ **A**; their *trace* [of an *square* matrix (operator)

The *symmetric projector π*{*ψ*} = *π*[Ψ({*ψ*})] (orthogonal) on the one–dimensional *subspace* Ψ({*ψ*}) of the space Ψ is nonnegative quantity for which Tr(*π*{*ψ*}) = 1 is valid. The projector *π*{*ψ*} *represents* [on the set of quantities **A** ⊂ *L*(Ψ)] the pure state *ψ* of the system **Ψ**. Thus, an arbitrary state of the system **Ψ** can be defined as a nonnegative quantity *θ* ∈ **Θ** ⊂ **A** for which Tr(*θ*) = 1 is valid. For the *pure state θ* is then valid that *θ*<sup>2</sup> = *θ* and the state space **Θ** of the

= sup *I*(*X*;*Y*) (6)

Information Capacity of Quantum Transfer Channels and Thermodynamic Analogies 85

<sup>X</sup> *pX*(*x*)ln *pX* <sup>d</sup>*x*, (7)

<sup>Y</sup> *pY*(*y*)ln *pY*(*y*) <sup>d</sup>*<sup>y</sup>* (8)

<sup>2</sup> This way enables the *mathematical* formulation of the

*αi*,*<sup>i</sup>* and, for *α* ≡ *θ* is valid that Tr(*θ*) = 1. (9)

<sup>Y</sup> *pX*,*Y*(*x*, *<sup>y</sup>*)ln *pX*|*Y*(*x*|*y*) <sup>d</sup>*x*d*y*,

<sup>Y</sup> *pX*,*Y*(*x*, *<sup>y</sup>*)ln *<sup>p</sup>*(*y*|*x*) <sup>d</sup>*x*d*<sup>y</sup>*

*<sup>C</sup>* Def

*H*(*X*) = −

*H*(*Y*) = −

*H*(*X*|*Y*) = −

*H*(*Y*|*X*) = −

**3. Representation of physical transfer channels**

term (*physical*) *state* and, generally, the term (*physical*) *quantity*.

changes Δ*H*(·)[= *H*(·)] and Δ*H*(·|·)[= *H*(·|·)].

their states expressed as *linear operators*.

Tr(*α*) Def = *n* ∑ *i*=1

system **Ψ** is defined as the set of all states *θ* of the system **Ψ**.

<sup>2</sup> The motivation is the *axiomatic theory of algebraic representation of physical systems* [7].

 X 

 X 

equation

*output* message *y*.

*x* ∈ X , *y* ∈ Y is

this system **Ψ**.

*α*] is defined





[the maximal probability of erroneously receiving *y* ∈ *B* (inappropriate for *x*) in an output message *b* ∈ Y],





The structure (*X*, K, *Y*) or (X , K, Y) is termed a *transfer (Shannon) chain*. The symbols *H*(*X*) and *H*(*Y*) respectively denote the *input information (Shannon) entropy* and the *output information (Shannon) entropy* of channel K, *discrete* for this while,

$$H(\mathbf{X}) \stackrel{\text{Def}}{=} - \sum\_{\mathbf{x} \in A} p\_{\mathbf{X}}(\mathbf{x}) \ln p\_{\mathbf{X}}(\mathbf{x}) \prime \quad H(\mathbf{Y}) \stackrel{\text{Def}}{=} - \sum\_{\mathbf{y} \in B} p\_{\mathbf{Y}}(\mathbf{y}) \ln p\_{\mathbf{Y}}(\mathbf{y}) \tag{2}$$

The symbol *H*(*X*|*Y*) denotes the *loss entropy* and the symbol *H*(*Y*|*X*) denotes the *noise entropy* of channel K. These entropies are defined as follows,

$$H(X|Y) \stackrel{\text{Def}}{=} -\sum\_{A} \sum\_{B} p\_{X,Y}(\mathbf{x}, y) \ln p\_{X|Y}(\mathbf{x}|y), \quad H(Y|X) \stackrel{\text{Def}}{=} -\sum\_{A} \sum\_{B} p\_{X,Y}(\mathbf{x}, y) \ln p\_{Y|X}(y|\mathbf{x}) \tag{3}$$

where the symbol *<sup>p</sup>*·|·(·|·) denotes the *condition* and the symbol *<sup>p</sup>*·,·(·, ·) denotes the *simultaneous* probabilities. For *mutual (transferred) usable* information, *transinformation T*(*X*;*Y*) or *T*(*Y*; *X*) is valid that

$$T(\mathbf{X};\mathbf{Y}) = H(\mathbf{X}) - H(\mathbf{X}|\mathbf{Y}) \text{ and } T(\mathbf{Y};\mathbf{X}) = H(\mathbf{Y}) - H(\mathbf{Y}|\mathbf{X}) \tag{4}$$

From (2) and (3), together with the definitions of *<sup>p</sup>*·|·(·|·) and *<sup>p</sup>*·,·(·, ·), is provable prove that the transinformation is symmetric. Then the *equation of entropy (information) conservation* is valid

$$H(X) - H(X|Y) = H(Y) - H(Y|X) \tag{5}$$

The *information capacity* of the channel K (both discrete an continuous) is defined by the equation

$$\mathbb{C} \stackrel{\text{Def}}{=} \sup I(X;Y) \tag{6}$$

over all possible probability distributions *q*(·), *p*(·).It is the maximum (supremum) of the medium value of the usable amount of information about the input message *x* within the *output* message *y*.

**Remark:** For continuous distributions (densities) *<sup>p</sup>*[·](·), *<sup>p</sup>*[·|·](·|·) on intervals <sup>X</sup> , Y ∈ **<sup>R</sup>**, *x* ∈ X , *y* ∈ Y is

$$H(\mathbf{X}) = -\int\_{\mathcal{X}} p\_{\mathbf{X}}(\mathbf{x}) \ln p\_{\mathbf{X}} \, d\mathbf{x},\tag{7}$$

$$H(Y) = -\int\_{\mathcal{Y}} p\_Y(y) \ln p\_Y(y) \, dy \tag{8}$$

$$\begin{aligned} H(X|Y) &= -\int\_{\mathcal{X}} \int\_{\mathcal{Y}} p\_{X,Y}(\mathbf{x}, \mathbf{y}) \ln p\_{X|Y}(\mathbf{x}|\mathbf{y}) \, \mathbf{dx} d\mathbf{y} \\ H(Y|X) &= -\int\_{\mathcal{X}} \int\_{\mathcal{Y}} p\_{X,Y}(\mathbf{x}, \mathbf{y}) \ln p(y|\mathbf{x}) \, \mathbf{dx} d\mathbf{y} \end{aligned}$$

Equations (4), (5 are valid for both the quantities *H*(·) and *H*(·|·), as well as for their respective changes Δ*H*(·)[= *H*(·)] and Δ*H*(·|·)[= *H*(·|·)].

#### **3. Representation of physical transfer channels**

2 Will-be-set-by-IN-TECH

regarded, but by opening the whole transfer chain for covering these *return energy needs from its environment* not counting them in - *not within the transfer chain only* as we do now. *The result is the corrected capacity formula for wide–band photonic (B–E) channels being used for an information*

<sup>=</sup> [*A*, *pX*(·)] , *<sup>Y</sup>* Def



[the maximal probability of erroneously receiving *y* ∈ *B* (inappropriate for *x*) in an output


*pX*(*x*) ln *pX*(*x*), *<sup>H</sup>*(*Y*) Def

*pX*,*Y*(*x*, *<sup>y</sup>*) ln *pX*|*<sup>Y</sup>* (*x*|*y*), *<sup>H</sup>*(*Y*|*X*) Def

The symbol *H*(*X*|*Y*) denotes the *loss entropy* and the symbol *H*(*Y*|*X*) denotes the *noise entropy*

where the symbol *<sup>p</sup>*·|·(·|·) denotes the *condition* and the symbol *<sup>p</sup>*·,·(·, ·) denotes the *simultaneous* probabilities. For *mutual (transferred) usable* information, *transinformation T*(*X*;*Y*)

From (2) and (3), together with the definitions of *<sup>p</sup>*·|·(·|·) and *<sup>p</sup>*·,·(·, ·), is provable prove that the transinformation is symmetric. Then the *equation of entropy (information) conservation*

= − ∑ *y*∈*B*

> = − ∑ *A* ∑ *B*

*H*(*X*) − *H*(*X*|*Y*) = *H*(*Y*) − *H*(*Y*|*X*) (5)

*T*(*X*;*Y*) = *H*(*X*) − *H*(*X*|*Y*) and *T*(*Y*; *X*) = *H*(*Y*) − *H*(*Y*|*X*) (4)

= [*B*, *pY*(·)] and (1)

= X , a *transceiver*,

= Y, a *receiver*,

*pY*(*y*) ln *pY*(*y*) (2)

*pX*,*Y*(*x*, *<sup>y</sup>*)ln *pY*|*X*(*y*|*x*) (3)

An *information*, *transfer* channel K is defined as an arranged *tri–partite* structure [5]

= [*X*, *<sup>ε</sup>*, *<sup>Y</sup>*] where *<sup>X</sup>* Def




*transfer organized cyclicaly*.

message *a* ∈ X

message *b* ∈ Y],

*<sup>H</sup>*(*X*|*Y*) Def

is valid

= − ∑ *A* ∑ *B*

or *T*(*Y*; *X*) is valid that

**2. Information transfer channel**

received by the receiver of messages, *Y*,

*<sup>H</sup>*(*X*) Def

*(Shannon) entropy* of channel K, *discrete* for this while,

= − ∑ *x*∈*A*

of channel K. These entropies are defined as follows,

<sup>K</sup> Def

The most simple way of description of stationary physical systems is an *eucleidian* space Ψ of their states expressed as *linear operators*. <sup>2</sup> This way enables the *mathematical* formulation of the term (*physical*) *state* and, generally, the term (*physical*) *quantity*.

Physical quantities *α*, associated with a physical system **Ψ** represented by the Eucleidian space Ψ are expressed by *symmetric* operators from the linear space *L*(Ψ) of operators on Ψ, *α* ∈ **A** ⊂ *L*(Ψ) [7]. The supposition is that any physical quantity can achieve only those *real* values *α* which are the *eigenvalues* of the associated *symmetric operator α* (symmetric *matrix* [*αi*,*j*]*n*,*n*, *n* = dim Ψ). They are elements of the *spectrum* **S**(*α*) of the *operator α*. The eigenvalues *α* ∈ **S**(*α*) ⊂ **R** of the quantity *α* being measured on the system **Ψ** ∼= Ψ depend on the (inner) states *θ* of this system **Ψ**.

• The *pure states* of the system **Ψ** ∼= Ψ are represented by *eigenvectors ψ* ∈ Ψ. It is valid that the *scalar project* (*ψ*, *ψ*) = 1; in *quantum physics* they are called normalized wave functions.

• The *mixed states* are nonnegative quantities *θ* ∈ **A**; their *trace* [of an *square* matrix (operator) *α*] is defined

$$\operatorname{Tr}(\mathfrak{a}) \stackrel{\text{Def}}{=} \sum\_{i=1}^{n} \mathfrak{a}\_{i,i} \quad \text{and, for } \mathfrak{a} \equiv \mathfrak{e} \text{ is valid that } \operatorname{Tr}(\mathfrak{e}) = 1. \tag{9}$$

The *symmetric projector π*{*ψ*} = *π*[Ψ({*ψ*})] (orthogonal) on the one–dimensional *subspace* Ψ({*ψ*}) of the space Ψ is nonnegative quantity for which Tr(*π*{*ψ*}) = 1 is valid. The projector *π*{*ψ*} *represents* [on the set of quantities **A** ⊂ *L*(Ψ)] the pure state *ψ* of the system **Ψ**. Thus, an arbitrary state of the system **Ψ** can be defined as a nonnegative quantity *θ* ∈ **Θ** ⊂ **A** for which Tr(*θ*) = 1 is valid. For the *pure state θ* is then valid that *θ*<sup>2</sup> = *θ* and the state space **Θ** of the system **Ψ** is defined as the set of all states *θ* of the system **Ψ**.

<sup>2</sup> The motivation is the *axiomatic theory of algebraic representation of physical systems* [7].

#### **3.1. Probabilities and information on physical systems**

#### 3.1.0.1. Theorem:

For any state *θ* ∈ **Θ** the pure states *θ<sup>i</sup>* = *π*{*ψi*} and numbers *q*(*i*|*θ*) ≥ 0 exists, that

$$q(i|\boldsymbol{\theta}) \ge 0, \; \boldsymbol{\theta} = \sum\_{i=1}^{n} q(i|\boldsymbol{\theta}) \; \boldsymbol{\theta}\_i \quad \text{where} \quad \sum\_{i=1}^{n} q(i|\boldsymbol{\theta}) = 1 \tag{10}$$

• The special case of the *p-distribution* is that *for measuring values of α*≡*θ*, *θ* ∈ **Θ**,

*<sup>p</sup>*(*α*|*α*|*θ*) = ∑

∑ S(*α*)

*α* Tr(*θπα*) = Tr

physical entropy H(*θ*) of the system **Ψ** *in the state θ* is defined by the equality

<sup>H</sup>(*θ*) Def

*E*(*α*) = ∑

*θ*ln *θ* = ∑

*n* ∑ *i*=1

*d*-distribution for the state *θ*, *n* = dim Ψ .

*<sup>I</sup>*[*p*(·|*θ*|*θ*) � *<sup>d</sup>*(·|*θ*)] = ∑

*θ*∈**S**(*θ*)

For a physical system **Ψ** in any state *θ* ∈ **Θ** is valid that

*α*∈**S**(*α*)

is valid

that

(*αψi*, *ψi*) = *αii*.

3.1.0.3. Theorem:

3.1.0.4. Proof:

H(*θ*) = −

It is the case of *measuring the θ itself*. Thus, by the equation (14), the term *measuring* **means just the spectral decomposition of the unit operator 1 within the spectral relation with** *α* [4, 20]. The act of measuring of the quantity *α* ∈ **A** in (the system) state *θ* gives the value *α* from the spectrum **S**(*α*) with the probability *p*(*α*|*α*|*θ*) = Tr(*θπα*) given by the *p*-distribution,

S(*α*)

*απα*

⎞

*θ*∈**S**(*θ*)

= ln *n* − *I*[*p*(·|*θ*|*θ*) � *d*(·|*θ*)] (21)

*<sup>θ</sup>*dim <sup>Ψ</sup>(*θ*|*θ*) · ln *<sup>n</sup><sup>θ</sup>* · dim <sup>Ψ</sup>(*θ*|*θ*)

dim Ψ(*θ*|*θ*)

*θ*dim Ψ(*θ*|*θ*) · ln *n* − H(*θ*) = ln *n* − H(*θ*) (22)

*θ*∈**S**(*θ*)

The *measured* **quantity** *α* **in the state** *θ* **is a** *stochastic* **quantity with its values [occuring with probabilities** *p*(·|*α*|*θ*)**] from its spectrum S**(*α*). For its *mathematical expectation, medium value*

Nevertheless, in the pure state *θ* = *π*{*ψ*} the values *i* = *α* ∈ **S**(*α*) are measured, Tr(*θiα*) =

Let **Ψ** is an *arbitrary* stationary physical system and *θ* ∈ **Θ** ⊂ **A** is its arbitrary state. The

When {*π<sup>θ</sup>* : *θ* ∈ **S**(*θ*)} is the decomposition of **1** spectral equivalent with *θ*, then it is valid

*<sup>θ</sup>* ln *<sup>θ</sup> <sup>π</sup><sup>θ</sup>* and H(*θ*) = − ∑

*<sup>q</sup>*(*i*|*θ*) · ln *<sup>q</sup>*(*i*|*θ*) = *<sup>H</sup>*[*q*(·|*θ*)] = ln *<sup>n</sup>* − ∑

where *H*(·) is the Shannon entropy, *I*(·�·) is the information divergence, *p*(·|*θ*|*θ*) is the *p*-distribution for the state *θ*, *q*(·|*θ*) is the *q*-distribution for the state *θ* and *d*(·|*θ*) is the

The relations in (21) follows from (20) and from definition (10) of the distribution *q*(·|*θ*). From

definitions (13) and (16) of the other two distributions follows that [12, 38]

*θ*∈**S**(*θ*)

= ∑ *θ*∈**S**(*θ*) ⎛ <sup>⎝</sup>*<sup>θ</sup>* ∑ *α*∈**S**(*α*)

*p*(*θ*|*θ*|*θ*) = Tr(*θπ<sup>θ</sup>* ) = *θ*dim Ψ(*θ*|*θ*), *θ* ∈ **S**(*θ*) (16)

Information Capacity of Quantum Transfer Channels and Thermodynamic Analogies 87

Tr(*θπα*) = 1 (17)

⎠ = Tr(*θα*)=(*αψ*, *ψ*) (18)

*θ* ln *θ* · dim Ψ(*θ*|*θ*) (20)

*<sup>p</sup>*(*θ*|*θ*|*θ*) · ln *<sup>p</sup>*(*θ*|*θ*|*θ*)

*d*(*θ*|*θ*)

= −Tr(*θ*ln *θ*) (19)

3.1.0.2. Proof:

Let *D*(*θ*) = {*D<sup>θ</sup>* : *θ* ∈ **S**(*θ*)} is a *disjoint decomposition* of the set {1, 2, ..., *n*} of indexes of the base {*ψ*1, *ψ*2, ..., *ψn*} of Ψ. The set {*ψ<sup>i</sup>* : *i* ∈ *Dθ*} is an orthogonal basis of the *eigenspace* Ψ(*θ*|*θ*) ⊂ Ψ of the operator *θ* [for its eigenvalue *θ* ∈ **S**(*θ*)]. Then

$$\text{card } D\_{\theta} = \dim \mathbb{P}(\theta | \theta) \ge 1 \text{ and } \; \pi[\mathbb{1}(\theta | \theta)] = \sum\_{i \in D\_{\theta}} \pi\{\psi\_{i}\}, \; \forall \theta \in \mathbb{S}(\theta) \tag{11}$$

Let *q*(*i*|*θ*) = *θ* is taken for all *i* ∈ *D<sup>θ</sup>* , *θ* ∈ **S**(*θ*). By the *spectral decomposition theorem* [9],

$$\theta = \sum\_{\theta \in \mathbf{S}(\theta)} \theta \,\, \pi\_{\theta} = \sum\_{\theta \in \mathbf{S}(\theta)} \theta \,\pi[\mathbf{Y}(\theta|\theta)] = \sum\_{\theta \in \mathbf{S}(\theta)} \theta \sum\_{i \in D\_{\theta}} \pi\{\psi\_{i}\} = \sum\_{i=1}^{n} q(i|\theta) \,\pi\{\psi\_{i}\} \tag{12}$$

$$\text{Tr}(\theta) = \sum\_{i=1}^{n} q(i|\theta) \cdot \text{Tr}(\pi\{\psi\_{i}\}) = \sum\_{i=1}^{n} q(i|\theta) = 1$$

The symbol *q*(·|*θ*) denotes the *probability distribution* into *pure, canonic components θ<sup>i</sup>* of *θ* is called:

• canonic distribution (*q*-distribution) of the state *θ* ∈ **Θ**. 3

Further the two distribution defined on spectras of *α* ∈ **A** and *θ* ∈ **Θ** ⊂ **A** will be dealt:

• dimensional distribution (*d*-distribution) of the state *<sup>θ</sup>* <sup>∈</sup> **<sup>Θ</sup>**<sup>4</sup>

$$d(\theta|\boldsymbol{\theta}) \stackrel{\text{Def}}{=} \frac{\dim \mathbb{Y}(\boldsymbol{\theta}|\boldsymbol{\theta})}{\dim \mathbb{Y}} = \frac{\dim \mathbb{Y}(\boldsymbol{\theta}|\boldsymbol{\theta})}{n}, \quad \boldsymbol{\theta} \in \mathbf{S}(\boldsymbol{\theta}) \tag{13}$$

• distribution of measuring (*p*-distribution) of the quantity *α* ∈ **A** in the state *θ* ∈ **Θ**,

$$p(\mathfrak{a}|\mathfrak{a}|\mathfrak{e}) \stackrel{\text{Def}}{=} \text{Tr}(\theta \pi\_{\mathfrak{a}}), \quad \mathfrak{a} \in \mathbf{S}(\mathfrak{a}) \tag{14}$$

where {*π<sup>α</sup>* : *α* ∈ **S**(*α*)} is the *spectral decomposition* of the *unit operator* **1**. Due the nonnegativity of *θπ<sup>α</sup>* is *p*(*α*|*α*|*θ*) = Tr(*απα*) ≥ 0. By spectral decomposition of **1** and by definition of the *trace* Tr(·) is

$$\sum\_{\mathfrak{a}\in\mathsf{S}(\mathfrak{a})} p(\mathfrak{a}|\mathfrak{a}|\mathfrak{g}) = \operatorname{Tr}\left(\theta \sum\_{\mathfrak{a}\in\mathsf{S}(\mathfrak{a})} \pi\_{\mathfrak{a}}\right) = \operatorname{Tr}(\theta \mathbf{1}) = 1 \tag{15}$$

Thus the relation (14) defines the probability distribution on the spectrum of the operator **S**(*α*).

<sup>3</sup> Or, the *system spectral distribution* SSD [11].

<sup>4</sup> Or, the *system distribution* of the system *spectral dimension* SDSD [11].

• The special case of the *p-distribution* is that *for measuring values of α*≡*θ*, *θ* ∈ **Θ**,

$$p(\theta|\theta|\theta) = \text{Tr}(\theta \pi\_{\theta}) = \theta \text{dim}\,\Psi(\theta|\theta), \quad \theta \in \mathbf{S}(\theta) \tag{16}$$

It is the case of *measuring the θ itself*. Thus, by the equation (14), the term *measuring* **means just the spectral decomposition of the unit operator 1 within the spectral relation with** *α* [4, 20]. The act of measuring of the quantity *α* ∈ **A** in (the system) state *θ* gives the value *α* from the spectrum **S**(*α*) with the probability *p*(*α*|*α*|*θ*) = Tr(*θπα*) given by the *p*-distribution,

$$\sum\_{\mathcal{S}(a)} p(a|\mathfrak{a}|\mathfrak{e}) = \sum\_{\mathcal{S}(a)} \text{Tr}(\theta \pi\_{\mathfrak{a}}) = 1 \tag{17}$$

The *measured* **quantity** *α* **in the state** *θ* **is a** *stochastic* **quantity with its values [occuring with probabilities** *p*(·|*α*|*θ*)**] from its spectrum S**(*α*). For its *mathematical expectation, medium value* is valid

$$E(\mathfrak{a}) = \sum\_{\mathfrak{a} \in \mathbf{S}(\mathfrak{a})} \mathfrak{a} \operatorname{Tr}(\mathfrak{\theta} \pi\_{\mathfrak{a}}) = \operatorname{Tr}\left(\mathfrak{\theta} \sum\_{\mathfrak{a} \in \mathbf{S}(\mathfrak{a})} \mathfrak{a} \pi\_{\mathfrak{a}}\right) = \operatorname{Tr}(\mathfrak{\theta}\mathfrak{a}) = (\mathfrak{a}\psi, \psi) \tag{18}$$

Nevertheless, in the pure state *θ* = *π*{*ψ*} the values *i* = *α* ∈ **S**(*α*) are measured, Tr(*θiα*) = (*αψi*, *ψi*) = *αii*.

Let **Ψ** is an *arbitrary* stationary physical system and *θ* ∈ **Θ** ⊂ **A** is its arbitrary state. The physical entropy H(*θ*) of the system **Ψ** *in the state θ* is defined by the equality

$$\mathcal{H}(\boldsymbol{\theta}) \stackrel{\text{Def}}{=} -\text{Tr}(\boldsymbol{\theta} \ln \boldsymbol{\theta}) \tag{19}$$

When {*π<sup>θ</sup>* : *θ* ∈ **S**(*θ*)} is the decomposition of **1** spectral equivalent with *θ*, then it is valid that

$$\theta \ln \theta = \sum\_{\theta \in \mathbf{S}(\theta)} \theta \ln \theta \,\, \pi\_{\theta} \text{ and } \, \mathcal{H}(\theta) = -\sum\_{\theta \in \mathbf{S}(\theta)} \theta \ln \theta \cdot \dim \mathbb{Y}(\theta|\theta) \tag{20}$$

3.1.0.3. Theorem:

4 Will-be-set-by-IN-TECH

*q*(*i*|*θ*) *θ<sup>i</sup>* where

Let *D*(*θ*) = {*D<sup>θ</sup>* : *θ* ∈ **S**(*θ*)} is a *disjoint decomposition* of the set {1, 2, ..., *n*} of indexes of the base {*ψ*1, *ψ*2, ..., *ψn*} of Ψ. The set {*ψ<sup>i</sup>* : *i* ∈ *Dθ*} is an orthogonal basis of the *eigenspace*

Let *q*(*i*|*θ*) = *θ* is taken for all *i* ∈ *D<sup>θ</sup>* , *θ* ∈ **S**(*θ*). By the *spectral decomposition theorem* [9],

*<sup>θ</sup>π*[Ψ(*θ*|*θ*)] = ∑

*q*(*i*|*θ*) · Tr(*π*{*ψi*}) =

*θ*∈**S**(*θ*)

The symbol *q*(·|*θ*) denotes the *probability distribution* into *pure, canonic components θ<sup>i</sup>* of *θ* is

dim <sup>Ψ</sup> <sup>=</sup> dim <sup>Ψ</sup>(*θ*|*θ*)

where {*π<sup>α</sup>* : *α* ∈ **S**(*α*)} is the *spectral decomposition* of the *unit operator* **1**. Due the nonnegativity of *θπ<sup>α</sup>* is *p*(*α*|*α*|*θ*) = Tr(*απα*) ≥ 0. By spectral decomposition of **1** and by definition of the

Thus the relation (14) defines the probability distribution on the spectrum of the operator

*πα* ⎞

⎛ <sup>⎝</sup>*<sup>θ</sup>* ∑ *α*∈**S**(*α*)

Further the two distribution defined on spectras of *α* ∈ **A** and *θ* ∈ **Θ** ⊂ **A** will be dealt:

• distribution of measuring (*p*-distribution) of the quantity *α* ∈ **A** in the state *θ* ∈ **Θ**,

*n* ∑ *i*=1

*θ* ∑ *i*∈*D<sup>θ</sup>*

*n* ∑ *i*=1

*i*∈*D<sup>θ</sup>*

*q*(*i*|*θ*) = 1

3

*π*{*ψi*} =

*n* ∑ *i*=1

*<sup>n</sup>* , *<sup>θ</sup>* <sup>∈</sup> **<sup>S</sup>**(*θ*) (13)

⎠ = Tr(*θ***1**) = 1 (15)

= Tr(*θπα*), *α* ∈ **S**(*α*) (14)

*q*(*i*|*θ*) = 1 (10)

*π*{*ψi*}, ∀*θ* ∈ **S**(*θ*) (11)

*q*(*i*|*θ*) *π*{*ψi*} (12)

For any state *θ* ∈ **Θ** the pure states *θ<sup>i</sup>* = *π*{*ψi*} and numbers *q*(*i*|*θ*) ≥ 0 exists, that

*n* ∑ *i*=1

**3.1. Probabilities and information on physical systems**

*q*(*i*|*θ*) ≥ 0, *θ* =

Ψ(*θ*|*θ*) ⊂ Ψ of the operator *θ* [for its eigenvalue *θ* ∈ **S**(*θ*)]. Then

*θ π<sup>θ</sup>* = ∑

Tr(*θ*) =

*θ*∈**S**(*θ*)

*n* ∑ *i*=1

• canonic distribution (*q*-distribution) of the state *θ* ∈ **Θ**.

*<sup>d</sup>*(*θ*|*θ*) Def

∑ *α*∈**S**(*α*)

<sup>4</sup> Or, the *system distribution* of the system *spectral dimension* SDSD [11].

<sup>3</sup> Or, the *system spectral distribution* SSD [11].

• dimensional distribution (*d*-distribution) of the state *<sup>θ</sup>* <sup>∈</sup> **<sup>Θ</sup>**<sup>4</sup>

<sup>=</sup> dim <sup>Ψ</sup>(*θ*|*θ*)

*<sup>p</sup>*(*α*|*α*|*θ*) Def

*p*(*α*|*α*|*θ*) = Tr

card *<sup>D</sup><sup>θ</sup>* = dim <sup>Ψ</sup>(*θ*|*θ*) ≥ 1 and *<sup>π</sup>*[Ψ(*θ*|*θ*)] = ∑

3.1.0.1. Theorem:

3.1.0.2. Proof:

called:

*trace* Tr(·) is

**S**(*α*).

*θ* = ∑ *θ*∈**S**(*θ*)

For a physical system **Ψ** in any state *θ* ∈ **Θ** is valid that

$$\begin{split} \mathcal{H}(\boldsymbol{\theta}) &= -\sum\_{i=1}^{n} q(i|\boldsymbol{\theta}) \cdot \ln q(i|\boldsymbol{\theta}) = H[q(\cdot|\boldsymbol{\theta})] = \ln n - \sum\_{\boldsymbol{\theta} \in \mathbf{S}(\boldsymbol{\theta})} p(\boldsymbol{\theta}|\boldsymbol{\theta}|\boldsymbol{\theta}) \cdot \ln \frac{p(\boldsymbol{\theta}|\boldsymbol{\theta}|\boldsymbol{\theta})}{d(\boldsymbol{\theta}|\boldsymbol{\theta})} \\ &= \ln n - I[p(\cdot|\boldsymbol{\theta}|\boldsymbol{\theta}) \parallel d(\cdot|\boldsymbol{\theta})] \end{split} \tag{21}$$

where *H*(·) is the Shannon entropy, *I*(·�·) is the information divergence, *p*(·|*θ*|*θ*) is the *p*-distribution for the state *θ*, *q*(·|*θ*) is the *q*-distribution for the state *θ* and *d*(·|*θ*) is the *d*-distribution for the state *θ*, *n* = dim Ψ .

#### 3.1.0.4. Proof:

The relations in (21) follows from (20) and from definition (10) of the distribution *q*(·|*θ*). From definitions (13) and (16) of the other two distributions follows that [12, 38]

$$\begin{split} I[p(\cdot|\theta|\theta) \parallel d(\cdot|\theta)] &= \sum\_{\theta \in \mathbf{S}(\theta)} \theta \text{dim } \mathbb{Y}(\theta|\theta) \cdot \ln \frac{n\theta \cdot \text{dim } \mathbb{Y}(\theta|\theta)}{\text{dim } \mathbb{Y}(\theta|\theta)} \\ &= \sum\_{\theta \in \mathbf{S}(\theta)} \theta \text{dim } \mathbb{Y}(\theta|\theta) \cdot \ln n - \mathcal{H}(\theta) = \ln n - \mathcal{H}(\theta) \end{split} \tag{22}$$

Due to the quantities *X*, *Y*, *X*|*Y*, *Y*|*X* describing the information transfer are in our algebraic description denoted as follows, *X* � = *θ*, *Y* � = *α* or *Y* � = (*α*�*θ*), (*X*|*Y*) � = (*θ*|*α*), (*Y*|*X*) � = (*α*|*θ*). The laws of information transfer are writable in this way too:

$$\mathbb{C} = \sup\_{\mathfrak{a}, \mathfrak{b}} I(\mathfrak{a}; \mathfrak{o}), \quad I(\cdot || \cdot) \equiv T(\cdot; \cdot) \tag{23}$$

$$I(\mathfrak{o}; \mathfrak{a}) = I(\mathfrak{a}; \mathfrak{o}) = \mathcal{H}(\mathfrak{o}) - H(\mathfrak{o} | \mathfrak{a}) = H(\mathfrak{a} \| \mathfrak{b}) - H(\mathfrak{a} | \mathfrak{o}) = \mathcal{H}(\mathfrak{o}) + H(\mathfrak{a} \| \mathfrak{b}) - H(\mathfrak{o}, \mathfrak{a})$$

$$H(\mathfrak{o}, \mathfrak{a}) = H(\mathfrak{a}, \mathfrak{o}) = \mathcal{H}(\mathfrak{o}) + H(\mathfrak{a} | \mathfrak{o}) = H(\mathfrak{a} \| \mathfrak{b}) + H(\mathfrak{b} | \mathfrak{a}) \tag{24}$$

## **4. Narrow-band quantum transfer channels**

Let the symmetric operator *ε* of energy of quantum particle is considered, the spectrum of which eigenvalues *ε<sup>i</sup>* is **S**(*ε*). Now the *equidistant* energy levels are supposed. In a pure state *θ<sup>i</sup>* of the measured (observed) system **Ψ** the eigenvalue *ε<sup>i</sup>* = *i*·*ε*, *ε* > 0. Further, the *output* quantity *α* of the *observed* system **Ψ** is supposed (the system is *cell of the phase space B–E or F–D*) with the spectrum of eigenvalues **S**(*α*) = {*α*0, *α*1, ..., } being measured with probability distribution Pr(·) = {*p*(0), *p*(1), *p*(2), ...}

$$p(\mathfrak{a}\_k|\mathfrak{a}|\mathfrak{e}\_l) = \begin{cases} p(k-i) \text{ pro } k \ge i \\ 0 & k < i \end{cases} \tag{25}$$

observed on the place where the *output* message is *decoded* - on the *output of the channel* **Ψ** ∼= K are arrangable in such a way that in a given *i*-th pure state *θ<sup>i</sup>* of the system **Ψ** ∼= K only the values *α<sup>k</sup>* ∈ S(*α*), *k* = *i* + *j* are measurable and, that the probability of measuring the *k*-th value is Pr(*j*) = *Pr*(*k* − *i*). This probability distribution describes the *additive noise* in the given channel **Ψ** ∼= K. Just it is this noise which creates observed values from the output spectrum

The pure states with energy level *ε<sup>i</sup>* = *i* · *ε* are achievable by sending *i* particles with energy *ε* of each. When the environment, through which these particles are going, generates a bundle of *j* particles with probability Pr(*j*) then, with the same probability the energy *εi*+*<sup>j</sup>* = *k*·*ε* is

It is supposed, also, the infinite number of states *θi*, infinite spectrum **S**(*α*) of the measured

The narrow–band, memory-less (quantum) channel, additive (with additive noise) operating

Let now *<sup>θ</sup><sup>i</sup>* ≡ *<sup>i</sup>*, *<sup>i</sup>* = 0, 1, ... are pure states of a system **<sup>Ψ</sup>**B−E,*<sup>ε</sup>* ∼= K and let *<sup>α</sup>* is output quantity

Thus the distribution *p*(·|*α*|*θi*) is determined by the *forced* (inner–input) state *θ<sup>i</sup>* = *π*{*ψi*} representing the coded input energy at the value *ε<sup>i</sup>* = *i* · *ε* ∈ **S**(*ε*), *ε* = const. For the *medium*

For the medium value of the number of particles *j* = *α* − *i* ≥ 0 with B–E statistics is valid

*α*∈**S**(*α*)

∑ *i*=0

where *p*(*α*|*α*|*θ*) is the probability of measuring the eigenvalue *α* = *k* of the output variable *α*.

*j pj*−<sup>1</sup> = (<sup>1</sup> <sup>−</sup> *<sup>p</sup>*) · *<sup>p</sup>* · <sup>d</sup>

*n* ∑ *i*=0

∞ ∑ *j*=0

*E*(*α*) = ∑

*q*(*i*|*θ*) · *p*(*α*|*α*|*θi*)=(1 − *p*) ·

= **S**, *α<sup>k</sup>*

*<sup>p</sup>*(*α*|*α*|*θi*)=(<sup>1</sup> <sup>−</sup> *<sup>p</sup>*) *<sup>p</sup>α*−*<sup>i</sup>* (30)

*i* · *q*(*i*|*θ*) = *E*(*θ*), *W* = *ε* · W (31)

d*p*

*<sup>q</sup>*(*i*|*θ*) *<sup>θ</sup><sup>i</sup>* of the system **<sup>Ψ</sup>**B−E,*<sup>ε</sup>* ∼= K

 1 1 − *p*

*α* · *p*(*α*|*α*|*θ*) (32)

 <sup>=</sup> *<sup>p</sup>* 1 − *p*

*<sup>q</sup>*(*i*|*θ*) · *<sup>p</sup>α*−*<sup>i</sup>* [= Tr(*θπα*)] (33)

K*<sup>ε</sup>* = {[S, *q*(*i*|*θ*)], *p*(*α*|*α*|*θi*), [S, *p*(*α*|*α*|*θ*)]}. (29)

Information Capacity of Quantum Transfer Channels and Thermodynamic Analogies 89

� = *α*].

S(*α*) = {*αi*, *αi*<sup>+</sup>1, ...} being the selecting space of the stochastic quantity *α*.

quantity *<sup>α</sup>*, then **<sup>S</sup>**(*α*) = {0, 1, 2, ...}, **<sup>S</sup>**(*α*) = *<sup>D</sup>*(*θ*); [**S**(*α*) �

**4.1. Capacity of Bose–Einstein narrow–band channel**

W =

*<sup>j</sup>* · (<sup>1</sup> <sup>−</sup> *<sup>p</sup>*) *<sup>p</sup><sup>j</sup>* = (<sup>1</sup> <sup>−</sup> *<sup>p</sup>*) · *<sup>p</sup>* ·

This probability is defined by the state *<sup>θ</sup>* <sup>=</sup> *<sup>n</sup>*

*n* ∑ *i*=0

*p*(*α*|*α*|*θ*) =

∞ ∑ *i*=0

The quantity *E*(*α*) is the medium value of the output quantity *α* and

The quantity *W* is the medium value of the energy coding the input signal *i*.

taking on in the state *θ<sup>i</sup>* values *α* ∈ **S** with probabilities

on the energy level *ε* ∈ S(*ε*) is defined by the tri–partite structure (1),

decoded on the output.

*value* W of the input *i* is valid

∞ ∑ *j*=0

Such a situation arises when a particle with energy *ε<sup>i</sup>* is *excited by an impact* from the output environment. The jump of energy level of the impacted particle is from *ε<sup>i</sup>* up to *εi*+*j*, *i* + *j* = *k*. The output *εi*+*<sup>j</sup>* for the excited particle is measured (it is the value on the output of the channel K ∼= **Ψ**. This transition *j* occurs with the probability distribution

$$\Pr(j), \ j \in \{0, 1, 2, \dots\} \tag{26}$$

Let be considered the *narrow–band* systems (with one *constant* level of a particle energy) **Ψ** of B–E or F–D type [27] (denoted further by **Ψ**B−E,*ε*, **Ψ**F−D,*ε*).

• In the B–E system, bosonic, e.g. the photonic gas the B–E distribution is valid

$$\Pr(j) = (1 - p) \cdot p^j, \quad j \in \{0, 1, \ldots\}, \ p \in (0, 1), \ p^{-\#} \tag{27}$$

• In the F–D system, fermionic, e.g. electron gas the F–D distribution is valid

$$\Pr(j) = \frac{p^j}{1+p'} \quad j \in \{0, 1\}, \ p \in (0, 1), \ p^{-\frac{\xi}{16}} \tag{28}$$

where parameter *p* is variable with *absolute temperature* Θ > 0; k is the **Boltzman** constant.

Also a collision with a bundle of *j* particles with constant energies *ε* of each and absorbing the energy *j* · *ε* of the bundle is considerable. E.g., by **Ψ** (e.g. **Ψ**B−E,*<sup>ε</sup>* is the photonic gas) the monochromatic impulses with amplitudes *i* ∈ S are transferred, nevertheless generated from the environment of the same type but at the temperature *TW*, *TW* > *T*<sup>0</sup> where *T*<sup>0</sup> is the temperature of the transfer system **Ψ** ∼= K (the *noise* temperature).

It is supposed that both pure states *θ<sup>i</sup>* in the place where the input message is being *coded* - on the *input of the channel* **Ψ** ∼= K and, also, the measurable values of the quantity *α* being observed on the place where the *output* message is *decoded* - on the *output of the channel* **Ψ** ∼= K are arrangable in such a way that in a given *i*-th pure state *θ<sup>i</sup>* of the system **Ψ** ∼= K only the values *α<sup>k</sup>* ∈ S(*α*), *k* = *i* + *j* are measurable and, that the probability of measuring the *k*-th value is Pr(*j*) = *Pr*(*k* − *i*). This probability distribution describes the *additive noise* in the given channel **Ψ** ∼= K. Just it is this noise which creates observed values from the output spectrum S(*α*) = {*αi*, *αi*<sup>+</sup>1, ...} being the selecting space of the stochastic quantity *α*.

The pure states with energy level *ε<sup>i</sup>* = *i* · *ε* are achievable by sending *i* particles with energy *ε* of each. When the environment, through which these particles are going, generates a bundle of *j* particles with probability Pr(*j*) then, with the same probability the energy *εi*+*<sup>j</sup>* = *k*·*ε* is decoded on the output.

It is supposed, also, the infinite number of states *θi*, infinite spectrum **S**(*α*) of the measured quantity *<sup>α</sup>*, then **<sup>S</sup>**(*α*) = {0, 1, 2, ...}, **<sup>S</sup>**(*α*) = *<sup>D</sup>*(*θ*); [**S**(*α*) � = **S**, *α<sup>k</sup>* � = *α*].

The narrow–band, memory-less (quantum) channel, additive (with additive noise) operating on the energy level *ε* ∈ S(*ε*) is defined by the tri–partite structure (1),

$$\mathcal{K}\_{\varepsilon} = \{ [\mathcal{S}, q(i|\theta)], \ p(a|\mathfrak{a}|\theta\_{i}), [\mathcal{S}, \ p(a|\mathfrak{a}|\theta)] \}. \tag{29}$$

#### **4.1. Capacity of Bose–Einstein narrow–band channel**

6 Will-be-set-by-IN-TECH

Due to the quantities *X*, *Y*, *X*|*Y*, *Y*|*X* describing the information transfer are in our algebraic

= *α* or *Y* �

*I*(*θ*; *α*) = *I*(*α*; *θ*) = H(*θ*) − *H*(*θ*|*α*) = *H*(*α*�*θ*) − *H*(*α*|*θ*) = H(*θ*) + *H*(*α*�*θ*) − *H*(*θ*, *α*) *H*(*θ*, *α*) = *H*(*α*, *θ*) = H(*θ*) + *H*(*α*|*θ*) = *H*(*α*�*θ*) + *H*(*θ*|*α*) (24)

Let the symmetric operator *ε* of energy of quantum particle is considered, the spectrum of which eigenvalues *ε<sup>i</sup>* is **S**(*ε*). Now the *equidistant* energy levels are supposed. In a pure state *θ<sup>i</sup>* of the measured (observed) system **Ψ** the eigenvalue *ε<sup>i</sup>* = *i*·*ε*, *ε* > 0. Further, the *output* quantity *α* of the *observed* system **Ψ** is supposed (the system is *cell of the phase space B–E or F–D*) with the spectrum of eigenvalues **S**(*α*) = {*α*0, *α*1, ..., } being measured with probability

Such a situation arises when a particle with energy *ε<sup>i</sup>* is *excited by an impact* from the output environment. The jump of energy level of the impacted particle is from *ε<sup>i</sup>* up to *εi*+*j*, *i* + *j* = *k*. The output *εi*+*<sup>j</sup>* for the excited particle is measured (it is the value on the output of the channel

Let be considered the *narrow–band* systems (with one *constant* level of a particle energy) **Ψ** of

where parameter *p* is variable with *absolute temperature* Θ > 0; k is the **Boltzman** constant. Also a collision with a bundle of *j* particles with constant energies *ε* of each and absorbing the energy *j* · *ε* of the bundle is considerable. E.g., by **Ψ** (e.g. **Ψ**B−E,*<sup>ε</sup>* is the photonic gas) the monochromatic impulses with amplitudes *i* ∈ S are transferred, nevertheless generated from the environment of the same type but at the temperature *TW*, *TW* > *T*<sup>0</sup> where *T*<sup>0</sup> is the

It is supposed that both pure states *θ<sup>i</sup>* in the place where the input message is being *coded* - on the *input of the channel* **Ψ** ∼= K and, also, the measurable values of the quantity *α* being

• In the B–E system, bosonic, e.g. the photonic gas the B–E distribution is valid

• In the F–D system, fermionic, e.g. electron gas the F–D distribution is valid

1 + *p*

temperature of the transfer system **Ψ** ∼= K (the *noise* temperature).

*<sup>p</sup>*(*<sup>k</sup>* <sup>−</sup> *<sup>i</sup>*) pro *<sup>k</sup>* <sup>≥</sup> *<sup>i</sup>*

, *<sup>j</sup>* ∈ {0, 1, ...}, *<sup>p</sup>* <sup>∈</sup> (0, 1), *<sup>p</sup>*<sup>−</sup> *<sup>ε</sup>*

, *<sup>j</sup>* ∈ {0, 1}, *<sup>p</sup>* <sup>∈</sup> (0, 1), *<sup>p</sup>*<sup>−</sup> *<sup>ε</sup>*

<sup>0</sup> *<sup>k</sup>* <sup>&</sup>lt; *<sup>i</sup>* (25)

<sup>k</sup><sup>Θ</sup> (27)

<sup>k</sup><sup>Θ</sup> (28)

Pr(*j*), *j* ∈ {0, 1, 2, ...} (26)

= (*α*�*θ*), (*X*|*Y*) �

*I*(*α*; *θ*), *I*(·||·)≡*T*(·; ·) (23)

= (*θ*|*α*), (*Y*|*X*) �

= (*α*|*θ*).

= *θ*, *Y* �

*p*(*αk*|*α*|*θi*) =

K ∼= **Ψ**. This transition *j* occurs with the probability distribution

B–E or F–D type [27] (denoted further by **Ψ**B−E,*ε*, **Ψ**F−D,*ε*).

Pr(*j*)=(<sup>1</sup> <sup>−</sup> *<sup>p</sup>*) · *<sup>p</sup><sup>j</sup>*

Pr(*j*) = *<sup>p</sup><sup>j</sup>*

The laws of information transfer are writable in this way too:

**4. Narrow-band quantum transfer channels**

distribution Pr(·) = {*p*(0), *p*(1), *p*(2), ...}

description denoted as follows, *X* �

*C* = sup *α*,*θ*

> Let now *<sup>θ</sup><sup>i</sup>* ≡ *<sup>i</sup>*, *<sup>i</sup>* = 0, 1, ... are pure states of a system **<sup>Ψ</sup>**B−E,*<sup>ε</sup>* ∼= K and let *<sup>α</sup>* is output quantity taking on in the state *θ<sup>i</sup>* values *α* ∈ **S** with probabilities

$$p(\mathfrak{a}|\mathfrak{a}|\mathfrak{e}\_i) = (1-p)\,p^{\mathfrak{a}-i} \tag{30}$$

Thus the distribution *p*(·|*α*|*θi*) is determined by the *forced* (inner–input) state *θ<sup>i</sup>* = *π*{*ψi*} representing the coded input energy at the value *ε<sup>i</sup>* = *i* · *ε* ∈ **S**(*ε*), *ε* = const. For the *medium value* W of the input *i* is valid

$$\mathcal{W} = \sum\_{i=0}^{\infty} i \cdot q(i|\theta) = E(\theta)\_{\prime} \quad \mathcal{W} = \varepsilon \cdot \mathcal{W} \tag{31}$$

The quantity *W* is the medium value of the energy coding the input signal *i*.

For the medium value of the number of particles *j* = *α* − *i* ≥ 0 with B–E statistics is valid

$$\sum\_{j=0}^{\infty} j \cdot (1-p) \, p^j = (1-p) \cdot p \cdot \sum\_{j=0}^{\infty} j \, p^{j-1} = (1-p) \cdot p \cdot \frac{\mathbf{d}}{\mathbf{d}p} \left[ \frac{1}{1-p} \right] = \frac{p}{1-p}$$

The quantity *E*(*α*) is the medium value of the output quantity *α* and

$$E(\mathfrak{a}) = \sum\_{\mathfrak{a} \in \mathbf{S}(\mathfrak{a})} \mathfrak{a} \cdot p(\mathfrak{a}|\mathfrak{a}|\mathfrak{e}) \tag{32}$$

where *p*(*α*|*α*|*θ*) is the probability of measuring the eigenvalue *α* = *k* of the output variable *α*. This probability is defined by the state *<sup>θ</sup>* <sup>=</sup> *<sup>n</sup>* ∑ *i*=0 *<sup>q</sup>*(*i*|*θ*) *<sup>θ</sup><sup>i</sup>* of the system **<sup>Ψ</sup>**B−E,*<sup>ε</sup>* ∼= K

$$p(\mathfrak{a}|\mathfrak{a}|\mathfrak{e}) = \sum\_{i=0}^{n} q(i|\mathfrak{e}) \cdot p(\mathfrak{a}|\mathfrak{a}|\mathfrak{e}) = (1-p) \cdot \sum\_{i=0}^{n} q(i|\mathfrak{e}) \cdot p^{a-i} \quad [= \operatorname{Tr}(\mathfrak{e}\pi\_{a})] \tag{33}$$

#### 8 Will-be-set-by-IN-TECH 90 Thermodynamics – Fundamentals and Its Application in Science Information Capacity of Quantum Transfer Channels and Thermodynamic Analogies <sup>9</sup>

From the differential equation with the *condition* for *α* = 0

$$p(a|\mathfrak{a}|\mathfrak{e}) = p(a-1|\mathfrak{a}|\mathfrak{e}) \cdot p + (1-p) \cdot q(a|\mathfrak{e}|\mathfrak{e}), \forall a \ge 1; \quad p(0|\mathfrak{a}|\mathfrak{e}) = (1-p) \cdot q(0|\mathfrak{e}) \tag{34}$$

The Lagrange function

<sup>−</sup>1−*λ*<sup>1</sup> · *<sup>e</sup>*

of <sup>W</sup> only. For *<sup>λ</sup>*<sup>2</sup> <sup>=</sup> *<sup>ε</sup>*

depending only on *<sup>ε</sup>*

*p*(·)], is valid that

*<sup>q</sup>*(*α*|*θ*) = <sup>1</sup> <sup>−</sup> *<sup>p</sup>*(W)

valid that

*<sup>E</sup>*(*α*) = <sup>∑</sup>*<sup>α</sup>*

= −*e*

By (35), (36) and for the parametr *p*=*e*

*geometric* probability distribution

*p<sup>α</sup>* = *e*

*<sup>L</sup>* <sup>=</sup> <sup>−</sup> <sup>∑</sup>*<sup>α</sup>*

which gives the condition for the extreme, *<sup>∂</sup><sup>L</sup>*

*<sup>p</sup><sup>α</sup>* · ln *<sup>p</sup><sup>α</sup>* <sup>−</sup> *<sup>λ</sup>*<sup>1</sup> · <sup>∑</sup>*<sup>α</sup>*

<sup>−</sup>*λ*2*<sup>α</sup>* <sup>=</sup> *<sup>p</sup>*(*α*|*α*|*θ*) and further, in <sup>∑</sup>*<sup>α</sup>*

*α* · *e*

 1 <sup>1</sup> − *<sup>e</sup>*−*λ*<sup>2</sup>

*<sup>E</sup>*(*α*) = *<sup>p</sup>*(W)

1 − *p*

<sup>1</sup> <sup>−</sup> *<sup>p</sup>* , *<sup>α</sup>* <sup>=</sup> 0 and *<sup>q</sup>*(*α*|*θ*) = <sup>1</sup> <sup>−</sup> *<sup>p</sup>*(W)

sup *<sup>θ</sup>*∈**Θ**<sup>0</sup>

1 − *p*(W)

+ W, resp.

From (35) and (46) is visible that *p*(W) or *TW* respectively is the *only one* root of the equation

From (34) and (45) follows that for state *θ* ∈ **Θ**<sup>0</sup> or, for the *q*-distribution *q*(·|*θ*) respectively, in which the value *H*(*α*�*θ*) is maximal [that state in which *α* achieves the distribution *q*(·|*θ*) =

For the *effective temperatutre TW of coding input messages* the distribution (45) supremizes

From (39) and (49) follows [12, 37] the capacity *C*B−E,*<sup>ε</sup>* of the narrow–band channel K ∼= **Ψ**B−<sup>E</sup>

*<sup>H</sup>*(*α*�*θ*) = *<sup>h</sup>*[*p*(W)]

<sup>1</sup> <sup>−</sup> *<sup>p</sup>*(W) <sup>−</sup> *<sup>h</sup>*(*p*)

(maximizes) the *p*-entropy *H*(*α*�*θ*) of *α* and, by using (37) with *p*(W), is gained that

*<sup>C</sup>*B−E,*<sup>ε</sup>* <sup>=</sup> *<sup>h</sup>*[*p*(W)]

<sup>−</sup> *<sup>ε</sup>*

<sup>−</sup>1−*λ*<sup>1</sup> · *<sup>e</sup>*

Then for the medium value *E*(*α*) the following result is obtained

*<sup>α</sup>* · *<sup>p</sup><sup>α</sup>* <sup>=</sup> <sup>∑</sup>*<sup>α</sup>*

*∂λ*2

<sup>−</sup>1−*λ*<sup>1</sup> · *<sup>∂</sup>*

k*TW*

k*TW*

*p*(W) <sup>1</sup> <sup>−</sup> *<sup>p</sup>*(W) <sup>=</sup> *<sup>p</sup>* *<sup>p</sup><sup>α</sup>* <sup>+</sup> *<sup>λ</sup>*<sup>1</sup> <sup>−</sup> *<sup>λ</sup>*<sup>2</sup> · <sup>∑</sup>*<sup>α</sup>*

<sup>−</sup>*λ*2*<sup>α</sup>* <sup>=</sup> <sup>−</sup>*<sup>e</sup>*

<sup>=</sup> *<sup>e</sup>*−1−*λ*<sup>1</sup> (<sup>1</sup> <sup>−</sup> *<sup>e</sup>*−*λ*<sup>2</sup> )<sup>2</sup> · *<sup>e</sup>*

*<sup>p</sup><sup>α</sup>* <sup>=</sup> <sup>∑</sup>*<sup>α</sup>*

Information Capacity of Quantum Transfer Channels and Thermodynamic Analogies 91

*∂p<sup>α</sup>*

*α* · *p<sup>α</sup>* + *λ*<sup>2</sup> *E*(*α*) (42)

<sup>1</sup> − *<sup>e</sup>*−*λ*<sup>2</sup>

= 1 (43)

<sup>−</sup>*λ*2*<sup>α</sup>* (44)

= − ln *p<sup>α</sup>* − 1 − *λ*<sup>1</sup> − *λ*<sup>2</sup> · *α* = 0, yielding in

*∂λ*<sup>2</sup> <sup>∑</sup>*<sup>α</sup> e*

<sup>−</sup>*λ*<sup>2</sup> <sup>=</sup> *<sup>e</sup>*−*λ*<sup>2</sup>

<sup>k</sup><sup>Θ</sup> = const. (*ε* = const. Θ = const.) *E*(*α*) is a function

<sup>1</sup> − *<sup>e</sup>*−*λ*<sup>2</sup>

<sup>k</sup>*TW* (46)

+ W (47)

*<sup>e</sup>λ*2*<sup>α</sup>* <sup>=</sup> *<sup>e</sup>*−1−*λ*<sup>1</sup>

*e*−1−*λ*<sup>1</sup>

<sup>−</sup>1−*λ*<sup>1</sup> · *<sup>∂</sup>*

is *<sup>e</sup>*−*λ*<sup>2</sup> <sup>=</sup> *<sup>p</sup>*(W) and *<sup>E</sup>*(*α*) is the medium value for *<sup>α</sup>* with the

*<sup>p</sup>*(·) = *<sup>p</sup>*(·|*α*|*θ*)=[<sup>1</sup> <sup>−</sup> *<sup>p</sup>*(W)] · *<sup>p</sup>*(W)*α*, *<sup>α</sup>* <sup>∈</sup> **<sup>S</sup>**(*α*) (45)

, *p*(W) = *e*

*e* <sup>−</sup> *<sup>ε</sup>* k*TW*

1 − *e* <sup>−</sup> *<sup>ε</sup>* k*TW*

or, on the absolute temperature *TW* respectively. Thus for *E*(*α*) is

<sup>−</sup> *<sup>ε</sup>*

<sup>=</sup> *<sup>e</sup>*

1 − *e* <sup>−</sup> *<sup>ε</sup>* k*T*0

<sup>−</sup> *<sup>ε</sup>* k*T*0

<sup>1</sup> <sup>−</sup> *<sup>p</sup>* [*p*(W) <sup>−</sup> *<sup>p</sup>*] · *<sup>p</sup>*(W)*α*−1, *<sup>α</sup>* <sup>&</sup>gt; 0 (48)

<sup>1</sup> <sup>−</sup> *<sup>p</sup>*(W) (49)

<sup>1</sup> <sup>−</sup> *<sup>p</sup>* (50)

follows, for the medium value *E*(*α*) of the output stochastic variable *α*, that

$$E(\mathfrak{a}) = \sum\_{a \ge 1} \mathfrak{a} \cdot p(\mathfrak{a} - 1 | \mathfrak{a} | \mathfrak{G}) \cdot p + \sum\_{a \ge 1} \mathfrak{a} \cdot (1 - p) \cdot q(\mathfrak{a} | \mathfrak{G}) \tag{35}$$

$$= \sum\_{a \ge 1} \mathfrak{a} \cdot p(\mathfrak{a} - 1 | \mathfrak{a} | \mathfrak{G}) \cdot p + \mathcal{W} \cdot (1 - p) = p \cdot E(\mathfrak{a}) + p \cdot \sum\_{a \ge 1} p(\mathfrak{a} - 1 | \mathfrak{a} | \mathfrak{G}) + \mathcal{W} \cdot (1 - p)$$

*α*≥1 *α*≥1

$$E(\mathfrak{a}) \cdot (1 - p) = p + \mathcal{W} \cdot (1 - p) \longrightarrow E(\mathfrak{a}) = \frac{p}{1 + p} + \mathcal{W}\_{\prime} \quad \mathcal{W} = E(\mathfrak{a}) > 0, \quad \mathfrak{G} \in \Theta\_0 \tag{36}$$

The quantity *H*(*α*�*θi*) is the *p*-entropy of measuring *α* for the input *i* ∈ **S** being represented by the pure state *θ<sup>i</sup>* of the system **Ψ**B−E,*<sup>ε</sup>*

$$-H(\mathfrak{a} \| \theta\_i) = \sum\_{j \in \mathbf{S}} (1 - p) \, p^j \cdot \ln[(1 - p) \, p^j] \tag{37}$$

$$= (1 - p) \cdot \ln(1 - p) \cdot \sum\_{j} p^j - (1 - p) \cdot p \cdot \ln p \cdot \sum\_{j} j \, p^{j - 1}$$

$$= \ln(1 - p) - (1 - p) \cdot p \cdot \ln p \cdot \frac{\mathbf{d}}{\mathbf{d}p} \left[ \frac{1}{1 - p} \right] = -\frac{h(p)}{1 - p}, \forall i \in \mathbf{S}$$

where *h*(*p*) � = −(1 − *p*) · ln(1 − *p*) − *p* · ln *p* is the Shannon entropy of Bernoulli distribution {*p*, 1 − *p*}.

The quantity *H*(*α*|*θ*) is the conditional Shannon entropy of the stochastic quantity *α* in the state *θ* of the system **Ψ**B−E,*ε*, not depending on the *θ* (the *noise* entropy)

$$H(\mathfrak{a}|\mathfrak{g}) = \sum\_{i=0}^{n} q(i|\mathfrak{g}) \cdot H(\mathfrak{a}|\mathfrak{f}\_{i}) = \sum\_{i=0}^{n} q(i|\mathfrak{g}) \cdot \frac{h(p)}{1-p} = \frac{h(p)}{1-p} \tag{38}$$

For capacity *<sup>C</sup>*B−E" of the channel K ∼= **<sup>Ψ</sup>**B−<sup>E</sup> is, following the capacity definition, valid

$$\mathsf{C}\_{\mathsf{B}-\mathsf{E}''} = \sup\_{\theta \in \mathsf{\Theta}\_{0}} H(\mathfrak{a} \| \theta) - H(\mathfrak{a} \| \theta) = \sup\_{\theta \in \mathsf{\Theta}\_{0}} H(\mathfrak{a} \| \theta) - \frac{h(p)}{1 - p} \tag{39}$$

where the set **Θ**<sup>0</sup> = {*θ* ∈ **Θ**, *E*(*θ*) = W ≥ 0} represents the coding procedure of the input *i* ∈ **S**. The quantity *H*(*α*�*θ*) = *H*(*p*(·|*α*|*θ*)) is the *p*-entropy of the output quantity *α*. Its supremum is determined by the **Lagrange** multipliers method:

$$H(\mathfrak{a} \| \mathfrak{\theta}) = -\sum\_{\mathfrak{a} \in \mathfrak{S}(\mathfrak{a})} p(\mathfrak{a}|\mathfrak{a}|\mathfrak{\theta}) \cdot \ln p(\mathfrak{a}|\mathfrak{a}|\mathfrak{\theta}) = -\sum\_{\mathfrak{a} \in \mathfrak{S}(\mathfrak{a})} p\_{\mathfrak{a}} \cdot \ln p\_{\mathfrak{a}\mathfrak{a}} \cdot p\_{\mathfrak{a}} \stackrel{\triangle}{=} p(\mathfrak{a}|\mathfrak{a}|\mathfrak{\theta}) \tag{40}$$

The conditions for determinating of the *bound* extreme are

$$\sum\_{\mathfrak{a}\in\mathsf{S}(\mathfrak{a})} p\_{\mathfrak{a}} = 1, \quad \sum\_{\mathfrak{a}\in\mathsf{S}(\mathfrak{a})} \mathfrak{a} \cdot p\_{\mathfrak{a}} = E(\mathfrak{a}) = \text{const.}\tag{41}$$

The Lagrange function

8 Will-be-set-by-IN-TECH

*p*(*α*|*α*|*θ*) = *p*(*α* − 1|*α*|*θ*) · *p* + (1 − *p*) · *q*(*α*|*θ*), ∀*α* ≥ 1; *p*(0|*α*|*θ*)=(1 − *p*) · *q*(0|*θ*) (34)

1 + *p*

*<sup>p</sup><sup>j</sup>* <sup>−</sup> (<sup>1</sup> <sup>−</sup> *<sup>p</sup>*) · *<sup>p</sup>* · ln *<sup>p</sup>* · ∑

 1 1 − *p*

= −(1 − *p*) · ln(1 − *p*) − *p* · ln *p* is the Shannon entropy of Bernoulli distribution

*n* ∑ *i*=0 *<sup>q</sup>*(*i*|*θ*) · *<sup>h</sup>*(*p*)

*<sup>θ</sup>*∈**Θ**<sup>0</sup>

*α*∈**S**(*α*)

d*p*

The quantity *H*(*α*|*θ*) is the conditional Shannon entropy of the stochastic quantity *α* in the

The quantity *H*(*α*�*θi*) is the *p*-entropy of measuring *α* for the input *i* ∈ **S** being represented

*α* · (1 − *p*) · *q*(*α*|*θ*) (35)

*p*(*α* − 1|*α*|*θ*) + W · (1 − *p*)

+ W, W = *E*(*θ*) > 0, *θ* ∈ **Θ**<sup>0</sup> (36)

, ∀*i* ∈ **S**

<sup>1</sup> <sup>−</sup> *<sup>p</sup>* (38)

<sup>1</sup> <sup>−</sup> *<sup>p</sup>* (39)

= *p*(*α*|*α*|*θ*) (40)

] (37)

*j pj*−<sup>1</sup>

<sup>1</sup> <sup>−</sup> *<sup>p</sup>* <sup>=</sup> *<sup>h</sup>*(*p*)

*<sup>H</sup>*(*α*�*θ*) <sup>−</sup> *<sup>h</sup>*(*p*)

*p<sup>α</sup>* · ln *pα*, *p<sup>α</sup>*

*α* · *p<sup>α</sup>* = *E*(*α*) = const. (41)

�

*α*≥1

*j*

<sup>=</sup> <sup>−</sup> *<sup>h</sup>*(*p*) 1 − *p*

From the differential equation with the *condition* for *α* = 0

*<sup>α</sup>* · *<sup>p</sup>*(*<sup>α</sup>* − <sup>1</sup>|*α*|*θ*) · *<sup>p</sup>* + ∑

*<sup>E</sup>*(*α*) · (<sup>1</sup> <sup>−</sup> *<sup>p</sup>*) = *<sup>p</sup>* <sup>+</sup> W · (<sup>1</sup> <sup>−</sup> *<sup>p</sup>*) −→ *<sup>E</sup>*(*α*) = *<sup>p</sup>*

= (<sup>1</sup> − *<sup>p</sup>*) · ln(<sup>1</sup> − *<sup>p</sup>*) · ∑

*H*(*α*|*θ*) =

*<sup>C</sup>*B−E" = sup

*α*∈**S**(*α*)

*<sup>H</sup>*(*α*�*θ*) = − ∑

(<sup>1</sup> <sup>−</sup> *<sup>p</sup>*) *<sup>p</sup><sup>j</sup>* · ln[(<sup>1</sup> <sup>−</sup> *<sup>p</sup>*) *<sup>p</sup><sup>j</sup>*

<sup>=</sup> ln(<sup>1</sup> <sup>−</sup> *<sup>p</sup>*) <sup>−</sup> (<sup>1</sup> <sup>−</sup> *<sup>p</sup>*) · *<sup>p</sup>* · ln *<sup>p</sup>* · <sup>d</sup>

*n* ∑ *i*=0

*<sup>θ</sup>*∈**Θ**<sup>0</sup>

supremum is determined by the **Lagrange** multipliers method:

The conditions for determinating of the *bound* extreme are

∑ *α*∈**S**(*α*)

by the pure state *θ<sup>i</sup>* of the system **Ψ**B−E,*<sup>ε</sup>*

*j*∈**S**

*E*(*α*) = ∑

*α*≥1

−*H*(*α*�*θi*) = ∑

where *h*(*p*) �

{*p*, 1 − *p*}.

= ∑ *α*≥1

follows, for the medium value *E*(*α*) of the output stochastic variable *α*, that

*<sup>α</sup>* · *<sup>p</sup>*(*<sup>α</sup>* − <sup>1</sup>|*α*|*θ*) · *<sup>p</sup>* + W · (<sup>1</sup> − *<sup>p</sup>*) = *<sup>p</sup>* · *<sup>E</sup>*(*α*) + *<sup>p</sup>* · ∑

*j*

state *θ* of the system **Ψ**B−E,*ε*, not depending on the *θ* (the *noise* entropy)

*q*(*i*|*θ*) · *H*(*α*�*θi*) =

For capacity *<sup>C</sup>*B−E" of the channel K ∼= **<sup>Ψ</sup>**B−<sup>E</sup> is, following the capacity definition, valid

*<sup>p</sup>*(*α*|*α*|*θ*) · ln *<sup>p</sup>*(*α*|*α*|*θ*) = − ∑

*α*∈**S**(*α*)

*p<sup>α</sup>* = 1, ∑

*H*(*α*�*θ*) − *H*(*α*|*θ*) = sup

where the set **Θ**<sup>0</sup> = {*θ* ∈ **Θ**, *E*(*θ*) = W ≥ 0} represents the coding procedure of the input *i* ∈ **S**. The quantity *H*(*α*�*θ*) = *H*(*p*(·|*α*|*θ*)) is the *p*-entropy of the output quantity *α*. Its

*α*≥1

$$L = -\sum\_{\mathfrak{a}} p\_{\mathfrak{a}} \cdot \ln p\_{\mathfrak{a}} - \lambda\_1 \cdot \sum\_{\mathfrak{a}} p\_{\mathfrak{a}} + \lambda\_1 - \lambda\_2 \cdot \sum\_{\mathfrak{a}} \mathfrak{a} \cdot p\_{\mathfrak{a}} + \lambda\_2 \to (\mathfrak{a}) \tag{42}$$

which gives the condition for the extreme, *<sup>∂</sup><sup>L</sup> ∂p<sup>α</sup>* = − ln *p<sup>α</sup>* − 1 − *λ*<sup>1</sup> − *λ*<sup>2</sup> · *α* = 0, yielding in

$$p\_d = e^{-1 - \lambda\_1} \cdot e^{-\lambda\_2 a} = p(a|a|\theta) \quad \text{and further, in } \sum\_a p\_a = \sum\_a \frac{e^{-1 - \lambda\_1}}{e^{\lambda\_2 a}} = \frac{e^{-1 - \lambda\_1}}{1 - e^{-\lambda\_2}} = 1 \tag{43}$$

Then for the medium value *E*(*α*) the following result is obtained

$$E(\mathfrak{a}) = \sum\_{\mathfrak{a}} \mathfrak{a} \cdot p\_{\mathfrak{a}} = \sum\_{\mathfrak{a}} \mathfrak{a} \cdot e^{-1 - \lambda\_{1}} \cdot e^{-\lambda\_{2}\mathfrak{a}} = -e^{-1 - \lambda\_{1}} \cdot \frac{\partial}{\partial \lambda\_{2}} \sum\_{\mathfrak{a}} e^{-\lambda\_{2}\mathfrak{a}} \tag{44}$$
 
$$= -e^{-1 - \lambda\_{1}} \cdot \frac{\partial}{\partial \lambda\_{2}} \left[ \frac{1}{1 - e^{-\lambda\_{2}}} \right] = \frac{e^{-1 - \lambda\_{1}}}{(1 - e^{-\lambda\_{2}})^{2}} \cdot e^{-\lambda\_{2}} = \frac{e^{-\lambda\_{2}}}{1 - e^{-\lambda\_{2}}}$$

By (35), (36) and for the parametr *p*=*e* <sup>−</sup> *<sup>ε</sup>* <sup>k</sup><sup>Θ</sup> = const. (*ε* = const. Θ = const.) *E*(*α*) is a function of <sup>W</sup> only. For *<sup>λ</sup>*<sup>2</sup> <sup>=</sup> *<sup>ε</sup>* k*TW* is *<sup>e</sup>*−*λ*<sup>2</sup> <sup>=</sup> *<sup>p</sup>*(W) and *<sup>E</sup>*(*α*) is the medium value for *<sup>α</sup>* with the *geometric* probability distribution

$$p(\cdot) = p(\cdot|\mathfrak{a}|\mathfrak{G}) = [1 - p(\mathcal{W})] \cdot p(\mathcal{W})^{\mathfrak{a}}, \; \mathfrak{a} \in \mathbf{S}(\mathfrak{a}) \tag{45}$$

depending only on *<sup>ε</sup>* k*TW* or, on the absolute temperature *TW* respectively. Thus for *E*(*α*) is valid that

$$E(\mathfrak{a}) = \frac{p(\mathcal{W})}{1 - p(\mathcal{W})}, \ p(\mathcal{W}) = e^{-\frac{\mathfrak{a}\cdot\mathfrak{F}}{\mathfrak{F}\mathfrak{F}}} \tag{46}$$

From (35) and (46) is visible that *p*(W) or *TW* respectively is the *only one* root of the equation

$$\frac{p(\mathcal{W})}{1 - p(\mathcal{W})} = \frac{p}{1 - p} + \mathcal{W} \text{, resp. } \frac{e^{-\frac{\ell}{kT\_W}}}{1 - e^{-\frac{\ell}{kT\_W}}} = \frac{e^{-\frac{\ell}{kT\_0}}}{1 - e^{-\frac{\ell}{kT\_0}}} + \mathcal{W} \tag{47}$$

From (34) and (45) follows that for state *θ* ∈ **Θ**<sup>0</sup> or, for the *q*-distribution *q*(·|*θ*) respectively, in which the value *H*(*α*�*θ*) is maximal [that state in which *α* achieves the distribution *q*(·|*θ*) = *p*(·)], is valid that

$$q(a|\theta) = \frac{1 - p(\mathcal{W})}{1 - p}, \ a = 0 \quad \text{and} \ \eta(a|\theta) = \frac{1 - p(\mathcal{W})}{1 - p} \left[ p(\mathcal{W}) - p \right] \cdot p(\mathcal{W})^{a - 1}, \ a > 0 \tag{48}$$

For the *effective temperatutre TW of coding input messages* the distribution (45) supremizes (maximizes) the *p*-entropy *H*(*α*�*θ*) of *α* and, by using (37) with *p*(W), is gained that

$$\sup\_{\theta \in \Theta\_0} H(\mathfrak{a} \| \theta) = \frac{h[p(\mathcal{W})]}{1 - p(\mathcal{W})} \tag{49}$$

From (39) and (49) follows [12, 37] the capacity *C*B−E,*<sup>ε</sup>* of the narrow–band channel K ∼= **Ψ**B−<sup>E</sup>

$$\mathcal{C}\_{\text{B}-\text{E},\varepsilon} = \frac{h[p(\mathcal{W})]}{1 - p(\mathcal{W})} - \frac{h(p)}{1 - p} \tag{50}$$

#### 10 Will-be-set-by-IN-TECH 92 Thermodynamics – Fundamentals and Its Application in Science Information Capacity of Quantum Transfer Channels and Thermodynamic Analogies <sup>11</sup>

By (35), (36) and (46) the medium value W of the input message *i* ∈ **S** is derived,

$$\mathcal{W} = \frac{p(\mathcal{W})}{1 - p(\mathcal{W})} - \frac{p}{1 - p} \tag{51}$$

where *p*(*α*|*α*|*θ*) is probability of realization of *α* ∈ **S** in the state *θ* of the system **Ψ** ≡

*<sup>q</sup>*(*i*|*θ*) · *<sup>p</sup>*(*α*|*α*|*θi*) = <sup>1</sup>

<sup>1</sup> <sup>+</sup> *<sup>p</sup>* · *<sup>q</sup>*(0|*θ*)

*<sup>q</sup>*(*<sup>α</sup>* <sup>−</sup> <sup>1</sup>|*θ*) + <sup>1</sup>

1 + *p*

<sup>1</sup> <sup>+</sup> *<sup>p</sup>* <sup>−</sup> *<sup>p</sup>*

<sup>1</sup> <sup>+</sup> *<sup>p</sup>* · ln *<sup>p</sup>*

 *p* 1 + *p*

*H*(*α*�*θ*) − *h*

1 + *p* ·

Information Capacity of Quantum Transfer Channels and Thermodynamic Analogies 93

*n* ∑ *i*=0

<sup>1</sup> <sup>+</sup> *<sup>p</sup>* , *<sup>α</sup>* <sup>≥</sup> <sup>1</sup> (62)

*α* · *q*(*α*|*θ*) (63)

1 + *p*

<sup>1</sup> <sup>+</sup> *<sup>p</sup>* · ln *<sup>p</sup>*

<sup>1</sup> <sup>+</sup> *<sup>p</sup>* <sup>=</sup> *<sup>h</sup>*

 = *h* · W

 *p* 1 + *p*

 *p* 1 + *p*

 *p* 1 + *p*

+ W, W = *E*(*θ*), *θ* ∈ **Θ**<sup>0</sup>

<sup>1</sup> <sup>+</sup> *<sup>p</sup>* (64)

, ∀*i* ∈ **S**

(65)

(66)

*<sup>q</sup>*(*i*|*θ*) · *<sup>p</sup>α*−*<sup>i</sup>* (61)

*n* ∑ *i*=0

*<sup>p</sup>*(*α*|*α*|*θ*) = *<sup>q</sup>*(*<sup>α</sup>* <sup>−</sup> <sup>1</sup>|*θ*) · *<sup>p</sup>* <sup>+</sup> *<sup>q</sup>*(*α*|*θ*)

*<sup>α</sup>* <sup>=</sup> 0, *<sup>p</sup>*(0|*α*|*θ*) = <sup>1</sup>

follows that for the medium value *E*(*α*) of the output stochastic variable *α* is valid that

<sup>1</sup> <sup>+</sup> *<sup>p</sup>* · ∑ *α*≥1

> <sup>1</sup> <sup>+</sup> *<sup>p</sup>* · ∑ *α*≥1

The quantity *H*(*α*�*θi*) is the *p*-entropy of measuring *α* for the input *i* ∈ **S** being represented

<sup>1</sup> <sup>+</sup> *<sup>p</sup>* · ln <sup>1</sup>

 <sup>−</sup> *<sup>p</sup>*

The quantity *H*(*α*|*θ*) is the conditional (the *noise*) Shannon entropy of the stochastic quantity

*i*

For capacity *C*F−D,*<sup>ε</sup>* of the channel K∼=**Ψ**F−D,*<sup>ε</sup>* is, by the capacity definition in (23)-(24), valid

The quantity *H*(*α*�*θ*) = *H*(*p*(·|*α*|*θ*)) is the *p*-entropy of the stochastic quantity *α* in the state *θ* of the system **Ψ**F−D,*ε*. Its supremum is determined by the **Lagrange** multipliers method in the same way as in B–E case and with the same results for the probility distribution *p*(·|*α*|*θ*)

*H*(*α*�*θ*) − *H*(*α*|*θ*) = sup

where the set **Θ**<sup>0</sup> = {*θ* ∈ **Θ** : *E*(*θ*) = W > 0} represents the coding procedure.

*q*(*i*|*θ*) · *h*

*<sup>θ</sup>*∈**Θ**<sup>0</sup>

· W −→ *<sup>E</sup>*(*α*) = *<sup>p</sup>*

**Ψ**F−D,*<sup>ε</sup>*

∼=K,

*θ* = ∑ *i*∈**S**

with the condition for

*<sup>E</sup>*(*α*) = *<sup>p</sup>*

<sup>=</sup> *<sup>p</sup>*

<sup>=</sup> *<sup>p</sup>* 1 + *p*

*H*(*α*�*θi*) = −

that

= − 

*H*(*α*|*θ*) =

From the differential equation

<sup>1</sup> <sup>+</sup> *<sup>p</sup>* · ∑ *α*≥1

<sup>1</sup> <sup>+</sup> *<sup>p</sup>* · ∑ *α*≥1

· W <sup>+</sup> *<sup>p</sup>*

by the pure state *θ<sup>i</sup>* of the system **Ψ**F−D,*<sup>ε</sup>*

1 ∑ *j*=0 1 + *p* + 1 1 + *p*

*pj* <sup>1</sup> <sup>+</sup> *<sup>p</sup>* · ln *<sup>p</sup><sup>j</sup>*

<sup>1</sup> <sup>−</sup> *<sup>p</sup>* 1 + *p*

*α* in the system state *θ*, but, independent on this *θ*,

*n* ∑ *i*=0

*<sup>C</sup>*F−D" = sup

(geometric) and the medium value *E*(*α*)

*<sup>θ</sup>*∈**Θ**<sup>0</sup>

 · ln

*q*(*i*|*θ*) *θi*, *p*(*α*|*α*|*θ*) =

*<sup>α</sup>* · *<sup>q</sup>*(*<sup>α</sup>* <sup>−</sup> <sup>1</sup>|*θ*) + <sup>1</sup>

(*<sup>α</sup>* <sup>−</sup> <sup>1</sup>) · *<sup>q</sup>*(*<sup>α</sup>* <sup>−</sup> <sup>1</sup>|*θ*) + *<sup>p</sup>*

<sup>1</sup> <sup>+</sup> *<sup>p</sup>* <sup>=</sup> <sup>−</sup> <sup>1</sup>

*<sup>q</sup>*(*i*|*θ*) · *<sup>H</sup>*(*α*�*θi*) = ∑

<sup>1</sup> <sup>−</sup> *<sup>p</sup>* 1 + *p*

By (31) the condition for the *minimal* average energy *WKrit* needed for coding the input message is

$$\mathcal{W} \ge 0 \quad \text{resp. } \mathcal{W} = \varepsilon \cdot \mathcal{W} \ge 0, \quad \varepsilon \in \mathbf{S}(\varepsilon) \tag{52}$$

$$\mathcal{W} \ge \mathcal{W}\_{\text{Krit}\prime} \quad \mathcal{W}\_{\text{Krit}} = \mathcal{0} \tag{53}$$

The relations (35), (36), (46) and (52), (53) yield in

$$E(\mathfrak{a}) = \frac{p(\mathcal{W})}{1 - p(\mathcal{W})} \ge \frac{p}{1 - p} \text{ and, then } \ p(\mathcal{W}) \ge p\_\prime \, 1 - p(\mathcal{W}) \le 1 - p \tag{54}$$

From (47) and (54) follows that for the defined direction of the signal (messaage) transmission at the temperature *TW* of its sending and decoding is valid that

$$\frac{p(\mathcal{W})}{1 - p(\mathcal{W})} > \frac{p}{1 - p'}, \ p(\mathcal{W}) > p, \ \mathcal{W} > 0 \tag{55}$$

$$p(\mathcal{W}) = e^{-\frac{\ell}{kT\_W}} \ge e^{-\frac{\ell}{kT\_0}} = p \quad \text{and thus} \quad T\_W \ge T\_0 \tag{56}$$

#### **4.2. Capacity of Fermi–Dirac narrow–band channel**

Let is now considered, in the same way as it was in the B–E system, the pure states *θ<sup>i</sup>* ≡ *i* of the system **Ψ** which are coding the input messages *i* = 0, 1, ... and the output stochastic quantity *α* having its selecting space **S**. On the spectrum **S** probabilities of realizations *α* ∈ **S** are defined,

$$p(\mathfrak{a}|\mathfrak{a}|\mathfrak{e}\_{i}) = \frac{p^{\mathfrak{a}-i}}{1+p^{\prime}} \quad p \in (0,1), \; i = 0, \; 1, \; \dots \tag{57}$$

expressing the additive stochastic transformation of an input *i* into the output *α* for wich is valid *α* = *i* or *α* = *i* + 1.<sup>5</sup> The uniform energy level *ε* = const. of particles is considered.

The quantity *W* is the mathematical expectation of the energy coding the input signal

$$\mathcal{W} = \varepsilon \cdot \mathcal{W}, \quad \mathcal{W} = \sum\_{i \in \mathbf{S}} i \cdot q(i|\boldsymbol{\theta}) = E(\boldsymbol{\theta}) \tag{58}$$

The medium value of a stochastic quantity with the F–D statistic is given by

$$\sum\_{j \in \{0, 1\}} j \cdot \frac{p^j}{1+p} = \frac{p}{1+p} \tag{59}$$

The quantity *E*(*α*) is the medium value of the output quantity *α*,

$$E(\mathfrak{a}) = \sum\_{\mathfrak{a} \in \mathbf{S}(\mathfrak{a})} \mathfrak{a} \cdot p(\mathfrak{a}|\mathfrak{a}|\mathfrak{g}) \tag{60}$$

<sup>5</sup> In accordance with *Pauli excluding principle* (valid for *fermions*) and a given energetic level *<sup>ε</sup>* <sup>∈</sup> **<sup>S</sup>**(*ε*).

where *p*(*α*|*α*|*θ*) is probability of realization of *α* ∈ **S** in the state *θ* of the system **Ψ** ≡ **Ψ**F−D,*<sup>ε</sup>* ∼=K,

$$\mathfrak{G} = \sum\_{i \in \mathbf{S}} q(i|\boldsymbol{\theta}) \,\, \mathfrak{G}\_{i\boldsymbol{\nu}} \cdot p(\boldsymbol{a}|\boldsymbol{a}|\boldsymbol{\theta}) = \sum\_{i=0}^{n} q(i|\boldsymbol{\theta}) \cdot p(\boldsymbol{a}|\boldsymbol{a}|\boldsymbol{\theta}\_{i}) = \frac{1}{1+p} \cdot \sum\_{i=0}^{n} q(i|\boldsymbol{\theta}) \cdot p^{a-i} \tag{61}$$

From the differential equation

$$p(\mathfrak{a}|\mathfrak{a}|\mathfrak{g}) = \frac{q(\mathfrak{a} - 1|\mathfrak{g}) \cdot p + q(\mathfrak{a}|\mathfrak{g})}{1 + p}, \quad \mathfrak{a} \ge 1 \tag{62}$$

with the condition for

10 Will-be-set-by-IN-TECH

<sup>1</sup> <sup>−</sup> *<sup>p</sup>*(W) <sup>−</sup> *<sup>p</sup>*

W ≥ 0 resp. *W* = *ε* ·W ≥ 0, *ε* ∈ **S**(*ε*) (52) *W* ≥ *WKrit*, *WKrit* = 0 (53)

and, then *p*(W) ≥ *p*, 1 − *p*(W) ≤ 1 − *p* (54)

, *p*(W) > *p*, W > 0 (55)

<sup>k</sup>*T*<sup>0</sup> = *p* and thus *TW* ≥ *T*<sup>0</sup> (56)

, *p* ∈ (0, 1), *i* = 0, 1, ... (57)

*i* · *q*(*i*|*θ*) = *E*(*θ*) (58)

<sup>1</sup> <sup>+</sup> *<sup>p</sup>* (59)

*α* · *p*(*α*|*α*|*θ*) (60)

By (31) the condition for the *minimal* average energy *WKrit* needed for coding the input

From (47) and (54) follows that for the defined direction of the signal (messaage) transmission

Let is now considered, in the same way as it was in the B–E system, the pure states *θ<sup>i</sup>* ≡ *i* of the system **Ψ** which are coding the input messages *i* = 0, 1, ... and the output stochastic quantity *α* having its selecting space **S**. On the spectrum **S** probabilities of realizations *α* ∈ **S**

expressing the additive stochastic transformation of an input *i* into the output *α* for wich is valid *α* = *i* or *α* = *i* + 1.<sup>5</sup> The uniform energy level *ε* = const. of particles is considered. The quantity *W* is the mathematical expectation of the energy coding the input signal

*i*∈**S**

<sup>1</sup> <sup>+</sup> *<sup>p</sup>* <sup>=</sup> *<sup>p</sup>*

*<sup>j</sup>* · *<sup>p</sup><sup>j</sup>*

*α*∈**S**(*α*)

1 − *p*

<sup>−</sup> *<sup>ε</sup>*

<sup>1</sup> <sup>−</sup> *<sup>p</sup>* (51)

By (35), (36) and (46) the medium value W of the input message *i* ∈ **S** is derived,

<sup>W</sup> <sup>=</sup> *<sup>p</sup>*(W)

The relations (35), (36), (46) and (52), (53) yield in

<sup>1</sup> <sup>−</sup> *<sup>p</sup>*(W) <sup>≥</sup> *<sup>p</sup>*

*p*(W) = *e*

**4.2. Capacity of Fermi–Dirac narrow–band channel**

at the temperature *TW* of its sending and decoding is valid that

*p*(W) <sup>1</sup> <sup>−</sup> *<sup>p</sup>*(W) <sup>&</sup>gt; *<sup>p</sup>*

> <sup>−</sup> *<sup>ε</sup>* <sup>k</sup>*TW* ≥ *e*

*<sup>p</sup>*(*α*|*α*|*θi*) = *<sup>p</sup>α*−*<sup>i</sup>*

1 + *p*

*<sup>W</sup>* = *<sup>ε</sup>* · W, W = ∑

The medium value of a stochastic quantity with the F–D statistic is given by

∑ *j*∈{0,1}

*E*(*α*) = ∑

<sup>5</sup> In accordance with *Pauli excluding principle* (valid for *fermions*) and a given energetic level *<sup>ε</sup>* <sup>∈</sup> **<sup>S</sup>**(*ε*).

The quantity *E*(*α*) is the medium value of the output quantity *α*,

1 − *p*

*<sup>E</sup>*(*α*) = *<sup>p</sup>*(W)

message is

are defined,

$$a = 0, \quad p(0|\mathfrak{a}|\mathfrak{g}) = \frac{1}{1+p} \cdot q(0|\mathfrak{g})$$

follows that for the medium value *E*(*α*) of the output stochastic variable *α* is valid that

$$E(\mathfrak{a}) = \frac{p}{1+p} \cdot \sum\_{a \ge 1} \mathfrak{a} \cdot q(\mathfrak{a} - 1|\mathfrak{g}) + \frac{1}{1+p} \cdot \sum\_{a \ge 1} \mathfrak{a} \cdot q(\mathfrak{a}|\mathfrak{g}) \tag{63}$$

$$= \frac{p}{1+p} \cdot \sum\_{a \ge 1} (\mathfrak{a} - 1) \cdot q(\mathfrak{a} - 1|\mathfrak{g}) + \frac{p}{1+p} \cdot \sum\_{a \ge 1} q(\mathfrak{a} - 1|\mathfrak{g}) + \frac{1}{1+p} \cdot \mathcal{W}$$

$$= \frac{p}{1+p} \cdot \mathcal{W} + \frac{p}{1+p} + \frac{1}{1+p} \cdot \mathcal{W} \quad \longrightarrow \quad E(\mathfrak{a}) = \frac{p}{1+p} + \mathcal{W}, \ \mathcal{W} = E(\mathfrak{e}), \ \mathfrak{e} \in \mathfrak{G}\_{0}$$

The quantity *H*(*α*�*θi*) is the *p*-entropy of measuring *α* for the input *i* ∈ **S** being represented by the pure state *θ<sup>i</sup>* of the system **Ψ**F−D,*<sup>ε</sup>*

$$H(\mathfrak{a} \| \boldsymbol{\theta}\_{i}) = -\sum\_{j=0}^{1} \frac{p^{j}}{1+p} \cdot \ln \frac{p^{j}}{1+p} = -\frac{1}{1+p} \cdot \ln \frac{1}{1+p} - \frac{p}{1+p} \cdot \ln \frac{p}{1+p} \tag{64}$$

$$= -\left(1 - \frac{p}{1+p}\right) \cdot \ln \left(1 - \frac{p}{1+p}\right) - \frac{p}{1+p} \cdot \ln \frac{p}{1+p} = h\left(\frac{p}{1+p}\right), \ \forall i \in \mathbf{S}$$

The quantity *H*(*α*|*θ*) is the conditional (the *noise*) Shannon entropy of the stochastic quantity *α* in the system state *θ*, but, independent on this *θ*,

$$H(\mathfrak{a}|\mathfrak{g}) = \sum\_{i=0}^{n} q(i|\mathfrak{e}) \cdot H(\mathfrak{a}|\mathfrak{f}\_{i}) = \sum\_{i} q(i|\mathfrak{e}) \cdot h\left(\frac{p}{1+p}\right) = h\left(\frac{p}{1+p}\right) \tag{65}$$

For capacity *C*F−D,*<sup>ε</sup>* of the channel K∼=**Ψ**F−D,*<sup>ε</sup>* is, by the capacity definition in (23)-(24), valid that

$$C\_{\mathcal{F}-\mathcal{D}''} = \sup\_{\theta \in \Theta\_0} H(\mathfrak{a} \| \theta) - H(\mathfrak{a} \| \theta) = \sup\_{\theta \in \Theta\_0} H(\mathfrak{a} \| \theta) - h\left(\frac{p}{1+p}\right) \tag{66}$$

where the set **Θ**<sup>0</sup> = {*θ* ∈ **Θ** : *E*(*θ*) = W > 0} represents the coding procedure.

The quantity *H*(*α*�*θ*) = *H*(*p*(·|*α*|*θ*)) is the *p*-entropy of the stochastic quantity *α* in the state *θ* of the system **Ψ**F−D,*ε*. Its supremum is determined by the **Lagrange** multipliers method in the same way as in B–E case and with the same results for the probility distribution *p*(·|*α*|*θ*) (geometric) and the medium value *E*(*α*)

$$p(\cdot) = p(\cdot|\mathfrak{a}|\mathfrak{G}) = [1 - p(\mathcal{W})] \cdot p(\mathcal{W})^{\mathfrak{a}}, \ \mathfrak{a} \in \mathbf{S}(\mathfrak{a}) \tag{67}$$

$$E(\mathfrak{a}) = \frac{p(\mathcal{W})}{1 - p(\mathcal{W})'} \ \ p(\mathcal{W}) = e^{-\frac{\mathfrak{e}}{kT\_{\mathcal{W}}}}$$

**5. Wide–band quantum transfer channels**

of a particle, having the spectrum of eigenavalues

**<sup>K</sup>** <sup>=</sup> <sup>×</sup>*<sup>ε</sup>*∈**S**(*ε*) <sup>K</sup>*<sup>ε</sup>* <sup>=</sup> <sup>×</sup>*<sup>ε</sup>*∈**S**(*ε*)

*<sup>i</sup>* <sup>=</sup> <sup>×</sup>*<sup>ε</sup>*∈**S**(*ε*) *<sup>i</sup><sup>ε</sup>* <sup>=</sup> <sup>×</sup>*<sup>ε</sup>*∈**S**(*ε*)

*<sup>α</sup>* <sup>=</sup> <sup>×</sup>*<sup>ε</sup>*∈**S**(*ε*)

independent stochastic quantities too.

∏ *ε*∈**S**(*ε*)

∏ *ε*∈**S**(*ε*)

*ε*∈**S**(*ε*)

individual narrow–band components K*ε*.

*αε*, *j*

� = ∑ *ε*∈**S**(*ε*)

**K**. In this channel the stochastic transformation of the input *i*

*jε*, *i*

Realizations of the stochastic systems *i*, *α*, *θ*, *j* are the *vectors (sequences) i*, *α*, *θ*, *j*

are

Values *i* � , *j* � , *α*�

> � +*i* �

; *α*� = ∑

*α*� = *j*

*<sup>τ</sup>* , ..., *nh*

(arranged) set of narrow–band, *independent* components K*ε*, *ε* ∈ **S**(*ε*),

*<sup>τ</sup>* , ...

0, *<sup>h</sup> <sup>τ</sup>* , <sup>2</sup>*<sup>h</sup>*

**<sup>S</sup>**(*ε*) =

Till now the narrow–band variant of an information transfer channel K*ε*, *ε* ∈ **S**(*ε*), card **S**(*ε*) = 1 has been dealt. Let is now considered the symmetric operator of energy *ε*

> = *rh τ*

where *τ* > 0 denotes the time length of the input signal and *h* denotes **Planck** constant. The multi–band physical transfer channel **K**, memory-less, with additive noise is defend by the

*iε*, *p*(*αε*|*αε*|*θε*,*iε*), *α<sup>ε</sup>*

*<sup>α</sup><sup>ε</sup>* <sup>=</sup> <sup>×</sup>*<sup>ε</sup>*∈**S**(*ε*) [**S**, *<sup>p</sup>*(*αε*|*αε*|*θε*)], *αε* <sup>∈</sup> **<sup>S</sup>** <sup>=</sup> {0, 1 , 2, ...}

Due the independency of narrow–band components K*<sup>ε</sup>* the vector quantities *iε*, *αε*, *θε*, *j<sup>ε</sup>* are

The simultaneous *q*-distribution of the input vector of *i<sup>ε</sup>* and the simultaneous *p*-distribution of measuring the output vector of values *αε* (of the individual narrow–band components K*ε*)

The system of quantities *θ<sup>ε</sup>* (the set of states of the narrow–band components K*ε*) is the state *θ* of the multi–band channel **K** in which the (canonic) *q*-distribution of the system **K** is defined.

> � = ∑ *ε*∈**S**(*ε*)

are the numbers of the input, output and additive (noise) particles of the *multi–band* channel

being determined by additive stochastic transformations of the input *i<sup>ε</sup>* into the output *αε* in

*<sup>i</sup>* = (*iε*)*ε*∈**S**(*ε*), *<sup>α</sup>* = (*αε*)*ε*∈**S**(*ε*), *<sup>θ</sup>* = (*θε*)*ε*∈**S**(*ε*), *<sup>j</sup>* = (*jε*)*ε*∈**S**(*ε*); *<sup>i</sup>*, *<sup>α</sup>*, *<sup>j</sup>* <sup>∈</sup> <sup>×</sup>*<sup>ε</sup>*∈**S**(*ε*)

*<sup>i</sup>ε*, *αε*, *<sup>j</sup><sup>ε</sup>* ∈ **<sup>S</sup>**, *θε* ∈ **<sup>S</sup>**(*θε*), *<sup>θ</sup><sup>ε</sup>* = ∑

*r*=0, 1, ..., *n*

Information Capacity of Quantum Transfer Channels and Thermodynamic Analogies 95

[**S**, *qε*(*iε*|*θε*)], *i<sup>ε</sup>* ∈ **S** = {0, 1, 2, ...}

=

*<sup>q</sup>ε*(*iε*|*θε*) = *<sup>q</sup>*(*i*|*θ*), *<sup>θ</sup>* <sup>=</sup> <sup>×</sup>*<sup>ε</sup>*∈**S**(*ε*) *<sup>θ</sup>ε*, *<sup>i</sup>* <sup>∈</sup> **<sup>S</sup>**(*i*) (77)

*<sup>p</sup>*(*αε*|*αε*|*θε*) = *<sup>p</sup>*(*α*|*α*|*θ*), *<sup>θ</sup>* <sup>=</sup> <sup>×</sup>*<sup>ε</sup>*∈**S**(*ε*) *<sup>θ</sup>ε*, *<sup>α</sup>* <sup>∈</sup> **<sup>S</sup>**(*α*) (78)

*iε*∈**S**

*i*, *p*(*α*|*α*|*θ<sup>i</sup>*

*iε*; *j<sup>ε</sup>* = *αε* − *i<sup>ε</sup>* ≥ 0, ∀*ε* ∈ **S**(*ε*), card**S***<sup>ε</sup>* > 1

*θε θε*,*i<sup>ε</sup>* , *θ* ∈ **S**(*θ*)

� into the output *α*� is performed,

, card **S**(*ε*) = *n* + 1 (75)

), *α*

(76)

(79)

**S**, (80)

Again , the value *<sup>E</sup>*(*α*) depends on *<sup>ε</sup>* k*TW* , or on absolute temperature *TW* respectively, only. By using *E*(*α*) in (63) it is seen that *p*(W) or *TW* respectively is the only one root of the equation [12, 30]

$$\frac{p(\mathcal{W})}{1 - p(\mathcal{W})} = \frac{p}{1 + p} + \mathcal{W}\_{\prime} \quad \text{resp.} \quad \frac{e^{-\frac{\ell}{kT\_{\mathcal{W}}}}}{1 - e^{-\frac{\ell}{kT\_{\mathcal{W}}}}} = \frac{e^{-\frac{\ell}{kT\_{0}}}}{1 + e^{-\frac{\ell}{kT\_{0}}}} + \mathcal{W} \tag{68}$$

For the *q*-distribution *q*(·|*θ*) = *p*(·) of states *θ* ∈ **Θ**0, for which the relation (62) and (67) is gained, follows that

$$\frac{q(a-1|\boldsymbol{\theta}) \cdot p + q(a|\boldsymbol{\theta})}{1+p} = [1 - p(\mathcal{W})] \cdot p(\mathcal{W})^a, \quad a \in \mathbf{S} \text{ with conditions} \quad \text{(69)}$$

$$\begin{aligned} q(0|\boldsymbol{\theta}) &= (1+p) \cdot [1 - p(\mathcal{W})] \quad \text{and} \; q(1|\boldsymbol{\theta}) = (1+p) \cdot [1 - p(\mathcal{W})] \cdot [p(\mathcal{W}) - p];\\ q(a|\boldsymbol{\theta}) &= (1+p) \cdot [1 - p(\mathcal{W})] \cdot \left[ \left(\sum\_{i=0}^{a-1} (-1)^i \cdot p(\mathcal{W})^{a-i} \cdot p^i \right) + (-1)^a \cdot p^a \right], a > 1 \end{aligned}$$

For the *effective temperatutre TW of coding the input messages* the distribution (67) supremises (maximizes) the *p*-entropy *H*(*α*�*θ*) of *α* is valid, in the same way as in (49), that

$$\sup\_{\theta \in \Theta\_0} H(\mathfrak{a} || \theta) = \frac{h[p(\mathcal{W})]}{1 - p(\mathcal{W})} \tag{70}$$

By using (70) in (66) the formula for the *C*F−D,*<sup>ε</sup>* capacity [12, 37] is gained

$$\mathcal{C}\_{\text{F}-\text{D},\varepsilon} = \frac{h[p(\mathcal{W})]}{1 - p(\mathcal{W})} - h\left(\frac{p}{1 + p}\right) \tag{71}$$

**The medium value** W **of the input** *i* = 0, 1, 2, ... **is limited by a minimal not-zero and positive 'bottom' value** W*Krit***.** From (58), (63) and (68) follows

$$E(\mathfrak{a}) = \frac{p(\mathcal{W})}{1 - p(\mathcal{W})} \ge \frac{p}{1 - p'}, \ p(\mathcal{W}) = e^{-\frac{\phi}{\mathsf{E}\_W^2}} \ge e^{-\frac{\phi}{\mathsf{E}\_0^2}} = p \quad \text{and thus} \quad T\_W \ge T\_0 \tag{72}$$

$$\mathcal{W} = \frac{p(\mathcal{W})}{1 - p(\mathcal{W})} - \frac{p}{1 + p} \ge 0,\ \mathcal{W} \ge \frac{2p^2}{1 - p^2} = \mathcal{W}\_{\text{Krit}},\\\text{resp. } \mathcal{W} = \varepsilon \cdot \mathcal{W} \ge \varepsilon \cdot \frac{2\varepsilon^{-2}\overline{\mathfrak{u}\_0^{\varepsilon}}}{1 - \varepsilon^{-2}\frac{\overline{\mathfrak{u}\_0^{\varepsilon}}}{\overline{\mathfrak{u}\_0^{\varepsilon}}}} \tag{73}$$

**For the average coding energy** *W***, when the channel** *C*F−D,*<sup>ε</sup>* **acts on a uniform energetic level** *ε***, is**

$$W \ge W\_{Krit} = \frac{2p^2}{1 - p^2} \tag{74}$$

For the F–D channel is then possible speak about the *effect of the not-zero capacity when the difference between the coding temperatures TW and the noise temperature T*<sup>0</sup> **is zero**. <sup>6</sup> This phennomenon is, by necessity, *given by properties of cells of the F–D phase space*.

<sup>6</sup> Not *not-zero capacity* for zero input power as was stated in [37]. The (74) also repares small missprint in [11].

#### **5. Wide–band quantum transfer channels**

12 Will-be-set-by-IN-TECH

1 − *p*(W)

using *E*(*α*) in (63) it is seen that *p*(W) or *TW* respectively is the only one root of the equation

For the *q*-distribution *q*(·|*θ*) = *p*(·) of states *θ* ∈ **Θ**0, for which the relation (62) and (67) is

*q*(0|*θ*)=(1 + *p*) · [1 − *p*(W)] and *q*(1|*θ*)=(1 + *p*) · [1 − *p*(W)] · [*p*(W) − *p*];

For the *effective temperatutre TW of coding the input messages* the distribution (67) supremises

<sup>1</sup> <sup>−</sup> *<sup>p</sup>*(W) <sup>−</sup> *<sup>h</sup>*

**The medium value** W **of the input** *i* = 0, 1, 2, ... **is limited by a minimal not-zero and**

<sup>−</sup> *<sup>ε</sup>* <sup>k</sup>*TW* ≥ *e*

**For the average coding energy** *W***, when the channel** *C*F−D,*<sup>ε</sup>* **acts on a uniform energetic level**

*<sup>W</sup>* <sup>≥</sup> *WKrit* <sup>=</sup> <sup>2</sup>*p*<sup>2</sup>

For the F–D channel is then possible speak about the *effect of the not-zero capacity when the difference between the coding temperatures TW and the noise temperature T*<sup>0</sup> **is zero**.

<sup>6</sup> Not *not-zero capacity* for zero input power as was stated in [37]. The (74) also repares small missprint in [11].

*<sup>H</sup>*(*α*�*θ*) = *<sup>h</sup>*[*p*(W)]

*e* <sup>−</sup> *<sup>ε</sup>* k*TW*

1 − *e* <sup>−</sup> *<sup>ε</sup>* k*TW*

(−1)*<sup>i</sup>* · *<sup>p</sup>*(W)*α*−*<sup>i</sup>* · *<sup>p</sup><sup>i</sup>*

 *p* 1 + *p*

<sup>−</sup> *<sup>ε</sup>*

<sup>1</sup>−*p*<sup>2</sup> <sup>=</sup>W*Krit*, resp. *<sup>W</sup>* <sup>=</sup>*<sup>ε</sup>* ·W ≥ *<sup>ε</sup>* · <sup>2</sup>*<sup>e</sup>*

*<sup>E</sup>*(*α*) = *<sup>p</sup>*(W)

+ W, resp.

*<sup>α</sup>*−<sup>1</sup> ∑ *i*=0

(maximizes) the *p*-entropy *H*(*α*�*θ*) of *α* is valid, in the same way as in (49), that

*<sup>C</sup>*F−D,*<sup>ε</sup>* <sup>=</sup> *<sup>h</sup>*[*p*(W)]

, *p*(W) = *e*

<sup>≥</sup> 0, W ≥ <sup>2</sup>*p*<sup>2</sup>

phennomenon is, by necessity, *given by properties of cells of the F–D phase space*.

sup *<sup>θ</sup>*∈**Θ**<sup>0</sup>

By using (70) in (66) the formula for the *C*F−D,*<sup>ε</sup>* capacity [12, 37] is gained

**positive 'bottom' value** W*Krit***.** From (58), (63) and (68) follows

1 − *p*

k*TW*

1 + *p*

*q*(*α* − 1|*θ*) · *p* + *q*(*α*|*θ*)

Again , the value *<sup>E</sup>*(*α*) depends on *<sup>ε</sup>*

*p*(W) <sup>1</sup> <sup>−</sup> *<sup>p</sup>*(W) <sup>=</sup> *<sup>p</sup>*

*q*(*α*|*θ*)=(1 + *p*) · [1 − *p*(W)] ·

<sup>1</sup> <sup>−</sup> *<sup>p</sup>*(W) <sup>≥</sup> *<sup>p</sup>*

<sup>−</sup> *<sup>p</sup>* 1+*p*

[12, 30]

gained, follows that

*<sup>E</sup>*(*α*) = *<sup>p</sup>*(W)

<sup>W</sup><sup>=</sup> *<sup>p</sup>*(W) 1−*p*(W)

*ε***, is**

*<sup>p</sup>*(·) = *<sup>p</sup>*(·|*α*|*θ*)=[<sup>1</sup> <sup>−</sup> *<sup>p</sup>*(W)] · *<sup>p</sup>*(W)*α*, *<sup>α</sup>* <sup>∈</sup> **<sup>S</sup>**(*α*) (67)

<sup>−</sup> *<sup>ε</sup>* k*TW*

<sup>=</sup> *<sup>e</sup>*

<sup>1</sup> <sup>+</sup> *<sup>p</sup>* = [<sup>1</sup> <sup>−</sup> *<sup>p</sup>*(W)] · *<sup>p</sup>*(W)*α*, *<sup>α</sup>* <sup>∈</sup> **<sup>S</sup>** with conditions (69)

1 + *e* <sup>−</sup> *<sup>ε</sup>* k*T*0

+ (−1)*<sup>α</sup>* · *<sup>p</sup><sup>α</sup>*

<sup>1</sup> <sup>−</sup> *<sup>p</sup>*(W) (70)

<sup>k</sup>*T*<sup>0</sup> = *p* and thus *TW* ≥ *T*<sup>0</sup> (72)

<sup>1</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup> (74)

, *α* > 1

<sup>−</sup><sup>2</sup> *<sup>ε</sup>* k*T*0

1 − *e* <sup>−</sup><sup>2</sup> *<sup>ε</sup>* k*T*0 (71)

(73)

<sup>6</sup> This

, or on absolute temperature *TW* respectively, only. By

<sup>−</sup> *<sup>ε</sup>* k*T*0

+ W (68)

, *p*(W) = *e*

Till now the narrow–band variant of an information transfer channel K*ε*, *ε* ∈ **S**(*ε*), card **S**(*ε*) = 1 has been dealt. Let is now considered the symmetric operator of energy *ε* of a particle, having the spectrum of eigenavalues

$$\mathbf{S}(\varepsilon) = \left\{ 0, \frac{h}{\tau}, \frac{2h}{\tau}, \dots, \frac{nh}{\tau}, \dots \right\} = \left\{ \frac{rh}{\tau} \right\}\_{r=0, 1, \dots, n} \text{ / card } \mathbf{S}(\varepsilon) = n+1 \tag{75}$$

where *τ* > 0 denotes the time length of the input signal and *h* denotes **Planck** constant. The multi–band physical transfer channel **K**, memory-less, with additive noise is defend by the (arranged) set of narrow–band, *independent* components K*ε*, *ε* ∈ **S**(*ε*),

$$\mathbf{K} = \underset{\varepsilon \in \mathbf{S}(\varepsilon)}{\underset{\varepsilon \in \mathbf{S}(\varepsilon)}{\times}} \quad \mathcal{K}\_{\varepsilon} = \underset{\varepsilon \in \mathbf{S}(\varepsilon)}{\underset{\varepsilon \in \mathbf{S}(\varepsilon)}{\left\{i\_{\varepsilon} \mid p\left(a\_{\varepsilon} \middle| \left.\mathbf{a}\_{\varepsilon} \middle| \left.\theta\_{\varepsilon,i\_{\varepsilon}} \right|\right), \left.\mathbf{a}\_{\varepsilon} \right\} \right\}} = \left\{i \mid p(\overline{\boldsymbol{\pi}} \middle| \left.\mathbf{a} \middle| \left.\theta\_{\overline{\boldsymbol{\pi}}} \right|), \mathbf{a} \right\} \right\} \tag{76}$$

$$\dot{\boldsymbol{a}} = \underset{\varepsilon \in \mathbf{S}(\varepsilon)}{\underset{\varepsilon \in \mathbf{S}(\varepsilon)}{\left\{}}} \quad \dot{\boldsymbol{a}}\_{\varepsilon} = \underset{\varepsilon \in \mathbf{S}(\varepsilon)}{\left\{}} \left[\mathbf{S}, \ p(\boldsymbol{a}\_{\varepsilon} | \boldsymbol{a}\_{\varepsilon} | \theta\_{\varepsilon})\right], \ a\_{\varepsilon} \in \mathbf{S} = \left\{0, \ 1, 2, \ldots \right\} \right.$$

$$\boldsymbol{a} = \underset{\varepsilon \in \mathbf{S}(\varepsilon)}{\left.\right|} \underbrace{\boldsymbol{a}\_{\varepsilon}}\_{=} = \underset{\varepsilon \in \mathbf{S}(\varepsilon)}{\left.\right|} \left[\mathbf{S}, \ p(\boldsymbol{a}\_{\varepsilon} | \boldsymbol{a}\_{\varepsilon} | \theta\_{\varepsilon}) \right], \ a\_{\varepsilon} \in \mathbf{S} = \left\{0, \ 1, 2, \ldots \right\} \right.$$

Due the independency of narrow–band components K*<sup>ε</sup>* the vector quantities *iε*, *αε*, *θε*, *j<sup>ε</sup>* are independent stochastic quantities too.

The simultaneous *q*-distribution of the input vector of *i<sup>ε</sup>* and the simultaneous *p*-distribution of measuring the output vector of values *αε* (of the individual narrow–band components K*ε*) are

$$\prod\_{\varepsilon \in \mathbf{S}(\varepsilon)} q\_{\varepsilon}(i\_{\varepsilon}|\theta\_{\varepsilon}) = q(\overline{i}|\theta), \qquad \theta = \underset{\varepsilon \in \mathbf{S}(\varepsilon)}{\underset{\varepsilon \in \mathbf{S}(\varepsilon)}{\text{g}}} \theta\_{\varepsilon}, \quad \overline{i} \in \mathbf{S}(i) \tag{77}$$

$$\prod\_{\varepsilon \in \mathbf{S}(\varepsilon)} p(\mathfrak{a}\_{\varepsilon}|\mathfrak{a}\_{\varepsilon}|\mathfrak{e}\_{\varepsilon}) = p(\overline{\mathfrak{a}}|\mathfrak{a}|\mathfrak{e}), \quad \mathfrak{G} = \underset{\varepsilon \in \mathbf{S}(\varepsilon)}{\underset{\varepsilon \in \mathbf{S}(\varepsilon)}{\gtrless}} \mathfrak{e}\_{\varepsilon}, \quad \overline{\mathfrak{a}} \in \mathbf{S}(\mathfrak{a}) \tag{78}$$

The system of quantities *θ<sup>ε</sup>* (the set of states of the narrow–band components K*ε*) is the state *θ* of the multi–band channel **K** in which the (canonic) *q*-distribution of the system **K** is defined. Values *i* � , *j* � , *α*�

$$\mathfrak{a}' = \mathfrak{j}' + \mathfrak{i}'; \ \mathfrak{a}' = \sum\_{\varepsilon \in \mathbf{S}(\varepsilon)} \mathfrak{a}\_{\varepsilon \iota} \mathfrak{j}' = \sum\_{\varepsilon \in \mathbf{S}(\varepsilon)} \mathfrak{j}\_{\varepsilon \iota} \mathfrak{i}' = \sum\_{\varepsilon \in \mathbf{S}(\varepsilon)} \mathfrak{i}\_{\varepsilon \iota} \mathfrak{j}\_{\varepsilon} = \mathfrak{a}\_{\varepsilon} - i\_{\varepsilon} \ge 0, \ \forall \varepsilon \in \mathbf{S}(\varepsilon), \ \text{card}\mathfrak{S}\_{\varepsilon} > 1 \tag{79}$$

are the numbers of the input, output and additive (noise) particles of the *multi–band* channel **K**. In this channel the stochastic transformation of the input *i* � into the output *α*� is performed, being determined by additive stochastic transformations of the input *i<sup>ε</sup>* into the output *αε* in individual narrow–band components K*ε*.

Realizations of the stochastic systems *i*, *α*, *θ*, *j* are the *vectors (sequences) i*, *α*, *θ*, *j*

$$\overline{\mathbf{i}} = \left(\mathbf{i}\_{\varepsilon}\right)\_{\varepsilon \in \mathbf{S}(\varepsilon)^{\varepsilon}} \quad \overline{\mathbf{a}} = \left(\mathbf{a}\_{\varepsilon}\right)\_{\varepsilon \in \mathbf{S}(\varepsilon)^{\varepsilon}} \quad \overline{\theta} = \left(\theta\_{\varepsilon}\right)\_{\varepsilon \in \mathbf{S}(\varepsilon)^{\varepsilon}} \quad \overline{\mathbf{j}} = \left(\mathbf{j}\_{\varepsilon}\right)\_{\varepsilon \in \mathbf{S}(\varepsilon)} ; \ \overline{\mathbf{i}} , \ \overline{\mathbf{a}} , \overline{\mathbf{j}} \in \underset{\varepsilon \in \mathbf{S}(\varepsilon)}{\operatorname{\mathbf{s}}} \quad \text{(80)}$$

$$\mathbf{i}\_{\varepsilon \wedge} \mathbf{a}\_{\varepsilon \wedge} \mathbf{j}\_{\varepsilon} \in \mathbf{S} , \ \theta\_{\varepsilon} \in \mathbf{S}(\theta\_{\varepsilon}) , \quad \theta\_{\varepsilon} = \sum\_{i\_{\varepsilon} \in \mathbf{S}} \theta\_{\varepsilon} \, \theta\_{\varepsilon \dot{j}\_{\varepsilon'}} \qquad \overline{\theta} \in \mathbf{S}(\theta)$$

#### 14 Will-be-set-by-IN-TECH 96 Thermodynamics – Fundamentals and Its Application in Science Information Capacity of Quantum Transfer Channels and Thermodynamic Analogies <sup>15</sup>

For the probability of the additive stochastic transformation (77), (78) of input *i* into the output *α* is valid

$$\prod\_{\varepsilon \in \mathbf{S}(\varepsilon)} p(a\_{\varepsilon}|\mathfrak{a}\_{\varepsilon}|\boldsymbol{\theta}\_{\varepsilon,\hat{\imath}\_{i}}) = p(\overline{\mathfrak{a}}|\mathfrak{a}|\boldsymbol{\theta}\_{\overline{\mathfrak{f}}}), \ \overline{\mathfrak{a}} = \overline{\mathfrak{f}} + \overline{\mathfrak{i}}, \ \overline{\mathfrak{i}} \in \mathbf{S}(\boldsymbol{i}), \ \overline{\mathfrak{j}} \in \mathbf{S}(\boldsymbol{j}), \ \overline{\mathfrak{a}} \in \mathbf{S}(\boldsymbol{\mathfrak{a}}) \tag{81}$$

The symbol *θε*,*i<sup>ε</sup>* denotes the pure state coding the input *i<sup>ε</sup>* ∈ **S** of a narrow–band component <sup>K</sup>*<sup>ε</sup>* and the state *<sup>θ</sup><sup>i</sup>* <sup>=</sup> ×*<sup>ε</sup>*∈**S**(*ε*)*θε*,*i<sup>ε</sup>* codes the input *<sup>i</sup>* for which

$$q(\tilde{i}|\boldsymbol{\theta}) = q\_{\overline{\boldsymbol{\theta}}} = \prod\_{\boldsymbol{\varepsilon} \in \mathbf{S}(\boldsymbol{\varepsilon})} q\_{\boldsymbol{\varepsilon}}(\boldsymbol{\theta}\_{\boldsymbol{\varepsilon}}) \tag{82}$$

**5.1. Transfer channels with continuous energy spectrum**

<sup>Δ</sup>*<sup>ε</sup>* <sup>=</sup> *<sup>n</sup>* <sup>+</sup> 1, *<sup>n</sup>* · <sup>Δ</sup>*<sup>ε</sup>* <sup>=</sup> *<sup>n</sup>* ·

*<sup>τ</sup>* <sup>=</sup> lim*τ*→<sup>∞</sup> *<sup>r</sup>* · <sup>Δ</sup>*<sup>ε</sup>*

*rh τ* 

lim*τ*→<sup>∞</sup> 1 *<sup>τ</sup>* <sup>=</sup> <sup>d</sup>*<sup>ε</sup>*

= *iε*, *j* �

the output *<sup>α</sup>* of the wide–band transfer channel **<sup>K</sup>**B−E|F−<sup>D</sup> is valid that

sup *θε*

1 *<sup>τ</sup>* ∑ *ε*∈**S**(*ε*)

**S**(*ε*) = {*εr*}*r*=0, 1, ..., *<sup>n</sup>* =

*rh*

and thus the *wide–band spectrum* **S**(*ε*) *of energies* is

� = *α*, *i* �

1 *<sup>τ</sup>* ∑ *ε*∈**S**(*ε*)

1 *<sup>τ</sup>* ∑ *ε*∈**S**(*ε*)

 ∑ *α*∈**S**

card **<sup>S</sup>**(*ε*) = *<sup>ε</sup><sup>n</sup>*

*band–width* equal to card **<sup>S</sup>**(*ε*) = *<sup>ε</sup><sup>n</sup>*

lim*τ*→<sup>∞</sup> *<sup>ε</sup><sup>r</sup>* <sup>=</sup> lim*τ*→<sup>∞</sup>

**S**(*ε*) = {*εr*}*r*=0, 1, ... =

considered

with. Then

sup *θ*

With the denotation *αε*

*<sup>H</sup>*(*α*�*θ*) = lim*τ*→<sup>∞</sup>

*<sup>H</sup>*(*α*�*θ*) = lim*τ*→<sup>∞</sup>

capacity is valid, by (85) and (86)

*<sup>C</sup>*(**K**B−E|F−D)== <sup>1</sup>

*<sup>H</sup>*(*α*|*θ*) = lim*τ*→<sup>∞</sup>

<sup>=</sup> <sup>1</sup> *h* ∞ 0

> *h* ∞ 0

<sup>=</sup> <sup>−</sup><sup>1</sup> *h* ∞ 0

Let a spectrum of energy with the finite cardinality *n* + 1 and a finite time interval *τ* > 0 are

*r*=0, 1, ..., *n*

<sup>=</sup> *<sup>r</sup>* <sup>d</sup> *<sup>ε</sup>* resp. lim*τ*→<sup>∞</sup>

<sup>=</sup> lim*τ*→<sup>∞</sup> *<sup>ε</sup><sup>r</sup>* <sup>=</sup> lim*τ*→<sup>∞</sup>

*<sup>τ</sup>* ∑ *ε*∈**S**(*ε*)

> d*ε*

1 *<sup>τ</sup>* ∑ *ε*∈**S**(*ε*)

 *h*(*pε*) 1 − *p<sup>ε</sup>*

*h <sup>τ</sup>* <sup>=</sup> *<sup>ε</sup>n*, *<sup>τ</sup>*

For a transfer channel with the continuous spectrum of energies of particles and with the

But the infinite wide–band and infinite number of particles (*τ* −→ ∞, *n* −→ ∞) will be dealt

, <sup>Δ</sup>*<sup>ε</sup>* <sup>=</sup> *<sup>h</sup>*

Information Capacity of Quantum Transfer Channels and Thermodynamic Analogies 97

*<sup>n</sup>* <sup>=</sup> *<sup>h</sup> εn*

> 1 *<sup>τ</sup>* <sup>=</sup> <sup>d</sup>*<sup>ε</sup>*

> > *rh <sup>τ</sup>* <sup>=</sup> lim *ε*→0

*<sup>h</sup>* , **<sup>S</sup>**(*ε*) = �0, <sup>∞</sup>) (90)

= *jε*, *iε*, *j<sup>ε</sup>* ∈ **S**, *αε* ∈ **S**, *ε* ∈ **S**(*ε*) For the *p*-entropy of

∑ *αε*∈**S**

*h*[*pε*(W)] <sup>1</sup> <sup>−</sup> *<sup>p</sup>ε*(W) <sup>d</sup>*<sup>ε</sup>*

> ∑ *i*∈**S**

> > *h p<sup>ε</sup>* 1 + *p<sup>ε</sup>*

 *h p<sup>ε</sup>* 1 + *p<sup>ε</sup>*

 *h*(*pε*) 1 − *p<sup>ε</sup>* *<sup>τ</sup>* , *<sup>ε</sup><sup>r</sup>* <sup>=</sup> *rh*

= const.

*<sup>τ</sup>* <sup>=</sup> *<sup>r</sup>* · <sup>Δ</sup>*ε*, (87)

*<sup>h</sup>* , **<sup>S</sup>**(*ε*) = �0,*εn*) (88)

*p*(*αε*|*αε*|*θε*) · ln *p*(*αε*|*αε*|*θε*)(91)

*q*(*i*|*θε*,*i*) · *H*(*αε*�*θε*,*i*) (92)

<sup>d</sup>*<sup>ε</sup>* (93)

<sup>d</sup>*<sup>ε</sup>*

*r* Δ*ε* = *r* d *ε* (89)

*rh τ* 

*<sup>h</sup>* , is valid that

�

*r*=0, 1, ...

*<sup>H</sup>*(*αε*�*θε*) = lim*τ*→<sup>∞</sup> <sup>−</sup> <sup>1</sup>

*p*(*α*|*αε*|*θε*) · ln *p*(*α*|*αε*|*θε*)

*h* ∞ 0

For conditional entropy of the wide–band transfer channel **<sup>K</sup>**B−E|F−<sup>D</sup> [entropy of the wide–band noise independent on the system (**K**B−E|F−D) state *<sup>θ</sup>*] and for its information

*<sup>H</sup>*(*αε*|*θε*) = lim*τ*→<sup>∞</sup>

*h* ∞ 0

<sup>d</sup>*<sup>ε</sup>* <sup>−</sup> <sup>1</sup> *h* ∞ 0

*<sup>H</sup>*(*αε*|*θε*) <sup>d</sup>*<sup>ε</sup>* <sup>=</sup> <sup>1</sup>

*h*[*pε*(W)] 1 − *pε*(W)

*<sup>H</sup>*(*αε*�*θε*) = <sup>1</sup>

For the multi–band channel **K** the following quantities are defined:<sup>7</sup>

• the *p-entropy of the output α*

$$\begin{split} H(\mathfrak{a} \| \boldsymbol{\theta}) &= \sum\_{\boldsymbol{\varepsilon} \in \mathbf{S}(\boldsymbol{\varepsilon})} H(\mathfrak{a}\_{\boldsymbol{\varepsilon}} \| \boldsymbol{\theta}\_{\boldsymbol{\varepsilon}}) = - \sum\_{\boldsymbol{\varepsilon} \in \mathbf{S}(\boldsymbol{\varepsilon})} \sum\_{\boldsymbol{\alpha}\_{\boldsymbol{\varepsilon}} \in \mathbf{S}} p(\mathfrak{a}\_{\boldsymbol{\varepsilon}} | \mathfrak{a}\_{\boldsymbol{\varepsilon}} | \boldsymbol{\theta}\_{\boldsymbol{\varepsilon}}) \cdot \ln p(\mathfrak{a}\_{\boldsymbol{\varepsilon}} | \mathfrak{a}\_{\boldsymbol{\varepsilon}} | \boldsymbol{\theta}\_{\boldsymbol{\varepsilon}}) \\ &\leq \sum\_{\boldsymbol{\varepsilon} \in \mathbf{S}(\boldsymbol{\varepsilon})} \sup\_{\boldsymbol{\theta}\_{\boldsymbol{\varepsilon}}} H(\mathfrak{a}\_{\boldsymbol{\varepsilon}} | \boldsymbol{\theta}\_{\boldsymbol{\varepsilon}}) \end{split} \tag{83}$$

for which, following the output narrow–band B–E and F–D components K*<sup>ε</sup>* ∈ **K**, is valid that

$$\sup\_{\theta \in \overline{\Theta}\_{0}} H(\mathfrak{a} \| \theta) = \sup\_{\theta \in \overline{\Theta}\_{0}} \sum\_{\varepsilon \in \mathbf{S}(\varepsilon)} H(\mathfrak{a}\_{\varepsilon} \| \theta\_{\varepsilon}) = \sum\_{\varepsilon \in \mathbf{S}(\varepsilon)} \sup\_{\theta\_{\varepsilon}} H(\mathfrak{a}\_{\varepsilon} \| \theta\_{\varepsilon}) = \sum\_{\varepsilon \in \mathbf{S}(\varepsilon)} \frac{h[p\_{\varepsilon}(\mathcal{W})]}{1 - p\_{\varepsilon}(\mathcal{W})} \tag{84}$$

• the conditional *noise entropy* (entropy of the *multi–band B–E* | *F–D noise*)

$$H(\mathfrak{a}|\mathfrak{\theta}) = \sum\_{\varepsilon \in \mathbf{S}(\varepsilon)} H(\mathfrak{a}\_{\varepsilon}|\mathfrak{\theta}\_{\varepsilon}) = \sum\_{\varepsilon \in \mathbf{S}(\varepsilon)} \sum\_{i \in \mathbf{S}} q(i | \mathfrak{f}\_{\varepsilon,i}) \cdot H(\mathfrak{a}\_{\varepsilon} | \mathfrak{f}\_{\varepsilon,i}) = \sum\_{\varepsilon \in \mathbf{S}(\varepsilon)} \left[ \frac{h(p\_{\varepsilon})}{1 - p\_{\varepsilon}} \left| h \left( \frac{p\_{\varepsilon}}{1 + p\_{\varepsilon}} \right) \right| \left| 85 \right) \right]$$
  $\text{where } p\_{\varepsilon}(W) = e^{-\frac{\mathfrak{f}}{kT\_{0W}}}, \text{ } p\_{\varepsilon} = e^{-\frac{\mathfrak{f}}{kT\_{0}}}, \text{ } T\_{W} \ge T\_{0} > 0 \text{ and } h(p) = -p \ln p - (1 - p) \ln(1 - p).$ 

• the *transinformation T*(*α*; *θ*) and the *information capacity C*(**K**),

$$\mathcal{C}(\mathbf{K}) = \sup\_{\theta \in \overline{\Theta}\_0} T\left(\mathfrak{a}; \theta\right) = \sup\_{\theta \in \overline{\Theta}\_0} H\left(\mathfrak{a} \|\left|\theta\right.\right) - H\left(\mathfrak{a} \|\theta\right.\right) \tag{86}$$

$$= \sum\_{\varepsilon \in \mathbf{S}(\varepsilon)} \frac{h[p\_\varepsilon(\mathcal{W})]}{1 - p\_\varepsilon(\mathcal{W})} - \sum\_{\varepsilon \in \mathbf{S}(\varepsilon)} \left[\frac{h(p\_\varepsilon)}{1 - p\_\varepsilon} \left|h\left(\frac{p\_\varepsilon}{1 + p\_\varepsilon}\right)\right|\right]$$

The set **<sup>Θ</sup>**<sup>0</sup> <sup>=</sup> <sup>×</sup>*<sup>ε</sup>*∈**S**(*ε*) {*θ<sup>ε</sup>* ∈ Θ*ε*; *E*(*θε*) = *W<sup>ε</sup>* ≥ 0} represents a coding procedure of the input *i* of the **K** into *θ<sup>i</sup>* , [by transforming each input *i<sup>ε</sup>* into pure state *θε*,*i<sup>ε</sup>* , ∀*ε* ∈ **S**(*ε*)].

<sup>7</sup> Using the *chain rule* for simultaneous probabilities it is found that for information entropy of an independent stochastic system −→*<sup>X</sup>* = (*X*1, *<sup>X</sup>*2, ..., *Xn*) is valid that *<sup>H</sup>*( −→*<sup>X</sup>* ) = ∑ *i <sup>H</sup>*(*Xi*|*X*1, ...*Xi*−<sup>1</sup>) = ∑ *i H*(*Xi*). Thus the physical entropy H(*θ*) of independent stochastic system, *θ* = {*θε*}*ε*, is the sum of *Hε*[*q*(·|*θε*)] over *ε* ∈ **S***<sup>ε</sup>* too.

#### **5.1. Transfer channels with continuous energy spectrum**

14 Will-be-set-by-IN-TECH

For the probability of the additive stochastic transformation (77), (78) of input *i* into the output

The symbol *θε*,*i<sup>ε</sup>* denotes the pure state coding the input *i<sup>ε</sup>* ∈ **S** of a narrow–band component

*ε*∈**S**(*ε*)

for which, following the output narrow–band B–E and F–D components K*<sup>ε</sup>* ∈ **K**, is valid that

*ε*∈**S**(*ε*)

*<sup>q</sup>*(*i*|*θε*,*i*) · *<sup>H</sup>*(*αε*�*θε*,*i*) = ∑

sup *θε*

*<sup>H</sup>*(*αε*�*θε*) = ∑

*ε*∈**S**(*ε*)

<sup>k</sup>*T*<sup>0</sup> , *TW* ≥ *T*<sup>0</sup> > 0 and *h*(*p*) = −*p* ln *p* − (1 − *p*)ln(1 −

*<sup>H</sup>*(*αε*�*θε*) = ∑

*ε*∈**S**(*ε*)

∑ *αε*∈**S**

*<sup>q</sup>*(*i*|*θ*) = *<sup>q</sup><sup>θ</sup>* = ∏

), *α* = *j* + *i*, *i* ∈ **S**(*i*), *j* ∈ **S**(*j*), *α* ∈ **S**(*α*) (81)

*qε*(*θε*) (82)

*p*(*αε*|*αε*|*θε*) · ln *p*(*αε*|*αε*|*θε*) (83)

*ε*∈**S**(*ε*)

 *h*(*pε*) 1 − *p<sup>ε</sup>*

*H* (*α*�*θ*) − *H* (*α*|*θ*) (86)

*H*(*Xi*). Thus the physical entropy H(*θ*)

 *h p<sup>ε</sup>* 1 + *p<sup>ε</sup>*

*h*[*pε*(W)] <sup>1</sup> <sup>−</sup> *<sup>p</sup>ε*(W) (84)

> (85)

*α* is valid

∏ *ε*∈**S**(*ε*)

• the *p-entropy of the output α*

sup *<sup>θ</sup>*∈**Θ**<sup>0</sup>

*<sup>H</sup>*(*α*|*θ*) = ∑

where *pε*(*W*) = *e*

The set **<sup>Θ</sup>**<sup>0</sup> <sup>=</sup> <sup>×</sup>*<sup>ε</sup>*∈**S**(*ε*)

of the **K** into *θ<sup>i</sup>*

*p*).

*ε*∈**S**(*ε*)

*<sup>H</sup>*(*α*�*θ*) = ∑

*H*(*α*�*θ*) = sup

*ε*∈**S**(*ε*)

≤ ∑ *ε*∈**S**(*ε*)

*<sup>θ</sup>*∈**Θ**<sup>0</sup>

*<sup>H</sup>*(*αε*|*θε*) = ∑

<sup>−</sup> *<sup>ε</sup>* <sup>k</sup>*T*0*<sup>W</sup>* , *<sup>p</sup><sup>ε</sup>* <sup>=</sup> *<sup>e</sup>*

*C*(**K**) = sup

system −→*<sup>X</sup>* = (*X*1, *<sup>X</sup>*2, ..., *Xn*) is valid that *<sup>H</sup>*(

*p*(*αε*|*αε*|*θε*,*iε*) = *p*(*α*|*α*|*θ<sup>i</sup>*

<sup>K</sup>*<sup>ε</sup>* and the state *<sup>θ</sup><sup>i</sup>* <sup>=</sup> ×*<sup>ε</sup>*∈**S**(*ε*)*θε*,*i<sup>ε</sup>* codes the input *<sup>i</sup>* for which

For the multi–band channel **K** the following quantities are defined:<sup>7</sup>

sup *θε*

∑ *ε*∈**S**(*ε*)

*ε*∈**S**(*ε*)

• the *transinformation T*(*α*; *θ*) and the *information capacity C*(**K**),

*<sup>θ</sup>*∈**Θ**<sup>0</sup>

= ∑ *ε*∈**S**(*ε*)

*<sup>H</sup>*(*αε*�*θε*) = − ∑

*H*(*αε*�*θε*)

• the conditional *noise entropy* (entropy of the *multi–band B–E* | *F–D noise*)

∑ *i*∈**S**

*T* (*α*; *θ*) = sup

*h*[*pε*(W)] <sup>1</sup> <sup>−</sup> *<sup>p</sup>ε*(W) <sup>−</sup> <sup>∑</sup>

*<sup>θ</sup>*∈**Θ**<sup>0</sup>

*ε*∈**S**(*ε*)

, [by transforming each input *i<sup>ε</sup>* into pure state *θε*,*i<sup>ε</sup>* , ∀*ε* ∈ **S**(*ε*)].

<sup>7</sup> Using the *chain rule* for simultaneous probabilities it is found that for information entropy of an independent stochastic

−→*<sup>X</sup>* ) = ∑ *i*

of independent stochastic system, *θ* = {*θε*}*ε*, is the sum of *Hε*[*q*(·|*θε*)] over *ε* ∈ **S***<sup>ε</sup>* too.

 *h*(*pε*) 1 − *p<sup>ε</sup>*

{*θ<sup>ε</sup>* ∈ Θ*ε*; *E*(*θε*) = *W<sup>ε</sup>* ≥ 0} represents a coding procedure of the input *i*

*<sup>H</sup>*(*Xi*|*X*1, ...*Xi*−<sup>1</sup>) = ∑

 *h p<sup>ε</sup>* 1 + *p<sup>ε</sup>*

*i*

− *ε*

Let a spectrum of energy with the finite cardinality *n* + 1 and a finite time interval *τ* > 0 are considered

$$\mathbf{S}(\varepsilon) = \{\varepsilon\_{\mathbf{r}}\}\_{\mathbf{r}=0,1,\ldots,n} = \left\{\frac{rh}{\tau}\right\}\_{\mathbf{r}=0,1,\ldots,n}, \quad \Delta\varepsilon = \frac{h}{\tau}, \ \varepsilon\_{\mathbf{r}} = \frac{rh}{\tau} = r \cdot \Delta\varepsilon,\tag{87}$$
 
$$\text{card } \mathbf{S}(\varepsilon) = \frac{\varepsilon\_{\mathbf{n}}}{\Delta\varepsilon} = n+1, \ n \cdot \Delta\varepsilon = n \cdot \frac{h}{\tau} = \varepsilon\_{\mathbf{n}\prime} \cdot \frac{\tau}{n} = \frac{h}{\varepsilon\_{\mathbf{n}}} = \text{const.}$$

For a transfer channel with the continuous spectrum of energies of particles and with the *band–width* equal to card **<sup>S</sup>**(*ε*) = *<sup>ε</sup><sup>n</sup> <sup>h</sup>* , is valid that

$$\lim\_{\tau \to \infty} \varepsilon\_{\tau} = \lim\_{\tau \to \infty} \frac{rh}{\tau} = \lim\_{\tau \to \infty} r \cdot \Delta \varepsilon \stackrel{\triangle}{=} r \,\text{d}\,\varepsilon \text{ resp. } \lim\_{\tau \to \infty} \frac{1}{\tau} = \frac{\text{d}\varepsilon}{h}, \quad \mathbf{S}(\varepsilon) = \langle 0, \varepsilon\_{n} \rangle \tag{88}$$

But the infinite wide–band and infinite number of particles (*τ* −→ ∞, *n* −→ ∞) will be dealt with. Then

$$\mathbf{S}(\varepsilon) = \{\varepsilon\_r\}\_{r=0,1,\ldots} = \left\{\frac{rh}{\tau}\right\}\_{r=0,1,\ldots} = \lim\_{\tau \to \infty} \varepsilon\_r = \lim\_{\tau \to \infty} \frac{rh}{\tau} = \lim\_{\varepsilon \to 0} r \,\Delta\varepsilon = r \,\text{d}\,\varepsilon \tag{89}$$

and thus the *wide–band spectrum* **S**(*ε*) *of energies* is

*θ*

$$\lim\_{\tau \to \infty} \frac{1}{\tau} = \frac{\mathbf{d}\varepsilon}{h}, \quad \mathbf{S}(\varepsilon) = \langle 0, \infty \rangle \tag{90}$$

With the denotation *αε* � = *α*, *i* � = *iε*, *j* � = *jε*, *iε*, *j<sup>ε</sup>* ∈ **S**, *αε* ∈ **S**, *ε* ∈ **S**(*ε*) For the *p*-entropy of the output *<sup>α</sup>* of the wide–band transfer channel **<sup>K</sup>**B−E|F−<sup>D</sup> is valid that

$$H(\mathfrak{a} \| \boldsymbol{\theta}) = \lim\_{\boldsymbol{\tau} \to \infty} \frac{1}{\boldsymbol{\tau}} \sum\_{\boldsymbol{\varepsilon} \in \mathbf{S}(\boldsymbol{\varepsilon})} H(\mathfrak{a}\_{\varepsilon} \| \boldsymbol{\theta}\_{\varepsilon}) = \lim\_{\boldsymbol{\tau} \to \infty} -\frac{1}{\boldsymbol{\tau}} \sum\_{\boldsymbol{\varepsilon} \in \mathbf{S}(\boldsymbol{\varepsilon})} \sum\_{\boldsymbol{\alpha}, \boldsymbol{\varepsilon} \in \mathbf{S}} p(\boldsymbol{\alpha}\_{\varepsilon} | \boldsymbol{\alpha}\_{\varepsilon} | \boldsymbol{\theta}\_{\varepsilon}) \cdot \ln p(\boldsymbol{\alpha}\_{\varepsilon} | \boldsymbol{\alpha}\_{\varepsilon} | \boldsymbol{\theta}\_{\varepsilon})$$

$$= -\frac{1}{h} \int\_{0}^{\infty} \left[ \sum\_{\boldsymbol{\alpha} \in \mathbf{S}} p(\boldsymbol{\alpha} | \boldsymbol{\alpha}\_{\varepsilon} | \boldsymbol{\theta}\_{\varepsilon}) \cdot \ln p(\boldsymbol{\alpha} | \boldsymbol{\alpha}\_{\varepsilon} | \boldsymbol{\theta}\_{\varepsilon}) \right] d\boldsymbol{\varepsilon}$$

$$\sup\_{\boldsymbol{\Theta}} H(\mathfrak{a} \| \boldsymbol{\Theta}) = \lim\_{\boldsymbol{\tau} \to \mathbf{S}} \frac{1}{\boldsymbol{\tau}} \sum\_{\boldsymbol{\varepsilon} \in \mathbf{S}(\boldsymbol{\varepsilon})} \sup\_{\boldsymbol{\Theta}\_{\varepsilon}} H(\mathfrak{a}\_{\varepsilon} \| \boldsymbol{\Theta}\_{\varepsilon}) = \frac{1}{h} \int\_{0}^{\infty} \frac{h[p\_{\varepsilon}(\mathcal{W})]}{1 - p\_{\varepsilon}(\mathcal{W})} \, d\boldsymbol{\varepsilon}$$

For conditional entropy of the wide–band transfer channel **<sup>K</sup>**B−E|F−<sup>D</sup> [entropy of the wide–band noise independent on the system (**K**B−E|F−D) state *<sup>θ</sup>*] and for its information capacity is valid, by (85) and (86)

$$H(\mathfrak{a}|\mathfrak{g}) = \lim\_{\tau \to \infty} \frac{1}{\mathfrak{T}} \sum\_{\varepsilon \in \mathbf{S}(\varepsilon)} H(\mathfrak{a}\_{\varepsilon}|\mathfrak{e}\_{\varepsilon}) = \lim\_{\tau \to \infty} \frac{1}{\mathfrak{T}} \sum\_{\varepsilon \in \mathbf{S}(\varepsilon)} \sum\_{i \in \mathbf{S}} q(i|\mathfrak{G}\_{\varepsilon i}) \cdot H(\mathfrak{a}\_{\varepsilon}|\mathfrak{G}\_{\varepsilon i}) \qquad \text{(92)}$$

$$= \frac{1}{h} \int\_{0}^{\infty} H(\mathfrak{a}\_{\varepsilon}|\mathfrak{e}\_{\varepsilon}) \, \mathrm{d}\varepsilon = \frac{1}{h} \int\_{0}^{\infty} \left[ \frac{h(p\_{\varepsilon})}{1 - p\_{\varepsilon}} \, \middle| \, h\left(\frac{p\_{\varepsilon}}{1 + p\_{\varepsilon}}\right) \right] \, \mathrm{d}\varepsilon$$

$$\mathrm{C}(\mathbf{K}\_{\mathsf{B} - \mathsf{E} \left| \mathbf{F} - \mathbf{D} \right|}) = -\frac{1}{h} \int\_{0}^{\infty} \frac{h[p\_{\varepsilon}(\mathcal{W})]}{1 - p\_{\varepsilon}(\mathcal{W})} \, \mathrm{d}\varepsilon - \frac{1}{h} \int\_{0}^{\infty} \left[ \frac{h(p\_{\varepsilon})}{1 - p\_{\varepsilon}} \, \middle| \, h\left(\frac{p\_{\varepsilon}}{1 + p\_{\varepsilon}}\right) \right] \, \mathrm{d}\varepsilon \tag{93}$$

By (52) and (74) the average number of particles on the input of a narrow–band component K*<sup>ε</sup>* is

$$\mathcal{W}\_{\varepsilon} = \sum\_{i \in \mathbf{S}} i \cdot q(i|\theta\_{\varepsilon}) \ge \left[ 0 \: \left| \frac{2p\_{\varepsilon}^2}{1 - p\_{\varepsilon}^2} \right. \right] \tag{94}$$

For *<sup>T</sup>*<sup>0</sup> → 0 the quantum aproximation of *<sup>C</sup>*(**K**B−E), independent on the heat noise energy

The classical approximation of *<sup>C</sup>*(**K**B−E) is gaind for temperatures *<sup>T</sup>*<sup>0</sup> � 0 (*T*<sup>0</sup> → <sup>∞</sup>

channel with the whole noise energy k*T*<sup>0</sup> and with the whole average input energy *W*. For *T*<sup>0</sup>

� 3*hW π*2k2*T*<sup>0</sup> 2 � <sup>=</sup> *<sup>W</sup>* k*T*<sup>0</sup>

�

<sup>d</sup>*<sup>ε</sup>* <sup>=</sup> *<sup>π</sup>*2k*T*<sup>0</sup>

*TW* <sup>−</sup> *<sup>T</sup>*<sup>0</sup> 2 �

2*TW* − 2*T*<sup>0</sup>

*W*

2 *h*

12*h*

<sup>2</sup> < 1.

� ∞ 0 *x*

2

<sup>d</sup>*<sup>ε</sup>* <sup>=</sup> k2*T*<sup>0</sup>

*<sup>t</sup>* <sup>+</sup> <sup>1</sup> <sup>d</sup>*<sup>t</sup>* <sup>=</sup> *<sup>π</sup>*2k2*T*<sup>0</sup>

*π*2k2*T*<sup>0</sup>

�

3*h*

*<sup>C</sup>*(**K**F−D) = *<sup>C</sup>*(**K**B−E)· <sup>2</sup>*TW* <sup>−</sup> *<sup>T</sup>*<sup>0</sup>

*<sup>ε</sup> <sup>p</sup>ε*(W)

1 + *e* <sup>−</sup> *<sup>ε</sup>* k*T*0

> � 1 0

2 *h*

�

<sup>1</sup> <sup>−</sup> *<sup>p</sup>ε*(W) <sup>d</sup>*<sup>ε</sup>* <sup>=</sup> *<sup>π</sup>*2k2*T*<sup>2</sup>

ln *t*

<sup>2</sup> *<sup>x</sup>* where *<sup>x</sup>* <sup>=</sup> <sup>6</sup>*hW*

<sup>d</sup>*<sup>ε</sup>* <sup>=</sup> *<sup>π</sup>*2k*TW*

<sup>3</sup>*<sup>h</sup>* <sup>−</sup> <sup>1</sup> *h* � ∞ 0 *h* � *p<sup>ε</sup>* 1 + *p<sup>ε</sup>*

<sup>6</sup>*<sup>h</sup>* (106)

<sup>6</sup>*<sup>h</sup>* (109)

*e*−*<sup>x</sup>*

<sup>1</sup> <sup>+</sup> *<sup>e</sup>*−*<sup>x</sup>* <sup>d</sup>*<sup>x</sup>* (110)

<sup>3</sup>*<sup>h</sup>* <sup>−</sup> *<sup>π</sup>*2k*T*<sup>0</sup> 3*h*

Information Capacity of Quantum Transfer Channels and Thermodynamic Analogies 99

⎞ ⎠ = *π*

, the Shannon capacity of the wide–band **Gaussian**

�2*W*

<sup>3</sup>*<sup>h</sup>* (103)

(104)

� d*ε* (105)

(107)

(108)

*π*4k2*T*<sup>0</sup> 2 <sup>9</sup>*h*<sup>2</sup> <sup>+</sup> *<sup>π</sup>*<sup>2</sup> <sup>2</sup>*<sup>W</sup>*

(deminishes whith temperture's aiming to absolute 0◦ *K*)

*T*0→0

*<sup>C</sup>*(**K**B−E) .

1 *h* � ∞ 0 *h* � *p<sup>ε</sup>* 1 + *p<sup>ε</sup>*

⎛ ⎝ �

k*T*<sup>0</sup>

*h* � ∞ 0 *h* � *p<sup>ε</sup>* 1 + *p<sup>ε</sup>*

By figuring (105) the capacity of the wide–band F–D channel **K**F−<sup>D</sup> is gained,

For the *whole average output energy* is valid the same as for the B–E case,

1 *h* � ∞ 0

*<sup>ε</sup> <sup>p</sup><sup>ε</sup>* 1 + *p<sup>ε</sup>*

<sup>2</sup> *<sup>x</sup>* <sup>−</sup> <sup>1</sup>

For the *whole average F–D wide–band noise* energy is being derived

<sup>8</sup> *<sup>x</sup>*<sup>2</sup> <sup>+</sup> ... .

<sup>=</sup> <sup>1</sup> *h* � ∞ 0 *ε e* <sup>−</sup> *<sup>ε</sup>* k*T*0

<sup>=</sup> <sup>−</sup>k2*T*<sup>0</sup>

= 1 + <sup>1</sup>

*<sup>C</sup>*(**K**F−D) = *<sup>π</sup>*2k

<sup>=</sup> *<sup>π</sup>*2k*T*<sup>0</sup> 3*h*

*<sup>C</sup>*(**K**B−E) = lim

**5.3. Fermi–Dirac wide–band channel capacity** By derivations (86) and (92), (93) is valid that [12]

> *h*[*pε*(W)] <sup>1</sup> <sup>−</sup> *<sup>p</sup>ε*(W) <sup>d</sup>*<sup>ε</sup>* <sup>−</sup> <sup>1</sup>

For the second integral obviously is valid

lim *T*0→0

*<sup>C</sup>*(**K**F−D) = <sup>1</sup>

*h* � ∞ 0

and for *TW* > *T*0is writable

lim*τ*→<sup>∞</sup> 1 *<sup>τ</sup>* ∑ *ε*∈**S**(*ε*)

<sup>8</sup> For <sup>|</sup>*x*<sup>|</sup> <sup>&</sup>lt; 1, <sup>√</sup><sup>1</sup> <sup>+</sup> *<sup>x</sup>* <sup>=</sup> <sup>1</sup> <sup>+</sup> <sup>1</sup>

respectively). It is near to value *W*

from (101), great enough, is gained that8

and then for the whole average number of input particles W� of the wide–band transfer channel **K** is obtained

$$\mathcal{W}' = \frac{1}{h} \int\_0^\infty \mathcal{W}\_\varepsilon \,\mathrm{d}\varepsilon \,, \; W = \frac{1}{h} \int\_0^\infty \varepsilon \,\mathcal{W}\_\varepsilon \,\mathrm{d}\varepsilon \tag{95}$$

where *W* is whole input energy and *TW* is the *effective coding temperature* being supposingly at the value *TW<sup>ε</sup>* , *TW* = *TW<sup>ε</sup>* , ∀*ε* ∈ **S**(*ε*).

#### **5.2. Bose–Einstein wide–band channel capacity**

By derivations (86) and (92), (93) is valid that [12]

$$\mathbb{C}(\mathbf{K}\_{\mathrm{B}-\mathrm{E}}) = \frac{1}{h} \int\_{0}^{\infty} \frac{h[p\_{\varepsilon}(\mathcal{W})]}{1 - p\_{\varepsilon}(\mathcal{W})} \, \mathrm{d}\varepsilon - \frac{1}{h} \int\_{0}^{\infty} \frac{h(p\_{\varepsilon})}{1 - p\_{\varepsilon}} \, \mathrm{d}\varepsilon \tag{96}$$

For the first or, for the second integral respectively, obviously is valid

$$\frac{1}{h} \int\_{0}^{\infty} \frac{h[p\_{\varepsilon}(\mathcal{W})]}{1 - p\_{\varepsilon}(\mathcal{W})} \, \mathrm{d}\varepsilon = \frac{\pi^{2} \mathrm{k}T\_{W}}{3h} \quad \mathrm{resp.} \quad \frac{1}{h} \int\_{0}^{\infty} \frac{h(p\_{\varepsilon})}{1 - p\_{\varepsilon}} \, \mathrm{d}\varepsilon = \frac{\pi^{2} \mathrm{k}T\_{0}}{3h} \tag{97}$$

Then, for the capacity of the wide–band B–E transfer channel **K**B−<sup>E</sup> is valid

$$\mathbf{C}(\mathbf{K}\_{\mathbf{B}-\mathbf{E}}) = \frac{\pi^2 \mathbf{k}}{3h} (T\_W - T\_0) = \frac{\pi^2 \mathbf{k} T\_W}{3h} \cdot \frac{T\_W - T\_0}{T\_W} \stackrel{\triangle}{=} \frac{\pi^2 \mathbf{k} T\_W}{3h} \cdot \eta\_{\text{max}} \cdot T\_W \ge T\_0 \tag{98}$$

and for the whole average output energy is valid

$$\lim\_{\tau \to \infty} \frac{1}{\tau} \sum\_{\varepsilon \in \mathbf{S}(\varepsilon)} \varepsilon \frac{p\_{\varepsilon}(\mathcal{W})}{1 - p\_{\varepsilon}(\mathcal{W})} = \frac{1}{h} \int\_{0}^{\infty} \varepsilon \frac{p\_{\varepsilon}(\mathcal{W})}{1 - p\_{\varepsilon}(\mathcal{W})} \, \mathrm{d}\varepsilon = -\frac{\mathbf{k}^{2} T\_{\mathcal{W}}^{2}}{h} \int\_{0}^{1} \frac{\ln(1 - t)}{t} \, \mathrm{d}t = \frac{\pi^{2} \mathbf{k}^{2} T\_{\mathcal{W}}^{2}}{6h} \tag{99}$$

For the whole average energy of the B–E noise must be valid

$$\lim\_{\tau \to \infty} \frac{1}{\tau} \sum\_{\varepsilon \in \mathbf{S}(\varepsilon)} \varepsilon \frac{p\_{\varepsilon}}{1 - p\_{\varepsilon}} = \frac{1}{h} \int\_{0}^{\infty} \varepsilon \frac{p\_{\varepsilon}}{1 - p\_{\varepsilon}} \, \mathrm{d}\varepsilon = \frac{\pi^2 \mathbf{k}^2 T\_0^2}{6h} \tag{100}$$

From the relations (79) among the energies of the output *α*� , of the noise *j* � and the input *i* � ,

$$\frac{\pi^2 \mathbf{k}^2 T\_W^2}{6\hbar} = \frac{\pi^2 \mathbf{k}^2 T\_0^2}{6\hbar} + W \tag{101}$$

the effective coding temperature *TW* is derivable, *TW* = *T*<sup>0</sup> · 1 + 6*hW π*2k2*T*<sup>0</sup> <sup>2</sup> . Using it in (98) gives

$$\mathcal{C}(\mathbf{K}\_{\rm B-E}) = \frac{\pi^2 \mathbf{k} T\_0}{3\hbar} \left( \sqrt{1 + \frac{6hW}{\pi^2 \mathbf{k}^2 T\_0^2}} - 1 \right) \tag{102}$$

For *<sup>T</sup>*<sup>0</sup> → 0 the quantum aproximation of *<sup>C</sup>*(**K**B−E), independent on the heat noise energy (deminishes whith temperture's aiming to absolute 0◦ *K*)

$$\lim\_{T\_0 \to 0} \mathbb{C}(\mathbf{K}\_{\mathbf{B} - \mathbf{E}}) = \lim\_{T\_0 \to 0} \left( \sqrt{\frac{\pi^4 \mathbf{k}^2 T\_0^2}{9h^2} + \pi^2 \frac{2W}{3h}} - \frac{\pi^2 \mathbf{k} T\_0}{3h} \right) = \pi \sqrt{\frac{2W}{3h}} \tag{103}$$

The classical approximation of *<sup>C</sup>*(**K**B−E) is gaind for temperatures *<sup>T</sup>*<sup>0</sup> � 0 (*T*<sup>0</sup> → <sup>∞</sup> respectively). It is near to value *W* k*T*<sup>0</sup> , the Shannon capacity of the wide–band **Gaussian** channel with the whole noise energy k*T*<sup>0</sup> and with the whole average input energy *W*. For *T*<sup>0</sup> from (101), great enough, is gained that8

$$\mathbf{C}(\mathbf{K}\_{\rm B-E}) \doteq \frac{\pi^2 \mathbf{k} T\_0}{3h} \left(\frac{3hW}{\pi^2 \mathbf{k}^2 T\_0^2}\right) = \frac{W}{\mathbf{k} T\_0} \tag{104}$$

#### **5.3. Fermi–Dirac wide–band channel capacity**

By derivations (86) and (92), (93) is valid that [12]

$$\mathbf{C}(\mathbf{K}\_{\rm F-D}) = \frac{1}{h} \int\_0^\infty \frac{h[p\_\varepsilon(\mathcal{W})]}{1 - p\_\varepsilon(\mathcal{W})} \, \mathrm{d}\varepsilon - \frac{1}{h} \int\_0^\infty h\left(\frac{p\_\varepsilon}{1 + p\_\varepsilon}\right) \, \mathrm{d}\varepsilon = \frac{\pi^2 \mathbf{k} T\_W}{\Im h} - \frac{1}{h} \int\_0^\infty h\left(\frac{p\_\varepsilon}{1 + p\_\varepsilon}\right) \, \mathrm{d}\varepsilon \tag{105}$$

For the second integral obviously is valid

$$\frac{1}{\hbar} \int\_{0}^{\infty} h \left( \frac{p\_{\varepsilon}}{1 + p\_{\varepsilon}} \right) \, \mathrm{d}\varepsilon = \frac{\pi^2 \mathrm{k}T\_0}{6\hbar} \tag{106}$$

By figuring (105) the capacity of the wide–band F–D channel **K**F−<sup>D</sup> is gained,

$$\mathcal{C}(\mathbf{K}\_{\rm F-D}) = \frac{\pi^2 \mathbf{k}}{3h} \left( T\_W - \frac{T\_0}{2} \right) \tag{107}$$

and for *TW* > *T*0is writable

16 Will-be-set-by-IN-TECH

By (52) and (74) the average number of particles on the input of a narrow–band component

and then for the whole average number of input particles W� of the wide–band transfer

<sup>W</sup>*<sup>ε</sup>* <sup>d</sup>*ε*, *<sup>W</sup>* <sup>=</sup> <sup>1</sup>

where *W* is whole input energy and *TW* is the *effective coding temperature* being supposingly at

*h*[*pε*(W)] <sup>1</sup> <sup>−</sup> *<sup>p</sup>ε*(W) <sup>d</sup>*<sup>ε</sup>* <sup>−</sup> <sup>1</sup>

<sup>3</sup>*<sup>h</sup>* resp.

3*h* ·

*<sup>ε</sup> <sup>p</sup>ε*(W)

<sup>=</sup> <sup>1</sup> *h* ∞ 0

 0 

> *h* ∞ 0

> > *h* ∞ 0

1 *h* ∞ 0

*TW* − *T*<sup>0</sup> *TW*

> *<sup>ε</sup> <sup>p</sup><sup>ε</sup>* 1 − *p<sup>ε</sup>*

> > 2

<sup>1</sup> <sup>−</sup> *<sup>p</sup>ε*(W) <sup>d</sup>*<sup>ε</sup>* <sup>=</sup> <sup>−</sup>k2*T*<sup>2</sup>

*h*(*pε*) 1 − *p<sup>ε</sup>*

<sup>d</sup>*<sup>ε</sup>* <sup>=</sup> *<sup>π</sup>*2k*T*<sup>0</sup>

ln(1 − *t*) *t*

2

<sup>6</sup>*<sup>h</sup>* <sup>+</sup> *<sup>W</sup>* (101)

6*hW π*2k2*T*<sup>0</sup>

*h*(*pε*) 1 − *p<sup>ε</sup>*

� <sup>=</sup> *<sup>π</sup>*2k*TW*

> *W h*

 1 0

<sup>d</sup>*<sup>ε</sup>* <sup>=</sup> *<sup>π</sup>*2k2*T*<sup>0</sup>

, of the noise *j*

 1 +

<sup>2</sup> − 1 

6*hW π*2k2*T*<sup>0</sup>

2*p*<sup>2</sup> *ε* <sup>1</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup> *ε* *ε* W*<sup>ε</sup>* d*ε* (95)

d*ε* (96)

<sup>3</sup>*<sup>h</sup>* (97)

<sup>d</sup>*<sup>t</sup>* <sup>=</sup> *<sup>π</sup>*2k2*T*<sup>2</sup>

<sup>6</sup>*<sup>h</sup>* (100)

� and the input *i*

<sup>2</sup> . Using it in (98)

*W* 6*h*

> � ,

(102)

(99)

<sup>3</sup>*<sup>h</sup>* · *<sup>η</sup>*max, *TW*≥*T*<sup>0</sup> (98)

(94)

*i* · *q*(*i*|*θε*) ≥

W*<sup>ε</sup>* = ∑ *i*∈**S**

<sup>W</sup>� <sup>=</sup> <sup>1</sup> *h* ∞ 0

**5.2. Bose–Einstein wide–band channel capacity**

*<sup>C</sup>*(**K**B−E) = <sup>1</sup>

*h*[*pε*(W)]

and for the whole average output energy is valid

*<sup>ε</sup> <sup>p</sup>ε*(W) <sup>1</sup> <sup>−</sup> *<sup>p</sup>ε*(W) <sup>=</sup> <sup>1</sup>

> lim*τ*→<sup>∞</sup> 1 *<sup>τ</sup>* ∑ *ε*∈**S**(*ε*)

*h* ∞ 0

For the first or, for the second integral respectively, obviously is valid

Then, for the capacity of the wide–band B–E transfer channel **K**B−<sup>E</sup> is valid

<sup>3</sup>*<sup>h</sup>* (*TW* <sup>−</sup> *<sup>T</sup>*0) = *<sup>π</sup>*2k*TW*

*<sup>ε</sup> <sup>p</sup><sup>ε</sup>* 1 − *p<sup>ε</sup>*

*π*2k2*T*<sup>2</sup> *W* <sup>6</sup>*<sup>h</sup>* <sup>=</sup> *<sup>π</sup>*2k2*T*<sup>0</sup>

*h* ∞ 0

For the whole average energy of the B–E noise must be valid

From the relations (79) among the energies of the output *α*�

the effective coding temperature *TW* is derivable, *TW* = *T*<sup>0</sup> ·

*<sup>C</sup>*(**K**B−E) = *<sup>π</sup>*2k*T*<sup>0</sup>

3*h*

 1 +

<sup>1</sup> <sup>−</sup> *<sup>p</sup>ε*(*W*) <sup>d</sup>*<sup>ε</sup>* <sup>=</sup> *<sup>π</sup>*2k*TW*

By derivations (86) and (92), (93) is valid that [12]

K*<sup>ε</sup>* is

channel **K** is obtained

the value *TW<sup>ε</sup>* , *TW* = *TW<sup>ε</sup>* , ∀*ε* ∈ **S**(*ε*).

1 *h* ∞ 0

*<sup>C</sup>*(**K**B−E) = *<sup>π</sup>*2k

lim*τ*→<sup>∞</sup> 1 *<sup>τ</sup>* ∑ *ε*∈**S**(*ε*)

gives

$$\mathcal{C}(\mathbf{K}\_{\rm F-D}) = \mathcal{C}(\mathbf{K}\_{\rm B-E}) \cdot \frac{2T\_W - T\_0}{2T\_W - 2T\_0} \tag{108}$$

For the *whole average output energy* is valid the same as for the B–E case,

$$\frac{1}{h} \int\_0^\infty \varepsilon \frac{p\_\varepsilon(\mathcal{W})}{1 - p\_\varepsilon(\mathcal{W})} \, \mathrm{d}\varepsilon = \frac{\pi^2 \mathbf{k}^2 T\_W^2}{6h} \tag{109}$$

For the *whole average F–D wide–band noise* energy is being derived

$$\lim\_{\tau \to \infty} \frac{1}{\tau} \sum\_{\varepsilon \in \mathbf{S}(\varepsilon)} \varepsilon \frac{p\_{\varepsilon}}{1 + p\_{\varepsilon}} = \frac{1}{h} \int\_{0}^{\infty} \varepsilon \frac{e^{-\frac{\tilde{\varepsilon}}{kT\_{0}}}}{1 + e^{-\frac{\tilde{\varepsilon}}{kT\_{0}}}} \, \mathrm{d}\varepsilon = \frac{\mathbf{k}^{2} T\_{0}^{2}}{h} \int\_{0}^{\infty} \mathbf{x} \, \frac{e^{-\mathbf{x}}}{1 + e^{-\mathbf{x}}} \, \mathrm{d}\mathbf{x} \tag{110}$$

$$= -\frac{\mathbf{k}^{2} T\_{0}^{2}}{h} \int\_{0}^{1} \frac{\ln t}{t + 1} \, \mathrm{d}t = \frac{\pi^{2} \mathbf{k}^{2} T\_{0}^{2}}{12h}$$
 $\text{For } |\mathbf{x}| < 1, \sqrt{1 + \mathbf{x}} = 1 + \frac{1}{2} \mathbf{x} - \frac{1}{3} \mathbf{x}^{2} + \dots \doteq 1 + \frac{1}{2} \mathbf{x} \text{ where } \mathbf{x} = \frac{6h \mathcal{W}}{\pi^{2} \mathbf{k}^{2} T\_{0}^{2}} < 1.$ 

#### 18 Will-be-set-by-IN-TECH 100 Thermodynamics – Fundamentals and Its Application in Science Information Capacity of Quantum Transfer Channels and Thermodynamic Analogies <sup>19</sup>

From the relation (79) among the whole output, input, and noise energy,

$$\frac{\pi^2 \mathbf{k}^2 T\_W^2}{6\hbar} = \frac{\pi^2 \mathbf{k}^2 T\_0^2}{12\hbar} + W \tag{111}$$

distribution *q*(*i*|*θ*�

system state, including itself.

*θ<sup>i</sup>* of *θ*<sup>+</sup> is *uniform* and thus

6.0.0.6. Proof:

= *n* ∑ *j*=1 ) = *n* ∑ *j*=1

quantity *<sup>I</sup>*(*p*�*d*) = <sup>H</sup>(*θ*+) − H(*θ*) where the state

*n* ∑ *i*=1

*<sup>θ</sup>*<sup>+</sup> <sup>=</sup> <sup>1</sup> *n*

*<sup>q</sup>*(·|*θ*) and *<sup>q</sup>*(·|*θ*+) of states (stochastic quantities)

and the equality arises for *θ* = *θ*� only [12, 38].

*f* ⎡ ⎣ *n* ∑ *j*=1

> *n* ∑ *j*=1

*n* ∑ *i*=1 *f* ⎡ ⎣ *n* ∑ *j*=1

*f* [*q*(*j*|*θ*)] =

6.0.0.5. H-Theorem, *I I*. *Second Principle of Thermodynamics*: Let for states *θ*, *θ*� ∈ **Θ** of the system **Ψ** is valid that *θ* → *θ*�

*p*(*i*|*j*) *q*(*j*|*θ*)

*p*(*i*|*j*) *q*(*j*|*θ*)

<sup>11</sup> It is the matrix of the *unitary operator* **u**(*t*) expressing the time evolution of the system **Ψ**.

*p*(*i*|*j*) *q*(*j*|*θ*), *p*(*i*|*j*)=(*ψ*�

From the relation *θ* −→ *θ*� also is visible that it is *reflexive* and *transitive* relation between states and, thus, it defines (a partial) *arrangment* on the set space **Θ**. The terminal, maximal state for this arrangement is the equilibrial state *θ*<sup>+</sup> of the system **Ψ**: it is the successor of an arbitrary

The *statistic, Shannon, information)* entropy *H*(·) is a generalization of the physical entropyH(*θ*). The quantity *I*-divergence *I*(·�·) is, by (21), a generalization of the physical

is the equlibrial state of the system **Ψ**. The probability distribution into the *canonic components*

Information divergence *I*(*p*�*d*) ≥ 0 expresses the *distance* of the two probability distributions

In the physical sense the divergence *I*(*p*�*d*) is a measure of a not-equilibriality of the state *θ* of the physical (let say a thermodynamic) system **Ψ**. Is maximized in the initial (starting),

not-equilibrium state of the (time) evolution of the **Ψ**. It is clear that *I*(*p*�*d*) ≡ *T*(*α*; *θ*)

H(*θ*�

(a) For a *strictly convex* function *f*(*u*) = *u* · ln *u* the **Jensen** inequality is valid [23]

*n* ∑ *j*=1

*n* ∑ *i*=1

*p*(*i*|*j*)

*q*(*j*|*θ*)ln *q*(*j*|*θ*) = −*H*[*q*(·|*θ*)] = −H(*θ*) due to

*n* ∑ *j*=1

*f* [*q*(*j*|*θ*)]

⎤ ⎦ ≤

⎤ ⎦ ≤

of the *transformation matrix* [*ui*,*j*] of a *base* of the space <sup>Ψ</sup> <sup>=</sup> {Ψ}*<sup>n</sup>*

*<sup>i</sup>*, *<sup>ψ</sup>j*)<sup>2</sup> <sup>=</sup> *<sup>u</sup>*<sup>2</sup>

Information Capacity of Quantum Transfer Channels and Thermodynamic Analogies 101

*ji* and *θ* −→ *θ*�

*θ<sup>i</sup>* ∈ **Θ** for *θ<sup>i</sup>* = *π*{*ψi*}, *i* = 1, 2, ..., *n* (116)

<sup>H</sup>(*θ*+) = ln *<sup>n</sup>* <sup>=</sup> ln dim (Ψ) (117)

*<sup>θ</sup>* = [**S**, *<sup>q</sup>*(·|*θ*)] and *<sup>θ</sup>*<sup>+</sup> = [**S**, *<sup>q</sup>*(·|*θ*+)] (118)

. Then

) ≥ H(*θ*) (119)

*p*(*i*|*j*) *f* [*q*(*j*|*θ*)], *i* = 1, 2 , ..., *n* (120)

*n* ∑ *i*=1

*p*(*i*|*j*) = 1

*<sup>i</sup>*=<sup>1</sup> into the base {Ψ�

, ensures existence

}*n <sup>i</sup>*=1. 11

follows the effective coding temperature *TW* = *T*<sup>0</sup> · � 1 2 + 6*hW π*2k2*T*<sup>0</sup> <sup>2</sup> . Using it in (107) the result is [24]

$$\mathbf{C}(\mathbf{K}\_{\rm F-D}) = \frac{\pi^2 \mathbf{k} T\_0}{3\hbar} \left( \sqrt{\frac{1}{2} + \frac{6h\mathcal{W}}{\pi^2 \mathbf{k}^2 T\_0^2}} - \frac{1}{2} \right) \tag{112}$$

For *<sup>T</sup>*<sup>0</sup> → 0 the quantum approximation capacity *<sup>C</sup>*(**K**F−D), independent on heat noise energy k*T*<sup>0</sup> is gaind (the same as in the B–E case (103),

$$\lim\_{T\_0 \to 0} \mathbb{C}(\mathbf{K}\_{\rm F-D}) = \lim\_{T\_0 \to 0} \left( \sqrt{\frac{\pi^4 \mathbf{k}^2 T\_0^2}{9h^2} \cdot \frac{1}{2} + \pi^2 \frac{2W}{3h}} - \frac{\pi^2 \mathbf{k} T\_0}{3h} \cdot \frac{1}{2} \right) = \pi \sqrt{\frac{2W}{3h}} \tag{113}$$

The classical approximation of the capacity *<sup>C</sup>*(**K**F−D) is gained for *<sup>T</sup>*<sup>0</sup> � 09

$$\mathbf{C}(\mathbf{K}\_{\mathbf{F}\to\mathbf{D}}) = \frac{\pi^2 \mathbf{k} T\_0}{3h} \left[ \frac{1}{\sqrt{2}} \sqrt{1 + \frac{12hW}{\pi^2 \mathbf{k}^2 T\_0^2}} - \frac{1}{2} \right] \doteq \frac{\pi^2 \mathbf{k} T\_0}{3h} \left[ \frac{1}{\sqrt{2}} \left( 1 + \frac{6hW}{\pi^2 \mathbf{k}^2 T\_0^2} \right) - \frac{1}{2} \right] \quad \text{(114)}$$

$$= \frac{\pi^2 \mathbf{k} T\_0}{6h} \left( \sqrt{2} - 1 \right) + \sqrt{2} \frac{W}{\mathbf{k} T\_0} \quad \left[ \underset{7\mathbf{q}\to\infty}{\longrightarrow} \frac{\pi^2 \mathbf{k} T\_0}{6h} \left( \sqrt{2} - 1 \right), \ W = \text{const.} \ge W\_{crit} \right]$$

By (74) the condition for the medium value of the input particles of a narrow–band component <sup>K</sup>*ε*, *<sup>ε</sup>* <sup>∈</sup> **<sup>S</sup>**(*ε*), of the channel **<sup>K</sup>**F−<sup>D</sup> is valid, <sup>W</sup>*<sup>ε</sup>* <sup>≥</sup> <sup>2</sup>*p*<sup>2</sup> *ε* <sup>1</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup> *ε* , from which the condition for the whole input energy of the wide–band channel **K**F−<sup>D</sup> follows. By (95) it is gaind, for *TW* <sup>≥</sup> *<sup>T</sup>*<sup>0</sup> <sup>&</sup>gt; 0, that10

$$\mathcal{W} \ge \lim\_{\tau \to \infty} \frac{1}{\tau} \sum\_{\varepsilon \in \mathbf{S}(\varepsilon)} \varepsilon \mathcal{W}\_{\varepsilon} \ge \lim\_{\tau \to \infty} \frac{1}{\tau} \sum\_{\varepsilon \in \mathbf{S}(\varepsilon)} \varepsilon \frac{2p\_{\varepsilon}^{2}}{1 - p\_{\varepsilon}^{2}} = \frac{2}{h} \int\_{0}^{\infty} \varepsilon \frac{e^{-2\frac{\varepsilon}{kT\_{0}}}}{1 - e^{\frac{2\varepsilon}{kT\_{0}}}} \,\mathrm{d}\varepsilon = \frac{\pi^{2} \mathbf{k}^{2} T\_{0}^{-2}}{12h} = \mathcal{W}\_{\mathrm{crit}} > 0 \tag{115}$$

#### **6. Physical information transfer and thermodynamics**

Whether the considered information transfers are narrow–band or wide–band, their algebraic-information description remains the same. So let be considered an arbitrary *stationary physical system* **Ψ** of these two band–types as usable for information transfer.

$$\begin{aligned} \text{Let a system state } \boldsymbol{\theta}^{\prime} &= \sum\_{i=1}^{n} q(i|\boldsymbol{\theta}^{\prime}) \, \pi \{\psi\_{i}^{\prime}\} \in \Theta \text{ of the system } \mathbf{Y} \text{ is the successor (followwer)}\\ \text{Let } \dots \dots \dots \dots \dots \quad \dots \dots \dots \quad \text{and} \quad \overset{\text{in}}{\dots} \quad (\text{'1:9}) \quad (\text{'1:1}) \quad \dots \quad \boldsymbol{\theta}^{\prime} \dots \quad \boldsymbol{\theta}^{\prime} \dots \quad \dots \dots \quad \dots \end{aligned}$$

equivocant) of the system state *θ* = ∑ *i*=1 *q*(*i*|*θ*) *π*{*ψi*} ∈ **Θ**, *θ* −→ *θ*� is written. The

<sup>9</sup> For <sup>√</sup><sup>1</sup> <sup>+</sup> *<sup>x</sup>* . = 1 + <sup>1</sup> <sup>2</sup> *<sup>x</sup>* when <sup>|</sup>*x*<sup>|</sup> <sup>&</sup>lt; 1; *<sup>x</sup>* <sup>=</sup> <sup>12</sup>*hW π*2k2*T*<sup>0</sup> 2 .

<sup>10</sup> If, in the special case of F–D channel, it is considered that the value *W* given by the number of electrons as the average energy of the *modulating current* entering into a wire, over a *time unit*, then it is the average power on the electric resistor *R* = 1Ω too.

distribution *q*(*i*|*θ*� ) = *n* ∑ *j*=1 *p*(*i*|*j*) *q*(*j*|*θ*), *p*(*i*|*j*)=(*ψ*� *<sup>i</sup>*, *<sup>ψ</sup>j*)<sup>2</sup> <sup>=</sup> *<sup>u</sup>*<sup>2</sup> *ji* and *θ* −→ *θ*� , ensures existence

of the *transformation matrix* [*ui*,*j*] of a *base* of the space <sup>Ψ</sup> <sup>=</sup> {Ψ}*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> into the base {Ψ� }*n <sup>i</sup>*=1. 11

From the relation *θ* −→ *θ*� also is visible that it is *reflexive* and *transitive* relation between states and, thus, it defines (a partial) *arrangment* on the set space **Θ**. The terminal, maximal state for this arrangement is the equilibrial state *θ*<sup>+</sup> of the system **Ψ**: it is the successor of an arbitrary system state, including itself.

The *statistic, Shannon, information)* entropy *H*(·) is a generalization of the physical entropyH(*θ*). The quantity *I*-divergence *I*(·�·) is, by (21), a generalization of the physical quantity *<sup>I</sup>*(*p*�*d*) = <sup>H</sup>(*θ*+) − H(*θ*) where the state

$$\boldsymbol{\theta}^{+} = \frac{1}{n} \sum\_{i=1}^{n} \boldsymbol{\theta}\_{i} \in \boldsymbol{\Theta} \quad \text{for} \quad \boldsymbol{\theta}\_{i} = \pi \{\boldsymbol{\psi}\_{i}\}, \quad i = 1, 2, \dots, n \tag{116}$$

is the equlibrial state of the system **Ψ**. The probability distribution into the *canonic components θ<sup>i</sup>* of *θ*<sup>+</sup> is *uniform* and thus

$$\mathcal{H}(\boldsymbol{\theta}^+) = \ln n = \ln \dim \left( \mathbb{Y} \right) \tag{117}$$

Information divergence *I*(*p*�*d*) ≥ 0 expresses the *distance* of the two probability distributions *<sup>q</sup>*(·|*θ*) and *<sup>q</sup>*(·|*θ*+) of states (stochastic quantities)

$$\boldsymbol{\theta} = [\mathbf{S}, \boldsymbol{\eta}(\cdot|\boldsymbol{\theta})] \quad \text{and} \quad \boldsymbol{\theta}^+ = [\mathbf{S}, \boldsymbol{\eta}(\cdot|\boldsymbol{\theta}^+)] \tag{118}$$

In the physical sense the divergence *I*(*p*�*d*) is a measure of a not-equilibriality of the state *θ* of the physical (let say a thermodynamic) system **Ψ**. Is maximized in the initial (starting), not-equilibrium state of the (time) evolution of the **Ψ**. It is clear that *I*(*p*�*d*) ≡ *T*(*α*; *θ*)

6.0.0.5. H-Theorem, *I I*. *Second Principle of Thermodynamics*:

Let for states *θ*, *θ*� ∈ **Θ** of the system **Ψ** is valid that *θ* → *θ*� . Then

$$\mathcal{H}(\mathfrak{g}') \ge \mathcal{H}(\mathfrak{g}) \tag{119}$$

and the equality arises for *θ* = *θ*� only [12, 38].

6.0.0.6. Proof:

18 Will-be-set-by-IN-TECH

�� 1 2 +

For *<sup>T</sup>*<sup>0</sup> → 0 the quantum approximation capacity *<sup>C</sup>*(**K**F−D), independent on heat noise energy

1

<sup>2</sup> <sup>+</sup> *<sup>π</sup>*<sup>2</sup> <sup>2</sup>*<sup>W</sup>*

2

� 1 2 +

6*hW π*2k2*T*<sup>0</sup>

<sup>3</sup>*<sup>h</sup>* <sup>−</sup> *<sup>π</sup>*2k*T*<sup>0</sup>

� 1 √2 � 1 +

�√ 2 − 1 �

*ε* <sup>1</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup> *ε*

> 1−*e* 2 *<sup>ε</sup>* k*T*0

*π*2k*T*<sup>0</sup> 6*h*

<sup>12</sup>*<sup>h</sup>* <sup>+</sup> *<sup>W</sup>* (111)

<sup>2</sup> . Using it in (107) the

�2*W*

, *W* = const. ≥ *Wcrit*

, from which the condition for

2

<sup>12</sup>*<sup>h</sup>* <sup>=</sup>*Wcrit*>0 (115)

<sup>d</sup>*ε*<sup>=</sup> *<sup>π</sup>*2k2*T*<sup>0</sup>

*<sup>i</sup>*} ∈ **Θ** of the system **Ψ** is the successor (follower,

*q*(*i*|*θ*) *π*{*ψi*} ∈ **Θ**, *θ* −→ *θ*� is written. The

<sup>3</sup>*<sup>h</sup>* (113)

(112)

(114)

�

6*hW π*2k2*T*<sup>0</sup>

<sup>2</sup> <sup>−</sup> <sup>1</sup> 2 �

3*h* ·

1 2 ⎞ ⎠ = *π*

6*hW π*2k2*T*<sup>0</sup> 2 � − 1 2 �

From the relation (79) among the whole output, input, and noise energy,

*<sup>C</sup>*(**K**F−D) = *<sup>π</sup>*2k*T*<sup>0</sup>

3*h*

*π*4k2*T*<sup>0</sup> 2 <sup>9</sup>*h*<sup>2</sup> ·

The classical approximation of the capacity *<sup>C</sup>*(**K**F−D) is gained for *<sup>T</sup>*<sup>0</sup> � 09

12*hW π*2k2*T*<sup>0</sup>

<sup>2</sup> <sup>−</sup> <sup>1</sup> 2 � . <sup>=</sup> *<sup>π</sup>*2k*T*<sup>0</sup> 3*h*

> � −→ *T*0→∞

By (74) the condition for the medium value of the input particles of a narrow–band component

the whole input energy of the wide–band channel **K**F−<sup>D</sup> follows. By (95) it is gaind, for

Whether the considered information transfers are narrow–band or wide–band, their algebraic-information description remains the same. So let be considered an arbitrary *stationary physical system* **Ψ** of these two band–types as usable for information transfer.

<sup>10</sup> If, in the special case of F–D channel, it is considered that the value *W* given by the number of electrons as the average energy of the *modulating current* entering into a wire, over a *time unit*, then it is the average power on the electric

follows the effective coding temperature *TW* = *T*<sup>0</sup> ·

k*T*<sup>0</sup> is gaind (the same as in the B–E case (103),

*<sup>C</sup>*(**K**F−D) = lim

� 1 √2

�√ 2 − 1 � <sup>+</sup> <sup>√</sup> <sup>2</sup> *<sup>W</sup>* k*T*<sup>0</sup>

*T*0→0

� 1 +

<sup>K</sup>*ε*, *<sup>ε</sup>* <sup>∈</sup> **<sup>S</sup>**(*ε*), of the channel **<sup>K</sup>**F−<sup>D</sup> is valid, <sup>W</sup>*<sup>ε</sup>* <sup>≥</sup> <sup>2</sup>*p*<sup>2</sup>

1 *<sup>τ</sup>* ∑ *ε*∈**S**(*ε*) *ε* 2*p*<sup>2</sup> *ε* <sup>1</sup>−*p*<sup>2</sup> *ε* . <sup>=</sup> <sup>2</sup> *h* � ∞ 0 *ε e* <sup>−</sup><sup>2</sup> *<sup>ε</sup>* k*T*0

**6. Physical information transfer and thermodynamics**

*q*(*i*|*θ*�

) *π*{*ψ*�

*π*2k2*T*<sup>0</sup> 2 .

*n* ∑ *i*=1

*<sup>ε</sup>*W*<sup>ε</sup>* <sup>≥</sup> lim*τ*→<sup>∞</sup>

*n* ∑ *i*=1

<sup>2</sup> *<sup>x</sup>* when <sup>|</sup>*x*<sup>|</sup> <sup>&</sup>lt; 1; *<sup>x</sup>* <sup>=</sup> <sup>12</sup>*hW*

⎛ ⎝ �

result is [24]

lim *T*0→0

*<sup>C</sup>*(**K**F−D) = *<sup>π</sup>*2k*T*<sup>0</sup>

*TW* <sup>≥</sup> *<sup>T</sup>*<sup>0</sup> <sup>&</sup>gt; 0, that10

*<sup>W</sup>*<sup>≥</sup> lim*τ*→<sup>∞</sup>

Let a system state *θ*� =

= 1 + <sup>1</sup>

<sup>9</sup> For <sup>√</sup><sup>1</sup> <sup>+</sup> *<sup>x</sup>* .

resistor *R* = 1Ω too.

equivocant) of the system state *θ* =

3*h*

<sup>=</sup> *<sup>π</sup>*2k*T*<sup>0</sup> 6*h*

> 1 *<sup>τ</sup>* ∑ *ε*∈**S**(*ε*)

*π*2k2*T*<sup>2</sup> *W* <sup>6</sup>*<sup>h</sup>* <sup>=</sup> *<sup>π</sup>*2k2*T*<sup>0</sup>

(a) For a *strictly convex* function *f*(*u*) = *u* · ln *u* the **Jensen** inequality is valid [23]

$$f\left[\sum\_{j=1}^{n} p(i|j) \, q(j|\theta) \right] \le \sum\_{j=1}^{n} p(i|j) \, f[q(j|\theta)], \quad i = 1, 2, \dots, n \tag{120}$$

$$\sum\_{i=1}^{n} f\left[\sum\_{j=1}^{n} p(i|j) \, q(j|\theta) \right] \le \sum\_{i=1}^{n} p(i|j) \sum\_{j=1}^{n} f[q(j|\theta)]$$

$$= \sum\_{j=1}^{n} f[q(j|\theta)] = \sum\_{j=1}^{n} q(j|\theta) \ln q(j|\theta) = -H[q(\cdot|\theta)] = -\mathcal{H}(\theta) \quad \text{due to} \quad \sum\_{i=1}^{n} p(i|j) = 1$$

<sup>11</sup> It is the matrix of the *unitary operator* **u**(*t*) expressing the time evolution of the system **Ψ**.

#### 20 Will-be-set-by-IN-TECH 102 Thermodynamics – Fundamentals and Its Application in Science Information Capacity of Quantum Transfer Channels and Thermodynamic Analogies <sup>21</sup>

and for distributions *q*(*i*|*θ*� ) is valid H(*θ*� ) ≥ H(*θ*):

$$\sum\_{i=1}^{n} f\left(\sum\_{j=1}^{n} p(i|j)\,q(j|\theta)\right) = \sum\_{i=1}^{n} q(i|\theta') \ln q(i|\theta') = -\mathcal{H}(\theta') \,\,[\le -\mathcal{H}(\theta)]$$

(b) The equality in (119) arises if and only if the index permutation [*i*(1), *i*(2), ..., *i*(*n*)] exists that *p*(*i*|*j*) = *δ*[*i*|*i*(*j*)], *j* = 1, 2, ..., *n*; then *q*[*i*(*j*)|*θ*� ] = *q*(*j*|*θ*), *j* = 1, 2, .... Let a fixed *j* is given. Then, when 0 = *p*(*i*|*j*)=(*ψ*� , *<sup>ψ</sup>j*)2, *<sup>i</sup>* �<sup>=</sup> *<sup>i</sup>*(*j*), the orthogonality is valid

*i*

$$\Psi(\boldsymbol{\psi}\_{\dot{\boldsymbol{\gamma}}}|\boldsymbol{\psi}\_{\dot{\boldsymbol{\gamma}}}) \perp \left[ \underset{i \neq i(j)}{\ominus} \,\Psi(\boldsymbol{\psi}\_{i}^{\prime}|\boldsymbol{\psi}\_{i}^{\prime}) \right], \quad \boldsymbol{\psi}\_{\dot{\boldsymbol{\gamma}}} = \pi \{\boldsymbol{\psi}\_{\dot{\boldsymbol{\gamma}}}\} = \boldsymbol{\theta}\_{\dot{\boldsymbol{\gamma}}}, \quad \boldsymbol{\psi}\_{i}^{\prime} = \pi \{\boldsymbol{\psi}\_{i}^{\prime}\} = \boldsymbol{\theta}\_{i}^{\prime} \tag{121}$$

and, consequently, *ψ<sup>j</sup>* ∈ Ψ[*ψ*� *i*(*j*) |*ψ*� *i*(*j*) ], *p*[*i*(*j*)|*j*]=(*ψ*� *i*(*j*) , *ψj*)<sup>2</sup> = 1. It results in *ψ<sup>j</sup>* = *ψ*� *i*(*j*) . This prooves that the equality H(*θ*� ) = H(*θ*) implies the equality *q*(*j*|*θ*) *π*{*ψj*} = *q*[*i*(*j*)|*θ*] *π*{*ψ*� *i*(*j*) } and *θ* = *θ*� .

H-theorem says, that a **reversible transition is not possible between any two different states** *θ* �= *θ*� . From the inequality (119) also follows that any state *θ* ∈ **Θ** of the system **Ψ** is the successor of itself, *θ* → *θ* and, that any **reversibility** of the **relation** *θ* → *θ*� **(the transition** *θ*� → *θ***) is not possible within the system only, it is not possible without openning this system Ψ**. The difference

$$\mathcal{H}(\boldsymbol{\theta}^{+}) - \mathcal{H}(\boldsymbol{\theta}) = \max\_{\boldsymbol{\theta}^{\prime} \in \boldsymbol{\Theta}} \mathcal{H}(\boldsymbol{\theta}^{\prime}) - \mathcal{H}(\boldsymbol{\theta}) = H\left[q(\cdot|\boldsymbol{\theta}^{+})\right] - H\left[q(\cdot|\boldsymbol{\theta})\right] \tag{122}$$

reppresents the information-theoretical expressing of the **Brillouin** (maximal) entropy defect Δ*H* (the Brillouin negentropic information principle [2, 30]). For the state *θ*<sup>+</sup> is valid that *<sup>θ</sup>* <sup>→</sup> *<sup>θ</sup>*+, <sup>∀</sup>*<sup>θ</sup>* <sup>∈</sup> **<sup>Θ</sup>**. It is also called the terminal state or the (atractor of the time evolution) of the system **Ψ**. 12

6.0.0.7. Gibbs Theorem:

For all *<sup>θ</sup>*, *<sup>θ</sup>*˜ <sup>∈</sup> **<sup>Θ</sup>** of the system **<sup>Ψ</sup>** is valid

$$\mathcal{H}(\boldsymbol{\theta}) \le -\text{Tr}(\boldsymbol{\theta} \ln \tilde{\boldsymbol{\theta}}) \tag{123}$$

operator *θ* in the base {*ψ*�

ln *θ*˜ =

valid that

H(*θ*�

*I*[*q*(·|*θ*�

follows that

*e* <sup>−</sup> *<sup>ε</sup>* <sup>k</sup>*TW* · *<sup>e</sup> <sup>ε</sup>*

operators ln *θ*˜ and Tr(*θ*ln *θ*˜) is valid that

*n* ∑ *i*=1

*n* ∑ *i*=1

= −

*I*[*q*(·|*θ*�

<sup>k</sup>*T*<sup>0</sup> ≥ 1, *e*

<sup>1</sup>, *ψ*�

)ln *<sup>q</sup>*(*i*|*θ*˜)

For the information divergence of the distributions *q*(·|*θ*�

*n* ∑ *i*=1

*q*(*i*|*θ*�

**(matematical-logical) way, the phenomenon of Gibbs paradox**.

transfer is running in the opposite direction (as for temperatures).

*<sup>C</sup>*B−E" from (50) is valid that *<sup>C</sup>*B−E" = 0. Then *<sup>W</sup>* = *WKrit* [= <sup>0</sup>] for *<sup>p</sup>*(W) = *<sup>p</sup>*.

ln *<sup>q</sup>*(*i*|*θ*˜) *<sup>π</sup>*{*ψ*�

*q*(*i*|*θ*�

)�*q*(·|*θ*˜)] =

)�*q*(·|*θ*˜)] = 0, <sup>H</sup>(*θ*�

*ε* k*T*0 *TW* <sup>−</sup>*T*<sup>0</sup> *TW* ≥ *e*

As for F–D channel; for the supposition W <

*Equivalence Principle of Thermodynamics in [16, 17, 19].*

By (119) for *θ* → *θ*� is writable that H(*θ*) ≤ H(*θ*�

) ≥ H(*θ*) are valid; the first equality is for

<sup>2</sup>, ..., *ψ*�

*<sup>n</sup>*} is obtained that *<sup>θ</sup>ij* <sup>=</sup> *<sup>n</sup>*

*<sup>i</sup>*} and <sup>−</sup> Tr(*θ*ln *<sup>θ</sup>*˜) = <sup>−</sup>

)ln *<sup>q</sup>*(*i*|*θ*�

) = <sup>H</sup>(*θ*˜), *<sup>q</sup>*(*i*|*θ*�

)

the second equality is for *θ*� = *θ*. The **Gibbs theorem expresses, in the deductive**

From formulas (47), (55), (56) and (68), (72), (73) for the narrow-band B–E and F–D capacities

and it is seen that the quantity temperature is decisive for studied information transfers. The last relation envokes, inevitably, such an opinion, that these transfers are able be modeled by <sup>a</sup> *direct* reversible **Carnot** cycle with efficiency *<sup>η</sup>max* <sup>∈</sup> (0, 1�). Conditions leading to *<sup>C</sup>*[·|·] <sup>&</sup>lt; <sup>0</sup> mean, in such a *direct* thermodynamic model, that its efficiency should be *η*max < 0. This is the contradiction with the *Equivalence Principle of Thermodynamics* [19]; expresses only that the

As for B–E channel; for the supposition *W* < 0 the inequalities *TW* < *T*<sup>0</sup> and *p*(W) < *p* would be gained which is the *contradiction* with (35), (36) and (47). It would be such a situation with the *information is transferred in a different direction and under a different operation mode*. Our sustaining on the meaning about the original organization of the transfer, for *TW* > *T*0, then leads to the contradiction mentioned above saying only that we are convinced mistakenly about the actual direction of the information transfer. In the case *TW* = *T*<sup>0</sup> for the capacity

gained which is the contradiction with (68). For *TW* = *<sup>T</sup>*<sup>0</sup> is for *<sup>C</sup>*F−<sup>D</sup> from (71) valid

<sup>13</sup> Derived by the information-thermodynamic way together with the *I*. and *I I*. *Thermodynamic Principle* and with the

0; *<sup>ε</sup>* <sup>&</sup>gt; 0, *<sup>T</sup>*<sup>0</sup> <sup>&</sup>gt; <sup>0</sup> −→ *TW* <sup>≥</sup> *<sup>T</sup>*<sup>0</sup> −→

2*p*<sup>2</sup>

*<sup>q</sup>*(*i*|*θ*˜) <sup>≥</sup> 0, −H(*θ*�

∑ *k*=1

*n* ∑ *i*=1

Information Capacity of Quantum Transfer Channels and Thermodynamic Analogies 103

 *n* ∑ *k*=1 *u*2 *ki q*(*k*|*θ*)

) ≥ *n* ∑ *i*=1

13

*uki ukj q*(*k*|*θ*) and thus for

ln *<sup>q</sup>*(*i*|*θ*˜) (124)

)ln *<sup>q</sup>*(*i*|*θ*˜). (125)

) is

), *<sup>q</sup>*(·|*θ*˜) and the entropy <sup>H</sup>(*θ*�

*q*(*i*|*θ*�

) ≤ −Tr(*θ*ln *<sup>θ</sup>*˜). By (123) <sup>−</sup>Tr(*θ*� ln *<sup>θ</sup>*˜) <sup>≥</sup>

) = *<sup>q</sup>*(*i*|*θ*˜), *<sup>i</sup>* <sup>=</sup> 1, 2, ..., *<sup>n</sup>*, *<sup>θ</sup>*� <sup>=</sup> *<sup>θ</sup>*˜ (126)

*TW* − *T*<sup>0</sup> *TW*

<sup>1</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup> *TW* <sup>&</sup>lt; *<sup>T</sup>*<sup>0</sup> and *<sup>p</sup>*(W) <sup>&</sup>lt; *<sup>p</sup>* is

�

= *η*max ≥ 0

(127)

and the equality arises only for *θ* = *θ*˜ [38].

6.0.0.8. Proof:

Let for *<sup>θ</sup>*, *<sup>θ</sup>*˜ <sup>∈</sup> **<sup>Θ</sup>** is valid that *<sup>θ</sup>* <sup>=</sup> *<sup>n</sup>* ∑ *i*=1 *<sup>q</sup>*(*i*|*θ*) *<sup>π</sup>*{*ψi*}, *<sup>θ</sup>*˜ <sup>=</sup> *<sup>n</sup>* ∑ *i*=1 *<sup>q</sup>*(*i*|*θ*˜) *<sup>π</sup>*{*ψ*� *i* } and let the operators *α*, *θ* are *commuting αθ* = *θα*, *D*(*α*) = {*D<sup>α</sup>* : *α* ∈ **S**(*α*)}, *D*(*θ*) = {*D<sup>θ</sup>* : *θ* ∈ **S**(*θ*)}, are their spectral decompositions. Let be the state *θ*� the successor of *θ*, *θ* → *θ*� and relations *<sup>p</sup>*(*α*|*α*|*θ*) = ∑ *i*∈*D<sup>α</sup> q*(*i*|*θ*� ) = *p*(*α*|*α*|*θ*� ) and H(*θ*� ) ≥ H(*θ*) are valid. For the matrix (*θij*) of the

<sup>12</sup> In this sense, the physical entropy <sup>H</sup>(*θ*) (19), (21) determines the direction of the thermodynamic time arrow [2], H(*θ*� ) − H(*θ*) <sup>Δ</sup>*<sup>t</sup>* <sup>=</sup> *<sup>∂</sup>*<sup>H</sup> *<sup>∂</sup><sup>t</sup>* <sup>≥</sup> 0, <sup>Δ</sup>*<sup>t</sup>* <sup>=</sup> *<sup>t</sup>θ*� <sup>−</sup> *<sup>t</sup><sup>θ</sup>* <sup>&</sup>gt; 0. The equality occurs in the equlibrial (stationary) state *<sup>θ</sup>*<sup>+</sup> of the system **Ψ** and its environment.

operator *θ* in the base {*ψ*� <sup>1</sup>, *ψ*� <sup>2</sup>, ..., *ψ*� *<sup>n</sup>*} is obtained that *<sup>θ</sup>ij* <sup>=</sup> *<sup>n</sup>* ∑ *k*=1 *uki ukj q*(*k*|*θ*) and thus for operators ln *θ*˜ and Tr(*θ* ln *θ*˜) is valid that

20 Will-be-set-by-IN-TECH

) ≥ H(*θ*):

*q*(*i*|*θ*�

(b) The equality in (119) arises if and only if the index permutation [*i*(1), *i*(2), ..., *i*(*n*)] exists

], *p*[*i*(*j*)|*j*]=(*ψ*�

H-theorem says, that a **reversible transition is not possible between any two different states**

reppresents the information-theoretical expressing of the **Brillouin** (maximal) entropy defect Δ*H* (the Brillouin negentropic information principle [2, 30]). For the state *θ*<sup>+</sup> is valid that *<sup>θ</sup>* <sup>→</sup> *<sup>θ</sup>*+, <sup>∀</sup>*<sup>θ</sup>* <sup>∈</sup> **<sup>Θ</sup>**. It is also called the terminal state or the (atractor of the time evolution)

*<sup>q</sup>*(*i*|*θ*) *<sup>π</sup>*{*ψi*}, *<sup>θ</sup>*˜ <sup>=</sup> *<sup>n</sup>*

*α*, *θ* are *commuting αθ* = *θα*, *D*(*α*) = {*D<sup>α</sup>* : *α* ∈ **S**(*α*)}, *D*(*θ*) = {*D<sup>θ</sup>* : *θ* ∈ **S**(*θ*)}, are their spectral decompositions. Let be the state *θ*� the successor of *θ*, *θ* → *θ*� and relations

<sup>12</sup> In this sense, the physical entropy <sup>H</sup>(*θ*) (19), (21) determines the direction of the thermodynamic time arrow [2],

∑ *i*=1

*<sup>∂</sup><sup>t</sup>* <sup>≥</sup> 0, <sup>Δ</sup>*<sup>t</sup>* <sup>=</sup> *<sup>t</sup>θ*� <sup>−</sup> *<sup>t</sup><sup>θ</sup>* <sup>&</sup>gt; 0. The equality occurs in the equlibrial (stationary) state *<sup>θ</sup>*<sup>+</sup> of the system

. From the inequality (119) also follows that any state *θ* ∈ **Θ** of the system **Ψ** is the successor of itself, *θ* → *θ* and, that any **reversibility** of the **relation** *θ* → *θ*� **(the transition** *θ*� → *θ***) is not possible within the system only, it is not possible without openning this**

) − H(*θ*) = *<sup>H</sup>* �

*i*

)ln *q*(*i*|*θ*�

⎦ , *ψ<sup>j</sup>* = *π*{*ψj*} = *θj*, *ψ*�

*i*(*j*)

) = −H(*θ*�

] = *q*(*j*|*θ*), *j* = 1, 2, ....

) = H(*θ*) implies the equality *q*(*j*|*θ*) *π*{*ψj*} = *q*[*i*(*j*)|*θ*] *π*{*ψ*�

*<sup>q</sup>*(·|*θ*+) �

<sup>H</sup>(*θ*) ≤ −Tr(*θ*ln *<sup>θ</sup>*˜) (123)

*<sup>q</sup>*(*i*|*θ*˜) *<sup>π</sup>*{*ψ*�

*i*

) ≥ H(*θ*) are valid. For the matrix (*θij*) of the

) [≤ −H(*θ*)]

, *<sup>ψ</sup>j*)2, *<sup>i</sup>* �<sup>=</sup> *<sup>i</sup>*(*j*), the orthogonality is valid

*<sup>i</sup>* = *π*{*ψ*�

, *ψj*)<sup>2</sup> = 1. It results in *ψ<sup>j</sup>* = *ψ*�

*<sup>i</sup>*} = *θ*�

− *H* [*q*(·|*θ*)] (122)

} and let the operators

*<sup>i</sup>* (121)

*i*(*j*) . This

> *i*(*j*) }

) is valid H(*θ*�

Ψ(*ψ*� *<sup>i</sup>* |*ψ*� *i* ) ⎤

*i*(*j*) |*ψ*� *i*(*j*)

<sup>H</sup>(*θ*+) − H(*θ*) = max

*θ*� <sup>∈</sup>**Θ**

> ∑ *i*=1

) and H(*θ*�

H(*θ*�

⎞ ⎠ =

*n* ∑ *i*=1

*p*(*i*|*j*) *q*(*j*|*θ*)

that *p*(*i*|*j*) = *δ*[*i*|*i*(*j*)], *j* = 1, 2, ..., *n*; then *q*[*i*(*j*)|*θ*�

Let a fixed *j* is given. Then, when 0 = *p*(*i*|*j*)=(*ψ*�

)⊥ ⎡ ⎣ ⊕ *i*�=*i*(*j*)

and for distributions *q*(*i*|*θ*�

*n* ∑ *i*=1 *f* ⎛ ⎝ *n* ∑ *j*=1

Ψ(*ψj*|*ψ<sup>j</sup>*

and, consequently, *ψ<sup>j</sup>* ∈ Ψ[*ψ*�

prooves that the equality H(*θ*�

.

**system Ψ**. The difference

12

For all *<sup>θ</sup>*, *<sup>θ</sup>*˜ <sup>∈</sup> **<sup>Θ</sup>** of the system **<sup>Ψ</sup>** is valid

and the equality arises only for *θ* = *θ*˜ [38].

Let for *<sup>θ</sup>*, *<sup>θ</sup>*˜ <sup>∈</sup> **<sup>Θ</sup>** is valid that *<sup>θ</sup>* <sup>=</sup> *<sup>n</sup>*

*q*(*i*|*θ*�

) = *p*(*α*|*α*|*θ*�

of the system **Ψ**.

6.0.0.8. Proof:

*<sup>p</sup>*(*α*|*α*|*θ*) = ∑

) − H(*θ*) <sup>Δ</sup>*<sup>t</sup>* <sup>=</sup> *<sup>∂</sup>*<sup>H</sup>

**Ψ** and its environment.

H(*θ*�

*i*∈*D<sup>α</sup>*

6.0.0.7. Gibbs Theorem:

and *θ* = *θ*�

*θ* �= *θ*�

$$\begin{split} \ln \tilde{\boldsymbol{\theta}} &= \sum\_{i=1}^{n} \ln q(i|\tilde{\boldsymbol{\theta}}) \, \mathop{\mathrm{tr}} \{ \psi\_{i}^{\prime} \} \quad \text{and} \quad -\operatorname{Tr}(\boldsymbol{\theta} \ln \tilde{\boldsymbol{\theta}}) = -\sum\_{i=1}^{n} \left[ \sum\_{k=1}^{n} u\_{ki}^{2} q(k|\boldsymbol{\theta}) \right] \ln q(i|\tilde{\boldsymbol{\theta}}) \qquad (124) \\ &= -\sum\_{i=1}^{n} q(i|\boldsymbol{\theta}^{\prime}) \ln q(i|\boldsymbol{\theta}) \end{split}$$

For the information divergence of the distributions *q*(·|*θ*� ), *<sup>q</sup>*(·|*θ*˜) and the entropy <sup>H</sup>(*θ*� ) is valid that

$$I[q(\cdot|\theta')|\|q(\cdot|\tilde{\theta})] = \sum\_{i=1}^{n} q(i|\theta') \ln \frac{q(i|\theta')}{q(i|\tilde{\theta})} \ge 0, \quad -\mathcal{H}(\theta') \ge \sum\_{i=1}^{n} q(i|\theta') \ln q(i|\tilde{\theta}).\tag{125}$$

By (119) for *θ* → *θ*� is writable that H(*θ*) ≤ H(*θ*� ) ≤ −Tr(*θ*ln *<sup>θ</sup>*˜). By (123) <sup>−</sup>Tr(*θ*� ln *<sup>θ</sup>*˜) <sup>≥</sup> H(*θ*� ) ≥ H(*θ*) are valid; the first equality is for

$$\mathcal{H}[q(\cdot|\theta')\|q(\cdot|\tilde{\theta})] = 0, \ \mathcal{H}(\theta') = \mathcal{H}(\tilde{\theta}), \ \mathcal{q}(i|\theta') = \mathcal{q}(i|\tilde{\theta}), \quad i = 1, 2, \dots, n, \ \theta' = \tilde{\theta} \tag{126}$$

the second equality is for *θ*� = *θ*. The **Gibbs theorem expresses, in the deductive (matematical-logical) way, the phenomenon of Gibbs paradox**. 13

From formulas (47), (55), (56) and (68), (72), (73) for the narrow-band B–E and F–D capacities follows that

$$e^{-\frac{\ell}{kT\_W}} \cdot e^{\frac{\ell}{kT\_0}} \ge 1, \ e^{\frac{\ell}{kT\_0}} \left(\frac{T\_W - T\_0}{T\_W}\right) \ge e^0; \; e > 0, \ T\_0 > 0 \implies T\_W \ge T\_0 \implies \frac{T\_W - T\_0}{T\_W} \stackrel{\triangle}{=} \eta\_{\max} \ge 0 \tag{127}$$

and it is seen that the quantity temperature is decisive for studied information transfers. The last relation envokes, inevitably, such an opinion, that these transfers are able be modeled by <sup>a</sup> *direct* reversible **Carnot** cycle with efficiency *<sup>η</sup>max* <sup>∈</sup> (0, 1�). Conditions leading to *<sup>C</sup>*[·|·] <sup>&</sup>lt; <sup>0</sup> mean, in such a *direct* thermodynamic model, that its efficiency should be *η*max < 0. This is the contradiction with the *Equivalence Principle of Thermodynamics* [19]; expresses only that the transfer is running in the opposite direction (as for temperatures).

As for B–E channel; for the supposition *W* < 0 the inequalities *TW* < *T*<sup>0</sup> and *p*(W) < *p* would be gained which is the *contradiction* with (35), (36) and (47). It would be such a situation with the *information is transferred in a different direction and under a different operation mode*. Our sustaining on the meaning about the original organization of the transfer, for *TW* > *T*0, then leads to the contradiction mentioned above saying only that we are convinced mistakenly about the actual direction of the information transfer. In the case *TW* = *T*<sup>0</sup> for the capacity *<sup>C</sup>*B−E" from (50) is valid that *<sup>C</sup>*B−E" = 0. Then *<sup>W</sup>* = *WKrit* [= <sup>0</sup>] for *<sup>p</sup>*(W) = *<sup>p</sup>*.

As for F–D channel; for the supposition W < 2*p*<sup>2</sup> <sup>1</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup> *TW* <sup>&</sup>lt; *<sup>T</sup>*<sup>0</sup> and *<sup>p</sup>*(W) <sup>&</sup>lt; *<sup>p</sup>* is gained which is the contradiction with (68). For *TW* = *<sup>T</sup>*<sup>0</sup> is for *<sup>C</sup>*F−<sup>D</sup> from (71) valid

<sup>13</sup> Derived by the information-thermodynamic way together with the *I*. and *I I*. *Thermodynamic Principle* and with the *Equivalence Principle of Thermodynamics in [16, 17, 19].*

22 Will-be-set-by-IN-TECH 104 Thermodynamics – Fundamentals and Its Application in Science Information Capacity of Quantum Transfer Channels and Thermodynamic Analogies <sup>23</sup>

$$\mathcal{C}\_{\mathcal{F}-\mathcal{D}} = \frac{h(p)}{1-p} + \frac{p}{1+p} \cdot \ln p - \ln(1+p).^{14}$$

Let be noticed yet the relations between the wide–band B–E and F–D capacities and the model heat efficiency *ηmax*. For the B–E capacity (98) is gained that

$$\mathbb{C}(\mathbf{K}\_{\mathrm{B}-\mathrm{E}}) = \frac{\pi^2 \mathbf{k} T\_W}{3h} \left( \frac{T\_W - T\_0}{T\_W} \right) = \frac{\pi^2 \mathbf{k} T\_W}{3h} \eta\_{\mathrm{max}} \underset{\eta\_{\mathrm{max}} \to 1}{\longrightarrow} \frac{\pi^2 \mathbf{k} T\_W}{3h} = \mathcal{C}^{\mathrm{max}}(\mathbf{K}\_{\mathrm{B}-\mathrm{E}}) \quad \text{(128)}$$

$$\mathcal{C}^{\mathrm{max}}(\mathbf{K}\_{\mathrm{B}-\mathrm{E}}) = \sup\_{\theta} H(\mathfrak{a} \| \theta) = H(\mathfrak{i}) = \mathcal{H}(\theta)$$

*Nevertheless, it will be shown that all these processes themselves are not organized cyclically 'by*

Further the relation between the information transfer in a wide–band B–E (photonic) channel *organized in a cyclical way* and a relevant (reverse) heat cycle will be dealt with. But, firstly, the way in which the capacity formula for an information transfer system of photons was derived

A transfer channel is now created by the electromagnetic radiation of a system L ∼= **K**L−<sup>L</sup> of photons being emitted from an *absolute black body* at temperature *T*<sup>0</sup> and within a frequency bandwidth of Δ*ν* = *R*+, where *ν* is the frequency. Then the energy of such radiation is the *energy* of *noise*. A source of *input messages, signals* transmits monochromatic electromagnetic impulses (numbers *ai* of photons) into this environment with an average *input energy W*.

superposition of the input signal and the noise signal. The input signal *ai*, within a frequency *ν*, is represented by the *occupation* number *m* = *m*(*ν*), which equates to the number of photons of an input field with an energy level *ε*(*ν*) = *hν*. The output signal is represented by the occupation number *l* = *l*(*ν*). The noise signal, created by the number of photons emitted by absolute black body radiation at temperature *T*0, is represented by the occupation number *n* = *n*(*ν*). The medium values of these quantities (spectral densities of the input, noise and output photonic stream) are denoted as *m*, *n* and *l*. In accordance with the *Planck radiation* law, the spectral density *r* of a photonic stream of absolute black body radiation at temperature

Thus, for the average energy *P* of radiation at temperature Θ within the bandwidth Δ*ν* = *R*<sup>+</sup>

Then, for the average noise energy *P*<sup>1</sup> at temperature *T*0, and for the average output energy *P*<sup>2</sup>

The entropy *H* of radiation at temperature *ϑ* is derived from *Clausius definition* of *heat entropy*

<sup>15</sup> To distinguish between two frequencies mutually deferring at an infinitesimally small d*ν* is needed, in accordance with *Heisenberg uncertainty principle*, a time interval spanning the infinite length of time, Δ*t* −→ ∞; analog of the

<sup>d</sup><sup>Θ</sup> <sup>d</sup><sup>Θ</sup> <sup>=</sup> *<sup>π</sup>*2k*<sup>ϑ</sup>*

at temperature *TW*, both of which occur within the bandwidth Δ*ν* = *R*<sup>+</sup> is valid that

2

d*P*(Θ)

, *<sup>l</sup>*(*ν*) = *<sup>p</sup>*(*ν*, *TW*)

6¯*<sup>h</sup>* where *<sup>ε</sup>*(*ν*, <sup>Θ</sup>) = *<sup>r</sup>*(*ν*)*h<sup>ν</sup>* and <sup>d</sup>*P*(Θ)

6¯*<sup>h</sup>* , *<sup>P</sup>*2(*TW*) = *<sup>π</sup>*2k2*TW*

1 − *p*(*ν*, *TW*)

2

3¯*<sup>h</sup>* <sup>=</sup> <sup>2</sup>*P*(*ϑ*)

, *p*(*ν*, Θ) = *e*

<sup>d</sup><sup>Θ</sup> <sup>=</sup> *<sup>π</sup>*2k2<sup>Θ</sup>

6¯*<sup>h</sup>* . (133)

<sup>k</sup>*<sup>ϑ</sup>* (134)

*<sup>i</sup>*=1, with a probability

<sup>−</sup> *<sup>h</sup><sup>ν</sup>*

*k*Θ (131)

3¯*<sup>h</sup>* . (132)

<sup>15</sup> The *output (whole, received)* signal is created by *additive*

Information Capacity of Quantum Transfer Channels and Thermodynamic Analogies 105

**6.1. Thermodynamic derivation of wide–band photonic capacity**

This source is defined by an alphabet of input messages, signals {*ai*}*<sup>n</sup>*

Θ and within frequency *ν*, is given by the *Planck distribution*,

, *<sup>n</sup>*(*ν*) = *<sup>p</sup>*(*ν*, *<sup>T</sup>*0)

*<sup>P</sup>*1(*T*0) = *<sup>π</sup>*2k2*T*<sup>0</sup>

1 kΘ

*H* = *ϑ* 0

*<sup>ε</sup>*(*ν*, <sup>Θ</sup>)d*<sup>ν</sup>* <sup>=</sup> *<sup>π</sup>*2k2Θ<sup>2</sup>

1 − *p*(*ν*, *T*0)

*themselves'.*

in [30] will be reviewed.

*<sup>r</sup>*(*ν*) = *<sup>p</sup>*(*ν*, <sup>Θ</sup>)

is gained that

and thus

*<sup>P</sup>*(Θ) = <sup>∞</sup>

0

*thermodynamic stationarity*.

1 − *p*(*ν*, Θ)

distribution *pi* = *p*(*ai*), *i* = 1, 2, ... , *n*.

*<sup>C</sup>*(**K**B−E) > 0, *TW* > *<sup>T</sup>*0, *<sup>C</sup>*(**K**B−E) −→ *TW*→*T*<sup>0</sup> (*η*max→0) 0, *TW* −→ *T*<sup>0</sup>

It is the information capacity for such a direct Carnot cycle where *<sup>H</sup>*(*X*) = *<sup>π</sup>*2k*TW* <sup>3</sup>*<sup>h</sup>* <sup>=</sup> *<sup>C</sup>*max(**K**B−E).

For the wide-band F–D capacity from (105) is valid

$$\begin{split} \mathbf{C}(\mathbf{K}\_{\mathrm{F-D}}) &= \frac{\pi^2 \mathbf{k} T\_W}{3h} - \frac{\pi^2 \mathbf{k} T\_0}{6h}, \quad T\_W \ge T\_0 \quad \text{and for } T\_W > T\_{0\prime} \\ \mathbf{C}(\mathbf{K}\_{\mathrm{F-D}}) &= \frac{\pi^2 \mathbf{k} T\_W}{3h} \cdot \frac{2T\_W - T\_0}{2T\_W} = \frac{\pi^2 \mathbf{k} T\_W}{3h} \cdot \frac{2T\_W - T\_0}{2(T\_W - T\_0)} \cdot \eta\_{\max} \\ &= \mathbf{C}(\mathbf{K}\_{\mathrm{B-E}}) \cdot \frac{2T\_W - T\_0}{2(T\_W - T\_0)} \end{split} \tag{129}$$

Due to 1 <sup>−</sup> *<sup>η</sup>*max <sup>=</sup> *<sup>T</sup>*<sup>0</sup> *TW* is valid *<sup>T</sup>*<sup>0</sup> <sup>=</sup> *TW*(<sup>1</sup> <sup>−</sup> *<sup>η</sup>*max) and also *<sup>C</sup>*(**K**F−D) = *<sup>π</sup>*2k*TW* <sup>6</sup>*<sup>h</sup>* · (<sup>1</sup> <sup>+</sup> *<sup>η</sup>*max). Then,

$$\mathbf{C}\left(\mathbf{K}\_{\mathrm{F-D}}\right) = \underset{\eta\_{\mathrm{max}} \to 1}{\longrightarrow} \frac{\pi^2 \mathbf{k} T\_W}{3h},\tag{130}$$

$$\mathbf{C}\left(\mathbf{K}\_{\mathrm{F-D}}\right) \underset{\left(\eta\_{\mathrm{max}} \to \mathbf{I}\_0\right)}{\underset{\left(\eta\_{\mathrm{max}} \to \mathbf{0}\right)}{\right2}} \frac{1}{2} H(i) = \frac{1}{2} \left.\mathcal{H}(\boldsymbol{\theta}) = \frac{\pi^2 \mathbf{k} T\_W}{6h}$$

$$\mathbf{C}\left(\mathbf{K}\_{\mathrm{F-D}}\right) \in \left\langle \frac{\pi^2 \mathbf{k} T\_W}{6\mathbf{h}}, \frac{\pi^2 \mathbf{k} T\_W}{3\mathbf{h}} \right\rangle = \left\langle \frac{1}{2} \left.\mathcal{H}(\boldsymbol{\theta}), \mathcal{H}(\boldsymbol{\theta}) \right\rangle$$

**Again the phenomenon of the not-zero capacity is seen here when the difference between the coding temperature** *TW* **and the noise temperature** *<sup>T</sup>*<sup>0</sup> **is zero.** Capacities *<sup>C</sup>*(**K**F−D) ≥ <sup>0</sup> are, surely, considerable for *TW* ∈ � *<sup>T</sup>*<sup>0</sup> <sup>2</sup> , *<sup>T</sup>*0� and being given by the property of the F–D phase space cells. Capacities *<sup>C</sup>*(**K**B−E) < 0 and *<sup>C</sup>*(**K**F−D) < 0 are without sense for the given direction of information transfer.

<sup>14</sup> Nevertheless the capacity *<sup>C</sup>*F−<sup>D</sup> for this case *<sup>W</sup>* <sup>&</sup>lt; *<sup>W</sup>*Krit is set in [12, 13]. *Similar* results as this one and (74) are gained for the **Maxwell–Boltzman** (M–B) system in [13].

*Nevertheless, it will be shown that all these processes themselves are not organized cyclically 'by themselves'.*

Further the relation between the information transfer in a wide–band B–E (photonic) channel *organized in a cyclical way* and a relevant (reverse) heat cycle will be dealt with. But, firstly, the way in which the capacity formula for an information transfer system of photons was derived in [30] will be reviewed.

#### **6.1. Thermodynamic derivation of wide–band photonic capacity**

22 Will-be-set-by-IN-TECH

Let be noticed yet the relations between the wide–band B–E and F–D capacities and the model

<sup>=</sup> *<sup>π</sup>*2k*TW*

*θ*

It is the information capacity for such a direct Carnot cycle where *<sup>H</sup>*(*X*) = *<sup>π</sup>*2k*TW*

<sup>3</sup>*<sup>h</sup> <sup>η</sup>*max −→

*η*max→1

*H*(*α*�*θ*) = *H*(*i*) = H(*θ*)

*TW*→*T*<sup>0</sup> (*η*max→0)

<sup>=</sup> *<sup>π</sup>*2k*TW*

is valid *<sup>T</sup>*<sup>0</sup> <sup>=</sup> *TW*(<sup>1</sup> <sup>−</sup> *<sup>η</sup>*max) and also *<sup>C</sup>*(**K**F−D) = *<sup>π</sup>*2k*TW*

 = 1

<sup>2</sup> <sup>H</sup>(*θ*) = *<sup>π</sup>*2k*TW*

6*h*

<sup>2</sup> <sup>H</sup>(*θ*), <sup>H</sup>(*θ*)

<sup>2</sup> , *<sup>T</sup>*0� and being given by the property of the F–D phase

*π*2k*TW*

0, *TW* −→ *T*<sup>0</sup>

<sup>6</sup>*<sup>h</sup>* , *TW*≥*T*<sup>0</sup> and for *TW* <sup>&</sup>gt; *<sup>T</sup>*0, (129)

<sup>2</sup>(*TW* <sup>−</sup> *<sup>T</sup>*0) · *<sup>η</sup>*max

<sup>3</sup>*<sup>h</sup>* , (130)

<sup>3</sup>*<sup>h</sup>* · <sup>2</sup>*TW* <sup>−</sup> *<sup>T</sup>*<sup>0</sup>

<sup>3</sup>*<sup>h</sup>* <sup>=</sup> *<sup>C</sup>*max(**K**B−E) (128)

<sup>3</sup>*<sup>h</sup>* <sup>=</sup>

<sup>6</sup>*<sup>h</sup>* · (<sup>1</sup> <sup>+</sup> *<sup>η</sup>*max).

14

*<sup>C</sup>*max(**K**B−E) = sup

*<sup>C</sup>*(**K**B−E) > 0, *TW* > *<sup>T</sup>*0, *<sup>C</sup>*(**K**B−E) −→

<sup>3</sup>*<sup>h</sup>* <sup>−</sup> *<sup>π</sup>*2k*T*<sup>0</sup>

<sup>=</sup> *<sup>C</sup>*(**K**B−E) · <sup>2</sup>*TW* <sup>−</sup> *<sup>T</sup>*<sup>0</sup>

*η*max→1

*π*2kTW

1 2

*TW*→*T*<sup>0</sup> (*η*max→0)

2*TW* − *T*<sup>0</sup> 2*TW*

2(*TW* − *T*0)

*π*2k*TW*

*<sup>H</sup>*(*i*) = <sup>1</sup>

6h , *<sup>π</sup>*2kTW 3h

**Again the phenomenon of the not-zero capacity is seen here when the difference between the coding temperature** *TW* **and the noise temperature** *<sup>T</sup>*<sup>0</sup> **is zero.** Capacities *<sup>C</sup>*(**K**F−D) ≥ <sup>0</sup>

space cells. Capacities *<sup>C</sup>*(**K**B−E) < 0 and *<sup>C</sup>*(**K**F−D) < 0 are without sense for the given

<sup>14</sup> Nevertheless the capacity *<sup>C</sup>*F−<sup>D</sup> for this case *<sup>W</sup>* <sup>&</sup>lt; *<sup>W</sup>*Krit is set in [12, 13]. *Similar* results as this one and (74) are gained

3*h* ·

*<sup>C</sup>* (**K**F−D) = −→

*<sup>C</sup>* (**K**F−D) −→

*<sup>C</sup>* (**K**F−D) ∈

*<sup>C</sup>*F−<sup>D</sup> <sup>=</sup> *<sup>h</sup>*(*p*)

*<sup>C</sup>*max(**K**B−E).

Due to 1 <sup>−</sup> *<sup>η</sup>*max <sup>=</sup> *<sup>T</sup>*<sup>0</sup>

Then,

1 − *p*

<sup>+</sup> *<sup>p</sup>*

*<sup>C</sup>*(**K**B−E) = *<sup>π</sup>*2k*TW*

3*h*

For the wide-band F–D capacity from (105) is valid

*<sup>C</sup>*(**K**F−D) = *<sup>π</sup>*2k*TW*

*<sup>C</sup>*(**K**F−D) = *<sup>π</sup>*2k*TW*

*TW*

are, surely, considerable for *TW* ∈ � *<sup>T</sup>*<sup>0</sup>

for the **Maxwell–Boltzman** (M–B) system in [13].

direction of information transfer.

<sup>1</sup> <sup>+</sup> *<sup>p</sup>* · ln *<sup>p</sup>* <sup>−</sup> ln(<sup>1</sup> <sup>+</sup> *<sup>p</sup>*).

heat efficiency *ηmax*. For the B–E capacity (98) is gained that

 *TW* <sup>−</sup> *<sup>T</sup>*<sup>0</sup> *TW*

> A transfer channel is now created by the electromagnetic radiation of a system L ∼= **K**L−<sup>L</sup> of photons being emitted from an *absolute black body* at temperature *T*<sup>0</sup> and within a frequency bandwidth of Δ*ν* = *R*+, where *ν* is the frequency. Then the energy of such radiation is the *energy* of *noise*. A source of *input messages, signals* transmits monochromatic electromagnetic impulses (numbers *ai* of photons) into this environment with an average *input energy W*. This source is defined by an alphabet of input messages, signals {*ai*}*<sup>n</sup> <sup>i</sup>*=1, with a probability distribution *pi* = *p*(*ai*), *i* = 1, 2, ... , *n*. <sup>15</sup> The *output (whole, received)* signal is created by *additive* superposition of the input signal and the noise signal. The input signal *ai*, within a frequency *ν*, is represented by the *occupation* number *m* = *m*(*ν*), which equates to the number of photons of an input field with an energy level *ε*(*ν*) = *hν*. The output signal is represented by the occupation number *l* = *l*(*ν*). The noise signal, created by the number of photons emitted by absolute black body radiation at temperature *T*0, is represented by the occupation number *n* = *n*(*ν*). The medium values of these quantities (spectral densities of the input, noise and output photonic stream) are denoted as *m*, *n* and *l*. In accordance with the *Planck radiation* law, the spectral density *r* of a photonic stream of absolute black body radiation at temperature Θ and within frequency *ν*, is given by the *Planck distribution*,

$$\overline{r}(\nu) = \frac{p(\nu, \Theta)}{1 - p(\nu, \Theta)}, \ \overline{n}(\nu) = \frac{p(\nu, T\_0)}{1 - p(\nu, T\_0)}, \ \overline{l}(\nu) = \frac{p(\nu, T\_W)}{1 - p(\nu, T\_W)}, \ \ p(\nu, \Theta) = \varepsilon^{-\frac{h\nu}{k\Theta}} \tag{131}$$

Thus, for the average energy *P* of radiation at temperature Θ within the bandwidth Δ*ν* = *R*<sup>+</sup> is gained that

$$P(\Theta) = \int\_0^\infty \overline{\varepsilon}(\nu, \Theta) d\nu = \frac{\pi^2 \mathbf{k}^2 \Theta^2}{6\hbar} \text{ where } \overline{\varepsilon}(\nu, \Theta) = \overline{\tau}(\nu)h\nu \text{ and } \frac{\mathrm{d}P(\Theta)}{\mathrm{d}\Theta} = \frac{\pi^2 \mathbf{k}^2 \Theta}{3\hbar}. \tag{132}$$

Then, for the average noise energy *P*<sup>1</sup> at temperature *T*0, and for the average output energy *P*<sup>2</sup> at temperature *TW*, both of which occur within the bandwidth Δ*ν* = *R*<sup>+</sup> is valid that

$$P\_1(T\_0) = \frac{\pi^2 \mathbf{k}^2 T\_0^2}{6\hbar},\ \ P\_2(T\_W) = \frac{\pi^2 \mathbf{k}^2 T\_W^2}{6\hbar}.\tag{133}$$

The entropy *H* of radiation at temperature *ϑ* is derived from *Clausius definition* of *heat entropy* and thus

$$H = \int\_0^{\theta} \frac{1}{\mathbf{k}\Theta} \frac{\mathbf{d}P(\Theta)}{\mathbf{d}\Theta} \mathbf{d}\Theta = \frac{\pi^2 \mathbf{k}\theta}{\mathfrak{M}} = \frac{2P(\theta)}{\mathbf{k}\theta} \tag{134}$$

<sup>15</sup> To distinguish between two frequencies mutually deferring at an infinitesimally small d*ν* is needed, in accordance with *Heisenberg uncertainty principle*, a time interval spanning the infinite length of time, Δ*t* −→ ∞; analog of the *thermodynamic stationarity*.

#### 24 Will-be-set-by-IN-TECH 106 Thermodynamics – Fundamentals and Its Application in Science Information Capacity of Quantum Transfer Channels and Thermodynamic Analogies <sup>25</sup>

Thus, for the entropy *H*<sup>1</sup> of the noise signal and for the entropy *H*<sup>2</sup> of the ouptut signal on the channel L is

$$H\_1 = \frac{\pi^2 \text{k}T\_0}{3\hbar} = \frac{2P\_1}{\text{k}T\_0}, \quad H\_2 = \frac{\pi^2 \text{k}T\_W}{3\hbar} = \frac{2P\_2}{\text{k}T\_W} \tag{135}$$

But the other *information arrangement, description* of a revese Carnot cycle will be used further,

In a general (reversible) *discrete* heat cycle O (with temperatures of its heat reservoires changing in a discrete way) considered as a model of the information transfer process in an transfer channel **K** ∼= L [17, 19] is, for the *elementary* changes *H*(Θ*k*) · *η*[*maxk* ] of information

In a general (reversible) *continuous* cycle O [with temperatures changing continuously, *n* −→ ∞ in the previous discrete system, at Θ will be Δ*Q*(Θ)] considered as an information transfer

0

 *TW T*0

<sup>2</sup> · *<sup>T</sup>*<sup>0</sup> <sup>=</sup> <sup>2</sup>Δ*QW*

· *η*max = *H*(*Y*) · *η*max = *H*(*X*) = Δ*I*

*δA*(Θ) <sup>k</sup><sup>Θ</sup> <sup>=</sup>

*TW* ,*T*<sup>0</sup> = *H*(*Y*)

 *T*<sup>0</sup> 0

 *TW T*0

*δQ*(*θ*)

<sup>d</sup>*H*(Θ) = <sup>2</sup>Δ*QW*

<sup>d</sup>*H*(Θ) = *<sup>T</sup>*<sup>0</sup>

k*TW*

<sup>d</sup>*H*(Θ) = *TW*

Δ*Q*<sup>0</sup> ∼ *H*(*X*), Δ*QW* ∼ *H*(*Y*) and Δ*A* ∼ *H*(*Y*|*X*), *H*(*X*|*Y*) = 0 (142)

Information Capacity of Quantum Transfer Channels and Thermodynamic Analogies 107

· *η*[*maxk* ], *k* = 1, 2, ..., *n* (143)

<sup>k</sup>*<sup>θ</sup>* <sup>d</sup>(*θ*); <sup>Δ</sup>*Q*(Θ)= <sup>Θ</sup>

k*TW*

d*H*(Θ)

= *H*(*X*) · *β*

d*H*(Θ)

*δQW*(Θ)

<sup>2</sup>*<sup>π</sup>* and *<sup>h</sup>* is *Planck constant*.

<sup>d</sup>*H*(Θ) = <sup>2</sup>Δ*QW*

k*TW*

0

· *T*0 *TW*

*T*0

 *TW T*0

0

<sup>18</sup> The change of heat

*δQ*(*θ*)d*θ* (144)

<sup>2</sup> · (*TW* − *T*0) (145)

<sup>k</sup><sup>Θ</sup> (146)

[*H*(*X*) = *H*(*Y*) · *η*max]

given by

entropies of <sup>L</sup>, valid that17

<sup>d</sup>*H*(Θ) �<sup>=</sup> *<sup>δ</sup>Q*(Θ)

*<sup>H</sup>*(*X*) = *<sup>S</sup>*(*TW*)

*<sup>H</sup>*(*Y*) = *<sup>S</sup>*(*TW*)

= *T*<sup>0</sup> 0

=  *H*(Θ*k*) · *η*[*maxk* ]

*∂Q*(Θ) *<sup>∂</sup>*<sup>Θ</sup> <sup>d</sup><sup>Θ</sup>

<sup>k</sup> <sup>=</sup>

<sup>k</sup><sup>Θ</sup> <sup>=</sup> *<sup>S</sup>*(*T*0)

<sup>k</sup><sup>Θ</sup> <sup>=</sup> <sup>2</sup>Δ*QW* k*TW*

*CTW* ,*T*<sup>0</sup> <sup>=</sup> *<sup>T</sup>*(*X*;*Y*) = *<sup>H</sup>*(*X*) =

<sup>17</sup> In reality for the least elementary heat change *<sup>δ</sup><sup>Q</sup>* <sup>=</sup> *<sup>h</sup>*¯ *<sup>ν</sup>* is right where ¯*<sup>h</sup>* <sup>=</sup> *<sup>h</sup>*

<sup>=</sup> <sup>2</sup>Δ*QW* k*TW*

<sup>18</sup> It is provable that the Carnot cycle itself is elenentary, *not dividible* [18].

 *TW* 0

For the whole cycle O*rrev*, *TW* > *T*<sup>0</sup> > 0, let be *H*(*X*|*Y*) = 0 and then

O*rrev*

*δQW*(Θ*W*) kΘ*<sup>W</sup>*

0

0

of the system L at temperatures Θ*<sup>k</sup>* is Δ*Q*(Θ*k*).

process in a transfer channel **K** ∼= L is valid that

<sup>k</sup><sup>Θ</sup> <sup>=</sup>

<sup>k</sup> <sup>−</sup> *<sup>S</sup>*(*T*0)

*∂QW*(Θ)

*δA*(Θ)

<sup>k</sup> <sup>=</sup>

*<sup>T</sup>*(*Y*; *<sup>X</sup>*) = *<sup>H</sup>*(*Y*) <sup>−</sup> *<sup>H</sup>*(*Y*|*X*) = *TW*

capacity *CTW* ,*T*<sup>0</sup> of the channel **K** ∼= L too,

O*rrev*

*<sup>H</sup>*(*Y*|*X*) = *<sup>H</sup>*(*Y*) <sup>−</sup> *<sup>H</sup>*(*X*) = *TW*

� <sup>=</sup> <sup>Δ</sup>*Q*(Θ*k*) kΘ*<sup>k</sup>*

<sup>k</sup><sup>Θ</sup> and*H*(Θ)= <sup>Θ</sup>

*δA*(Θ) <sup>k</sup><sup>Θ</sup> <sup>=</sup>

> = *TW* 0

d*H*(Θ) −

<sup>k</sup> <sup>=</sup> <sup>2</sup>Δ*QW* k*TW*

d*H*(Θ) −

<sup>2</sup> · (*TW* <sup>−</sup> *<sup>T</sup>*0), *<sup>C</sup>*max

Obviously, *T*(*X*;*Y*) = *H*(*X*) − *H*(*X*|*Y*) = *T*(*Y*|*X*). Further it is obvious that *T*(*X*;*Y*) is the

O*rrev*

where *n* ≥ 2 is the maximal number of its *elementary* Carnot cycles O*k*.

The information capacity *CTW* ,*T*<sup>0</sup> of the wide-band photonic transfer channel L is given by the maximal *entropy defect* [2, 30]) by

$$\mathbf{C}\_{T\_{\Theta}, T\_{W}} = H\_{2} - H\_{1} = \int\_{T\_{0}}^{T\_{W}} \frac{1}{\mathbf{k}\Theta} \frac{\mathbf{d}P(\Theta)}{\mathbf{d}\Theta} \mathbf{d}\Theta = \frac{\pi^{2}\mathbf{k}}{3\hbar} \int\_{T\_{0}}^{T\_{W}} \mathbf{d}\Theta = \frac{\pi^{2}\mathbf{k}}{3\hbar} \cdot \left(T\_{W} - T\_{0}\right) \,. \tag{136}$$

For *P*<sup>2</sup> = *P*<sup>1</sup> + *W*, where *W* is the average energy of the input signal is then valid that

$$\frac{\pi^2 \mathbf{k}^2 T\_W^2}{6\hbar} = \frac{\pi^2 \mathbf{k}^2 T\_0^2}{6\hbar} + W \quad \longrightarrow \quad T\_W = T\_0 \cdot \sqrt{1 + \frac{6\hbar \cdot W}{\pi^2 \mathbf{k}^2 T\_0^2}}\tag{137}$$

Then, in accordance with (102), (103), (104) [30]

$$\mathbf{C}\_{T\_0, W}(\mathbf{K}\_{\rm L-L}) = \frac{\pi^2 \mathbf{k} T\_0}{3\hbar} \cdot \left(\sqrt{1 + \frac{6\hbar \cdot W}{\pi^2 \mathbf{k}^2 T\_0^2}} - 1\right) \tag{138}$$

#### **7. Reverse heat cycle and transfer channel**

A *reverse* and reversible Carnot cycle O*rrev* starts with the *isothermal expansion* at temperature *T*<sup>0</sup> (the *diathermic* contact [31] is made between the system L and the cooler B) when L is receiving the *pumped out, transferred* heat Δ*Q*<sup>0</sup> from the B. During the *isothermal compression*, when the temperature of both the system L and the heater A is at the same value *TW*, *TW* > *T*<sup>0</sup> > 0, the *output* heat Δ*QW* is being delivered to the A

$$
\Delta Q\_W = \Delta Q\_\emptyset + \Delta A \tag{139}
$$

where Δ*A* is the *input* mechanical energy (work) delivered into L during this isothermal compression. It follows from [2, 8, 28] that when an average amount of information Δ*I* is being *recorded, transmitted*, *computed*, etc. at temperature Θ, there is a need for the average

energy <sup>Δ</sup>*<sup>W</sup>* <sup>≥</sup> *<sup>k</sup>* · <sup>Θ</sup> · <sup>Δ</sup>*I*; at this case <sup>Δ</sup>*<sup>W</sup>* � = Δ*A*. Thus O*rrev* is considerable as a *thermodynamic model* of information transfer process in the channel K ∼= L [14]. The following values are *changes* of the information entropies defined on <sup>K</sup>16:

$$H(\mathbf{Y}) \stackrel{\triangle}{=} \frac{\Delta \mathbf{Q}\_W}{\mathbf{k} T\_W} \text{ output } (\stackrel{\triangle}{=} \Delta I), \text{ } H(\mathbf{X}) \stackrel{\triangle}{=} \frac{\Delta \mathbf{A}}{\mathbf{k} T\_W} \text{ input, } \ H(\mathbf{Y}|\mathbf{X}) \stackrel{\triangle}{=} \frac{\Delta \mathbf{Q}\_0}{\mathbf{k} T\_W} \text{ noise} \tag{140}$$

where k is Boltzman constant. The information transfer in K ∼= L is *without losses* caused by the *friction, noise heat* (Δ*Q*0*<sup>x</sup>* = 0) and thus *H*(*X*|*Y*) = 0.

By assuming that for the changes (140) and *H*(*X*|*Y*) = 0 the channel equation (4), (5) and (23) is valid The result is

$$T(X;Y) = \frac{\Delta A}{\mathbf{k}T\_W} - 0 = \frac{\Delta Q\_W}{\mathbf{k}T\_W} \cdot \eta\_{\text{max}} = H(X) \tag{141}$$

$$T(Y;X) = \frac{\Delta Q\_0 + \Delta A}{\mathbf{k}T\_W} - \frac{\Delta Q\_0}{\mathbf{k}T\_W} = \frac{\Delta A}{\mathbf{k}T\_W} = H(X).$$

<sup>16</sup> In *information* units *Hartley*, *nat*, *bit*; *<sup>H</sup>*(·) = <sup>Δ</sup>*H*(·), *<sup>H</sup>*(·|·) = <sup>Δ</sup>*H*(·|·).

But the other *information arrangement, description* of a revese Carnot cycle will be used further, given by

$$
\Delta Q\_0 \sim H(X), \ \Delta Q\_W \sim H(Y) \text{ and } \Delta A \sim H(Y|X), \ H(X|Y) = 0 \tag{142}
$$

In a general (reversible) *discrete* heat cycle O (with temperatures of its heat reservoires changing in a discrete way) considered as a model of the information transfer process in an transfer channel **K** ∼= L [17, 19] is, for the *elementary* changes *H*(Θ*k*) · *η*[*maxk* ] of information entropies of <sup>L</sup>, valid that17

$$H(\Theta\_k) \cdot \eta\_{[\max\_k]} \stackrel{\triangle}{=} \frac{\Delta Q(\Theta\_k)}{\mathbf{k}\Theta\_k} \cdot \eta\_{[\max\_k]}, \ k = 1, 2, \dots, n \tag{143}$$

where *n* ≥ 2 is the maximal number of its *elementary* Carnot cycles O*k*. <sup>18</sup> The change of heat of the system L at temperatures Θ*<sup>k</sup>* is Δ*Q*(Θ*k*).

In a general (reversible) *continuous* cycle O [with temperatures changing continuously, *n* −→ ∞ in the previous discrete system, at Θ will be Δ*Q*(Θ)] considered as an information transfer process in a transfer channel **K** ∼= L is valid that

$$\mathrm{d}H(\Theta) \stackrel{\triangle}{=} \frac{\delta Q(\Theta)}{\mathbf{k}\Theta} = \frac{\frac{\partial Q(\Theta)}{\partial \Theta}\mathrm{d}\Theta}{\mathbf{k}\Theta} \text{ and} \\ H(\Theta) = \int\_0^{\Theta} \frac{\delta Q(\theta)}{\mathbf{k}\theta} \mathrm{d}(\theta); \ \Delta Q(\Theta) = \int\_0^{\Theta} \delta Q(\theta) \mathrm{d}\theta \text{ (144)}$$

For the whole cycle O*rrev*, *TW* > *T*<sup>0</sup> > 0, let be *H*(*X*|*Y*) = 0 and then

24 Will-be-set-by-IN-TECH

Thus, for the entropy *H*<sup>1</sup> of the noise signal and for the entropy *H*<sup>2</sup> of the ouptut signal on the

The information capacity *CTW* ,*T*<sup>0</sup> of the wide-band photonic transfer channel L is given by the

<sup>d</sup><sup>Θ</sup> <sup>d</sup><sup>Θ</sup> <sup>=</sup> *<sup>π</sup>*2k

<sup>6</sup>*<sup>h</sup>* <sup>+</sup> *<sup>W</sup>* −→ *TW* <sup>=</sup> *<sup>T</sup>*<sup>0</sup> ·

 1 +

3*h* ·

A *reverse* and reversible Carnot cycle O*rrev* starts with the *isothermal expansion* at temperature *T*<sup>0</sup> (the *diathermic* contact [31] is made between the system L and the cooler B) when L is receiving the *pumped out, transferred* heat Δ*Q*<sup>0</sup> from the B. During the *isothermal compression*, when the temperature of both the system L and the heater A is at the same value *TW*, *TW* >

where Δ*A* is the *input* mechanical energy (work) delivered into L during this isothermal compression. It follows from [2, 8, 28] that when an average amount of information Δ*I* is being *recorded, transmitted*, *computed*, etc. at temperature Θ, there is a need for the average

*model* of information transfer process in the channel K ∼= L [14]. The following values are

where k is Boltzman constant. The information transfer in K ∼= L is *without losses* caused by

By assuming that for the changes (140) and *H*(*X*|*Y*) = 0 the channel equation (4), (5) and (23)

<sup>−</sup> <sup>0</sup> <sup>=</sup> <sup>Δ</sup>*QW* k*TW*

> <sup>−</sup> <sup>Δ</sup>*Q*<sup>0</sup> k*TW*

<sup>=</sup> <sup>Δ</sup>*<sup>A</sup>* k*TW*

<sup>=</sup> <sup>Δ</sup>*<sup>A</sup>* k*TW*

For *P*<sup>2</sup> = *P*<sup>1</sup> + *W*, where *W* is the average energy of the input signal is then valid that

, *<sup>H</sup>*<sup>2</sup> <sup>=</sup> *<sup>π</sup>*2k*TW*

3¯*h*

 *TW T*0

3¯*<sup>h</sup>* <sup>=</sup> <sup>2</sup>*P*<sup>2</sup>

k*TW*

<sup>d</sup><sup>Θ</sup> <sup>=</sup> *<sup>π</sup>*2k

 1 +

6*h* · *W π*2k2*T*<sup>0</sup>

<sup>2</sup> − 1 

Δ*QW* = Δ*Q*<sup>0</sup> + Δ*A* (139)

= Δ*A*. Thus O*rrev* is considerable as a *thermodynamic*

<sup>=</sup> <sup>Δ</sup>*Q*<sup>0</sup> k*TW*

· *ηmax* = *H*(*X*) (141)

= *H*(*X*).

*noise* (140)

*input*, *<sup>H</sup>*(*Y*|*X*) �

6*h* · *W π*2k2*T*<sup>0</sup> (135)

(138)

3¯*<sup>h</sup>* · (*TW* <sup>−</sup> *<sup>T</sup>*0). (136)

<sup>2</sup> (137)

*<sup>H</sup>*<sup>1</sup> <sup>=</sup> *<sup>π</sup>*2k*T*<sup>0</sup>

 *TW T*0

1 kΘ

*CT*0,*W*(**K**L−L) = *<sup>π</sup>*2k*T*<sup>0</sup>

3¯*<sup>h</sup>* <sup>=</sup> <sup>2</sup>*P*<sup>1</sup> k*T*<sup>0</sup>

d*P*(Θ)

2

channel L is

maximal *entropy defect* [2, 30]) by

*CT*0,*TW* = *H*<sup>2</sup> − *H*<sup>1</sup> =

*π*2k2*TW* 2 <sup>6</sup>*<sup>h</sup>* <sup>=</sup> *<sup>π</sup>*2k2*T*<sup>0</sup>

Then, in accordance with (102), (103), (104) [30]

**7. Reverse heat cycle and transfer channel**

*T*<sup>0</sup> > 0, the *output* heat Δ*QW* is being delivered to the A

energy <sup>Δ</sup>*<sup>W</sup>* <sup>≥</sup> *<sup>k</sup>* · <sup>Θ</sup> · <sup>Δ</sup>*I*; at this case <sup>Δ</sup>*<sup>W</sup>* �

*H*(*Y*) �

is valid The result is

<sup>=</sup> <sup>Δ</sup>*QW* k*TW*

*changes* of the information entropies defined on <sup>K</sup>16:

*output* (

the *friction, noise heat* (Δ*Q*0*<sup>x</sup>* = 0) and thus *H*(*X*|*Y*) = 0.

�

*<sup>T</sup>*(*X*;*Y*) = <sup>Δ</sup>*<sup>A</sup>*

<sup>16</sup> In *information* units *Hartley*, *nat*, *bit*; *<sup>H</sup>*(·) = <sup>Δ</sup>*H*(·), *<sup>H</sup>*(·|·) = <sup>Δ</sup>*H*(·|·).

k*TW*

k*TW*

*<sup>T</sup>*(*Y*; *<sup>X</sup>*) = <sup>Δ</sup>*Q*<sup>0</sup> <sup>+</sup> <sup>Δ</sup>*<sup>A</sup>*

= Δ*I*), *H*(*X*) �

*<sup>H</sup>*(*X*) = *<sup>S</sup>*(*TW*) <sup>k</sup> <sup>−</sup> *<sup>S</sup>*(*T*0) <sup>k</sup> <sup>=</sup> O*rrev δA*(Θ) <sup>k</sup><sup>Θ</sup> <sup>=</sup> *TW T*0 <sup>d</sup>*H*(Θ) = <sup>2</sup>Δ*QW* k*TW* <sup>2</sup> · (*TW* − *T*0) (145) *<sup>H</sup>*(*Y*) = *<sup>S</sup>*(*TW*) <sup>k</sup> <sup>=</sup> *TW* 0 *δQW*(Θ*W*) kΘ*<sup>W</sup>* = *TW* 0 <sup>d</sup>*H*(Θ) = <sup>2</sup>Δ*QW* k*TW* [*H*(*X*) = *H*(*Y*) · *η*max] *<sup>H</sup>*(*Y*|*X*) = *<sup>H</sup>*(*Y*) <sup>−</sup> *<sup>H</sup>*(*X*) = *TW* 0 d*H*(Θ) − *TW T*0 <sup>d</sup>*H*(Θ) = *<sup>T</sup>*<sup>0</sup> 0 d*H*(Θ) = *T*<sup>0</sup> 0 *∂QW*(Θ) <sup>k</sup><sup>Θ</sup> <sup>=</sup> *<sup>S</sup>*(*T*0) <sup>k</sup> <sup>=</sup> <sup>2</sup>Δ*QW* k*TW* <sup>2</sup> · *<sup>T</sup>*<sup>0</sup> <sup>=</sup> <sup>2</sup>Δ*QW* k*TW* · *T*0 *TW* = *H*(*X*) · *β <sup>T</sup>*(*Y*; *<sup>X</sup>*) = *<sup>H</sup>*(*Y*) <sup>−</sup> *<sup>H</sup>*(*Y*|*X*) = *TW* 0 d*H*(Θ) − *T*<sup>0</sup> 0 <sup>d</sup>*H*(Θ) = *TW T*0 d*H*(Θ) = O*rrev δA*(Θ) <sup>k</sup><sup>Θ</sup> <sup>=</sup> <sup>2</sup>Δ*QW* k*TW* · *η*max = *H*(*Y*) · *η*max = *H*(*X*) = Δ*I*

Obviously, *T*(*X*;*Y*) = *H*(*X*) − *H*(*X*|*Y*) = *T*(*Y*|*X*). Further it is obvious that *T*(*X*;*Y*) is the capacity *CTW* ,*T*<sup>0</sup> of the channel **K** ∼= L too,

$$\mathbf{C}\_{T\_W, T\_0} = T(\mathbf{X}; Y) = H(\mathbf{X}) = \oint\_{\mathcal{O}\_{mv}} \frac{\delta A(\Theta)}{\mathbf{k}\Theta} = \int\_{T\_0}^{T\_W} \frac{\delta Q\_W(\Theta)}{\mathbf{k}\Theta} \tag{146}$$

$$= \frac{2\Delta Q\_W}{\mathbf{k}T\_W^2} \cdot (T\_W - T\_0), \quad \mathbf{C}\_{T\_W, T\_0}^{\text{max}} = H(Y)$$

<sup>17</sup> In reality for the least elementary heat change *<sup>δ</sup><sup>Q</sup>* <sup>=</sup> *<sup>h</sup>*¯ *<sup>ν</sup>* is right where ¯*<sup>h</sup>* <sup>=</sup> *<sup>h</sup>* <sup>2</sup>*<sup>π</sup>* and *<sup>h</sup>* is *Planck constant*.

<sup>18</sup> It is provable that the Carnot cycle itself is elenentary, *not dividible* [18].

#### **7.1. Triangular heat cycle**

Elementary change dΘ of temperature Θ of the *environment* of the general continuous cycle O and thus of its working *medium* L (both are in the *diathermic contact at* Θ) causes the elementary reversible change of the heat *Q*∗(Θ) *delivered, (radiated)* into L, just about the value *δQ*∗(Θ),

$$\delta \mathcal{Q}\*(\Theta) = \frac{\partial \mathcal{Q}\*(\Theta)}{\partial \Theta} \mathrm{d}\Theta, \quad \mathcal{Q}\*(\Theta\_W) = \int\_0^{\Theta\_W} \frac{\partial \mathcal{Q}\*(\Theta)}{\partial \Theta} \mathrm{d}\Theta \tag{147}$$

For the change <sup>Δ</sup>*S*∗L of the thermodynamic entropy *<sup>S</sup>*∗L of the system L, at the tememperature Θ running through the interval �*T*0, *TW*�, by gaining heat from its environment

By (149) for the entropy *<sup>S</sup>*∗L(Θ*W*) of L at variable temperature <sup>Θ</sup> ∈ �0, <sup>Θ</sup>*W*�, <sup>Θ</sup>*<sup>W</sup>* ≤ *TW*, is

1 2

 Θ*<sup>W</sup>* 0

By the result of derivation (155)-(156) for an arbitrary temperature Θ of medium L is valid

<sup>Δ</sup>*S*∗L <sup>=</sup> *<sup>l</sup>* · (*TW* <sup>−</sup> *<sup>T</sup>*0) = *<sup>l</sup>* · *TW* · (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*), *<sup>β</sup>* <sup>=</sup> *<sup>T</sup>*<sup>0</sup>

Let such a reverse cycle is given that the medium L of which takes, through the elementary

*T*0

*S*∗L(*T*0)

<sup>=</sup> <sup>Θ</sup>*W*(*T*0,*TW* ) · [*S*∗L(*TW*) <sup>−</sup> *<sup>S</sup>*∗L(*T*0)] <sup>=</sup> *TW* <sup>+</sup> *<sup>T</sup>*<sup>0</sup>

continuous heat cycle O, drawing up the same heat Δ*Q*0, consumpting the same mechanical

*<sup>l</sup>*Θd<sup>Θ</sup> <sup>=</sup> *<sup>l</sup>*

2 · *TW* <sup>2</sup> <sup>−</sup> *<sup>T</sup>*<sup>2</sup> 0 

<sup>Θ</sup>d*S*∗L(Θ)

*<sup>S</sup>*∗L(Θ) · <sup>Θ</sup> 2

<sup>L</sup>(Θ)d<sup>Θ</sup> +

dΘ <sup>Θ</sup> <sup>=</sup>  *<sup>Q</sup>*∗*<sup>W</sup> TW*

dΘ

 Θ*<sup>W</sup>* 0

<sup>−</sup> *<sup>Q</sup>*∗<sup>0</sup> *T*0

<sup>Θ</sup> <sup>=</sup> <sup>2</sup> ·

*<sup>S</sup>*∗L(Θ)

<sup>d</sup>*S*∗L(Θ)

<sup>Θ</sup><sup>2</sup> −→ *<sup>Q</sup>*∗(Θ) = *<sup>λ</sup>* · <sup>Θ</sup>2, *<sup>λ</sup>* <sup>=</sup> *<sup>l</sup>*

 = *TW T*0

Information Capacity of Quantum Transfer Channels and Thermodynamic Analogies 109

1 2

dΘ Θ

<sup>Θ</sup> <sup>=</sup> <sup>d</sup>*S*∗L(Θ) (156)

 Θ*<sup>W</sup>* 0

<sup>=</sup> <sup>2</sup>*Q*∗(Θ*W*) Θ*<sup>W</sup>*

*TW*

<sup>2</sup> · <sup>2</sup>

<sup>2</sup> · (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*2), *<sup>β</sup>* <sup>=</sup> *<sup>T</sup>*<sup>0</sup>

*rrev*, equivalent with the just considered general

*<sup>x</sup>* , then ln <sup>|</sup> *<sup>f</sup>*(*x*)<sup>|</sup> <sup>=</sup> ln <sup>|</sup>*x*<sup>|</sup> <sup>+</sup> ln *<sup>L</sup>*, *<sup>L</sup>* <sup>&</sup>gt; 0, ln <sup>|</sup> *<sup>f</sup>*(*x*)<sup>|</sup> <sup>=</sup> ln(*<sup>L</sup>* · |*x*|), *<sup>f</sup>*(*x*) =

 *Q*∗*<sup>W</sup> TW*

*δQ*∗(Θ)

<sup>Θ</sup> (154)

d*S*∗*L*(Θ) (155)

<sup>2</sup> (157)

= *Q*∗*<sup>W</sup>* − *Q*∗<sup>0</sup>

*TW*

<sup>−</sup> *<sup>Q</sup>*∗<sup>0</sup> *T*0

(158)

(159)

(the environment of the cycle O), is valid

*<sup>S</sup>*∗L(Θ*W*) = <sup>Θ</sup>*<sup>W</sup>*

*<sup>S</sup>*∗L(Θ*W*) = <sup>Θ</sup>*<sup>W</sup>*

and thus *<sup>S</sup>*∗L(Θ)

gained that

that20

and then

<sup>Δ</sup>*S*∗L = *<sup>S</sup>*∗L(*TW*) − *<sup>S</sup>*∗L(*T*0) = <sup>2</sup>

 *∂ ∂*Θ 

*S*∗�

*<sup>S</sup>*∗L(Θ)

dΘ

isothermal expansions at temperatures Θ ∈ �*T*0, *TW*�, the whole heat Δ*Q*<sup>0</sup>

*<sup>δ</sup>Q*∗(Θ) = *TW*

Δ*Q*<sup>0</sup> = (*TW* + *T*0) · *λ* · [*TW* − *T*0] = *λTW*

*<sup>δ</sup>Q*∗(Θ) = *<sup>S</sup>*∗L(*TW* )

0

0

*<sup>S</sup>*∗L(Θ) = *<sup>l</sup>* · <sup>Θ</sup>, *<sup>l</sup>* <sup>=</sup> <sup>2</sup>*Q*∗(Θ)

 *TW T*0

 *TW T*0

Obviously, from (154) for Θ ∈ �*T*0, *TW*� is derivable that

Δ*Q*<sup>0</sup> =

Δ*Q*<sup>0</sup> =

For a reverse reversible Carnot cycle O�

d*f*(*x*) *<sup>f</sup>*(*x*) <sup>=</sup> <sup>d</sup>*<sup>x</sup>*

d*f*(*x*), or

or, with medium values

and thus equivalently

<sup>20</sup> If *<sup>f</sup>*(*x*)

*l* · *x*, *l* ∈ **R**.

*<sup>x</sup>* <sup>d</sup>*<sup>x</sup>* <sup>=</sup>   Θ*<sup>W</sup>* 0

<sup>=</sup> <sup>1</sup> 2

The heat *Q*∗(Θ*W*) is the whole heat delivered (reversibly) into L (at the *end* temperature Θ*W*). For the infinitezimal heat *δQ*∗(Θ) delivered (reversibly) into L at temperature Θ and in accordance with the *Clausius* definition of heat entropy *<sup>S</sup>*∗L [22] is valid that

$$\delta \mathbb{Q} \* (\Theta) = \Theta \cdot \mathrm{d} \mathbb{S} \* \_{\mathcal{L}} (\Theta), \quad \mathrm{d} \mathbb{S} \* \_{\mathcal{L}} (\Theta) = \frac{\delta \mathbb{Q} \* (\Theta)}{\Theta} \tag{148}$$

For the whole change of entropy <sup>Δ</sup>*S*∗L(Θ*W*), or for the entropy *<sup>S</sup>*∗L(Θ*W*) respectively, delivered into the medium L by its heating within the temperature interval (0, Θ*W*�, is valid that

$$\Delta \mathbf{S} \*\_{\mathcal{L}} (\Theta\_W) = \int\_0^{\Theta\_W} \mathbf{d} \mathbf{S} \*\_{\mathcal{L}} (\Theta) = \int\_0^{\Theta\_W} \frac{\delta \mathbf{Q} \* (\Theta)}{\Theta} = \int\_0^{\Theta\_W} \frac{\frac{\delta \mathbf{Q} \* (\Theta)}{\delta \Theta} \mathbf{d} \Theta}{\Theta} = \mathbf{S} \*\_{\mathcal{L}} (\Theta\_W) \tag{149}$$

when *<sup>S</sup>*∗L(0) Def = 0 is set down. By (148) for the whole heat *Q*∗(Θ*W*) deliverd into L within the temperature interval Θ ∈ (0, Θ*W*� also is valid that

$$Q\*(\Theta\_W) = \int\_0^{\Theta\_W} \Theta \,\mathrm{d}S\*\_{\mathcal{L}}(\Theta) \tag{150}$$

Then, by *medium value* theorem19 is valid that <sup>Θ</sup>(0,Θ*<sup>W</sup>* ) <sup>=</sup> <sup>0</sup> <sup>+</sup> <sup>Θ</sup>*<sup>W</sup>* <sup>2</sup> <sup>=</sup> <sup>Θ</sup>*<sup>W</sup>* <sup>2</sup> and

$$Q\*(\Theta\_W) = \int\_{S\*\mathcal{L}(0)}^{S\*\mathcal{L}(\Theta\_W)} \Theta \mathrm{d}S\*\_{\mathcal{L}}(\Theta) = [\mathcal{S}\*\_{\mathcal{L}}(\Theta\_W) - \mathcal{S}\*\_{\mathcal{L}}(0)] \cdot \overline{\Theta\_{(0:\Theta\_W)}}\tag{151}$$

For the extremal values *T*<sup>0</sup> a *TW* of the cooler temperature Θ of O and by (151)

$$Q\*\_{0} \stackrel{\triangle}{=} Q\*(T\_{0}) = \int\_{0}^{T\_{0}} \delta Q\*\_{W}(\Theta) \quad \text{and} \quad Q\*\_{W} \stackrel{\triangle}{=} Q\*(T\_{W}) = \int\_{0}^{T\_{W}} \delta Q\*\_{W}(\Theta) \tag{152}$$

$$Q\*\_{0} = \int\_{S\*\_{\mathcal{L}}(0)}^{S\*\_{\mathcal{L}}(T\_{0})} \Theta \mathbf{d} \\ S\*\_{\mathcal{L}}(\Theta) = \left[S\*\_{\mathcal{L}}(T\_{0}) - S\*\_{\mathcal{L}}(0)\right] \cdot \overline{\Theta\_{(0,\mathbb{T}\_{0})'}} \quad \overline{\Theta\_{(0,\mathbb{T}\_{0})}} = \frac{T\_{0}}{2}$$

$$Q\*\_{W} = \int\_{S\*\_{\mathcal{L}}(0)}^{S\*\_{\mathcal{L}}(T\_{W})} \Theta \mathbf{d} \\ S\*\_{\mathcal{L}}(\Theta) = \left[S\*\_{\mathcal{L}}(T\_{W}) - S\*\_{\mathcal{L}}(0)\right] \cdot \overline{\Theta\_{(0,\mathbb{T}\_{W})'}} \quad \overline{\Theta\_{(0,\mathbb{T}\_{0})}} = \frac{T\_{W}}{2}$$

With *<sup>S</sup>*∗L(0) = 0 for the (end) temperatures <sup>Θ</sup>, *<sup>T</sup>*0, *TW* of L and the relevant heats and their entropies is valid

$$Q\*(\Theta) = S\*\_{\mathcal{L}}(\Theta) \cdot \frac{\Theta}{2} \qquad \text{and then} \quad S\*\_{\mathcal{L}}(\Theta) = \frac{2Q\*(\Theta)}{\Theta} \tag{153}$$

$$Q\*\_{W} = S\*\_{\mathcal{L}}(T\_{W}) \cdot \frac{T\_{W}}{2} \quad \text{and then} \quad S\*\_{\mathcal{L}}(T\_{W}) = \frac{2Q\*\_{W}}{T\_{W}}$$

$$Q\*\_{0} = S\*\_{\mathcal{L}}(T\_{0}) \cdot \frac{T\_{0}}{2} \quad \text{and then} \quad S\*\_{\mathcal{L}}(T\_{0}) = \frac{2Q\*\_{0}}{T\_{0}}$$

<sup>19</sup> Of Integral Calculus.

For the change <sup>Δ</sup>*S*∗L of the thermodynamic entropy *<sup>S</sup>*∗L of the system L, at the tememperature Θ running through the interval �*T*0, *TW*�, by gaining heat from its environment (the environment of the cycle O), is valid

$$
\Delta \mathbf{S} \*\_{\mathcal{L}} = \mathbf{S} \*\_{\mathcal{L}} (T\_W) - \mathbf{S} \*\_{\mathcal{L}} (T\_0) = 2 \left( \frac{\mathbf{Q} \*\_{W}}{T\_W} - \frac{\mathbf{Q} \*\_{0}}{T\_0} \right) = \int\_{T\_0}^{T\_W} \frac{\delta \mathbf{Q} \* (\Theta)}{\Theta} \tag{154}
$$

By (149) for the entropy *<sup>S</sup>*∗L(Θ*W*) of L at variable temperature <sup>Θ</sup> ∈ �0, <sup>Θ</sup>*W*�, <sup>Θ</sup>*<sup>W</sup>* ≤ *TW*, is gained that

$$\begin{split} S\*\_{\mathcal{L}}(\Theta\_{W}) &= \int\_{0}^{\Theta\_{W}} \left( \frac{\partial}{\partial \Theta} \left[ \frac{S\*\_{\mathcal{L}}(\Theta) \cdot \Theta}{2} \right] \right) \frac{d\Theta}{\Theta} = 2 \cdot \frac{1}{2} \int\_{0}^{\Theta\_{W}} \mathrm{d}S\*\_{L}(\Theta) \\ &= \frac{1}{2} \int\_{0}^{\Theta\_{W}} S\*'\_{\mathcal{L}}(\Theta) \mathrm{d}\Theta + \frac{1}{2} \int\_{0}^{\Theta\_{W}} \mathrm{S}\*\_{\mathcal{L}}(\Theta) \frac{\mathrm{d}\Theta}{\Theta} \end{split} \tag{155}$$

and then

26 Will-be-set-by-IN-TECH

Elementary change dΘ of temperature Θ of the *environment* of the general continuous cycle O and thus of its working *medium* L (both are in the *diathermic contact at* Θ) causes the elementary reversible change of the heat *Q*∗(Θ) *delivered, (radiated)* into L, just about the value *δQ*∗(Θ),

The heat *Q*∗(Θ*W*) is the whole heat delivered (reversibly) into L (at the *end* temperature Θ*W*). For the infinitezimal heat *δQ*∗(Θ) delivered (reversibly) into L at temperature Θ and in

*<sup>δ</sup>Q*∗(Θ) = <sup>Θ</sup> · <sup>d</sup>*S*∗L(Θ), d*S*∗L(Θ) = *<sup>δ</sup>Q*∗(Θ)

For the whole change of entropy <sup>Δ</sup>*S*∗L(Θ*W*), or for the entropy *<sup>S</sup>*∗L(Θ*W*) respectively, delivered into the medium L by its heating within the temperature interval (0, Θ*W*�, is valid

> Θ*<sup>W</sup>* 0

> > �

With *<sup>S</sup>*∗L(0) = 0 for the (end) temperatures <sup>Θ</sup>, *<sup>T</sup>*0, *TW* of L and the relevant heats and their

= *Q*∗(*TW*) =

<sup>2</sup> and then *<sup>S</sup>*∗L(Θ) = <sup>2</sup>*Q*∗(Θ)

<sup>2</sup> and then *<sup>S</sup>*∗L(*TW*) = <sup>2</sup>*Q*∗*<sup>W</sup>*

<sup>2</sup> and then *<sup>S</sup>*∗L(*T*0) = <sup>2</sup>*Q*∗<sup>0</sup>

<sup>Θ</sup>d*S*∗L(Θ) = [*S*∗L(*T*0) <sup>−</sup> *<sup>S</sup>*∗L(0)] · <sup>Θ</sup>(0,*T*0) , <sup>Θ</sup>(0,*T*0) <sup>=</sup> *<sup>T</sup>*<sup>0</sup>

<sup>Θ</sup>d*S*∗L(Θ) = [*S*∗L(*TW*) <sup>−</sup> *<sup>S</sup>*∗L(0)] · <sup>Θ</sup>(0,*TW* ) , <sup>Θ</sup>(0,*TW* ) <sup>=</sup> *TW*

*δQ*∗(Θ) <sup>Θ</sup> <sup>=</sup>

= 0 is set down. By (148) for the whole heat *Q*∗(Θ*W*) deliverd into L within

 Θ*<sup>W</sup>* 0

*Q*∗(Θ*W*) =

For the extremal values *T*<sup>0</sup> a *TW* of the cooler temperature Θ of O and by (151)

*δQ*∗*W*(Θ) and *Q*∗*<sup>W</sup>*

Θ

*TW*

*T*0

 Θ*<sup>W</sup>* 0

> Θ*<sup>W</sup>* 0

*∂Q*∗(Θ)

*∂Q*∗(Θ) *<sup>∂</sup>*<sup>Θ</sup> dΘ

<sup>2</sup> <sup>=</sup> <sup>Θ</sup>*<sup>W</sup>*

<sup>Θ</sup>d*S*∗L(Θ) = [*S*∗L(Θ*W*) − *<sup>S</sup>*∗L(0)] · <sup>Θ</sup>(0,Θ*<sup>W</sup>* ) (151)

 *TW* 0

<sup>Θ</sup>dS∗L(Θ) (150)

<sup>2</sup> and

*<sup>∂</sup>*<sup>Θ</sup> <sup>d</sup><sup>Θ</sup> (147)

<sup>Θ</sup> (148)

<sup>Θ</sup> <sup>=</sup> *<sup>S</sup>*∗L(Θ*W*) (149)

*δQ*∗*W*(Θ) (152)

<sup>Θ</sup> (153)

*TW*

*T*0

2

2

*<sup>∂</sup>*<sup>Θ</sup> <sup>d</sup>Θ, *<sup>Q</sup>*∗(Θ*W*) =

accordance with the *Clausius* definition of heat entropy *<sup>S</sup>*∗L [22] is valid that

**7.1. Triangular heat cycle**

<sup>Δ</sup>*S*∗L(Θ*W*) =

when *<sup>S</sup>*∗L(0) Def

*Q*∗<sup>0</sup> �

entropies is valid

<sup>19</sup> Of Integral Calculus.

that

*<sup>δ</sup>Q*∗(Θ) = *<sup>∂</sup>Q*∗(Θ)

 Θ*<sup>W</sup>* 0

*Q*∗(Θ*W*) =

 *T*<sup>0</sup> 0

 *S*∗L(*T*0) *S*∗L(0)

 *S*∗L(*TW* ) *S*∗L(0)

*<sup>Q</sup>*∗(Θ) = *<sup>S</sup>*∗L(Θ) ·

*<sup>Q</sup>*∗*<sup>W</sup>* = *<sup>S</sup>*∗L(*TW*) ·

*<sup>Q</sup>*∗<sup>0</sup> = *<sup>S</sup>*∗L(*T*0) ·

= *Q*∗(*T*0) =

*Q*∗<sup>0</sup> =

*Q*∗*<sup>W</sup>* =

the temperature interval Θ ∈ (0, Θ*W*� also is valid that

<sup>d</sup>*S*∗L(Θ) =

Then, by *medium value* theorem19 is valid that <sup>Θ</sup>(0,Θ*<sup>W</sup>* ) <sup>=</sup> <sup>0</sup> <sup>+</sup> <sup>Θ</sup>*<sup>W</sup>*

 *S*∗L(Θ*<sup>W</sup>* ) *S*∗L(0)

$$\mathbf{S} \* \mathcal{L}(\Theta\_W) = \int\_0^{\Theta\_W} \mathbf{S} \* \mathcal{L}(\Theta) \frac{\mathbf{d}\Theta}{\Theta} = \int\_0^{\Theta\_W} \mathbf{d} \mathbf{S} \* \mathcal{L}(\Theta) \ \left[ = \frac{2\mathbf{Q} \* (\Theta\_W)}{\Theta\_W} \right]$$

$$\text{and thus} \qquad \mathbf{S} \* \mathcal{L}(\Theta) \frac{\mathbf{d}\Theta}{\Theta} = \mathbf{d} \mathbf{S} \* \mathcal{L}(\Theta) \tag{156}$$

By the result of derivation (155)-(156) for an arbitrary temperature Θ of medium L is valid that20

$$\mathcal{S} \*\_{\mathcal{L}}(\Theta) = l \cdot \Theta, \ l = \frac{2Q\*(\Theta)}{\Theta^2} \quad \longrightarrow \ Q\*(\Theta) = \lambda \cdot \Theta^2, \ \lambda = \frac{l}{2} \tag{157}$$

Obviously, from (154) for Θ ∈ �*T*0, *TW*� is derivable that

$$
\Delta S \*\_{\mathcal{L}} = l \cdot (T\_W - T\_0) = l \cdot T\_W \cdot (1 - \beta), \quad \beta = \frac{T\_0}{T\_W} \tag{158}
$$

Let such a reverse cycle is given that the medium L of which takes, through the elementary isothermal expansions at temperatures Θ ∈ �*T*0, *TW*�, the whole heat Δ*Q*<sup>0</sup>

$$\Delta Q\_0 = \int\_{T\_0}^{T\_W} \delta Q \* (\Theta) = \int\_{T\_0}^{T\_W} l \Theta d\Theta = \frac{l}{2} \cdot \left( T\_W^2 - T\_0^2 \right) = Q \*\_W - Q \*\_{\emptyset} \text{(159)}$$

or, with medium values

$$\begin{split} \Delta Q\_{0} &= \int\_{T\_{0}}^{T\_{W}} \delta Q \* (\Theta) = \int\_{S \ast \mathcal{L}(T\_{0})}^{S \ast \mathcal{L}(T\_{W})} \Theta \mathbf{d} \mathbf{S} \* \_{\mathcal{L}} (\Theta) \\ &= \overline{\Theta \mathbf{w}\_{(T\_{0}, T\_{W})}} \cdot [\mathbf{S} \* \_{\mathcal{L}}(T\_{W}) - \mathbf{S} \* \_{\mathcal{L}}(T\_{0})] = \frac{T\_{W} + T\_{0}}{2} \cdot 2 \left[ \frac{\mathbf{Q} \*\_{W}}{T\_{W}} - \frac{\mathbf{Q} \*\_{0}}{T\_{0}} \right] \end{split}$$

and thus equivalently

$$
\Delta Q\_0 = (T\_W + T\_0) \cdot \lambda \cdot [T\_W - T\_0] = \lambda T\_W^2 \cdot (1 - \beta^2), \quad \beta = \frac{T\_0}{T\_W}
$$

For a reverse reversible Carnot cycle O� *rrev*, equivalent with the just considered general continuous heat cycle O, drawing up the same heat Δ*Q*0, consumpting the same mechanical

$$\begin{aligned} \text{120 } \text{ If } \int \frac{f(\mathbf{x})}{\mathbf{x}} d\mathbf{x} = \int \mathbf{d}f(\mathbf{x}), \text{ or } \frac{\mathbf{d}f(\mathbf{x})}{f(\mathbf{x})} = \frac{\mathbf{d}\mathbf{x}}{\mathbf{x}}, \text{ then } \ln|f(\mathbf{x})| = \ln|\mathbf{x}| + \ln L, \; L > 0, \; \ln|f(\mathbf{x})| = \ln(L \cdot |\mathbf{x}|), \; f(\mathbf{x}) = \int \mathbf{d}f(\mathbf{x}) d\mathbf{x} \\\text{If } \mathbf{x}, \; L \in \mathbb{R}. \end{aligned}$$

work Δ*A* and giving, at its higher temperature *tW* (the average temperature of the heater of our general cycle), the same heat Δ*QW*, is valid that Δ*Q*<sup>0</sup> = Δ*QW* · *γ* where *γ* = *T*0+*TW* 2 *tW* is the *transform ratio*.

Then

$$
\Delta Q\_W = \Delta Q\_0 \cdot \frac{2t\_W}{T\_0 + T\_W} = \frac{1}{2} \cdot \left(T\_W^2 - T\_0^2\right) \cdot \frac{2t\_W}{T\_0 + T\_W} = l \cdot t\_W \cdot \left(T\_W - T\_0\right) \tag{160}
$$

For works *<sup>δ</sup>A*(Θ, dΘ; *<sup>T</sup>*0) of elementary Carnot cycles covering cycle O*rrev*� (166), the range of their working temperatures is dΘ and, for the given heater temperature Θ ∈ �*T*0, Θ*W*� is,

<sup>Θ</sup> <sup>=</sup> *<sup>l</sup>* · <sup>Θ</sup> · (<sup>Θ</sup> <sup>−</sup> *<sup>T</sup>*0) <sup>d</sup><sup>Θ</sup>

Δ*QW*(Θ) = *l*Θ (Θ − *T*0) and for *γ* used in (160) is gained that

For the whole heats Δ*Q*<sup>0</sup> a Δ*QW* being changed mutually between the working medium L of the whole triangular cycle O*rrev*� and its environment (166), and for the work <sup>Δ</sup>*A*, in its

<sup>2</sup> · (1−*β*2) �=*<sup>W</sup>*

dΘ = *l* · (*TW* − *T*0)·*TW* = *l* · *TW*

(<sup>1</sup> <sup>+</sup> *<sup>β</sup>*)] = *<sup>l</sup>*

*rrev* with working temperatures *<sup>T</sup>*<sup>0</sup> <sup>+</sup> *TW*

<sup>2</sup> · (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*) <sup>−</sup> *<sup>l</sup>* · *TW*

<sup>2</sup> · *TW*

The average *output* energy *P*2(Θ*W*) of the message being received within interval (0, *TW*� of the temperature <sup>Θ</sup> of the medium L ∼= **<sup>K</sup>**L−<sup>L</sup> from [30], when 0 < <sup>Θ</sup><sup>0</sup> ≤ <sup>Θ</sup> ≤ <sup>Θ</sup>*<sup>W</sup>* and <sup>Θ</sup><sup>0</sup> ≤ *<sup>T</sup>*<sup>0</sup> and Θ*<sup>W</sup>* ≤ *TW* are valid, is given by the sum of the *input* average energy *W*(Θ*W*, Θ0) and the

The output message bears the *whole average output information H*2(Θ*W*). By the medium value theorem is possible, for a certain maximal temperature Θ*<sup>W</sup>* ≤ *TW* of the temperature Θ ∈ (0, Θ*W*�, consider that the receiving of the output message is performed at the *average*

= *<sup>H</sup>*2(Θ*W*) [the thermodynamic entropy *<sup>S</sup>*∗L(Θ*W*) in information units] is valid

�

<sup>Θ</sup> <sup>=</sup> *<sup>l</sup>* · (<sup>Θ</sup> <sup>−</sup> *<sup>T</sup>*0)d<sup>Θ</sup> (168)

*l*

<sup>2</sup> · (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*) (170)

*δA*

<sup>Θ</sup><sup>2</sup> <sup>−</sup> *<sup>T</sup>*<sup>0</sup> 2 

<sup>2</sup> and *TW*, will be valid

<sup>2</sup> · (*TW* <sup>−</sup>*T*0)<sup>2</sup>

 (169)

Θ + *T*<sup>0</sup> <sup>2</sup><sup>Θ</sup> <sup>=</sup> *<sup>λ</sup>* ·

Information Capacity of Quantum Transfer Channels and Thermodynamic Analogies 111

=*l* · (*TW* −*T*0)·*T*0+

<sup>2</sup> · (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*)

O*rrev*�

= *H*1(Θ*W*, Θ0) + *H*[*W*(Θ*W*, Θ0)] (172)

<sup>2</sup> · (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*)<sup>2</sup> <sup>=</sup>

*P*2(Θ*W*) = *P*1(Θ0) + *W*(Θ*W*, Θ0) (171)

<sup>2</sup> . Then for the whole change of the output information

dΘ

= [*S*∗L(Θ) − *<sup>S</sup>*∗L(*T*0)] · <sup>d</sup><sup>Θ</sup>

Δ*Q*0(Θ) = Δ*QW*(Θ) · *γ*(Θ) = *l* · Θ · (Θ − *T*0) ·

<sup>2</sup> · *TW*

dΘ = ·*lTW*

2

**7.2. Capacity corections for wide–band photonic transfer channel**

<sup>=</sup> *<sup>P</sup>*1(Θ0) + *<sup>W</sup>*(Θ*W*, <sup>Θ</sup>0) kΘ<sup>W</sup>

6¯*<sup>h</sup>* , *<sup>l</sup>* <sup>=</sup> *<sup>π</sup>*2*k*<sup>2</sup>

3¯*<sup>h</sup>* <sup>=</sup> <sup>2</sup> · *<sup>λ</sup>*.

by (162)-(163) valid that

equivalent Carnot cycle O�

 *TW T*0

> *TW T*0

= *l* · *TW*

 *TW T*0

 Θ 0 *l*d*θ* <sup>d</sup>Θ<sup>=</sup> *<sup>l</sup>*

 *TW* 0

> *TW T*0

average energy *P*1(Θ0) of the *additive noise*

*(constant) temperature* <sup>Θ</sup>*<sup>W</sup>* <sup>=</sup> <sup>Θ</sup>*<sup>W</sup>*

kΘ<sup>W</sup>

�

*<sup>H</sup>*2(Θ*W*) = *<sup>P</sup>*2(Θ*W*)

<sup>21</sup> Further it willbe layed down *<sup>λ</sup>* <sup>=</sup> *<sup>π</sup>*2*k*<sup>2</sup>

entropy Δ*H*<sup>2</sup>

*l*d*θ* 

> *l*d*θ*

<sup>2</sup> · (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*) · [<sup>1</sup> <sup>−</sup> *<sup>l</sup>*

that21

Δ*Q*0=

Δ*QW* =

<sup>Δ</sup>*<sup>A</sup>* <sup>=</sup> <sup>1</sup> 2

*δA*(Θ, dΘ; *T*0) = Δ*QW*(Θ) ·

$$
\Delta A = \Delta Q\_W \cdot (1 - \gamma) = l \cdot t\_W \cdot (T\_W - T\_0) \cdot \left(1 - \frac{T\_0 + T\_W}{2t\_W}\right) \tag{161}
$$

$$=\frac{l}{2} \cdot (T\_W - T\_0) \cdot (2t\_W - T\_0 - T\_W)$$

For the elementary work *δA*(·, ·) corresponding with the heat *Q*∗(Θ) pumped out (reversibly) from L at the (end, output) temperature <sup>Θ</sup> of L and for the entropy *<sup>S</sup>*∗L(Θ) of the whole environment of O (including L with O) is valid

$$\mathbf{S} \*\_{\mathcal{L}}(\Theta) \frac{\mathbf{d}\Theta}{\Theta} = \frac{Q\*(\Theta)\frac{\mathbf{d}\Theta}{\Theta}}{\Theta} \stackrel{\triangle}{=} \frac{\delta A(\Theta, \mathbf{d}\Theta)}{\Theta} = \mathbf{d} \mathbf{S} \*\_{\mathcal{L}}(\Theta) = l \cdot \mathbf{d}\Theta \tag{162}$$

$$\delta A(\Theta, \mathrm{d}\Theta) = \mathrm{S} \ast\_{\mathcal{L}}(\Theta)\mathrm{d}\Theta = l \cdot \Theta \mathrm{d}\Theta \text{ and } \delta A(\mathrm{d}\Theta, \mathrm{d}\Theta) = l \cdot \mathrm{d}\Theta \mathrm{d}\Theta = \mathrm{d}\mathrm{S} \ast\_{\mathcal{L}}(\Theta)\mathrm{d}(\Theta) \stackrel{\triangle}{=} \delta A$$

$$\left(\int\_{T\_0}^{\Theta} l \mathrm{d}\theta\right) \mathrm{d}\Theta \;= \; l \cdot (\Theta - T\_0) \mathrm{d}\Theta \stackrel{\triangle}{=} \delta A(\Theta, \mathrm{d}\Theta; T\_0) \tag{163}$$

For the whole work Δ*A*(Θ*W*; *T*0) consumpted by the general reverse cycle O between temperatures *T*<sup>0</sup> and Θ*W*, being coverd by elementary cycles (162), is valid that

$$\begin{split} \Delta A(\Theta\_{\mathsf{W}};T\_{0}) &= \int\_{T\_{0}}^{\Theta\_{\mathsf{W}}} \left[ \int\_{T\_{0}}^{\Theta} \mathrm{d}S \ast\_{\mathcal{L}}(\theta) \right] \mathrm{d}\Theta = \int\_{T\_{0}}^{\Theta\_{\mathsf{W}}} \left[ \int\_{T\_{0}}^{\Theta} l \mathrm{d}\theta \right] \mathrm{d}\Theta = l \cdot \int\_{T\_{0}}^{\Theta\_{\mathsf{W}}} (\Theta - T\_{0}) \mathrm{d}\Theta(164) \\ &= \frac{l}{2} \cdot (\Theta\_{\mathsf{W}}^{2} - T\_{0}^{2}) - l \cdot T\_{0}(\Theta\_{\mathsf{W}} - T\_{0}) = \frac{l}{2} \cdot \Theta\_{\mathsf{W}}^{2} + \frac{l}{2} \cdot T\_{0}^{2} - \frac{2l}{2} \cdot T\_{0}\Theta\_{\mathsf{W}} = \lambda \cdot (\Theta\_{\mathsf{W}} - T\_{0})^{2} \\ &\overset{(a)}{=} \int\_{\mathcal{O}(\Theta\_{\mathsf{W}},t\_{0})} \delta A = \lambda \cdot T\_{\mathcal{W}}^{2} \cdot (1 - \beta)^{2}, \text{ when it is valid that} \quad \Theta\_{\mathsf{W}} = T\_{W\_{\mathsf{W}}} \cdot \beta = \frac{T\_{0}}{T\_{\mathcal{W}}} \end{split}$$

But, then for the results for Δ*A* in (160), (161) and (164) follows that

$$t\_W = \frac{T\_W + \Theta\_W}{2} = T\_W \quad \text{and then} \quad t\_W = T\_W = \text{const.} \tag{165}$$

Thus our general cycle <sup>O</sup> is of a triangle shape, <sup>O</sup> � = O*rrev*�with the *apexes*

$$\begin{bmatrix} \begin{bmatrix} \end{bmatrix} T\_{0\prime} \ T\_{0} \end{bmatrix}\_{\prime} \begin{bmatrix} \begin{bmatrix} \end{bmatrix} T\_{W\prime\prime} \ T\_{W} \end{bmatrix}\_{\prime} \begin{bmatrix} \begin{bmatrix} \end{bmatrix} T\_{0\prime} \ T\_{W} \end{bmatrix} \tag{166}$$

and its efficiency is 1 <sup>−</sup> *<sup>γ</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>β</sup>* <sup>2</sup> <sup>=</sup> <sup>1</sup> <sup>2</sup> · *<sup>η</sup>*max. Thus, the return to the *initial (starting) state* of the medium L is possible by using the oriented *abscissas* (in the *S* − *T* diagram)

$$\begin{array}{c} \overbrace{[lT\_{W\prime},T\_{W}]\_{\prime}[lT\_{0\prime},T\_{W}]}^{} \text{ and } \begin{array}{c} \overbrace{[lT\_{0\prime},T\_{W}]\_{\prime}[lT\_{0\prime},T\_{0}]}^{} \end{array} \tag{167}$$

For works *<sup>δ</sup>A*(Θ, dΘ; *<sup>T</sup>*0) of elementary Carnot cycles covering cycle O*rrev*� (166), the range of their working temperatures is dΘ and, for the given heater temperature Θ ∈ �*T*0, Θ*W*� is, by (162)-(163) valid that

28 Will-be-set-by-IN-TECH

work Δ*A* and giving, at its higher temperature *tW* (the average temperature of the heater of

For the elementary work *δA*(·, ·) corresponding with the heat *Q*∗(Θ) pumped out (reversibly) from L at the (end, output) temperature <sup>Θ</sup> of L and for the entropy *<sup>S</sup>*∗L(Θ) of the whole

<sup>=</sup> *<sup>δ</sup>A*(Θ, dΘ)

<sup>d</sup><sup>Θ</sup> <sup>=</sup> *<sup>l</sup>* · (<sup>Θ</sup> <sup>−</sup> *<sup>T</sup>*0)d<sup>Θ</sup> �

For the whole work Δ*A*(Θ*W*; *T*0) consumpted by the general reverse cycle O between

<sup>2</sup> · <sup>Θ</sup>*<sup>W</sup>*

 Θ*<sup>W</sup> T*0

> <sup>2</sup> + *l* <sup>2</sup> · *<sup>T</sup>*<sup>0</sup>

 Θ *T*0 *l*d*θ* 

*<sup>δ</sup>A*(Θ, dΘ) = *<sup>S</sup>*∗L(Θ)d<sup>Θ</sup> <sup>=</sup> *<sup>l</sup>* · <sup>Θ</sup>d<sup>Θ</sup> and *<sup>δ</sup>A*(dΘ, dΘ) = *<sup>l</sup>* · <sup>d</sup>Θd<sup>Θ</sup> <sup>=</sup> <sup>d</sup>*S*∗L(Θ)d(Θ) �

· <sup>2</sup>*tW T*<sup>0</sup> + *TW*

> <sup>1</sup> <sup>−</sup> *<sup>T</sup>*<sup>0</sup> <sup>+</sup> *TW* 2*tW*

<sup>Θ</sup> <sup>=</sup> dS∗L(Θ) = *<sup>l</sup>* · <sup>d</sup><sup>Θ</sup> (162)

dΘ = *l* ·

<sup>2</sup> <sup>−</sup> <sup>2</sup>*<sup>l</sup>*

<sup>2</sup> <sup>=</sup> *TW* and then *tW* <sup>=</sup> *TW* <sup>=</sup> const. (165)

= O*rrev*�with the *apexes* [*lT*0, *T*0], [*lTW*, *TW*], [*lT*0, *TW*] (166)

−−−−−−−−−−−−→ [*lTW*, *TW*], [*lT*0, *TW*] and −−−−−−−−−−−→ [*lT*0, *TW*], [*lT*0, *<sup>T</sup>*0] (167)

<sup>2</sup> · *<sup>η</sup>*max. Thus, the return to the *initial (starting) state* of

<sup>2</sup> , when it is valid that <sup>Θ</sup>*<sup>W</sup>* <sup>=</sup> *TW*, *<sup>β</sup>* <sup>=</sup> *<sup>T</sup>*<sup>0</sup>

= *δA*(Θ, dΘ; *T*0) (163)

 Θ*<sup>W</sup> T*0

<sup>2</sup> · *<sup>T</sup>*0Θ*<sup>W</sup>* <sup>=</sup> *<sup>λ</sup>* · (Θ*<sup>W</sup>* <sup>−</sup> *<sup>T</sup>*0)

*TW*

*T*0+*TW* 2 *tW*

= *l*·*tW* · (*TW* − *T*0) (160)

is

(161)

= *δA*

(Θ − *T*0)dΘ(164)

2

our general cycle), the same heat Δ*QW*, is valid that Δ*Q*<sup>0</sup> = Δ*QW* · *γ* where *γ* =

the *transform ratio*.

<sup>Δ</sup>*QW* <sup>=</sup> <sup>Δ</sup>*Q*<sup>0</sup> · <sup>2</sup>*tW*

= *l*

*<sup>S</sup>*∗L(Θ)

Δ*A*(Θ*W*; *T*0) =

<sup>2</sup> <sup>−</sup> *<sup>T</sup>*<sup>0</sup>

*δA* = *λ* · *TW*

and its efficiency is 1 <sup>−</sup> *<sup>γ</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>β</sup>*

<sup>2</sup> · (Θ*<sup>W</sup>*

O(Θ*<sup>W</sup>* ,*T*0)

<sup>=</sup> *<sup>l</sup>*

� = 

environment of O (including L with O) is valid

<sup>Θ</sup> <sup>=</sup> *<sup>Q</sup>*∗(Θ) <sup>d</sup><sup>Θ</sup>

 Θ *T*0 *l*d*θ* 

> Θ *T*0

<sup>2</sup>) <sup>−</sup> *<sup>l</sup>* · *<sup>T</sup>*0(Θ*<sup>W</sup>* <sup>−</sup> *<sup>T</sup>*0) = *<sup>l</sup>*

<sup>2</sup> · (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*)

*tW* <sup>=</sup> *TW* <sup>+</sup> <sup>Θ</sup>*<sup>W</sup>*

Thus our general cycle <sup>O</sup> is of a triangle shape, <sup>O</sup> �

dΘ

 Θ*<sup>W</sup> T*0

*T*<sup>0</sup> + *TW*

Δ*A* = Δ*QW* · (1 − *γ*) = *l*·*tW* · (*TW* − *T*0) ·

<sup>2</sup> · (*TW* <sup>−</sup> *<sup>T</sup>*0) · (2*tW* <sup>−</sup> *<sup>T</sup>*<sup>0</sup> <sup>−</sup> *TW*)

Θ Θ

�

temperatures *T*<sup>0</sup> and Θ*W*, being coverd by elementary cycles (162), is valid that

 dΘ =

<sup>d</sup>*S*∗L(*θ*)

But, then for the results for Δ*A* in (160), (161) and (164) follows that

<sup>2</sup> <sup>=</sup> <sup>1</sup>

the medium L is possible by using the oriented *abscissas* (in the *S* − *T* diagram)

= *l* 2 · *TW* <sup>2</sup> <sup>−</sup> *<sup>T</sup>*<sup>0</sup> 2 

Then

$$\delta A(\Theta, \mathrm{d\Theta}; T\_0) = \Delta Q\_W(\Theta) \cdot \frac{\mathrm{d\Theta}}{\Theta} = l \cdot \Theta \cdot (\Theta - T\_0) \cdot \frac{\mathrm{d\Theta}}{\Theta} = l \cdot (\Theta - T\_0) \mathrm{d\Theta} \tag{168}$$

$$= [\mathrm{S} \ast\_{\mathcal{L}}(\Theta) - \mathrm{S} \ast\_{\mathcal{L}}(T\_0)] \cdot \mathrm{d\Theta}$$

$$\Delta Q\_W(\Theta) = l \Theta \left(\Theta - T\_0\right) \text{ and for } \gamma \text{ used in (160) is gained that}$$

$$\Delta Q\_0(\Theta) = \Delta Q\_W(\Theta) \cdot \gamma(\Theta) = l \cdot \Theta \cdot (\Theta - T\_0) \cdot \frac{\Theta + T\_0}{2\Theta} = \lambda \cdot \left(\Theta^2 - T\_0^2\right)$$

For the whole heats Δ*Q*<sup>0</sup> a Δ*QW* being changed mutually between the working medium L of the whole triangular cycle O*rrev*� and its environment (166), and for the work <sup>Δ</sup>*A*, in its equivalent Carnot cycle O� *rrev* with working temperatures *<sup>T</sup>*<sup>0</sup> <sup>+</sup> *TW* <sup>2</sup> and *TW*, will be valid that21

$$
\Delta Q\_0 = \int\_{T\_0}^{T\_W} \left[ \int\_0^{\Theta} l \mathbf{d} \theta \right] d\Theta = \frac{l}{2} \cdot T\_W^2 \cdot (1 - \beta^2) \stackrel{\triangle}{=} W \left[ = l \cdot (T\_W - T\_0) \cdot T\_0 + \frac{l}{2} \cdot (T\_W - T\_0)^2 \right] (169)
$$

$$
\Delta Q\_W = \int\_{T\_0}^{T\_W} \left[ \int\_0^{T\_W} l \mathbf{d} \theta \right] d\Theta = l \cdot (T\_W - T\_0) \cdot T\_W = l \cdot T\_W^2 \cdot (1 - \beta)
$$

$$
\Delta A = \frac{1}{2} \int\_{T\_0}^{T\_W} \left[ \int\_{T\_0}^{T\_W} l \mathbf{d} \theta \right] d\Theta = \cdot l \cdot T\_W^2 \cdot (1 - \beta) - l \cdot T\_W^2 \cdot (1 - \beta) \tag{170}
$$

$$
= l \cdot T\_W^2 \cdot (1 - \beta) \cdot \left[ 1 - \frac{l}{2}(1 + \beta) \right] = \frac{l}{2} \cdot T\_W^2 \cdot (1 - \beta)^2 = \oint\_{\mathcal{O}\_{mo}} \delta A
$$

#### **7.2. Capacity corections for wide–band photonic transfer channel**

The average *output* energy *P*2(Θ*W*) of the message being received within interval (0, *TW*� of the temperature <sup>Θ</sup> of the medium L ∼= **<sup>K</sup>**L−<sup>L</sup> from [30], when 0 < <sup>Θ</sup><sup>0</sup> ≤ <sup>Θ</sup> ≤ <sup>Θ</sup>*<sup>W</sup>* and <sup>Θ</sup><sup>0</sup> ≤ *<sup>T</sup>*<sup>0</sup> and Θ*<sup>W</sup>* ≤ *TW* are valid, is given by the sum of the *input* average energy *W*(Θ*W*, Θ0) and the average energy *P*1(Θ0) of the *additive noise*

$$P\_2(\Theta\_W) = P\_1(\Theta\_0) + W(\Theta\_{W'}\Theta\_0) \tag{171}$$

The output message bears the *whole average output information H*2(Θ*W*). By the medium value theorem is possible, for a certain maximal temperature Θ*<sup>W</sup>* ≤ *TW* of the temperature Θ ∈ (0, Θ*W*�, consider that the receiving of the output message is performed at the *average (constant) temperature* <sup>Θ</sup>*<sup>W</sup>* <sup>=</sup> <sup>Θ</sup>*<sup>W</sup>* <sup>2</sup> . Then for the whole change of the output information entropy Δ*H*<sup>2</sup> � = *<sup>H</sup>*2(Θ*W*) [the thermodynamic entropy *<sup>S</sup>*∗L(Θ*W*) in information units] is valid

$$H\_2(\Theta\_W) = \frac{P\_2(\Theta\_W)}{\mathbf{k}\overline{\Theta\_W}} = \frac{P\_1(\Theta\_0) + W(\Theta\_W, \Theta\_0)}{\mathbf{k}\overline{\Theta\_W}} \stackrel{\triangle}{=} H\_1(\Theta\_W, \Theta\_0) + H[W(\Theta\_W, \Theta\_0)] \tag{172}$$

<sup>21</sup> Further it willbe layed down *<sup>λ</sup>* <sup>=</sup> *<sup>π</sup>*2*k*<sup>2</sup> 6¯*<sup>h</sup>* , *<sup>l</sup>* <sup>=</sup> *<sup>π</sup>*2*k*<sup>2</sup> 3¯*<sup>h</sup>* <sup>=</sup> <sup>2</sup> · *<sup>λ</sup>*.

#### 30 Will-be-set-by-IN-TECH 112 Thermodynamics – Fundamentals and Its Application in Science Information Capacity of Quantum Transfer Channels and Thermodynamic Analogies <sup>31</sup>

By (153) is valid that *<sup>Q</sup>*∗(Θ) = *<sup>λ</sup>*Θ<sup>2</sup> and *<sup>δ</sup>Q*∗(Θ) = *<sup>l</sup>*ΘdΘ. Thus for <sup>Θ</sup> <sup>∈</sup> (0, *<sup>T</sup>*0� <sup>a</sup> <sup>Θ</sup> <sup>∈</sup> (0, *TW*� is valid

$$\mathrm{d}H\_{\Theta\_{\mathrm{W}}}(Y) = \frac{\delta Q \* (\Theta\_{\mathrm{W}})}{\mathrm{k}\Theta\_{\mathrm{W}}} = \frac{l}{\mathrm{k}} \mathrm{d}\Theta\_{\mathrm{W}} \quad \mathrm{d}H\_{\Theta\_{\mathrm{W}}}(Y|X) = \frac{2l\Theta \mathrm{d}\Theta}{\mathrm{k}\Theta\_{\mathrm{W}}} \tag{173}$$

For the **sustaining, in the sense repeatable, cyclical information transfer, the renewal of the initial or starting state of the transfer channel K**L−<sup>L</sup> ∼= L**, after any** *individual information transfer act* **- the** *sending input and receiving output message* **has been accomplished, is**

Information Capacity of Quantum Transfer Channels and Thermodynamic Analogies 113

Nevertheless, in derivations of the formulas (102), (177) and (138) **this return of the physical medium** L, after accomplishing any individual information transfer act, into the starting state **is either not considered**, **or, on the contrary, is considered, but by that the whole transfer chain is opened** *to cover the energetic needs for this return transition from another, outer resources* **than from those ones within the transfer chain itself**. In both these two cases the channel equation is fulfilled. This enables any individual act of information transfer be realized by *external and forced out, repeated starting* of each this individual transfer act.22 23

If for the creation of a cycle the resources of the transfer chain are used only, the need for *another correction*, this time in (177) arises. To express it it will be used the *full cyclical*

modeled by the cyclical thermodynamic process O*rrev*� of reversible changes in the channel

Now the further correction for capacity formulas (102), (138) and (177) will be dealt with for that case that the *return* of the medium L into its initial, starting state is performed *within the transfer chain* only. It will be envisiged by a triangular reverse heat cycle O*rrev*� created by the oriented abscissas within the apexes in the *S* − *T* diagram (166), [*lT*0, *T*0], [*lTW*, *TW*], [*lT*0, *TW*]. The abstract experiment from [30] will be now, formally and as an analogy, realized by this

<sup>2</sup> · <sup>Θ</sup>2, *<sup>Q</sup>*∗*<sup>W</sup>* <sup>=</sup> *<sup>l</sup>*

The working temperature Θ<sup>0</sup> of cooling and Θ*<sup>W</sup>* of heating are changing by (157),

*<sup>δ</sup>Q*∗(*θ*) = <sup>Θ</sup>

0

<sup>22</sup> For these both cases is not possible to construct a construction-relevant heat cycles *described in a proper information way*. <sup>23</sup> But the modeling by the direct cycle such as in (128) is possible for the *I I*. *Principle of Thermodynamics* is valid in any

*∂Q*∗(*θ*) *∂θ* <sup>d</sup>*<sup>θ</sup>* <sup>=</sup>

and the heat entropy *<sup>S</sup>*∗L(Θ) of the medium L is changing by (155)-(156),

<sup>2</sup> · *TW*

*<sup>l</sup>* · *<sup>S</sup>*∗L(Θ0) ∈ �*T*0, *TW*� and <sup>Θ</sup>*<sup>W</sup>* <sup>=</sup> *TW* <sup>=</sup> const. (180)

<sup>Θ</sup> and then *<sup>S</sup>*∗L(Θ) = *<sup>l</sup>* · <sup>Θ</sup> <sup>=</sup> <sup>2</sup>*Q*∗(Θ)

 Θ 0

<sup>L</sup>−<sup>L</sup> ∼= L and without opening the transfer chain. (Also **K** ∼= **K**�

<sup>L</sup>−L. The information transfer will be

<sup>B</sup>−E).

*rrev*, described informationaly, and thought as

2, *<sup>Q</sup>*∗<sup>0</sup> <sup>=</sup> *<sup>l</sup>*

<sup>L</sup>−L|B−<sup>E</sup> ∼= L. Thus the denotation

<sup>2</sup> (179)

<sup>Θ</sup> (181)

*lθ*d*θ* (182)

<sup>2</sup> · *<sup>T</sup>*<sup>0</sup>

*thermodynamic analogy* **K** of **K**L−<sup>L</sup> used *cyclically*, **K**�

*reverse* and reversible heat cycle O*rrev*� ≡ O�

*<sup>Q</sup>*∗(Θ) = *<sup>π</sup>*2k2Θ<sup>2</sup>

<sup>Θ</sup><sup>0</sup> <sup>=</sup> <sup>1</sup>

<sup>d</sup>*S*∗L <sup>=</sup> *<sup>∂</sup>Q*∗(Θ)

Using integral (149) it is possible to write that

case and giving the possibility of the cycle description.

*<sup>Q</sup>*∗(Θ) = <sup>Θ</sup>

**<sup>K</sup>** ≡ **<sup>K</sup>**� is usable. By (153)-(157) it will be

*7.2.1. Return of transfer medium into initial state, second correction*

modeling information transfer process in a channel **K** ∼= **K**�

<sup>6</sup>*<sup>h</sup>* <sup>=</sup> *<sup>l</sup>*

*<sup>∂</sup>*<sup>Θ</sup> <sup>d</sup><sup>Θ</sup> · <sup>1</sup>

0

**needed**.

**K** ∼= **K**�

With <sup>Θ</sup>*<sup>W</sup>* <sup>=</sup> *TW* and <sup>Θ</sup><sup>0</sup> <sup>=</sup> *<sup>T</sup>*<sup>0</sup> and with the reducing temperature *TW* <sup>2</sup> is possible to write

$$\begin{aligned} P\_2 = W + P\_1 \stackrel{\triangle}{=} Q \*\_W = \lambda T\_W^2 \stackrel{\triangle}{=} Y\\ H\_2 \stackrel{\triangle}{=} H(Y) = \int\_0^{T\_W} \frac{l}{\mathbf{k}} d\Theta\_W = \frac{l}{\mathbf{k}} T\_W = \frac{P\_2}{\mathbf{k}\frac{T\_W}{2}} = \frac{2(W + P\_1)}{\mathbf{k}T\_W} = \frac{2\lambda T\_W^2}{\mathbf{k}T\_W} = \frac{lT\_W}{k} \end{aligned} \tag{174}$$

$$\begin{aligned} P\_1 \stackrel{\triangle}{=} Q \*\_0 = \lambda T\_0^2 \stackrel{\triangle}{=} Y|X\\ H\_1 \stackrel{\triangle}{=} H(Y|X) = \int\_0^{T\_0} \frac{2l\Theta d\Theta}{\mathbf{k}T\_W} = \frac{l}{\mathbf{k}T\_W} \cdot T\_0^2 = \frac{P\_1}{\mathbf{k}\frac{T\_W}{2}} = \frac{2\lambda T\_0^2}{\mathbf{k}T\_W} = \frac{lT\_0}{\mathbf{k}T\_W} \end{aligned}$$

$$\begin{aligned} W = Q \*\_W - Q\*\_0 \stackrel{\triangle}{=} X\\ H[W(T\_W, T\_0)] \stackrel{\triangle}{=} H(X) = \frac{W}{\mathbf{k}\frac{T\_W}{2}} = \frac{2\lambda}{\mathbf{k}T\_W} \cdot (T\_W^2 - T\_0^2) \end{aligned} \tag{175}$$

By the channel equation (4), (5) and by equations (23)-(24) and also by definitions (174)-(175) and with the loss entropy *H*(*X*|*Y*) = 0 it must be valid for the transinformation *T*(·; ·) that

$$T(Y;X) = H(Y) - H(Y|X) = \frac{l}{\mathbf{k}} \cdot (T\_W - T\_0) \cdot (1 + \boldsymbol{\beta}) = H(X) \tag{176}$$

$$T(X;Y) = H(X) - H(X|Y) = H(X) = T(X,Y) \text{ and by using } l = \frac{\pi^2 k^2}{3\hbar}, \ \boldsymbol{\beta} = \frac{T\_0}{T\_W},$$

$$T(Y;X) = \frac{\pi^2 \mathbf{k}}{3\hbar} \cdot T\_W \cdot (1 - \boldsymbol{\beta}^2) = \frac{\pi^2 k}{3\hbar} \cdot T\_W \cdot (1 - \boldsymbol{\beta}) \cdot (1 + \boldsymbol{\beta}) = \mathbb{C}\_{T\_0, W}(\mathbf{K}\_{\mathbf{L}-\mathbf{L}}) \cdot (1 + \boldsymbol{\beta})$$

For the given extremal temperatures *T*0, *TW* the value *T*(*X*;*Y*) stated this way is the only one, and thus also, it is the information capacity *C*� *<sup>T</sup>*0,*TW* (*W*) of the channel **<sup>K</sup>**L−<sup>L</sup> (the *first* correction)

$$\mathbf{C}'\_{T\_0, T\_W} = (\mathcal{W}) = T(\mathbf{X}; \mathbf{Y}) = \frac{\pi^2 \mathbf{k} T\_0}{\mathfrak{A}\mathfrak{h}} \cdot \left(\sqrt{1 + \frac{6\hbar \cdot \mathcal{W}}{\pi^2 \mathbf{k}^2 T\_0^2}} - 1\right) \cdot (1 + \mathfrak{f}) \tag{177}$$

The information capacity correction (177) of the wide–band photonic channel **K**L−<sup>L</sup> [30], stated this way, is (1 + *β*)-times *higher* than the formulas (102) and (138) say. The reason is in using two different information descriptions of the oriented abscissa −−−−−−−−−−→ [*l*0, 0], [*lTW*, *TW*] in derivation (138) and (177) which abscissa −−−−−−−−−−→ [*l*0, 0], [*lTW*, *TW*] is on one line in *<sup>S</sup>* <sup>−</sup> *<sup>T</sup>* diagram and is composed from two oriented abscissas,

$$\overline{[l0,0]}, \overline{[lT\_{0\prime}T\_{0}]} \text{ and } \left[\overline{lT\_{0\prime}T\_{0}}\right], \overline{[lT\_{W\prime}T\_{W}]} \tag{178}$$

The first abscissa represents the *phase of noise generation* and the second one the *phase of input signal generation*. The whole composed abscissa represents the *phase of whole output signal generation*.

For the **sustaining, in the sense repeatable, cyclical information transfer, the renewal of the initial or starting state of the transfer channel K**L−<sup>L</sup> ∼= L**, after any** *individual information transfer act* **- the** *sending input and receiving output message* **has been accomplished, is needed**.

Nevertheless, in derivations of the formulas (102), (177) and (138) **this return of the physical medium** L, after accomplishing any individual information transfer act, into the starting state **is either not considered**, **or, on the contrary, is considered, but by that the whole transfer chain is opened** *to cover the energetic needs for this return transition from another, outer resources* **than from those ones within the transfer chain itself**. In both these two cases the channel equation is fulfilled. This enables any individual act of information transfer be realized by *external and forced out, repeated starting* of each this individual transfer act.22 23

If for the creation of a cycle the resources of the transfer chain are used only, the need for *another correction*, this time in (177) arises. To express it it will be used the *full cyclical thermodynamic analogy* **K** of **K**L−<sup>L</sup> used *cyclically*, **K**� <sup>L</sup>−L. The information transfer will be modeled by the cyclical thermodynamic process O*rrev*� of reversible changes in the channel **K** ∼= **K**� <sup>L</sup>−<sup>L</sup> ∼= L and without opening the transfer chain. (Also **K** ∼= **K**� <sup>B</sup>−E).

#### *7.2.1. Return of transfer medium into initial state, second correction*

30 Will-be-set-by-IN-TECH

By (153) is valid that *<sup>Q</sup>*∗(Θ) = *<sup>λ</sup>*Θ<sup>2</sup> and *<sup>δ</sup>Q*∗(Θ) = *<sup>l</sup>*ΘdΘ. Thus for <sup>Θ</sup> <sup>∈</sup> (0, *<sup>T</sup>*0� <sup>a</sup> <sup>Θ</sup> <sup>∈</sup>

<sup>k</sup> *TW* <sup>=</sup> *<sup>P</sup>*<sup>2</sup> k *TW* 2

· *T*<sup>0</sup>

<sup>2</sup> <sup>−</sup> *<sup>T</sup>*<sup>0</sup>

<sup>2</sup> <sup>=</sup> *<sup>P</sup>*<sup>1</sup> k *TW* 2

<sup>=</sup> *<sup>l</sup>* k*TW*

· (*TW*

By the channel equation (4), (5) and by equations (23)-(24) and also by definitions (174)-(175) and with the loss entropy *H*(*X*|*Y*) = 0 it must be valid for the transinformation *T*(·; ·) that

For the given extremal temperatures *T*0, *TW* the value *T*(*X*;*Y*) stated this way is the only

3*h* ·

The information capacity correction (177) of the wide–band photonic channel **K**L−<sup>L</sup> [30], stated this way, is (1 + *β*)-times *higher* than the formulas (102) and (138) say. The reason is in using two different information descriptions of the oriented abscissa −−−−−−−−−−→ [*l*0, 0], [*lTW*, *TW*] in derivation (138) and (177) which abscissa −−−−−−−−−−→ [*l*0, 0], [*lTW*, *TW*] is on one line in *<sup>S</sup>* <sup>−</sup> *<sup>T</sup>* diagram and

The first abscissa represents the *phase of noise generation* and the second one the *phase of input signal generation*. The whole composed abscissa represents the *phase of whole output signal*

 1 +

<sup>k</sup>dΘ*W*, d*H*Θ*<sup>W</sup>* (*Y*|*X*) = <sup>2</sup>*l*Θd<sup>Θ</sup>

= *Y* (174)

<sup>=</sup> <sup>2</sup>(*<sup>W</sup>* <sup>+</sup> *<sup>P</sup>*1) k*TW*

<sup>k</sup> · (*TW* <sup>−</sup> *<sup>T</sup>*0) · (<sup>1</sup> <sup>+</sup> *<sup>β</sup>*) = *<sup>H</sup>*(*X*) (176)

<sup>3</sup>*<sup>h</sup>* ·*TW* · (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*) · (<sup>1</sup> <sup>+</sup> *<sup>β</sup>*) = *CT*0,*W*(**K**L−L) · (<sup>1</sup> <sup>+</sup> *<sup>β</sup>*)

6*h* · *W π*2k2*T*<sup>0</sup>

−−−−−−−−−→ [*l*0, 0],[*lT*0, *<sup>T</sup>*0] and −−−−−−−−−−−−→ [*lT*0, *<sup>T</sup>*0], [*lTW*, *TW*] (178)

<sup>2</sup> − 1 

<sup>=</sup> <sup>2</sup>*λT*<sup>0</sup> 2

k*TW*

<sup>2</sup>) (175)

*<sup>T</sup>*0,*TW* (*W*) of the channel **<sup>K</sup>**L−<sup>L</sup> (the *first*

kΘ*<sup>W</sup>*

<sup>2</sup> is possible to write

<sup>=</sup> <sup>2</sup>*λTW*

k*TW*

<sup>=</sup> *lT*<sup>0</sup> 2 k*TW*

3¯*<sup>h</sup>* , *<sup>β</sup>* <sup>=</sup> *<sup>T</sup>*<sup>0</sup>

*TW* ,

· (1 + *β*) (177)

2

<sup>=</sup> *lTW k*

(173)

= *l*

<sup>d</sup>*H*Θ*<sup>W</sup>* (*Y*) = *<sup>δ</sup>Q*∗(Θ*W*)

= *Q*∗*<sup>W</sup>* = *λTW*

= *Q*∗<sup>0</sup> = *λT*<sup>0</sup>

= *H*(*Y*|*X*) =

*W* = *Q*∗*<sup>W</sup>* − *Q*∗<sup>0</sup>

*<sup>T</sup>*(*Y*; *<sup>X</sup>*) = *<sup>H</sup>*(*Y*) <sup>−</sup> *<sup>H</sup>*(*Y*|*X*) = *<sup>l</sup>*

is composed from two oriented abscissas,

<sup>=</sup> *<sup>H</sup>*(*X*) = *<sup>W</sup>*

kΘ*<sup>W</sup>*

With <sup>Θ</sup>*<sup>W</sup>* <sup>=</sup> *TW* and <sup>Θ</sup><sup>0</sup> <sup>=</sup> *<sup>T</sup>*<sup>0</sup> and with the reducing temperature *TW*

2 �

*l*

2 � = *Y*|*X*

 *T*<sup>0</sup> 0

> � = *X*

k *TW* 2

<sup>3</sup>*<sup>h</sup>* ·*TW* · (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*2) = *<sup>π</sup>*2*<sup>k</sup>*

*<sup>T</sup>*0,*TW* = (*W*) = *<sup>T</sup>*(*X*;*Y*) = *<sup>π</sup>*2k*T*<sup>0</sup>

one, and thus also, it is the information capacity *C*�

<sup>k</sup>dΘ*<sup>W</sup>* <sup>=</sup> *<sup>l</sup>*

2*l*ΘdΘ k*TW*

<sup>=</sup> <sup>2</sup>*<sup>λ</sup>* k*TW*

*<sup>T</sup>*(*X*;*Y*) = *<sup>H</sup>*(*X*) <sup>−</sup> *<sup>H</sup>*(*X*|*Y*) = *<sup>H</sup>*(*X*) = *<sup>T</sup>*(*X*,*Y*) and by using *<sup>l</sup>* <sup>=</sup> *<sup>π</sup>*2*k*<sup>2</sup>

 *TW* 0

(0, *TW*� is valid

*P*<sup>2</sup> = *W* + *P*<sup>1</sup>

*H*[*W*(*TW*, *T*0)] �

*<sup>T</sup>*(*Y*; *<sup>X</sup>*) = *<sup>π</sup>*2k

*C*�

correction)

*generation*.

�

*H*<sup>2</sup> � = *H*(*Y*) =

*P*1 �

*H*<sup>1</sup> �

> Now the further correction for capacity formulas (102), (138) and (177) will be dealt with for that case that the *return* of the medium L into its initial, starting state is performed *within the transfer chain* only. It will be envisiged by a triangular reverse heat cycle O*rrev*� created by the oriented abscissas within the apexes in the *S* − *T* diagram (166), [*lT*0, *T*0], [*lTW*, *TW*], [*lT*0, *TW*]. The abstract experiment from [30] will be now, formally and as an analogy, realized by this *reverse* and reversible heat cycle O*rrev*� ≡ O� *rrev*, described informationaly, and thought as modeling information transfer process in a channel **K** ∼= **K**� <sup>L</sup>−L|B−<sup>E</sup> ∼= L. Thus the denotation **<sup>K</sup>** ≡ **<sup>K</sup>**� is usable. By (153)-(157) it will be

$$Q\*(\Theta) = \frac{\pi^2 \mathbf{k}^2 \Theta^2}{6\hbar} = \frac{l}{2} \cdot \Theta^2 \text{ . Q\*}\_{W} = \frac{l}{2} \cdot T\_W \text{ . Q\*}\_{0} = \frac{l}{2} \cdot T\_0^2 \tag{179}$$

The working temperature Θ<sup>0</sup> of cooling and Θ*<sup>W</sup>* of heating are changing by (157),

$$\Theta\_0 = \frac{1}{l} \cdot \mathbb{S} \*\_{\mathcal{L}}(\Theta\_0) \in \langle T\_{0\prime} T\_W \rangle \text{ and } \Theta\_W = T\_W = \text{const.} \tag{180}$$

and the heat entropy *<sup>S</sup>*∗L(Θ) of the medium L is changing by (155)-(156),

$$\mathrm{d}\mathbb{S}\*\_{\mathcal{L}} = \frac{\partial Q\*(\Theta)}{\partial \Theta} \mathrm{d}\Theta \cdot \frac{1}{\Theta} \text{ and then } \mathrm{S}\*\_{\mathcal{L}}(\Theta) = l \cdot \Theta = \frac{2Q\*(\Theta)}{\Theta} \tag{181}$$

Using integral (149) it is possible to write that

$$Q\*(\Theta) = \int\_0^{\Theta} \delta Q\*(\theta) = \int\_0^{\Theta} \frac{\partial Q\*(\theta)}{\partial \theta} d\theta = \int\_0^{\Theta} l\theta d\theta \tag{182}$$

<sup>22</sup> For these both cases is not possible to construct a construction-relevant heat cycles *described in a proper information way*.

<sup>23</sup> But the modeling by the direct cycle such as in (128) is possible for the *I I*. *Principle of Thermodynamics* is valid in any case and giving the possibility of the cycle description.

For the whole heats <sup>Δ</sup>*Q*<sup>0</sup> and <sup>Δ</sup>*QW* being changed mutually between L with the cycle O*rrev*� and its environment and, for the whole work Δ*A* for the equivalent Carnot cycle O� *rrev* with working temperatures *<sup>T</sup>*<sup>0</sup> <sup>+</sup> *TW* <sup>2</sup> and *TW* is valid, by (169)-(170), that

$$\Delta W = \Delta Q\_0 = \frac{l}{2} \cdot T\_W^2 \cdot (1 - \beta^2) \stackrel{\triangle}{=} X, \quad \Delta Q\_W = l \cdot T\_W^2 \cdot (1 - \beta) \stackrel{\triangle}{=} Y \tag{183}$$

$$\Delta Q\_W - \Delta Q\_0 = \Delta A = \frac{l}{2} \cdot T\_W^2 \cdot (1 - \beta)^2 \stackrel{\triangle}{=} Y |X$$

It is visible that the quantity *H*(*Y*) [= *H*(*X*) + *H*(*Y*|*X*)] is introduced correctly, for by (185) is

2k · *TW* · (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*2) = *<sup>l</sup>*

For the transinformation and the information capacity of the transfer organized this way is

2 *C*�

With the extremal temperatures *T*<sup>0</sup> and *TW* the information capacity *C*∗*T*0,*TW* (*W*) is given by

1 + *β*

6*hTW* · �� 1 +

<sup>=</sup> *<sup>C</sup>*∗*T*<sup>0</sup> (*W*) = *<sup>W</sup>*

<sup>1</sup> <sup>+</sup> *<sup>β</sup>*<sup>×</sup> *less* than (138).

For *T*<sup>0</sup> −→ 0 the quantum approximation *C*(*W*) of the capacity *C*∗*T*0,*TW* (*W*) is obtained,

The classical aproximation *CT*<sup>0</sup> (*W*) of *C*∗*T*0,*TW* (*W*) is gained for *T*<sup>0</sup> � 0. This value is near Shannon capacity of the wide–band *Gaussian* channel with noise energy k*T*<sup>0</sup> and with the

independent on the noise energy (the noise power deminishes near the abslute 0◦ *K*)

whole average input energy (energy) *W*; in the same way as in (104) is now gained

k*T*<sup>0</sup> ·

<sup>6</sup>*<sup>h</sup>* <sup>−</sup> *<sup>π</sup>*2k*T*<sup>0</sup> 6*h*

· (<sup>1</sup> <sup>+</sup> *<sup>β</sup>*) = *<sup>W</sup>*

� 1 +

⎞

2k*T*<sup>0</sup>

<sup>2</sup> <sup>=</sup> *<sup>π</sup>*2k*T*<sup>0</sup>

*T*0−→*TW*

= *W* = *Q*∗<sup>0</sup> − *Q*∗*<sup>W</sup>* (in L) follows that the temperature

Information Capacity of Quantum Transfer Channels and Thermodynamic Analogies 115

3*h* ·

6*h* · *W π*2k2*T*<sup>0</sup>

<sup>2</sup> − 1 �

> � = *C*(**K**�

6*h* · *W π*2k2*T*<sup>0</sup> 2

⎠ · (1 + *β*) = *π* ·

· (1 + *β*) −→

�� 1 +

*<sup>T</sup>*(*X*;*Y*) = <sup>1</sup>

2k · *TW* · (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*) · (<sup>1</sup> <sup>−</sup> *<sup>β</sup>* <sup>+</sup> <sup>1</sup> <sup>+</sup> *<sup>β</sup>*)(186)

*<sup>T</sup>*0,*TW* (*W*) (187)

*C*∗*T*0,*TW* (*W*) = *H*(*Y*) (188)

6*h* · *W π*2k2*T*<sup>0</sup>

<sup>2</sup> − 1 � · 1 + *β* 2 (189)

· (*T*<sup>0</sup> + *TW*) (190)

<sup>L</sup>−L|B−E)

<sup>6</sup>*<sup>h</sup>* (191)

(192)

�*W*

*W* k*T*<sup>0</sup>

valid that

valid (177),

*TW* = *T*<sup>0</sup> ·

*<sup>T</sup>*(*X*;*Y*) = *<sup>π</sup>*2*<sup>k</sup>*

*<sup>H</sup>*(*Y*) = *<sup>l</sup>*

= *l*

From the difference Δ*Q*<sup>0</sup>

6*h* · � <sup>1</sup> <sup>−</sup> *<sup>β</sup>*<sup>2</sup> � <sup>=</sup> *<sup>π</sup>*2*<sup>k</sup>*

� 1 +

2k · *TW* · (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*)<sup>2</sup> <sup>+</sup>

<sup>k</sup> · *TW* · (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*)

6*h* · *W π*2k2*T*<sup>0</sup>

2 .

*<sup>T</sup>*(*X*;*Y*) = *<sup>C</sup>*(**K**�) = *<sup>C</sup>*∗*T*0,*TW* (*W*) = *<sup>π</sup>*2k*T*<sup>0</sup>

k*TW*

<sup>=</sup> *<sup>C</sup>*∗*TW* (*W*) = *<sup>W</sup>*

which value is 2<sup>×</sup> *less* than (177) and <sup>2</sup>

*C*(*W*) = lim

*CT*<sup>0</sup> (*W*) .

*T*0→0

<sup>=</sup> *<sup>π</sup>*2k*T*<sup>0</sup> 6*h*

⎛ ⎝ �

*π*4k2*T*<sup>0</sup> 2 <sup>62</sup>*h*¯ <sup>2</sup> <sup>+</sup> *<sup>π</sup>*<sup>2</sup> *<sup>W</sup>*

� <sup>3</sup>*<sup>h</sup>* · *<sup>W</sup> π*2k2*T*<sup>0</sup> 2 �

*l*

*<sup>T</sup>*(*X*;*Y*) = *<sup>C</sup>*∗*T*0,*TW* (*W*) and then *<sup>C</sup>*max <sup>=</sup> lim

Then, for the transinformation, in the same way as in (187), is now valid

<sup>3</sup>*<sup>h</sup>* ·(*TW* <sup>−</sup> *<sup>T</sup>*0)·

The transiformation *<sup>T</sup>*(*X*;*Y*) is the capacity *<sup>C</sup>*(**K**�) and it is possible to write

�

For the whole work <sup>Δ</sup>*<sup>A</sup>* delivered into the cycle O*rrev*�, at the temperature *TW*, and the entropy *<sup>S</sup>*∗L of its working medium L is valid

$$\frac{\Delta A}{T\_W} = \oint\_{\mathcal{O}\_{\rm{WM}\triangle}} \frac{\delta A}{T\_W} = \int\_{T\_0}^{T\_W} l(\Theta - T\_0) \cdot \frac{\mathbf{d}\Theta}{T\_W} \tag{184}$$

$$\oint\_{\mathcal{O}\_{\rm{WM}\triangle}} \frac{\delta A}{T\_W} = \frac{1}{2} \int\_{T\_0}^{T\_W} \left[ \int\_{T\_0}^{T\_W} \mathbf{d}S \ast \mathcal{L}(\theta) \right] \frac{\mathbf{d}\Theta}{T\_W} = \int\_{T\_0}^{T\_W} \left[ S \ast \mathcal{L}(T\_W) - S \ast \mathcal{L}(T\_0) \right] \frac{\mathbf{d}\Theta}{2T\_W}$$

$$= \frac{l}{2T\_W} \cdot (T\_W - T\_0)^2 = \frac{l}{2} \cdot T\_W \cdot (1 - \beta)^2 = \frac{\Delta A}{T\_W}$$

Following (4), (5) and (23) and the triangular shape of the cycle O*rrev*�, the changes of information entropies by expressions (142), (169)-(170) are defined, valid for the equivalent O� *rrev*24, see (142),

*<sup>H</sup>*(*X*) Def <sup>=</sup> <sup>Δ</sup>*Q*<sup>0</sup> k*TW* = *TW T*0 Θ 0 *δQ*∗(*θ*) *θ* dΘ k*TW* = *TW T*0 Θ 0 *<sup>l</sup>*d` <sup>1</sup> k*TW* dΘ (185) *<sup>H</sup>*(*Y*) Def <sup>=</sup> <sup>Δ</sup>*QW* k*TW* = *TW T*0 *TW* 0 *δQ*∗(*θ*) *θ* dΘ k*TW* = *TW T*0 *TW* 0 *<sup>l</sup>*d` <sup>1</sup> k*TW* dΘ *<sup>H</sup>*(*Y*|*X*) Def <sup>=</sup> <sup>Δ</sup>*<sup>A</sup>* k*TW* <sup>=</sup> <sup>1</sup> 2 *TW T*0 *TW T*0 *δQ*∗(*θ*) *θ* dΘ k*TW* <sup>=</sup> <sup>1</sup> 2 *TW T*0 *TW T*0 *l*d*θ* 1 k*TW* dΘ *<sup>T</sup>*(*Y*; *<sup>X</sup>*) = *<sup>H</sup>*(*Y*) <sup>−</sup> *<sup>H</sup>*(*Y*|*X*) = *<sup>l</sup>* 2k*TW TW T*0 2 *TW* 0 d*θ* − *TW T*0 d*θ* dΘ

and by figguring these formulas with *<sup>l</sup>* <sup>=</sup> *<sup>π</sup>*2*<sup>k</sup>* <sup>3</sup>*<sup>h</sup>* is gained that

$$\begin{aligned} H(X) &= \frac{l}{2\mathbf{k}} \cdot T\_W \cdot (1 - \beta^2) = \frac{\pi^2 \mathbf{k} T\_W}{6h} \cdot (1 - \beta^2) \\ H(Y) &= \frac{l}{\mathbf{k}} \cdot T\_W \cdot (1 - \beta) = \frac{\pi^2 \mathbf{k} T\_W}{3h} \cdot (1 - \beta) \\ H(Y|X) &= \frac{l}{2\mathbf{k}} \cdot T\_W \cdot (1 - \beta)^2 = \frac{\pi^2 \mathbf{k} T\_W}{6h} \cdot (1 - \beta)^2 = \oint\_{Om} \frac{\delta A}{\mathbf{k} T\_W} \\ T(Y;X) &= \frac{l}{\mathbf{k}} \cdot T\_W \cdot (1 - \beta) - \frac{l}{2\mathbf{k}} \cdot T\_W \cdot (1 - \beta)^2 = \frac{l}{2\mathbf{k}} \cdot T\_W \cdot (1 - \beta) \cdot (2 - 1 + \beta) \\ &= \frac{l}{2\mathbf{k}} \cdot T\_W \cdot (1 - \beta^2) = \frac{\pi^2 \mathbf{k} T\_W}{6h} \cdot (1 - \beta^2) = H(X) = T(X;Y) \\ H(X|Y) &= 0 \end{aligned}$$

<sup>24</sup> In accordance with the input energy delivered and the extremal temperatures used in [30].

It is visible that the quantity *H*(*Y*) [= *H*(*X*) + *H*(*Y*|*X*)] is introduced correctly, for by (185) is valid that

$$\begin{split}H(Y) &= \frac{l}{2\mathbf{k}} \cdot T\_W \cdot (1-\beta)^2 + \frac{l}{2\mathbf{k}} \cdot T\_W \cdot (1-\beta^2) = \frac{l}{2\mathbf{k}} \cdot T\_W \cdot (1-\beta) \cdot (1-\beta+1+\beta) \\ &= \frac{l}{\mathbf{k}} \cdot T\_W \cdot (1-\beta) \end{split} \tag{186}$$

For the transinformation and the information capacity of the transfer organized this way is valid (177),

$$T(X;Y) = \frac{1}{2} \mathbb{C}'\_{T\_0, T\_W}(W) \tag{187}$$

With the extremal temperatures *T*<sup>0</sup> and *TW* the information capacity *C*∗*T*0,*TW* (*W*) is given by

$$T(X;Y) = \mathbb{C} \*\_{T\_0, T\_W}(W) \text{ and then } \mathbb{C}^{\text{max}} = \lim\_{T\_0 \longrightarrow T\_W} \mathbb{C} \*\_{T\_0, T\_W}(W) = H(Y) \tag{188}$$

From the difference Δ*Q*<sup>0</sup> � = *W* = *Q*∗<sup>0</sup> − *Q*∗*<sup>W</sup>* (in L) follows that the temperature *TW* = *T*<sup>0</sup> · � 1 + 6*h* · *W π*2k2*T*<sup>0</sup> 2 .

Then, for the transinformation, in the same way as in (187), is now valid

32 Will-be-set-by-IN-TECH

For the whole heats <sup>Δ</sup>*Q*<sup>0</sup> and <sup>Δ</sup>*QW* being changed mutually between L with the cycle O*rrev*�

<sup>2</sup> and *TW* is valid, by (169)-(170), that

For the whole work <sup>Δ</sup>*<sup>A</sup>* delivered into the cycle O*rrev*�, at the temperature *TW*, and the

*l*(Θ − *T*0) ·

 dΘ *TW* = *TW T*0

Following (4), (5) and (23) and the triangular shape of the cycle O*rrev*�, the changes of information entropies by expressions (142), (169)-(170) are defined, valid for the equivalent

> dΘ k*TW* = *TW T*0

> > dΘ k*TW* = *TW T*0

 dΘ k*TW*

> 2 *TW* 0

<sup>6</sup>*<sup>h</sup>* · (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*2)

<sup>6</sup>*<sup>h</sup>* · (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*)<sup>2</sup> <sup>=</sup>

2k · *TW* · (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*)<sup>2</sup> <sup>=</sup> *<sup>l</sup>*

<sup>3</sup>*<sup>h</sup>* · (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*)

<sup>=</sup> <sup>1</sup> 2

d*θ* −

<sup>6</sup>*<sup>h</sup>* · (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*2) = *<sup>H</sup>*(*X*) = *<sup>T</sup>*(*X*;*Y*)

O*rrev*

2

= *X*, Δ*QW* = *l* · *TW*

<sup>2</sup> · *TW*

dΘ *TW*

·*TW* · (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*)<sup>2</sup> <sup>=</sup> <sup>Δ</sup>*<sup>A</sup>*

<sup>2</sup> · (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*) �

= *Y*|*X*

[*S*∗L(*TW*) <sup>−</sup> *<sup>S</sup>*∗L(*T*0)] <sup>d</sup><sup>Θ</sup>

<sup>2</sup> · (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*)<sup>2</sup> �

*TW*

 Θ 0 *l*d` 1 k*TW*

 *TW T*0

 *TW* 0

> *TW T*0 d*θ* dΘ

*l*d` 1 k*TW* dΘ

<sup>3</sup>*<sup>h</sup>* is gained that

*δA* k*TW*

2k · *TW* · (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*) · (<sup>2</sup> <sup>−</sup> <sup>1</sup> <sup>+</sup> *<sup>β</sup>*)

*l*d*θ* 1 k*TW* dΘ

 *TW T*0

*rrev* with

(184)

= *Y* (183)

2*TW*

dΘ (185)

and its environment and, for the whole work Δ*A* for the equivalent Carnot cycle O�

<sup>2</sup> · (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*2) �

<sup>Δ</sup>*QW* <sup>−</sup> <sup>Δ</sup>*Q*<sup>0</sup> <sup>=</sup> <sup>Δ</sup>*<sup>A</sup>* <sup>=</sup> *<sup>l</sup>*

<sup>d</sup>*S*∗L(*θ*)

*δQ*∗(*θ*) *θ*

2k*TW*

and by figguring these formulas with *<sup>l</sup>* <sup>=</sup> *<sup>π</sup>*2*<sup>k</sup>*

*δQ*∗(*θ*) *θ*

> *δQ*∗(*θ*) *θ*

> > *TW T*0

· (*TW* <sup>−</sup> *<sup>T</sup>*0)<sup>2</sup> <sup>=</sup> *<sup>l</sup>*

 Θ 0

> *TW* 0

> > *TW T*0

working temperatures *<sup>T</sup>*<sup>0</sup> <sup>+</sup> *TW*

Δ*A TW* = 

*δA TW*

O�

O*rrev*�

*rrev*24, see (142),

*<sup>H</sup>*(*X*) Def

*<sup>H</sup>*(*Y*) Def

*<sup>H</sup>*(*Y*|*X*) Def

*<sup>H</sup>*(*X*) = *<sup>l</sup>*

*<sup>H</sup>*(*Y*) = *<sup>l</sup>*

*<sup>H</sup>*(*Y*|*X*) = *<sup>l</sup>*

*<sup>T</sup>*(*Y*; *<sup>X</sup>*) = *<sup>l</sup>*

*H*(*X*|*Y*) = 0

<sup>=</sup> *<sup>l</sup>*

<sup>=</sup> <sup>Δ</sup>*Q*<sup>0</sup> k*TW* = *TW T*0

<sup>=</sup> <sup>Δ</sup>*QW* k*TW*

<sup>=</sup> <sup>Δ</sup>*<sup>A</sup>* k*TW* = *TW T*0

<sup>=</sup> <sup>1</sup> 2

*<sup>T</sup>*(*Y*; *<sup>X</sup>*) = *<sup>H</sup>*(*Y*) <sup>−</sup> *<sup>H</sup>*(*Y*|*X*) = *<sup>l</sup>*

 *TW T*0

2k · *TW* · (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*2) = *<sup>π</sup>*2k*TW*

<sup>k</sup> · *TW* · (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*) = *<sup>π</sup>*2k*TW*

2k · *TW* · (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*)<sup>2</sup> <sup>=</sup> *<sup>π</sup>*2k*TW*

2k · *TW* · (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*2) = *<sup>π</sup>*2k*TW*

<sup>24</sup> In accordance with the input energy delivered and the extremal temperatures used in [30].

<sup>k</sup> · *TW* · (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*) <sup>−</sup> *<sup>l</sup>*

*<sup>W</sup>* <sup>=</sup> <sup>Δ</sup>*Q*<sup>0</sup> <sup>=</sup> *<sup>l</sup>*

entropy *<sup>S</sup>*∗L of its working medium L is valid

<sup>=</sup> <sup>1</sup> 2

<sup>=</sup> *<sup>l</sup>* 2*TW*

O*rrev*�

 *TW T*0

<sup>2</sup> · *TW*

*δA TW* = *TW T*0

 *TW T*0

$$T(\mathbf{X};\mathbf{Y}) = \frac{\pi^2 \mathbf{k}}{6h} \cdot \left(1 - \boldsymbol{\beta}^2\right) = \frac{\pi^2 \mathbf{k}}{3h} \cdot (T\_W - T\_0) \cdot \frac{1 + \boldsymbol{\beta}}{2} = \frac{\pi^2 \mathbf{k} T\_0}{3h} \cdot \left(\sqrt{1 + \frac{6h \cdot W}{\pi^2 \mathbf{k}^2 T\_0^2}} - 1\right) \cdot \frac{1 + \boldsymbol{\beta}}{2} \tag{189}$$

The transiformation *<sup>T</sup>*(*X*;*Y*) is the capacity *<sup>C</sup>*(**K**�) and it is possible to write

$$T(\mathbf{X};\mathbf{Y}) = \mathbb{C}(\mathbf{K}\_{\triangle}) = \mathbb{C} \ast\_{T\_0, T\_W}(W) = \frac{\pi^2 \mathbf{k} T\_0}{6hT\_W} \cdot \left(\sqrt{1 + \frac{6h \cdot W}{\pi^2 \mathbf{k}^2 T\_0^2}} - 1\right) \cdot (T\_0 + T\_W) \tag{190}$$

$$= \mathbb{C} \ast\_{T\_W}(W) = \frac{W}{\mathbf{k} T\_W} = \mathbb{C} \ast\_{T\_0}(W) = \frac{W}{\underbrace{\int\_{\mathbf{I}\_{\text{max}}} \frac{\triangle}{\triangle} \cdot \mathbf{G} \mathbf{k} \cdot W}} \stackrel{\triangle}{=} \mathbb{C}(\mathbf{K}\_{\triangle - \mathbf{I}\_i \big|\mathbf{B} - \mathbf{E}})$$

k*T*<sup>0</sup> · *π*2k2*T*<sup>0</sup> which value is 2<sup>×</sup> *less* than (177) and <sup>2</sup> <sup>1</sup> <sup>+</sup> *<sup>β</sup>*<sup>×</sup> *less* than (138).

For *T*<sup>0</sup> −→ 0 the quantum approximation *C*(*W*) of the capacity *C*∗*T*0,*TW* (*W*) is obtained, independent on the noise energy (the noise power deminishes near the abslute 0◦ *K*)

$$\mathcal{C}(\mathcal{W}) = \lim\_{T\_0 \to 0} \left( \sqrt{\frac{\pi^4 \mathbf{k}^2 T\_0^2}{6^2 \hbar^2}} + \pi^2 \frac{\mathcal{W}}{6\hbar} - \frac{\pi^2 \mathbf{k} T\_0}{6\hbar} \right) \cdot (1 + \beta) = \pi \cdot \sqrt{\frac{\mathcal{W}}{6\hbar}} \tag{191}$$

1 +

6*h* · *W*

2

The classical aproximation *CT*<sup>0</sup> (*W*) of *C*∗*T*0,*TW* (*W*) is gained for *T*<sup>0</sup> � 0. This value is near Shannon capacity of the wide–band *Gaussian* channel with noise energy k*T*<sup>0</sup> and with the whole average input energy (energy) *W*; in the same way as in (104) is now gained

$$\mathcal{C}\_{T\_0}(W) \doteq \frac{\pi^2 \text{k}T\_0}{6\hbar} \left(\frac{3\hbar \cdot W}{\pi^2 \text{k}^2 T\_0^2}\right) \cdot (1+\beta) = \frac{W}{2\text{k}T\_0} \cdot (1+\beta) \longrightarrow \frac{W}{\text{k}T\_0} \tag{192}$$

#### 34 Will-be-set-by-IN-TECH 116 Thermodynamics – Fundamentals and Its Application in Science Information Capacity of Quantum Transfer Channels and Thermodynamic Analogies <sup>35</sup>

The mutual difference of results (189) and and (102), (138) [12, 30] is given by **the necessity of the returning the transfer medium, the channel K**� ∼= **K**� <sup>L</sup>−L|*B*−*<sup>E</sup>* ∼= L **into its initial state after each individual information transfer act has been accomplished and, by the relevant temperatutre reducing of the heat** Δ*Q*<sup>0</sup> [by *TW* in (183)-(189)]. Thus, our thermodynamic cyclical model **<sup>K</sup>**� ∼= O*rrev*� for the repeatible information transfer through the channel **K**� <sup>L</sup>−L|B−<sup>E</sup> is of the information capacity (189), while in [12, 30] the information capacity of the *one-act* information transfer is stated.25 By (189) the *whole energy costs* for the cyclical information transfer considered is countable.26

[8] Gershenfeld, N. Signal entropy and the thermodynamics of computation. *IBM Systems*

Information Capacity of Quantum Transfer Channels and Thermodynamic Analogies 117

[10] Hašek, O.; Nožiˇcka, J. *Technická mechanika pro elektrotechnické obory II.*; SNTL: Praha, 1968. [11] Hejna, B.; Vajda, I. Information transmission in stationary stochastic systems. *AIP Conf.*

[12] Hejna, B. Informaˇcní kapacita stacionárních fyzikálních systému. Ph.D. Dissertation,

[13] Hejna, B. Generalized Formula of Physical Channel Capacities. *International Journal of*

[14] Hejna, B. Thermodynamic Model of Noise Information Transfer. In *AIP Conference Proceedings*, Computing Anticipatory Systems: CASYS'07 – Eighth International Conference; Dubois, D., Ed.; American Institute of Physics: Melville, New York, 2008;

[15] Hejna, B. Proposed Correction to Capacity Formula for a Wide-Band Photonic Transfer Channel. In *Proceedings of International Conference Automatics and Informatics'08*; Atanasoff, J., Ed.; Society of Automatics and Informatics: Sofia, Bulgaria, 2008; pp VII-1–VIII-4. [16] Hejna, B. Gibbs Paradox as Property of Observation, Proof of II. Principle of Thermodynamics. In *AIP Conf. Proc.*, Computing Anticipatory Systems: CASYS'09: Ninth International Conference on Computing, Anticipatory Systems, 3–8 August 2009; Dubois, D., Ed.; American Institute of Physics: Melville, New York, 2010; pp 131–140.

[17] Hejna, B. *Informaˇcní termodynamika I.: Rovnovážná termodynamika pˇrenosu informace*;

[18] Hejna, B. *Informaˇcní termodynamika II.: Fyzikální systémy pˇrenosu informace*; VŠCHT Praha:

[19] Hejna, B. Information Thermodynamics, *Thermodynamics - Physical Chemistry of Aqueous Systems*, Juan Carlos Moreno-Pirajaán (Ed.), ISBN: 978-953-307-979-0, InTech, 2011

http://www.intechopen.com/articles/show/title/information-thermodynamics [20] Helstroem, C. W. *Quantum Detection and Estimation Theory*; Pergamon Press: London,

[23] Jaglom, A. M.; Jaglom, I. M. *Pravdˇepodobnost a teorie informace*; Academia: Praha, 1964. [24] Chodaseviˇc, M. A.; Sinitsin, G. V.; Jasjukeviˇc, A. S. *Ideal fermionic communication channel in the number-state model;*Division for Optical Problems in Information technologies, Belarus

[26] Karpuško, F. V.; Chodaseviˇc, M. A. *Sravnitel'noje issledovanije skorostnych charakteristik bozonnych i fermionnych kommunikacionnych kanalov*; Vesšˇc NAN B, The National Academy

[27] Landau, L. D.; Lifschitz, E. M. *Statistical Physics*, 2nd ed.; Pergamon Press: Oxford, 1969. [28] Landauer, M. Irreversibility and Heat Generation in the Computing Process. *IBM J. Res.*

[9] Halmos, P. R. *Koneˇcnomernyje vektornyje prostranstva*; Nauka: Moskva, 1963.

*Journal* 1996, *35* (3/4), 577–586. DOI 10.1147/sj.353.0577.

*Proc.* 1999, *465* (1), 405–418. DOI: 10.1063/1.58272.

pp 67–75. ISBN 978-0-7354-0579-0. ISSN 0094-243X.

VŠCHT Praha: Praha, 2010. ISBN 978-80-7080-747-7.

[21] Horák, Z.; Krupka, F. *Technická fyzika*; SNTL/ALFA: Praha, 1976. [22] Horák, Z.; Krupka, F. *Technická fysika*; SNTL: Praha, 1961.

[25] Kalˇcík, J.; Sýkora, K. *Technická termomechanika*; Academia: Praha, 1973.

[29] Lavenda, B. H. *Statistical Physics*; Wiley: New York, 1991.

ÚTIA AV CR, Praha, FJFI ˇ CVUT, Praha, 2000. ˇ

*Control, Automation, and Systems* 2003, 15.

ISBN 978-0-7354-0858-6. ISSN 0094-243X.

Praha, 2011. ISBN 978-80-7080-774-3.

Available from:

Academy of Sciences, 2000.

of Sciences of Belarus, 1998.

*Dev.* 2000, *44* (1/2), 261.

1976.

## **8. Conclusion**

After each completed 'transmission of an input message and receipt of an output message' ('one-act' transfer) the transferring system must be reverted to its starting state, otherwise the constant (in the sense repeatable) flow of information could not exist. The author believes that either the opening of the chain was presupposed in the original derivation in [30], or that the return of transferring system to its starting state was not considered at all, it was not counted-in. In our derivations this needed state transition is considered be powered within the transfer chain itself, without its openning. Although our derivation of the information capacity for a cyclical case (using the cyclic thermodynamic model) results in a lower value than the original one it seems to be more exact and its result as more precise from the *theoretic point of view*, extending and not ceasing the previous, *original* result [12, 30] which *remains* of its *technology-drawing value*. Also it forces us in being aware and respecting of the *global costs* for (any) communication and its evaluation and, as such, it is of a *gnoseologic character*.

## **Author details**

Bohdan Hejna

*Institute of Chemical Technology Prague, Department of Mathematics, Studentská 6, 166 28 Prague 6, Czech Republic*

## **9. References**


<sup>1</sup> <sup>+</sup> *<sup>β</sup>* <sup>&</sup>gt; 1 is valid.

<sup>25</sup> For one-act information transfer the choose between two information descriptions is possible which result in capacity (138) or (177).

<sup>26</sup> For the energy *<sup>T</sup>*(*X*;*Y*)·*TW* on the output the energy <sup>2</sup> <sup>1</sup> <sup>+</sup> *<sup>β</sup>* ·*TW* <sup>×</sup> *greater* is needed on the input of the transfer channel which is in accordance with the *I I*. *Principle of thermodynamics* for <sup>2</sup>


34 Will-be-set-by-IN-TECH

The mutual difference of results (189) and and (102), (138) [12, 30] is given by **the necessity of**

**after each individual information transfer act has been accomplished and, by the relevant temperatutre reducing of the heat** Δ*Q*<sup>0</sup> [by *TW* in (183)-(189)]. Thus, our thermodynamic cyclical model **<sup>K</sup>**� ∼= O*rrev*� for the repeatible information transfer through the channel

<sup>L</sup>−L|B−<sup>E</sup> is of the information capacity (189), while in [12, 30] the information capacity of the *one-act* information transfer is stated.25 By (189) the *whole energy costs* for the cyclical

After each completed 'transmission of an input message and receipt of an output message' ('one-act' transfer) the transferring system must be reverted to its starting state, otherwise the constant (in the sense repeatable) flow of information could not exist. The author believes that either the opening of the chain was presupposed in the original derivation in [30], or that the return of transferring system to its starting state was not considered at all, it was not counted-in. In our derivations this needed state transition is considered be powered within the transfer chain itself, without its openning. Although our derivation of the information capacity for a cyclical case (using the cyclic thermodynamic model) results in a lower value than the original one it seems to be more exact and its result as more precise from the *theoretic point of view*, extending and not ceasing the previous, *original* result [12, 30] which *remains* of its *technology-drawing value*. Also it forces us in being aware and respecting of the *global costs* for (any) communication and its evaluation and, as such, it is of a *gnoseologic character*.

*Institute of Chemical Technology Prague, Department of Mathematics, Studentská 6, 166 28 Prague 6,*

[3] Cholevo, A. S. On the Capacity of Quantum Communication Channel. *Problems of*

[4] Cholevo, A. S. *Verojatnostnyje i statistiˇceskije aspekty kvantovoj teorii*; Nauka: Moskva, 1980. [5] Cover, T. M.; Thomas, J. B. *Elements of Information Theory*; Wiley: New York, 1991.

[7] Emch, G. G. *Algebraiˇceskije metody v statistiˇceskoj mechanike i kvantovoj tˇeorii polja*; Mir:

<sup>25</sup> For one-act information transfer the choose between two information descriptions is possible which result in capacity

<sup>1</sup> <sup>+</sup> *<sup>β</sup>* ·*TW* <sup>×</sup> *greater* is needed on the input of the transfer channel

<sup>1</sup> <sup>+</sup> *<sup>β</sup>* <sup>&</sup>gt; 1 is valid.

[2] Brillouin, L. *Science and Information Theory*; Academia Press: New York, 1963.

<sup>L</sup>−L|*B*−*<sup>E</sup>* ∼= L **into its initial state**

**the returning the transfer medium, the channel K**� ∼= **K**�

information transfer considered is countable.26

[1] Bell, D. A. *Teorie informace*; SNTL: Praha, 1961.

*Information Transmission* 1979, *15* (4), 3–11.

<sup>26</sup> For the energy *<sup>T</sup>*(*X*;*Y*)·*TW* on the output the energy <sup>2</sup>

[6] Davydov, A. S. *Kvantová mechanika*; SPN: Praha, 1978.

which is in accordance with the *I I*. *Principle of thermodynamics* for <sup>2</sup>

**K**�

**8. Conclusion**

**Author details**

Bohdan Hejna

*Czech Republic*

**9. References**

Moskva, 1976.

(138) or (177).


http://www.intechopen.com/articles/show/title/information-thermodynamics

	- [30] Lebedev, D.; Levitin, L. B. Information Transmission by Electromagnetic Field. *Information and Control* 1966, *9*, 1–22.

**Thermodynamics' Microscopic Connotations**

**Chapter 5**

Thermodynamics is the science of energy conversion. It involves heat and other forms of energy, mechanical one being the foremost one. Potential energy is the capacity of doing work because of the position of something. Kinetic energy is due to movement, depending upon mass and speed. Since all objects have structure, they possess some internal energy that holds such structure together, a kind of strain energy. As for work, there are to kinds of it: internal and external. The later is work done on "something". The former is work effected within something, being a capacity. Heat is another king of energy, the leit-motif of thermodynamics. Thermodynamics studies and interrelates the macroscopic variables, such as temperature, volume, and pressure that are employed to describe thermal systems and

In thermodynamics one is usually interested in special system's states called equilibrium ones. Such states are steady ones reached after a system has stabilized itself to such an extent that it no longer keeps changing with the passage of time, as far as its macroscopic variables are concerned. From a thermodynamics point of view a system is defined by its being prepared in a certain, specific way. The system will always reach, eventually, a unique state of thermodynamic equilibrium, univocally determined by the preparation-manner. Empiric reproducibility is a fundamental requirement for physics in general an thermodynamics in particular. The main source of the strength, or robustness, of thermodynamics, lies on the fact

Historically, thermodynamics developed out the need for increasing the efficiency of early steam engines, particularly through the work of the French physicist Nicolas Sadi-Carnot (1824) who believed that a heat engine's efficiency was to play an important role in helping France win the Napoleonic Wars. Scottish physicist Lord Kelvin was the first to formulate a succint definition of thermodynamics in 1854: "Thermodynamics is the subject of the relation of heat to forces acting between contiguous parts of bodies, and the relation of heat to electrical agency". Chemical thermodynamics studies the role of entropy in the process of chemical reactions and provides the main body of knowledge of the field. Since Boltzmann in the 1870's,

and reproduction in any medium, provided the original work is properly cited.

©2012 Plastino et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

concerns itself with phenomena that can be experimentally reproducible.

does it deals only with phenomena that are experimentally reproducible.

cited.

A. Plastino, Evaldo M. F. Curado and M. Casas

http://dx.doi.org/10.5772/51370

**1. Introduction**

Additional information is available at the end of the chapter


## **Thermodynamics' Microscopic Connotations**

A. Plastino, Evaldo M. F. Curado and M. Casas

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/51370

**1. Introduction**

36 Will-be-set-by-IN-TECH

[30] Lebedev, D.; Levitin, L. B. Information Transmission by Electromagnetic Field.

[35] Shannon, C. E. A Mathematical Theory of Communication. *The Bell Systems Technical*

[36] Sinitsin, G. V.; Chodaseviˇc, M. A.; Jasjukeviˇc, A. S. *Elektronnyj kommunikacionnyj kanal v modeli svobodnogo vyroždennogo fermi-gaza*; Belarus Academy of Sciences, 1999.

[37] Vajda, I. *Teória informácie a štatistického rozhodovania*; Alfa: Bratislava, 1982.

ˇ

*Information and Control* 1966, *9*, 1–22.

*Journal* 1948, *27*, 379–423, 623–656.

[31] Maršák, Z. *Termodynamika a statistická fyzika*; CVUT: Praha, 1995.

[32] Marx, G. *Úvod do kvantové mechaniky*; SNTL: Praha, 1965.

[38] Watanabe, S. *Knowing and Guessing*; Wiley: New York, 1969.

[33] Moore, W. J. *Fyzikální chemie*; SNTL: Praha, 1981. [34] Prchal, J. *Signály a soustavy*; SNTL/ALFA: Praha, 1987.

> Thermodynamics is the science of energy conversion. It involves heat and other forms of energy, mechanical one being the foremost one. Potential energy is the capacity of doing work because of the position of something. Kinetic energy is due to movement, depending upon mass and speed. Since all objects have structure, they possess some internal energy that holds such structure together, a kind of strain energy. As for work, there are to kinds of it: internal and external. The later is work done on "something". The former is work effected within something, being a capacity. Heat is another king of energy, the leit-motif of thermodynamics. Thermodynamics studies and interrelates the macroscopic variables, such as temperature, volume, and pressure that are employed to describe thermal systems and concerns itself with phenomena that can be experimentally reproducible.

> In thermodynamics one is usually interested in special system's states called equilibrium ones. Such states are steady ones reached after a system has stabilized itself to such an extent that it no longer keeps changing with the passage of time, as far as its macroscopic variables are concerned. From a thermodynamics point of view a system is defined by its being prepared in a certain, specific way. The system will always reach, eventually, a unique state of thermodynamic equilibrium, univocally determined by the preparation-manner. Empiric reproducibility is a fundamental requirement for physics in general an thermodynamics in particular. The main source of the strength, or robustness, of thermodynamics, lies on the fact does it deals only with phenomena that are experimentally reproducible.

> Historically, thermodynamics developed out the need for increasing the efficiency of early steam engines, particularly through the work of the French physicist Nicolas Sadi-Carnot (1824) who believed that a heat engine's efficiency was to play an important role in helping France win the Napoleonic Wars. Scottish physicist Lord Kelvin was the first to formulate a succint definition of thermodynamics in 1854: "Thermodynamics is the subject of the relation of heat to forces acting between contiguous parts of bodies, and the relation of heat to electrical agency". Chemical thermodynamics studies the role of entropy in the process of chemical reactions and provides the main body of knowledge of the field. Since Boltzmann in the 1870's,

©2012 Plastino et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

statistical thermodynamics, or statistical mechanics, that are microscopic theories, began to explain macroscopic thermodynamics via statistical predictions on the collective motion of atoms.

content. The derivative of the function *f* becomes the argument to the function *fT*.

*I*(*A*1,..., *AM*) = *α* +

*α*(*λ*1,..., *λM*) = *I* −

*∂I <sup>∂</sup>*�*Ak*� <sup>=</sup> *<sup>λ</sup><sup>k</sup>* ;

The Legendre transform its own inverse. It is used to get from Lagrangians the Hamiltonian

Legendre' reciprocity relations constitute thermodynamics' essential formal ingredient [2]. In

with the *Ai* extensive variables and the *λ<sup>i</sup>* intensive ones. Obviously, the Legendre transform main goal is that of changing the identity of our relevant independent variables. For *α* we

Thermodynamics can be regarded as a formal logical structure whose *axioms* are empirical facts [2], which gives it a unique status among the scientific disciplines [1]. The four axioms given below are equivalent to the celebrated laws of thermodynamics of the prevous

• For every system there exists a quantity *E*, the internal energy, such that a unique *Es*−value is associated to each and every state *s*. The difference *Es*<sup>1</sup> − *Es*<sup>2</sup> for two different states *s*<sup>1</sup> and *s*<sup>2</sup> in a closed system is equal to the work required to bring the system, while

• There exist particular states of a system, the equilibrium ones, that are uniquely determined by *E* and by a set of extensive macroscopic parameters *A<sup>ξ</sup>* , *ξ* = 1, . . . , *M*.

• For every system there exists a state function *S*(*E*, ∀*A<sup>ξ</sup>* ) that (i) always grows if internal constraints are removed and (ii) is a monotonously (growing) function of *E*. *S* remains

The number and characteristics of the *A<sup>ξ</sup>* depends on the nature of the system.

*M* ∑ *k*=1

*M* ∑ *k*=1

> *∂I ∂λ<sup>i</sup>* = *M* ∑ *k λk ∂*�*Ak*� *∂λ<sup>i</sup>*

(*x*) ⇒ reciprocity. (1)

Thermodynamics' Microscopic Connotations 121

*λkAk*, (2)

*λ<sup>k</sup>* �*Ak*�. (3)

, (4)

*fT*(*y*) = *xy* − *f*(*x*); *y* = *f* �

formulation of classical mechanics.

have

Subsection [2].

general, for two functions *I* and *α* one has

The three operative reciprocity relations become [2]

= −�*Ak*� ;

adiabatically enclosed, from one state to the other.

constant in quasi-static adiabatic changes.

the last one being the so-called Euler theorem.

**1.3. The axioms of thermodynamics**

*∂α ∂λ<sup>k</sup>*

## **1.1. Thermodynamics' laws**

The laws of physics are established scientific regularities regarded as universal and invariable facts of the universe. A "law" differs from hypotheses, theories, postulates, principles, etc., in that it constitutes an analytic statement. A theory starts from a set of axioms from which all laws and phenomena should arise via adequate mathematical treatment. The principles of thermodynamics, often called "its laws", count themselves amongst the most fundamental regularities of Nature [1]. These laws define fundamental physical quantities, such as temperature, energy, and entropy, to describe thermodynamic systems and they account for the transfer of energy as heat and work in thermodynamic processes. An empirically reproducible distinction between heat and work constitutes the "hard-core" of thermodynamics. For processes in which this distinction cannot be made, thermodynamics remains silent. One speaks of four thermodynamics' laws:


Classical thermodynamics accounts for the exchange of work and heat between systems with emphasis in states of thermodynamic equilibrium. Thermal equilibrium is a condition *sine qua non* for *macroscopically specified systems* only. It shoul be noted that, at the microscopic (atomic) level all physical systems undergo random fluctuations. Every finite system will exhibit statistical fluctuations in its thermodynamic variables of state (entropy, temperature, pressure, etc.), but these are negligible for macroscopically specified systems. Fluctuations become important for microscopically specified systems. Exceptionally, for macroscopically specified systems found at critical states, fluctuations are of the essence.

## **1.2. The Legendre transform**

The Legendre transform is an operation that transforms one real-valued function of *f* a real variable *x* into another *fT*, of a different variable *y*, maintaining constant its information content. The derivative of the function *f* becomes the argument to the function *fT*.

$$f\_{\mathsf{T}}(y) = xy - f(\mathsf{x});\ y = f'(\mathsf{x}) \Rightarrow \text{ reciprocity.} \tag{1}$$

The Legendre transform its own inverse. It is used to get from Lagrangians the Hamiltonian formulation of classical mechanics.

Legendre' reciprocity relations constitute thermodynamics' essential formal ingredient [2]. In general, for two functions *I* and *α* one has

$$I(A\_1, \ldots, A\_M) = \alpha + \sum\_{k=1}^{M} \lambda\_k A\_{k\prime} \tag{2}$$

with the *Ai* extensive variables and the *λ<sup>i</sup>* intensive ones. Obviously, the Legendre transform main goal is that of changing the identity of our relevant independent variables. For *α* we have

$$\left(\alpha(\lambda\_1, \ldots, \lambda\_M)\right) = I - \sum\_{k=1}^{M} \lambda\_k \left< A\_k \right> . \tag{3}$$

The three operative reciprocity relations become [2]

$$\frac{\partial \mathfrak{a}}{\partial \lambda\_k} = - \langle A\_k \rangle \; ; \qquad \frac{\partial I}{\partial \langle A\_k \rangle} = \lambda\_k \; ; \qquad \frac{\partial I}{\partial \lambda\_i} = \sum\_k^M \lambda\_k \frac{\partial \langle A\_k \rangle}{\partial \lambda\_i} \, \tag{4}$$

the last one being the so-called Euler theorem.

#### **1.3. The axioms of thermodynamics**

2 Will-be-set-by-IN-TECH

statistical thermodynamics, or statistical mechanics, that are microscopic theories, began to explain macroscopic thermodynamics via statistical predictions on the collective motion of

The laws of physics are established scientific regularities regarded as universal and invariable facts of the universe. A "law" differs from hypotheses, theories, postulates, principles, etc., in that it constitutes an analytic statement. A theory starts from a set of axioms from which all laws and phenomena should arise via adequate mathematical treatment. The principles of thermodynamics, often called "its laws", count themselves amongst the most fundamental regularities of Nature [1]. These laws define fundamental physical quantities, such as temperature, energy, and entropy, to describe thermodynamic systems and they account for the transfer of energy as heat and work in thermodynamic processes. An empirically reproducible distinction between heat and work constitutes the "hard-core" of thermodynamics. For processes in which this distinction cannot be made, thermodynamics

• The zeroth law of thermodynamics allows for the assignment of a unique temperature to

• The first law postulates the existence of a quantity called the internal energy of a system and shows how it is related to the distinction between energy transfer as work and energy transfer as heat. The internal energy is conserved but work and heat are not defined as separately conserved quantities. Alternatively, one can reformulate the first law as stating

• The second law of thermodynamics expresses the existence of a quantity called the entropy *S* and states that for an isolated macroscopic system *S* never decreases, or, alternatively,

• The third law of thermodynamics refers to the entropy of a system at absolute zero temperature (*T* = 0) and states that it is impossible to lower *T* in such a manner that

Classical thermodynamics accounts for the exchange of work and heat between systems with emphasis in states of thermodynamic equilibrium. Thermal equilibrium is a condition *sine qua non* for *macroscopically specified systems* only. It shoul be noted that, at the microscopic (atomic) level all physical systems undergo random fluctuations. Every finite system will exhibit statistical fluctuations in its thermodynamic variables of state (entropy, temperature, pressure, etc.), but these are negligible for macroscopically specified systems. Fluctuations become important for microscopically specified systems. Exceptionally, for macroscopically

The Legendre transform is an operation that transforms one real-valued function of *f* a real variable *x* into another *fT*, of a different variable *y*, maintaining constant its information

atoms.

**1.1. Thermodynamics' laws**

reaches the limit *T* = 0.

**1.2. The Legendre transform**

remains silent. One speaks of four thermodynamics' laws:

systems that are in thermal equilibrium with each other.

that perpetual motion machines of the first kind can not exist.

that perpetual motion machines of the second kind are impossible.

specified systems found at critical states, fluctuations are of the essence.

Thermodynamics can be regarded as a formal logical structure whose *axioms* are empirical facts [2], which gives it a unique status among the scientific disciplines [1]. The four axioms given below are equivalent to the celebrated laws of thermodynamics of the prevous Subsection [2].

	- *S* and the temperature *T* = [ *<sup>∂</sup><sup>E</sup> <sup>∂</sup><sup>S</sup>* ]*A*1,...,*AM* vanish for the state of minimum energy and are non-negative for all other states.

From the second and 3rd. Postulates one extracts the following two essential assertions

1. **Statement 3a)** for every system there exists a state function *S*, a function of *E* and the *A<sup>ξ</sup>*

$$S = S(E, A\_1, \dots, A\_M). \tag{5}$$

system's variables, called the *reciprocity relations* (RR), that are crucial for the microscopic

In 1903 Gibbs formulated the first axiomatic theory for statistical mechanics [1, 3], revolving around the concept of phase space. The phase space (PS) precise location is given by generalized coordinates and momenta. Gibbs' postulates properties of an imaginary (Platonic) ad-hoc notion: the "ensemble" (a mental picture). The ensemble consists of extremely many (*N*) independent systems, all identical in nature with the one of actual physical interest, but differing in PS-location. That is, the original system is to be mentally repeated many times, each with a different arrangement of generalized coordinates and momenta. Here Liouville's theorem of volume conservation in phase space for Hamiltonian motion plays a crucial role. The ensemble amounts to a distribution of *N* PS-points, representative of the actual system. *N* is large enough that one can properly speak of a density *D* at any PS-point *φ* = *q*1,..., *qN*; *p*1,..., *pN*, with *D* = *D*(*q*1,..., *qN*; *p*1,..., *pN*, *t*) ≡ *D*(*φ*), with *t* the time, and,

> *N* =

*N*<sup>12</sup> =

*∂D ∂pi*

*D*˙ + *N* ∑ *i*

> *N* ∑ *i*

*∂D ∂pi* *p*˙*<sup>i</sup>* + *N* ∑ *i*

*dN*<sup>12</sup>

Randomly extracting a system from the ensemble, the probability of selecting it being located

Liouville's theorem follows from the fact that, since phase-space points can not be

 *φ*<sup>2</sup> *φ*1

An appropriate analytical manipulation involving Hamilton's canonical equations of motion

*p*˙*<sup>i</sup>* + *N* ∑ *i*

*∂D ∂qi*

*∂D ∂qi*

*dφ D*; ∀t. (11)

Thermodynamics' Microscopic Connotations 123

*P*(*φ*) = *D*(*φ*)/*N*. (12)

*P dφ* = 1. (13)

*dt* <sup>=</sup> 0. (15)

*D dφ*, (14)

*q*˙*<sup>i</sup>* = 0, (16)

*q*˙*<sup>i</sup>* = 0. (17)

discussion of Thermodynamics.

if we call *dφ* the volume element,

in a neighborhood of *φ* would yield

then yields the theorem in the form [1]

entailing the PS-conservation of density. **Equilibrium** means simply *D*˙ = 0, i. e.,

Consequently,

"destroyed", if

then

**2. Classical statistical mechanics**

2. **Statement 3b)** *S* is a monotonous (growing) function of *E*, so that one can interchange the roles of *E* and *S* in (5) and write

$$E = E(S, A\_1, \dots, A\_M)\_\prime \tag{6}$$

Eq. (6) clearly indicates that

$$dE = \frac{\partial E}{\partial S}dS + \sum\_{\tilde{\xi}} \frac{\partial E}{\partial A\_{\tilde{\xi}}} dA\_{\tilde{\xi}} \Rightarrow dE = TdS + \sum\_{\tilde{\xi}} P\_{\tilde{\xi}} dA\_{\tilde{\xi}\prime} \tag{7}$$

with *P<sup>ξ</sup>* generalized pressures and the temperature *T* defined as [2]

$$T = \left(\frac{\partial E}{\partial \mathcal{S}}\right)\_{\left[\mathbb{V}\,A\_{\mathcal{S}}\right]}.\tag{8}$$

Eq. (7) will play a key-role in our future considerations. If we know *S*(*E*, *A*1,..., *An*) or, equivalently because of monotonicity, *E*(*S*, *A*1,..., *An*)) we have a *complete* thermodynamic description of a system [2]. For experimentalists, it is often more convenient to work with *intensive* variables defined as follows [2].

Let *S* ≡ *A*0. The intensive variable associated to the extensive *Ai*, to be called *Pi* are the derivatives

$$P\_0 \equiv T = [\frac{\partial E}{\partial S}]\_{A\_1, \dots, A\_{n'}} \ 1/T = \beta. \tag{9}$$

$$P\_{\rangle} \equiv \lambda\_{\rangle} / T = [\frac{\partial E}{\partial A\_{\rangle}}]\_{S\_{\prime}A\_{1}, \dots, A\_{\slash -1}, A\_{\slash +1}, \dots, A\_{\hbar}}.\tag{10}$$

Any one of the Legendre transforms that replaces any *s* extensive variables by their associated intensive ones (*β*, *λ*'s will be Lagrange multipliers in SM)

$$L\_{r\_1,\ldots,r\_s} = E - \sum\_{j} P\_j \, A\_{j\prime} \quad (j = r\_{1\prime}, \ldots, r\_s)$$

contains the same information as either *S* or *E*. The transform *Lr*1,...,*rs* is a function of *n* − *s* extensive and *s* intensive variables. This is called the *Legendre invariant structure of thermodynamics*. As we saw above, this implies certain relationships amongst the relevant system's variables, called the *reciprocity relations* (RR), that are crucial for the microscopic discussion of Thermodynamics.

### **2. Classical statistical mechanics**

In 1903 Gibbs formulated the first axiomatic theory for statistical mechanics [1, 3], revolving around the concept of phase space. The phase space (PS) precise location is given by generalized coordinates and momenta. Gibbs' postulates properties of an imaginary (Platonic) ad-hoc notion: the "ensemble" (a mental picture). The ensemble consists of extremely many (*N*) independent systems, all identical in nature with the one of actual physical interest, but differing in PS-location. That is, the original system is to be mentally repeated many times, each with a different arrangement of generalized coordinates and momenta. Here Liouville's theorem of volume conservation in phase space for Hamiltonian motion plays a crucial role. The ensemble amounts to a distribution of *N* PS-points, representative of the actual system. *N* is large enough that one can properly speak of a density *D* at any PS-point *φ* = *q*1,..., *qN*; *p*1,..., *pN*, with *D* = *D*(*q*1,..., *qN*; *p*1,..., *pN*, *t*) ≡ *D*(*φ*), with *t* the time, and, if we call *dφ* the volume element,

$$N = \int d\phi \, D; \quad \forall \mathbf{t}. \tag{11}$$

Randomly extracting a system from the ensemble, the probability of selecting it being located in a neighborhood of *φ* would yield

$$P(\phi) = D(\phi) / N. \tag{12}$$

Consequently,

$$\int P \, d\phi = 1.\tag{13}$$

Liouville's theorem follows from the fact that, since phase-space points can not be "destroyed", if

$$N\_{12} = \int\_{\phi\_1}^{\phi\_2} D \, d\phi\_{\prime} \tag{14}$$

then

4 Will-be-set-by-IN-TECH

From the second and 3rd. Postulates one extracts the following two essential assertions

1. **Statement 3a)** for every system there exists a state function *S*, a function of *E* and the *A<sup>ξ</sup>*

2. **Statement 3b)** *S* is a monotonous (growing) function of *E*, so that one can interchange the

*dA<sup>ξ</sup>* ⇒ *dE* = *TdS* + ∑

[∀ *A<sup>ξ</sup>* ]

*<sup>∂</sup><sup>S</sup>* ]*A*1,...,*AM* vanish for the state of minimum energy and are

*S* = *S*(*E*, *A*1,..., *AM*). (5)

*E* = *E*(*S*, *A*1,..., *AM*), (6)

*ξ*

*<sup>∂</sup><sup>S</sup>* ]*A*1,...,*An* , 1/*<sup>T</sup>* <sup>=</sup> *<sup>β</sup>*. (9)

]*S*,*A*1,...,*Aj*−1,*Aj*+1,...,*An* . (10)

*PξdA<sup>ξ</sup>* , (7)

. (8)

• *S* and the temperature *T* = [ *<sup>∂</sup><sup>E</sup>*

non-negative for all other states.

roles of *E* and *S* in (5) and write

*dE* <sup>=</sup> *<sup>∂</sup><sup>E</sup>*

*intensive* variables defined as follows [2].

derivatives

*<sup>∂</sup><sup>S</sup> dS* <sup>+</sup> ∑ *ξ*

with *P<sup>ξ</sup>* generalized pressures and the temperature *T* defined as [2]

*∂E ∂A<sup>ξ</sup>*

*T* =

*<sup>P</sup>*<sup>0</sup> <sup>≡</sup> *<sup>T</sup>* = [ *<sup>∂</sup><sup>E</sup>*

*Pj* <sup>≡</sup> *<sup>λ</sup>j*/*<sup>T</sup>* = [ *<sup>∂</sup><sup>E</sup>*

*Lr*1,...,*rs* = *<sup>E</sup>* − ∑

intensive ones (*β*, *λ*'s will be Lagrange multipliers in SM)

 *∂E ∂S* 

Eq. (7) will play a key-role in our future considerations. If we know *S*(*E*, *A*1,..., *An*) or, equivalently because of monotonicity, *E*(*S*, *A*1,..., *An*)) we have a *complete* thermodynamic description of a system [2]. For experimentalists, it is often more convenient to work with

Let *S* ≡ *A*0. The intensive variable associated to the extensive *Ai*, to be called *Pi* are the

*∂Aj*

*j*

Any one of the Legendre transforms that replaces any *s* extensive variables by their associated

contains the same information as either *S* or *E*. The transform *Lr*1,...,*rs* is a function of *n* − *s* extensive and *s* intensive variables. This is called the *Legendre invariant structure of thermodynamics*. As we saw above, this implies certain relationships amongst the relevant

*Pj Aj*, (*j* = *r*1,...,*rs*)

Eq. (6) clearly indicates that

$$\frac{dN\_{12}}{dt} = 0.\tag{15}$$

An appropriate analytical manipulation involving Hamilton's canonical equations of motion then yields the theorem in the form [1]

$$\dot{D} + \sum\_{i}^{N} \frac{\partial D}{\partial p\_{i}} \dot{p}\_{i} + \sum\_{i}^{N} \frac{\partial D}{\partial q\_{i}} \dot{q}\_{i} = 0,\tag{16}$$

entailing the PS-conservation of density.

**Equilibrium** means simply *D*˙ = 0, i. e.,

$$
\sum\_{i}^{N} \frac{\partial D}{\partial p\_i} \not{p}\_i + \sum\_{i}^{N} \frac{\partial D}{\partial q\_i} \not{q}\_i = 0. \tag{17}
$$

6 Will-be-set-by-IN-TECH 124 Thermodynamics – Fundamentals and Its Application in Science Thermodynamics' Microscopic Connotations <sup>7</sup>

## **2.1. The classical axioms**

Gibbs refers to PS-location as the "phase" of the system [1, 3]. The following statements completely explain in microscopic fashion the corpus of classical equilibrium thermodynamics [1].

*<sup>I</sup>*(*p*) = *<sup>I</sup>*(*p*1) + ∑

*S* = −

information terms. Since IT's central concept is that of information measure (IM)

gives us the only way of complying with Kinchin's axioms.

**4. Statistical mechanics and information theory**

measure

[1, 12–15].

be considered [6]:

*i*

An important consequence is that, out of the four Kinchin axioms one finds that Shannons's

It has been argued [9] that the statistical mechanics (SM) of Gibbs is a juxtaposition of subjective, probabilistic ideas on the one hand and objective, mechanical ideas on the other. From the mechanical viewpoint, the vocables "statistical mechanics" suggest that for solving physical problems we ought to acknowledge a degree of uncertainty as to the experimental conditions. Turning this problem around, it also appears that the purely statistical arguments are incapable of yielding any physical insight unless some mechanical information is a priori assumed [9]. This is the conceptual origin of the link SM-IT pioneered by Jaynes in 1957 via his Maximum Entropy Principle (MaxEnt) [5, 6, 10] which allowed for reformulating SM in

Descartes' scientific methodology considers that truth is established via the agreement between two *independent* instances that can neither suborn nor bribe each other: analysis (purely mental) and experiment [11]. The analytic part invokes mathematical tools and concepts: Mathematics' world ⇔ Laboratory. The mathematical realm is called Plato's Topos Uranus (TP). Science in general, and physics in particular, may thus be seen as a [TP ⇔ "Experiment"] two-way bridge. TP concepts are related to each other in the form of "laws" that adequately describe the relationships obtaining among suitable chosen variables that describe the phenomenon at hand. In many cases these laws are integrated into a comprehensive theory (e.g., classical electromagnetism, based upon Maxwell's equations)

Jaynes' MaxEnt ideas describe thermodynamics via the link [IT as a part of TP]⇔ [Thermal experiment], or in a more general scenario: [IT] ⇔ [Phenomenon at hand]. It is clear that the relation between an information measure and entropy is [IM] ⇔ [Entropy *S*]. One can then assert that an IM is not necessarily an entropy, since the first belongs to the Topos Uranus and the later to the laboratory. Of course, in some special cases an association *IM* ⇔ entropy *S* can be established. Such association is both useful and proper in very many situations [5].

If, in a given scenario, *N* distinct outcomes (*i* = 1, . . . , *N*) are possible, three alternatives are to

2. Maximum ignorance: Nothing can be said in advance. The *N* outcomes are equally likely.

1. Zero ignorance: predict with certainty the actual outcome.

*N* ∑ *i*=1 *p*1(*i*) *I Q*(*j*|*i*) 

. (19)

Thermodynamics' Microscopic Connotations 125

*p*(*i*)ln [*p*(*i*)], (20)


## **3. Information**

Information theory (IT) treats information as data communication, with the primary goal of concocting efficient manners of encoding and transferring data. IT is a branch of applied mathematics and electrical engineering, involving the quantification of information, developed by Claude E. Shannon [4] in order to i) find fundamental limits on signal processing operations such as compressing data and ii) finding ways of reliably storing and communicating data. Since its 1948-inception it has considerably enlarged its scope and found applications in many areas that include statistical inference, natural language processing, cryptography, and networks other than communication networks. A key information-measure (IM) was originally called (by Shannon) entropy, in principle unrelated to thermodynamic entropy. It is usually expressed by the average number of bits needed to store or communicate one symbol in a message and quantifies the uncertainty involved in predicting the value of a random variable.Thus, a degree of knowledge (or ignorance) is associated to any normalized probability distribution *p*(*i*), (*i* = 1, . . . , *N*), determined by a functional *I*[{*pi*}] of the {*pi*} [4–7] which is precisely Shannon's entropy. IT was la axiomatized in 1950 by Kinchin [8], on the basis of four axioms, namely,


As for the last axiom, consider two sub-systems [Σ1, {*p*1(*i*)}] and [Σ2, {*p*2(*j*)}] of a composite system [Σ, {*p*(*i*, *<sup>j</sup>*)}] with *<sup>p</sup>*(*i*, *<sup>j</sup>*) = *<sup>p</sup>*1(*i*) *<sup>p</sup>*2(*j*). Assume further that the conditional probability distribution (PD) *Q*(*j*|*i*) of realizing the event *j* in system 2 for a fixed *i*−event in system 1. To this PD one associates the information measure *I*[*Q*]. Clearly,

$$p(i,j) = p^1(i) \, Q(j|i) . \tag{18}$$

Then Kinchin's fourth axiom states that

$$I(p) = I(p^1) + \sum\_{i} p^1(i) \, I(Q(j|i)). \tag{19}$$

An important consequence is that, out of the four Kinchin axioms one finds that Shannons's measure

$$S = -\sum\_{i=1}^{N} p(i) \ln \left[ p(i) \right],\tag{20}$$

gives us the only way of complying with Kinchin's axioms.

6 Will-be-set-by-IN-TECH

Gibbs refers to PS-location as the "phase" of the system [1, 3]. The following statements completely explain in microscopic fashion the corpus of classical equilibrium

• The probability that at time *t* the system will be found in the dynamical state characterized by *φ* equals the probability *P*(*φ*) that a system randomly selected from the ensemble shall

• The time-average of a dynamical quantity *F* equals its average over the ensemble,

Information theory (IT) treats information as data communication, with the primary goal of concocting efficient manners of encoding and transferring data. IT is a branch of applied mathematics and electrical engineering, involving the quantification of information, developed by Claude E. Shannon [4] in order to i) find fundamental limits on signal processing operations such as compressing data and ii) finding ways of reliably storing and communicating data. Since its 1948-inception it has considerably enlarged its scope and found applications in many areas that include statistical inference, natural language processing, cryptography, and networks other than communication networks. A key information-measure (IM) was originally called (by Shannon) entropy, in principle unrelated to thermodynamic entropy. It is usually expressed by the average number of bits needed to store or communicate one symbol in a message and quantifies the uncertainty involved in predicting the value of a random variable.Thus, a degree of knowledge (or ignorance) is associated to any normalized probability distribution *p*(*i*), (*i* = 1, . . . , *N*), determined by a functional *I*[{*pi*}] of the {*pi*} [4–7] which is precisely Shannon's entropy. IT was la

As for the last axiom, consider two sub-systems [Σ1, {*p*1(*i*)}] and [Σ2, {*p*2(*j*)}] of a composite system [Σ, {*p*(*i*, *<sup>j</sup>*)}] with *<sup>p</sup>*(*i*, *<sup>j</sup>*) = *<sup>p</sup>*1(*i*) *<sup>p</sup>*2(*j*). Assume further that the conditional probability distribution (PD) *Q*(*j*|*i*) of realizing the event *j* in system 2 for a fixed *i*−event

*<sup>p</sup>*(*i*, *<sup>j</sup>*) = *<sup>p</sup>*1(*i*) *<sup>Q</sup>*(*j*|*i*). (18)

possess the phase *φ* will be given by Eq. (12) above.

• *D* depends only upon the system's Hamiltonian.

• All phase-space neighborhoods (cells) have the same a priori probability.

axiomatized in 1950 by Kinchin [8], on the basis of four axioms, namely,

• *I* is an absolute maximum for the uniform probability distribution, • *I* is not modified if an *N* + 1 event of probability zero is added,

in system 1. To this PD one associates the information measure *I*[*Q*]. Clearly,

**2.1. The classical axioms**

thermodynamics [1].

evaluated using *D*.

• *I* is a function ONLY of the *p*(*i*),

Then Kinchin's fourth axiom states that

• Composition law.

**3. Information**

#### **4. Statistical mechanics and information theory**

It has been argued [9] that the statistical mechanics (SM) of Gibbs is a juxtaposition of subjective, probabilistic ideas on the one hand and objective, mechanical ideas on the other. From the mechanical viewpoint, the vocables "statistical mechanics" suggest that for solving physical problems we ought to acknowledge a degree of uncertainty as to the experimental conditions. Turning this problem around, it also appears that the purely statistical arguments are incapable of yielding any physical insight unless some mechanical information is a priori assumed [9]. This is the conceptual origin of the link SM-IT pioneered by Jaynes in 1957 via his Maximum Entropy Principle (MaxEnt) [5, 6, 10] which allowed for reformulating SM in information terms. Since IT's central concept is that of information measure (IM)

Descartes' scientific methodology considers that truth is established via the agreement between two *independent* instances that can neither suborn nor bribe each other: analysis (purely mental) and experiment [11]. The analytic part invokes mathematical tools and concepts: Mathematics' world ⇔ Laboratory. The mathematical realm is called Plato's Topos Uranus (TP). Science in general, and physics in particular, may thus be seen as a [TP ⇔ "Experiment"] two-way bridge. TP concepts are related to each other in the form of "laws" that adequately describe the relationships obtaining among suitable chosen variables that describe the phenomenon at hand. In many cases these laws are integrated into a comprehensive theory (e.g., classical electromagnetism, based upon Maxwell's equations) [1, 12–15].

Jaynes' MaxEnt ideas describe thermodynamics via the link [IT as a part of TP]⇔ [Thermal experiment], or in a more general scenario: [IT] ⇔ [Phenomenon at hand]. It is clear that the relation between an information measure and entropy is [IM] ⇔ [Entropy *S*]. One can then assert that an IM is not necessarily an entropy, since the first belongs to the Topos Uranus and the later to the laboratory. Of course, in some special cases an association *IM* ⇔ entropy *S* can be established. Such association is both useful and proper in very many situations [5].

If, in a given scenario, *N* distinct outcomes (*i* = 1, . . . , *N*) are possible, three alternatives are to be considered [6]:


#### 8 Will-be-set-by-IN-TECH 126 Thermodynamics – Fundamentals and Its Application in Science Thermodynamics' Microscopic Connotations <sup>9</sup>

3. Partial ignorance: we are given the probability distribution {*Pi*}; *i* = 1, . . . , *N*.

If our state of knowledge is appropriately represented by a set of, say, *M* expectation values, then the "best", least unbiased probability distribution is the one that [6]

**Axiom (2)** If there are *W* microscopic accessible states labelled by *i*, whose microscopic

Thus, we are actually taking as a postulate something that is actually known from both

**Axiom (3)** The internal energy *E* and the external parameters *A<sup>ν</sup>* are to be considered as the expectation values of suitable operators, that is, the hamiltonian *H* and the hermitian operators R*<sup>ν</sup>* (i.e., *A<sup>ν</sup>* ≡< R*<sup>ν</sup>* >). Thus, the *A<sup>ν</sup>* (and also *E*) will depend on the eigenvalues of these operators *and* on the probability set. (Note that energy eigenvalues depend of course

The reader will immediately realize that Axiom (2) is just a way of re-expressing Boltzmann's "atomic" conjecture. Thus, macroscopic quantities become statistical averages evaluated using a microscopic probability distribution [16]. Our present three new axioms are statements of fact. What do we mean? That they are borrowed from either experiment or pre-existent theories. Somewhat surprisingly, our three axioms do not actually incorporate any knew knowledge at all. The merely re-express known previous notions. Ockham's razor at its best!

We need now to prove that the above three postulates allow one to reconstruct the imposing edifice of statistical mechanics. We will tackle this issue by showing below that they our axioms are equivalent to those of Jaynes' [17]. At this point we need to recall the main goal of statistical mechanics, namely, finding the probability distribution (or the density operator) that best describes our physical system. In order to do so Jaynes appealed to his MaxEnt

*MaxEnt axiom*: assume your prior knowledge about the system is given by the values of M

Then, *ρ* is uniquely fixed by extremizing the information measure *I*(*ρ*) subject to *ρ*−normalization plus the constraints given by the *M* conditions constituting our assumed

This leads, after a Lagrange-constrained extremizing process, to the introduction of *M* Lagrange multipliers *λν*, that one assimilates to the generalized pressures *Pν*. The truth, the whole truth, nothing but the truth [6]. Jaynes rationale asserts that if the entropic measure that reflects our ignorance were not of maximal character, we would actually be *inventing*

While working through his variational process, Jaynes discovers that, after multiplying by Boltzmann's constant *kB* the right-hand-side of his expression for the information measure, it converts itself into an entropy, *I* ≡ *S*, the equilibrium thermodynamic one, with the caveat that *A*<sup>1</sup> =< R<sup>1</sup> >,..., *AM* =< R*<sup>M</sup>* > refer to extensive quantities. Having *ρ*, his universal form *I*(*ρ*) *yields complete microscopic information with respect to the system of interest*. To achieve our ends one needs now just to prove that the new axiomatics, with (21) and (22), is equivalent

*A*<sup>1</sup> ≡< R<sup>1</sup> >,..., *AR* ≡< R*<sup>M</sup>* > . (23)

*A<sup>ν</sup>* =< R*<sup>ν</sup>* >= *Tr*[*ρ* R*ν*]. (24)

*S* = *S*(*p*1, *p*2,..., *pW*). (22)

Thermodynamics' Microscopic Connotations 127

probability we call *pi*, then

upon the R*ν*.

expectation values

foreknowledge

to MaxEnt.

information not at hand.

quantum and classical mechanics.

Our theory could no be more economical.

postulate, that we restate below for the sake of fixing notation.


Such is the MaxEnt rationale. In using MaxEnt, one is not maximizing a physical entropy, but only maximizing ignorance in order to obtain the least biased distribution compatible with the a priori knowledge.

Statistical mechanics and thereby thermodynamics can be formulated on an information theory basis if the density operator *ρ*ˆ is obtained by appealing to Jaynes' maximum entropy principle (MaxEnt), that can be stated as follows:

Assume that your prior knowledge about the system is given by the values of *M* expectation values < *A*<sup>1</sup> >,..., < *AM* >. In such circumstances *ρ*ˆ is uniquely determined by extremizing *I*(*ρ*ˆ) subject to *M* constraints given, namely, the *M* conditions < *Aj* >= *Tr*[*ρ*ˆ *A*ˆ*j*], a procedure that entails introducing *M* Lagrange multipliers *λi*. Additionally, since normalization of *ρ*ˆ is necessary, a normalization Lagrange multiplier *ξ* should be invoked. The procedure immediately leads one [6] to realizing that *I* ≡ *S*, the equilibrium Boltzmann's entropy, if the a priori knowledge < *A*<sup>1</sup> >,..., < *AM* > refers only to extensive quantities. Of course, *I*, once determined, *affords for complete thermodynamical information for the system of interest* [6].

## **5. A new micro-macroscopic way of accounting for thermodynamics**

Gibbs' and MaxEnt approaches satisfactorily describe equilibrium thermodynamics. We will here search for a new, different alternative able to account for thermodynamics from first principles. Our idea is to give axiom-status to Eq. (7), *which is an empirical statement*. Why? Because neither in Gibbs' nor in MaxEnt's axioms we encounter a direct connection with actual thermal data. By appealing to Eq. (7) we would instead be actually employing empirical information. This is our rationale.

Consequently, we will concoct a new SM-axiomatics by giving postulate status to the following macroscopic statement:

#### **Axiom (1)**

$$dE = TdS + \sum\_{\nu} P\_{\nu} dA\_{\nu}.\tag{21}$$

This is a macroscopic postulate to be inserted into a microscopic axiomatics' corpus.

We still need *some* amount of microscopic information, since we are building up a microscopic theory. We wish to add as little as possible, of course (Ockham's razor). At this point it is useful to remind the reader of Kinchin's postulates, recounted above. We will content ourselves with borrowing for our theoretical concerns just his his first axiom. Thus, we conjecture at this point, and will prove below, that the following assertion suffices for our theoretical purposes: **Axiom (2)** If there are *W* microscopic accessible states labelled by *i*, whose microscopic probability we call *pi*, then

8 Will-be-set-by-IN-TECH

If our state of knowledge is appropriately represented by a set of, say, *M* expectation values,

• reflects just what we know, without "inventing" unavailable pieces of knowledge [5, 6]

Such is the MaxEnt rationale. In using MaxEnt, one is not maximizing a physical entropy, but only maximizing ignorance in order to obtain the least biased distribution compatible with

Statistical mechanics and thereby thermodynamics can be formulated on an information theory basis if the density operator *ρ*ˆ is obtained by appealing to Jaynes' maximum entropy

Assume that your prior knowledge about the system is given by the values of *M* expectation values < *A*<sup>1</sup> >,..., < *AM* >. In such circumstances *ρ*ˆ is uniquely determined by extremizing *I*(*ρ*ˆ) subject to *M* constraints given, namely, the *M* conditions < *Aj* >= *Tr*[*ρ*ˆ *A*ˆ*j*], a procedure that entails introducing *M* Lagrange multipliers *λi*. Additionally, since normalization of *ρ*ˆ is necessary, a normalization Lagrange multiplier *ξ* should be invoked. The procedure immediately leads one [6] to realizing that *I* ≡ *S*, the equilibrium Boltzmann's entropy, if the a priori knowledge < *A*<sup>1</sup> >,..., < *AM* > refers only to extensive quantities. Of course, *I*, once determined, *affords for complete thermodynamical information for the system of interest* [6].

**5. A new micro-macroscopic way of accounting for thermodynamics**

Gibbs' and MaxEnt approaches satisfactorily describe equilibrium thermodynamics. We will here search for a new, different alternative able to account for thermodynamics from first principles. Our idea is to give axiom-status to Eq. (7), *which is an empirical statement*. Why? Because neither in Gibbs' nor in MaxEnt's axioms we encounter a direct connection with actual thermal data. By appealing to Eq. (7) we would instead be actually employing

Consequently, we will concoct a new SM-axiomatics by giving postulate status to the

We still need *some* amount of microscopic information, since we are building up a microscopic theory. We wish to add as little as possible, of course (Ockham's razor). At this point it is useful to remind the reader of Kinchin's postulates, recounted above. We will content ourselves with borrowing for our theoretical concerns just his his first axiom. Thus, we conjecture at this point, and will prove below, that the following assertion suffices for our theoretical purposes:

*PνdAν*. (21)

*dE* <sup>=</sup> *TdS* <sup>+</sup> <sup>∑</sup>*<sup>ν</sup>*

This is a macroscopic postulate to be inserted into a microscopic axiomatics' corpus.

3. Partial ignorance: we are given the probability distribution {*Pi*}; *i* = 1, . . . , *N*.

then the "best", least unbiased probability distribution is the one that [6]

• maximizes ignorance: the truth, all the truth, *nothing but* the truth [6].

and, additionally,

the a priori knowledge.

principle (MaxEnt), that can be stated as follows:

empirical information. This is our rationale.

following macroscopic statement:

**Axiom (1)**

$$S = S(p\_1, p\_2, \dots, p\_W). \tag{22}$$

Thus, we are actually taking as a postulate something that is actually known from both quantum and classical mechanics.

**Axiom (3)** The internal energy *E* and the external parameters *A<sup>ν</sup>* are to be considered as the expectation values of suitable operators, that is, the hamiltonian *H* and the hermitian operators R*<sup>ν</sup>* (i.e., *A<sup>ν</sup>* ≡< R*<sup>ν</sup>* >). Thus, the *A<sup>ν</sup>* (and also *E*) will depend on the eigenvalues of these operators *and* on the probability set. (Note that energy eigenvalues depend of course upon the R*ν*.

The reader will immediately realize that Axiom (2) is just a way of re-expressing Boltzmann's "atomic" conjecture. Thus, macroscopic quantities become statistical averages evaluated using a microscopic probability distribution [16]. Our present three new axioms are statements of fact. What do we mean? That they are borrowed from either experiment or pre-existent theories. Somewhat surprisingly, our three axioms do not actually incorporate any knew knowledge at all. The merely re-express known previous notions. Ockham's razor at its best! Our theory could no be more economical.

We need now to prove that the above three postulates allow one to reconstruct the imposing edifice of statistical mechanics. We will tackle this issue by showing below that they our axioms are equivalent to those of Jaynes' [17]. At this point we need to recall the main goal of statistical mechanics, namely, finding the probability distribution (or the density operator) that best describes our physical system. In order to do so Jaynes appealed to his MaxEnt postulate, that we restate below for the sake of fixing notation.

*MaxEnt axiom*: assume your prior knowledge about the system is given by the values of M expectation values

$$A\_1 \equiv <\mathcal{R}\_1>, \dots, A\_R \equiv <\mathcal{R}\_M>. \tag{23}$$

Then, *ρ* is uniquely fixed by extremizing the information measure *I*(*ρ*) subject to *ρ*−normalization plus the constraints given by the *M* conditions constituting our assumed foreknowledge

$$A\_{\vee} = <\mathcal{R}\_{\vee}> = \text{Tr}[\rho \,\mathcal{R}\_{\vee}].\tag{24}$$

This leads, after a Lagrange-constrained extremizing process, to the introduction of *M* Lagrange multipliers *λν*, that one assimilates to the generalized pressures *Pν*. The truth, the whole truth, nothing but the truth [6]. Jaynes rationale asserts that if the entropic measure that reflects our ignorance were not of maximal character, we would actually be *inventing* information not at hand.

While working through his variational process, Jaynes discovers that, after multiplying by Boltzmann's constant *kB* the right-hand-side of his expression for the information measure, it converts itself into an entropy, *I* ≡ *S*, the equilibrium thermodynamic one, with the caveat that *A*<sup>1</sup> =< R<sup>1</sup> >,..., *AM* =< R*<sup>M</sup>* > refer to extensive quantities. Having *ρ*, his universal form *I*(*ρ*) *yields complete microscopic information with respect to the system of interest*. To achieve our ends one needs now just to prove that the new axiomatics, with (21) and (22), is equivalent to MaxEnt.

### **6. New connection between macroscopic and microscopic approaches**

In establishing our new connections between the micro- and macro-scenarios we shall work with the classical instance only, since the corresponding quantum treatment constitute in this sense just a straightforward extension.

Our main idea is to pay attention to the generic change *pi* → *pi* + *dpi* as constrained by Eq. ( 21). In other word, we insist on studying the change *dpi* that takes place in such a manner that (21) holds. Our main macroscopic quantities *S*, *Aj*, and *E* will vary with *dpi*. These changes are not arbitrary but are constrained by (21). Note here an important advantage to be of our approach. We need *not* specify beforehand the information measure employed.

Since several possibilities exist (see for instance Gell-Mann and Tsallis [18]), this entails that the choice of information nature is not predetermined by macroscopic thermodynamics. For a detailed discussion of this issue see Ferri, Martinez, and Plastino [19].

The pertinent ingredients at hand are

• an arbitrary, smooth function *f*(*p*) permitting one expressing the information measure via

$$I \equiv S(\{p\_i\}) = \sum\_i p\_i f(p\_i)\_\prime \tag{25}$$

*pi* → *pi* + *dpi*, (29)

Thermodynamics' Microscopic Connotations 129

*<sup>i</sup>* ]*dpi* ≡ ∑*<sup>i</sup> Kidpi* = 0, (30)

(*β* ≡ 1/*kT*), (31)

*<sup>i</sup>* = 0; (*f or any i*), (32)

and ii) expand the resulting equation up to first order in the *dpi*.

<sup>∑</sup>*i*[*C*(1)

*C*(1) *<sup>i</sup>* = [∑*<sup>M</sup>*

*C*(2)

*<sup>i</sup>* <sup>+</sup> *<sup>C</sup>*(2)

*T*(1)

*<sup>i</sup>* <sup>=</sup> <sup>−</sup>*β*[(∑*<sup>M</sup>*

*T*(1) *<sup>i</sup>* <sup>+</sup> *<sup>T</sup>*(2)

an expression whose importance will become manifest later on.

[*S* − *β*�*H*� −

of Eq. (7), one thus encounter, after a little algebra [20–26],

*T*(2)

and we are in a position to recast (30) in the fashion

*Aν*, and normalization. For details see [20–26].

*δpi*

(30) arise then from two different approaches:

• following the well known MaxEnt route.

• our methodology, based on Eqs. (21) and (22), and

so that, appropriately rearranging things

distribution is not required.

Setting *λ*<sup>1</sup> ≡ *β* = 1/*T* one has

fact

Remembering that the Lagrange multipliers *λν* are identical to the generalized pressures *P<sup>ν</sup>*

*<sup>ν</sup>*=<sup>1</sup> *λν <sup>a</sup><sup>ν</sup>*

*∂pi*

*<sup>i</sup>* + *�i*) *g*�

*<sup>i</sup>* <sup>=</sup> <sup>−</sup>*<sup>T</sup> <sup>∂</sup><sup>S</sup>*

*<sup>i</sup>* = *f*(*pi*) + *pi f* �

*<sup>ν</sup>*=<sup>1</sup> *λν <sup>a</sup><sup>ν</sup>*

Eqs. (30) or (32) yield one and just one *pi*−expression, as demonstrated in Refs. [20–26]. However, it will be realized below that, at this stage, an explicit expression for this probability

We pass now to traversing the opposite road that leads from Jaynes' MaxEnt procedure and ends up with *our* present equations. This entails extremization of *S* subject to constraints in *E*,

(normalization Lagrange multiplier *ξ*) is easily seen in the above cited references to yield as a solution the very set of Eqs. (30). The detailed proof is given in the forthcoming Section. Eqs.

Accordingly, we see that both MaxEnt and our axiomatics co-imply one another. They are indeed equivalent ways of constructing equilibrium statistical mechanics. As a really relevant

*λν*�R*ν*� − *<sup>ξ</sup>* ∑

*i*

*pi*] = 0, (33)

*M* ∑ *ν*=2 *<sup>i</sup>* + *�i*]

(*pi*)

(*pi*) − *K*],

such that *S*({*pi*}) is a concave function,


$$A\_{\boldsymbol{\nu}} \equiv \langle \mathcal{R}\_{\boldsymbol{\nu}} \rangle = \sum\_{i}^{W} a\_{i}^{\boldsymbol{\nu}} \, \mathcal{g}(p\_{i}); \; \boldsymbol{\nu} = \mathbf{2}, \ldots, M\_{\boldsymbol{\nu}} \tag{26}$$

$$E = \sum\_{i}^{W} \epsilon\_{i} \, g(p\_{i}),\tag{27}$$

where *�<sup>i</sup>* is the energy associated to the microstate *i*.

We take *A*<sup>1</sup> ≡ *E* and pass to a consideration of the probability variations *dpi* that should generate accompanying changes *dS*, *dAν*, and *dE* in, respectively, *S*, the *Aν*, and *E*.

The essential issue at hand is that of enforcing compliance with

$$dE - TdS + \sum\_{\nu=1}^{W} dA\_{\nu} \lambda\_{\nu} = 0,\tag{28}$$

with *T* the temperature and *λν* generalized pressures. By recourse to (25), (26), and (27) we i) recast now (28) for

$$p\_i \to p\_i + dp\_i.\tag{29}$$

and ii) expand the resulting equation up to first order in the *dpi*.

Remembering that the Lagrange multipliers *λν* are identical to the generalized pressures *P<sup>ν</sup>* of Eq. (7), one thus encounter, after a little algebra [20–26],

$$\mathsf{C}\_{i}^{(1)} = [\Sigma\_{\nu=1}^{M} \lambda\_{\nu} a\_{i}^{\nu} + \mathfrak{e}\_{i}]$$

$$\mathsf{C}\_{i}^{(2)} = -T \frac{\partial \mathsf{S}}{\partial p\_{i}}$$

$$\sum\_{i} [\mathsf{C}\_{i}^{(1)} + \mathsf{C}\_{i}^{(2)}] dp\_{i} \equiv \sum\_{i} \mathsf{K}\_{i} dp\_{i} = 0,\tag{30}$$

so that, appropriately rearranging things

10 Will-be-set-by-IN-TECH

In establishing our new connections between the micro- and macro-scenarios we shall work with the classical instance only, since the corresponding quantum treatment constitute in this

Our main idea is to pay attention to the generic change *pi* → *pi* + *dpi* as constrained by Eq. ( 21). In other word, we insist on studying the change *dpi* that takes place in such a manner that (21) holds. Our main macroscopic quantities *S*, *Aj*, and *E* will vary with *dpi*. These changes are not arbitrary but are constrained by (21). Note here an important advantage to be of our

Since several possibilities exist (see for instance Gell-Mann and Tsallis [18]), this entails that the choice of information nature is not predetermined by macroscopic thermodynamics. For

• an arbitrary, smooth function *f*(*p*) permitting one expressing the information measure via

• *M* quantities *A<sup>ν</sup>* representing values of extensive quantities �R*ν*�, that adopt, for a

• still another arbitrary smooth, monotonic function *g*(*pi*) (*g*(0) = 0; *g*(1) = 1). With the express purpose of employing generalized, non-Shannonian entropies, we slightly

*i*

*pi f*(*pi*), (25)

*<sup>i</sup> g*(*pi*); *ν* = 2, . . . , *M*, (26)

*�<sup>i</sup> g*(*pi*), (27)

*dAνλν* = 0, (28)

*<sup>I</sup>* ≡ *<sup>S</sup>*({*pi*}) = ∑

*<sup>i</sup>* with probability *pi*,

generalize here the expectation-value definitions by recourse to *g* via (26):

*W* ∑ *i aν*

*E* = *W* ∑ *i*

generate accompanying changes *dS*, *dAν*, and *dE* in, respectively, *S*, the *Aν*, and *E*.

*dE* − *TdS* +

We take *A*<sup>1</sup> ≡ *E* and pass to a consideration of the probability variations *dpi* that should

*W* ∑ *ν*=1

with *T* the temperature and *λν* generalized pressures. By recourse to (25), (26), and (27) we i)

*A<sup>ν</sup>* ≡ �R*ν*� =

where *�<sup>i</sup>* is the energy associated to the microstate *i*.

The essential issue at hand is that of enforcing compliance with

**6. New connection between macroscopic and microscopic approaches**

approach. We need *not* specify beforehand the information measure employed.

a detailed discussion of this issue see Ferri, Martinez, and Plastino [19].

sense just a straightforward extension.

The pertinent ingredients at hand are

micro-state *i*, the value *a<sup>ν</sup>*

recast now (28) for

such that *S*({*pi*}) is a concave function,

$$\begin{aligned} T\_i^{(1)} &= f(p\_i) + p\_i f'(p\_i) \\ T\_i^{(2)} &= -\beta [ (\sum\_{\nu=1}^M \lambda\_\nu a\_i^\nu + \varepsilon\_i) \lg'(p\_i) - \mathcal{K} ] \\ &\qquad (\beta \equiv 1/kT)\_\prime \end{aligned} \tag{31}$$

and we are in a position to recast (30) in the fashion

$$T\_i^{(1)} + T\_i^{(2)} = 0; \text{ (for any } i\text{)}, \tag{32}$$

an expression whose importance will become manifest later on.

Eqs. (30) or (32) yield one and just one *pi*−expression, as demonstrated in Refs. [20–26]. However, it will be realized below that, at this stage, an explicit expression for this probability distribution is not required.

We pass now to traversing the opposite road that leads from Jaynes' MaxEnt procedure and ends up with *our* present equations. This entails extremization of *S* subject to constraints in *E*, *Aν*, and normalization. For details see [20–26].

Setting *λ*<sup>1</sup> ≡ *β* = 1/*T* one has

$$\delta\_{p\_i}[\mathcal{S} - \beta \langle H \rangle - \sum\_{\nu=2}^{M} \lambda\_{\nu} \langle \mathcal{R}\_{\nu} \rangle - \xi \sum\_{i} p\_i] = 0,\tag{33}$$

(normalization Lagrange multiplier *ξ*) is easily seen in the above cited references to yield as a solution the very set of Eqs. (30). The detailed proof is given in the forthcoming Section. Eqs. (30) arise then from two different approaches:


Accordingly, we see that both MaxEnt and our axiomatics co-imply one another. They are indeed equivalent ways of constructing equilibrium statistical mechanics. As a really relevant fact

#### 12 Will-be-set-by-IN-TECH 130 Thermodynamics – Fundamentals and Its Application in Science Thermodynamics' Microscopic Connotations <sup>13</sup>

One does not need to know the analytic form of *S*[*pi*] neither in Eqs. (30) nor in (33).

## **7. Proof**

Here we prove that Eqs. (30) can be derived from the MaxEnt approach (33). One wishes to extremize *S* subject to the constraints of fixed valued for i) *U*, ii) the *M* values *A<sup>ν</sup>* (entailing Lagrange multipliers (1) *β* and (2) *M γν*), and iii) normalization (Lagrange multiplier *ξ*). One has also

$$A\_{\boldsymbol{\nu}} = \langle \mathcal{R}\_{\boldsymbol{\nu}} \rangle = \sum\_{i} p\_{i} a\_{i\prime}^{\boldsymbol{\nu}} \,\tag{34}$$

distribution (PD) that controls microstates-population [27]. Our present ideas yield a detailed picture, from a new perspective [20–26], of how changes in the independent external thermodynamic parameters affect the micro-state population and, consequently, the entropy

Thermodynamics' Microscopic Connotations 131

*Universidad Nacional de La Plata, Instituto de Física (IFLP-CCT-CONICET), C.C. 727, 1900 La Plata,*

*Physics Departament and IFISC-CSIC, University of Balearic Islands, 07122 Palma de Mallorca, Spain*

*Physics Departament and IFISC-CSIC, University of Balearic Islands, 07122 Palma de Mallorca, Spain*

[3] J. Willard Gibbs, *Elementary Principles in Statistical Mechanics*, New Haven, Yale

[5] E. T. Jaynes *Papers on probability, statistics and statistical physics*, edited by R. D.

[6] A. Katz, *Principles of Statistical Mechanics, The information Theory Approach*, San Francisco,

[8] A. Plastino and A. R. Plastino in *Condensed Matter Theories*, Volume 11, E. Ludeña (Ed.),

[9] D. M. Rogers, T. L. Beck, S. B. Rempe, *Information Theory and Statistical Mechanics Revisited*,

[10] D. J. Scalapino in *Physics and probability. Essays in honor of Edwin T. Jaynes* edited by W. T. Grandy, Jr. and P. W. Milonni (Cambridge University Press, NY, 1993), and references

[13] P. Duhem *The aim and structure of physical theory* (Princeton University Press, Princeton,

[14] R. B. Lindsay *Concepts and methods of theoretical physics* (Van Nostrand, NY, 1951).

[7] T. M. Cover and J. A. Thomas, *Elements of information theory*, NY, J. Wiley, 1991.

[11] B. Russell, *A history of western philosophy* (Simon & Schuster, NY, 1945).

[12] P. W. Bridgman *The nature of physical theory* (Dover, NY, 1936).

This work was partially supported by the MEC Grant FIS2005-02796 (Spain).

[1] R. B. Lindsay and H. Margenau, *Foundations of physics*, NY, Dover, 1957. [2] E. A. Desloge, *Thermal physics* NY, Holt, Rhinehart and Winston, 1968.

[4] C. E. Shannon, Bell System Technol. J. 27 (1948) 379-390.

Rosenkrantz, Dordrecht, Reidel, 1987.

Nova Science Publishers, p. 341 (1996).

*Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro, Brazil*

and the internal energy.

**Acknowledgement**

**Author details**

Evaldo M. F. Curado

**9. References**

University Press, 1902.

Freeman and Co., 1967.

ArXiv 1105.5662v1.

New Jersey, 1954).

therein.

A. Plastino

*Argentina*

M. Casas

with *a<sup>ν</sup> <sup>i</sup>* = �*i*|R*ν*|*i*� the matrix elements in the basis �*i*� of R*ν*. The ensuing variational problem one faces, with *U* = ∑*<sup>i</sup> pi�i*, is

$$\delta\_{\{p\_i\}} \left[ \mathcal{S} - \beta \mathcal{U} - \sum\_{\nu=1}^{M} \gamma\_{\nu} A\_{\nu} - \tilde{\xi} \sum\_{i} p\_i \right] = 0,\tag{35}$$

that immediately leads, for *γν* = *βλν*, to

$$\delta\_{p\_m} \sum\_{i} \left( p\_i f(p\_i) - \left[ \beta p\_i (\sum\_{\nu=1}^M \lambda\_\nu \ a\_i^\nu + \epsilon\_i) + \tilde{\xi} p\_i \right] \right) = 0,\tag{36}$$

so that the the following two quantities vanish

$$f(p\_i) + p\_i f'(p\_i) - \left[\beta(\sum\_{\nu=1}^M \lambda\_\nu a\_i^\nu + \epsilon\_i) + \xi\right]$$

$$\Rightarrow \quad \text{if } \xi \equiv \beta K,$$

$$f(p\_i) + p\_i f'(p\_i) - \beta(\sum\_{\nu=1}^M \lambda\_\nu a\_i^\nu + \epsilon\_i) + K\right]$$

$$\Rightarrow 0 = T\_i^{(1)} + T\_i^{(2)}.\tag{37}$$

We realize now that (32) and the last equality of (37) are one and the same equation. MaxEnt does lead to (32).

#### **8. Conclusions**

We have formally proved above that our axiomatics allows one to derive MaxEnt equations and viceversa. Thus, our treatment provides an alternative foundation for equilibrium statistical mechanics. We emphasized that, opposite to what happens with both Gibbs' and Jaynes' axioms, our postulates have zero new informational content. Why? Because they are borrowed either from experiment or from pre-existing theories, namely, information theory and quantum mechanics.

The first and second laws of thermodynamics are two of physics' most important empirical facts, constituting pillars to our present view of Nature. Statistical mechanics (SM) adds an underlying microscopic substratum able to explain not only these two laws but the whole of thermodynamics itself [2, 6, 27–30]. Basic SM-ingredient is a microscopic probability distribution (PD) that controls microstates-population [27]. Our present ideas yield a detailed picture, from a new perspective [20–26], of how changes in the independent external thermodynamic parameters affect the micro-state population and, consequently, the entropy and the internal energy.

## **Acknowledgement**

This work was partially supported by the MEC Grant FIS2005-02796 (Spain).

## **Author details**

A. Plastino

12 Will-be-set-by-IN-TECH

Here we prove that Eqs. (30) can be derived from the MaxEnt approach (33). One wishes to extremize *S* subject to the constraints of fixed valued for i) *U*, ii) the *M* values *A<sup>ν</sup>* (entailing Lagrange multipliers (1) *β* and (2) *M γν*), and iii) normalization (Lagrange multiplier *ξ*). One

*i*

*γνA<sup>ν</sup>* − *<sup>ξ</sup>* ∑

*λν a<sup>ν</sup>*

*<sup>ν</sup>*=<sup>1</sup> *λν <sup>a</sup><sup>ν</sup>*

*<sup>ν</sup>*=<sup>1</sup> *λνa<sup>ν</sup>*

*<sup>i</sup>* <sup>+</sup> *<sup>T</sup>*(2)

*<sup>i</sup>* = �*i*|R*ν*|*i*� the matrix elements in the basis �*i*� of R*ν*. The ensuing variational problem

*pi a<sup>ν</sup>*

*i pi* 

*<sup>i</sup>* + *�i*) + *ξ pi*]

*<sup>i</sup>* + *�i*) + *ξ*]

*<sup>i</sup>* + *�i*) + *K*]

*<sup>i</sup>* . (37)

*<sup>i</sup>* , (34)

= 0, (35)

= 0, (36)

*<sup>A</sup><sup>ν</sup>* = �R*ν*� = ∑

*M* ∑ *ν*=1

> *M* ∑ *ν*=1

(*pi*) <sup>−</sup> [*β*(∑*<sup>M</sup>*

(*pi*) <sup>−</sup> *<sup>β</sup>*(∑*<sup>M</sup>*

<sup>⇒</sup> <sup>0</sup> <sup>=</sup> *<sup>T</sup>*(1)

⇒ if *ξ* ≡ *βK*,

We realize now that (32) and the last equality of (37) are one and the same equation. MaxEnt

We have formally proved above that our axiomatics allows one to derive MaxEnt equations and viceversa. Thus, our treatment provides an alternative foundation for equilibrium statistical mechanics. We emphasized that, opposite to what happens with both Gibbs' and Jaynes' axioms, our postulates have zero new informational content. Why? Because they are borrowed either from experiment or from pre-existing theories, namely, information theory

The first and second laws of thermodynamics are two of physics' most important empirical facts, constituting pillars to our present view of Nature. Statistical mechanics (SM) adds an underlying microscopic substratum able to explain not only these two laws but the whole of thermodynamics itself [2, 6, 27–30]. Basic SM-ingredient is a microscopic probability

One does not need to know the analytic form of *S*[*pi*] neither in Eqs. (30) nor in (33).

**7. Proof**

has also

with *a<sup>ν</sup>*

does lead to (32).

**8. Conclusions**

and quantum mechanics.

one faces, with *U* = ∑*<sup>i</sup> pi�i*, is

that immediately leads, for *γν* = *βλν*, to

*δpm* ∑ *i*

so that the the following two quantities vanish

*<sup>δ</sup>*{*pi*} 

*f*(*pi*) + *pi f* �

*f*(*pi*) + *pi f* �

*S* − *βU* −

*pi f*(*pi*) − [*βpi*(

*Universidad Nacional de La Plata, Instituto de Física (IFLP-CCT-CONICET), C.C. 727, 1900 La Plata, Argentina*

*Physics Departament and IFISC-CSIC, University of Balearic Islands, 07122 Palma de Mallorca, Spain*

Evaldo M. F. Curado *Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro, Brazil*

M. Casas *Physics Departament and IFISC-CSIC, University of Balearic Islands, 07122 Palma de Mallorca, Spain*

## **9. References**

	- [15] H. Weyl *Philosophy of mathematics and natural science* (Princeton University Press, Princeton, New Jersey, 1949).

**Property Prediction and Thermodynamics** 


**Property Prediction and Thermodynamics** 

14 Will-be-set-by-IN-TECH

[15] H. Weyl *Philosophy of mathematics and natural science* (Princeton University Press,

[17] W.T. Grandy Jr. and P. W. Milonni (Editors), *Physics and Probability. Essays in Honor of*

[18] M. Gell-Mann and C. Tsallis, Eds. *Nonextensive Entropy: Interdisciplinary applications*,

[30] B. H. Lavenda, *Statistical Physics* (J. Wiley, New York, 1991); B. H. Lavenda,

[19] G. L. Ferri, S. Martinez, A. Plastino, Journal of Statistical Mechanics, P04009 (2005).

[22] A. Plastino, E. Curado, International Journal of Modern Physics B 21 (2007) 2557

[29] J. J.Sakurai, *Modern quantum mechanics* (Benjamin, Menlo Park, Ca., 1985).

Princeton, New Jersey, 1949).

[16] D. Lindley, *Boltzmann's atom*, NY, The free press, 2001.

[20] E. Curado, A. Plastino, Phys. Rev. E 72 (2005) 047103. [21] A. Plastino, E. Curado, Physica A 365 (2006) 24

[24] A. Plastino, E. Curado, M. Casas, Entropy A 10 (2008) 124 [25] International Journal of Modern Physics B 22, (2008) 4589 [26] E. Curado, F. Nobre, A. Plastino, Physica A 389 (2010) 970. [27] R.K. Pathria, *Statistical Mechanics* (Pergamon Press, Exeter, 1993). [28] F. Reif, *Statistical and thermal physics* (McGraw-Hill, NY, 1965).

*Thermodynamics of Extremes* (Albion, West Sussex, 1995).

[23] A. Plastino, E. Curado, Physica A 386 (2007) 155

Oxford, Oxford University Press, 2004.

*Edwin T. Jaynes*, NY, Cambridge University Press, 1993.

**Chapter 6** 

© 2012 Kolská et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**Group Contribution Methods for Estimation** 

Thermodynamic data play an important role in the understanding and design of chemical processes. To determine values of physico-chemical properties of compounds we can apply experimental or non-experimental techniques. Experimental techniques belong to the most correct, accurate and reliable. All experimental methods require relevant technical equipment, time necessary for experiment, sufficient amount of measured compounds of satisfactory purity. Compound must not affect technical apparatus and should not be decomposed during experiment. Other aspect is a valid legislation, which limits a usage of

If due to any of these conditions mentioned above causes the experiment cannot be realized,

If due to any of conditions results in that experimental determination cannot be realized and data on physico-chemical property are necessary, we have to employ some nonexperimental approaches, either calculation methods or estimation ones. Due to the lack of experimental data for several industrially important compounds, different estimation methods have been developed to provide missing data. Estimation methods include those based on theory (e.g. statistical thermodynamics or quantum mechanics), various empirical relationships (correlations of required property with variable, experimentally determined compound characteristics, e.g. number of carbon atoms in their molecule, molecular weight, normal boiling temperature, etc.), and several classes of "additivity-principle" methods

and reproduction in any medium, provided the original work is properly cited.

**2. Non-experimental approaches to determine physico-chemical** 

**of Selected Physico-Chemical Properties** 

Zdeňka Kolská, Milan Zábranský and Alena Randová

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/49998

dangerous compounds by any users.

**properties of compounds** 

some non-experimental approaches can be applied.

**1. Introduction** 

**of Organic Compounds** 

Zdeňka Kolská, Milan Zábranský and Alena Randová

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/49998

## **1. Introduction**

Thermodynamic data play an important role in the understanding and design of chemical processes. To determine values of physico-chemical properties of compounds we can apply experimental or non-experimental techniques. Experimental techniques belong to the most correct, accurate and reliable. All experimental methods require relevant technical equipment, time necessary for experiment, sufficient amount of measured compounds of satisfactory purity. Compound must not affect technical apparatus and should not be decomposed during experiment. Other aspect is a valid legislation, which limits a usage of dangerous compounds by any users.

If due to any of these conditions mentioned above causes the experiment cannot be realized, some non-experimental approaches can be applied.

## **2. Non-experimental approaches to determine physico-chemical properties of compounds**

If due to any of conditions results in that experimental determination cannot be realized and data on physico-chemical property are necessary, we have to employ some nonexperimental approaches, either calculation methods or estimation ones. Due to the lack of experimental data for several industrially important compounds, different estimation methods have been developed to provide missing data. Estimation methods include those based on theory (e.g. statistical thermodynamics or quantum mechanics), various empirical relationships (correlations of required property with variable, experimentally determined compound characteristics, e.g. number of carbon atoms in their molecule, molecular weight, normal boiling temperature, etc.), and several classes of "additivity-principle" methods

© 2012 Kolská et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

(Baum, 1989; Pauling et al., 2001). Estimation methods can be divided into several groups from many aspects, e.g. into methods based on theoretical, semi-theoretical relations and the empirical ones. Books and papers of last decades divide estimation methods depending on the required input data into QPPR or QSPR approaches (Baum, 1989). QPPR methods (Quantity-Property-Property-Relationship) are input data-intensive. They require for calculation of searched value property knowledge of other experimental data. We can use them successfully only when we have input data. On the other hand QSPR (Baum, 1989) methods (Quantity-Structure-Property-Relationship) need only knowledge of the chemical structure of a compound to predict the estimated property. QSPR methods use some structural characteristics, such as number of fragments (atoms, bonds or group of atoms in a molecule), topological indices or other structural information, molecular descriptors, to express the relation between the property and molecular structure of compound (Baum, 1989; Pauling et al., 2001; Gonzáles et al., 2007a). Empirical and group contribution methods seem to be the most suitable (Pauling et al., 2001; Majer et al., 1989) due to their simplicity, universality and fast usage.

Group Contribution Methods for Estimation of Selected Physico-Chemical Properties of Organic Compounds 137

CH3-, -CH2- and -OH, or: (ii) CH3- and –CH2OH. More complex compounds are described

**Figure 1.** Dependence of normal boiling temperature of *n*-alkanols in homological series C1-C12

**Figure 2.** Example of division of ethanol molecule into atomic, bond and group structural fragments

Group contribution methods are essentially empirical estimation methods. A large variety of these models have been designed during last centuries, differing in a field of their applicability and in the set of experimental data. They were developed to estimate, e.g. critical properties (Lydersen, 1955; Ambrose, 1978; Ambrose, 1979; Joback & Reid, 1987;

by more complex structural fragments.

## **2.1. Group contribution methods**

Group contribution methods are presented as empirical QSPR approaches. The easiest models were based on study of property on number of carbon atoms *n*C or methylen groups *n*CH2 in molecules of homological series. In Fig. 1 is presented dependence of normal boiling temperature *T*b on number of carbon atoms *n*C (bottom axis) or methylen groups *n*CH2 (top axis) in molecules of homological series *n*-alkanols C1-C12 (Majer & Svoboda, 1985; NIST database). As we can see, this dependence is clearly linear in some range of *n*C=C2-C10. But increasing discrepancy is evident either for low number of carbon atoms C1 or for higher one *n*C>C10. Due to these departures from linear behaviour some parameters covering structural effects on property were inclusive to these easy models (e.g. Chickos et al., 1996). From these approaches structural fragments and subsequently group contribution methods have been established.

Group contribution methods are based on the so called "additive principle". That means any compound can be divided into fragments, usually atoms, bonds or group of atoms, etc. All fragments have a partial value called a contribution. These contributions are calculated from known experimental data. Property of a compound is obtained by a summing up the values of all contributions presented in the molecule. Example of division of molecule of ethanol into atomic, bond and group fragments is presented in Fig. 2. When we divide this molecule into atomic fragments, the total value of property *X* of ethanol is given by summing up the values for two carbon atom contributions *X*(C), six hydrogen atom contributions *X*(H) and one oxygen atom contribution *X*(O). The second way is the division of ethanol molecules into the following bond fragments with their contribution: *X*(C-C), *X*(C-O), *X*(C-H) and *X*(O-H). Due to increasing quality and possibility of computer technique a fragmentation into more complex group structural fragments is applied in present papers (Baum, 1989; Pauling et al., 2001). Some of ways to divide the molecule ethanol into group structural fragments are presented in Fig. 2. Ethanol molecule can be divided either into: (i) CH3-, -CH2- and -OH, or: (ii) CH3- and –CH2OH. More complex compounds are described by more complex structural fragments.

136 Thermodynamics – Fundamentals and Its Application in Science

universality and fast usage.

have been established.

**2.1. Group contribution methods** 

(Baum, 1989; Pauling et al., 2001). Estimation methods can be divided into several groups from many aspects, e.g. into methods based on theoretical, semi-theoretical relations and the empirical ones. Books and papers of last decades divide estimation methods depending on the required input data into QPPR or QSPR approaches (Baum, 1989). QPPR methods (Quantity-Property-Property-Relationship) are input data-intensive. They require for calculation of searched value property knowledge of other experimental data. We can use them successfully only when we have input data. On the other hand QSPR (Baum, 1989) methods (Quantity-Structure-Property-Relationship) need only knowledge of the chemical structure of a compound to predict the estimated property. QSPR methods use some structural characteristics, such as number of fragments (atoms, bonds or group of atoms in a molecule), topological indices or other structural information, molecular descriptors, to express the relation between the property and molecular structure of compound (Baum, 1989; Pauling et al., 2001; Gonzáles et al., 2007a). Empirical and group contribution methods seem to be the most suitable (Pauling et al., 2001; Majer et al., 1989) due to their simplicity,

Group contribution methods are presented as empirical QSPR approaches. The easiest models were based on study of property on number of carbon atoms *n*C or methylen groups *n*CH2 in molecules of homological series. In Fig. 1 is presented dependence of normal boiling temperature *T*b on number of carbon atoms *n*C (bottom axis) or methylen groups *n*CH2 (top axis) in molecules of homological series *n*-alkanols C1-C12 (Majer & Svoboda, 1985; NIST database). As we can see, this dependence is clearly linear in some range of *n*C=C2-C10. But increasing discrepancy is evident either for low number of carbon atoms C1 or for higher one *n*C>C10. Due to these departures from linear behaviour some parameters covering structural effects on property were inclusive to these easy models (e.g. Chickos et al., 1996). From these approaches structural fragments and subsequently group contribution methods

Group contribution methods are based on the so called "additive principle". That means any compound can be divided into fragments, usually atoms, bonds or group of atoms, etc. All fragments have a partial value called a contribution. These contributions are calculated from known experimental data. Property of a compound is obtained by a summing up the values of all contributions presented in the molecule. Example of division of molecule of ethanol into atomic, bond and group fragments is presented in Fig. 2. When we divide this molecule into atomic fragments, the total value of property *X* of ethanol is given by summing up the values for two carbon atom contributions *X*(C), six hydrogen atom contributions *X*(H) and one oxygen atom contribution *X*(O). The second way is the division of ethanol molecules into the following bond fragments with their contribution: *X*(C-C), *X*(C-O), *X*(C-H) and *X*(O-H). Due to increasing quality and possibility of computer technique a fragmentation into more complex group structural fragments is applied in present papers (Baum, 1989; Pauling et al., 2001). Some of ways to divide the molecule ethanol into group structural fragments are presented in Fig. 2. Ethanol molecule can be divided either into: (i)

**Figure 1.** Dependence of normal boiling temperature of *n*-alkanols in homological series C1-C12

**Figure 2.** Example of division of ethanol molecule into atomic, bond and group structural fragments

Group contribution methods are essentially empirical estimation methods. A large variety of these models have been designed during last centuries, differing in a field of their applicability and in the set of experimental data. They were developed to estimate, e.g. critical properties (Lydersen, 1955; Ambrose, 1978; Ambrose, 1979; Joback & Reid, 1987;

Gani & Constantinou, 1996; Poling et al., 2001; Marrero & Gani, 2001; Brown et al., 2010; Monago & Otobrise, 2010; Sales-Cruz et al., 2010; Manohar & Udaya Sankar, 2011; Garcia et al., 2012), parameters of state equations (Pereda et al., 2010; Schmid & Gmehling, 2012), acentric factor (Constantinou & Gani, 1994; Brown et al., 2010; Monago & Otobrise, 2010), activity coefficients (Tochigi et al., 2005; Tochigi & Gmehling, 2011), vapour pressure (Poling et al., 2001; Miller, 1964), liquid viscosity (Joback & Reid, 1987; Conte et al., 2008; Sales-Cruz et al., 2010), gas viscosity (Reichenberg, 1975), heat capacity (Joback & Reid, 1987; Ruzicka & Domalski, 1993a; Ruzicka & Domalski, 1993b; Kolská et al., 2008), enthalpy of vaporization (e.g. Chickos et al., 1995; Chickos & Wilson, 1997; Marrero & Gani, 2001; Kolská et al. 2005, etc.), entropy of vaporization (Chiskos et al., 1998; Kolská et al. 2005), normal boiling temperature (Joback & Reid, 1987; Gani & Constantinou, 1996; Marrero & Gani, 2001), liquid thermal conductivity (Nagvekar & Daubert, 1987), gas thermal conductivity (Chung et al., 1984), gas permeability and diffusion coefficients (Yampolskii et al., 1998), liquid density (Campbell & Thodos, 1985; Sales-Cruz et al., 2010; Shahbaz et al., 2012), surface tension (Brock, 1955; Conte et al., 2008; Awasthi et al., 2010), solubility parameters of fatty acid methyl esters (Lu et al., 2011), flash temperatures (Liaw & Chiu, 2006; Liaw et al., 2011). Large surveys of group contribution methods for enthalpy of vaporization and liquid heat capacity have been presented in references (Zábranský et al., 2003; Kolská, 2004; Kolská et al., 2005; Kolská et. al, 2008; Zábranský et al, 2010a). Group-contribution-based property estimation methods ca be also used to predict the missing UNIFAC group-interaction parameters for the calculation of vaporliquid equilibrium (Gonzáles at al., 2007b).

Group Contribution Methods for Estimation of Selected Physico-Chemical Properties of Organic Compounds 139

In this chapter for most of estimations the modified group contribution method by Marrero and Gani (Marrero & Gani, 2001; Kolská et al., 2005; Kolská et. al. 2008) was applied, which has been originally developed for estimation of different thermodynamic properties at one temperature only (Constantinou & Gani, 1996; Marrero & Gani, 2001). Determination of group contribution parameters is performed in three levels, primary, secondary and third. At first, all compounds are divided into the primary (first) order group contributions. This primary level uses contributions from simple groups that allow description of a wide variety of organic compounds. Criteria for their creation and calculation have been described (Marrero & Gani, 2001; Kolská et al., 2005; Kolská et al., 2008). The primary level groups, however, are insufficient to capture a proximity effect (they do not implicate an influence of their surroundings) and differences between isomers. Using primary level groups enables to estimate correctly properties of only simple and monofunctional compounds, but the estimation errors for more complex substances are higher. The primary level contributions provide an initial approximation that is improved at the second level and further refined at the third level, if that is possible and necessary. The higher levels (second and third) involve polyfunctional and structural groups that provide more information about a molecular structure of more complex compounds. These higher levels are able to describe more correctly polyfunctional compounds with at least one ring in a molecule, or non-ring chains including more than four carbon atoms in a molecule, and multi-ring compounds with a fused or non-fused aromatic or non-aromatic rings. The differences between some isomers are also able to distinguish by these higher levels. Complex polycyclic compounds or systems of fused aromatic or nonaromatic rings are described by the third order contributions. They are still bigger and more complex than the first, even the second order ones. The multilevel scheme enhances the accuracy, reliability and the range of application of group contribution method for an almost all classes of organic compounds.

After these all three levels the total value of predicted property *X* is obtained by the summing up of all group contributions, which occur in the molecule. First order groups,

> <sup>0</sup> 11 1 *nm o X x NC M D z O E i xi j xj k xk ijk*

where *X* stands the estimated property, *x*0 is an adjustable parameter for the relevant property, *Cxi* is the first-order group contribution of type *i*, *Dxj* is the second-order group contribution of type *j*, *Exk* is the third-order group contributions of the type *k* and *Ni*, *Mj*, *Ok* denote the number of occurrences of individual group contributions. The more detail description of parameters calculation is mentioned in original papers (Marrero & Gani, 2001;

To develop reliable and accurate group contribution model three important steps should be realized: (i) to collect input database, rather of critically assessed experimental data, from which parameters, group contributions, would be calculated; (ii) to design structural

(1)

*2.1.1. Group contribution methods by Marrero-Gani* 

second and third order ones, if they are in.

Kolská et al., 2005; Kolská et al., 2008)

Group contribution methods can be used for pure compounds, even inorganic compounds (e.g. Williams, 1997; Briard et al., 2003), organometallic compounds (e.g. Nikitin et al., 2010) and also for mixtures (e.g. Awasthi et al., 2010; Papaioannou et al., 2010; Teixeira et al., 2011; Garcia et al., 2012). Also e.g. estimation of thermodynamic properties of polysacharides was presented (Lobanova et al., 2011). Discussion about determination of properties of polymers has been also published (Satyanarayana et al., 2007; Bogdanic, 2009; Oh & Bae, 2009). Property models based on the group contribution approach for lipid technology have been also presented (Díaz-Tovar et al., 2007).

During last years also models for ionic liquids and their variable properties were developed, e.g. for density, thermal expansion and viscosity of cholinium-derived ionic liquids (Costa et al., 2011; Costa et al., 2012), viscosity (Adamová et al., 2011), the glass-transition temperature and fragility (Gacino et al., 2011), experimental data of mixture with ionic liquid were compared with group contribution methods (Cehreli & Gmehling, 2010) or thermophysical properties were studied (Gardas et al., 2010).

Some of these group contribution methods were developed for only limited number of compounds, for some family of compounds, e.g. for fluorinated olefins (Brown et al., 2010), hydrocarbons (Chickos et al., 1995), fatty acid methyl esters (Lu et al., 2011), etc., most of approaches were established for a wide range of organic compounds.

#### *2.1.1. Group contribution methods by Marrero-Gani*

138 Thermodynamics – Fundamentals and Its Application in Science

liquid equilibrium (Gonzáles at al., 2007b).

also presented (Díaz-Tovar et al., 2007).

properties were studied (Gardas et al., 2010).

Gani & Constantinou, 1996; Poling et al., 2001; Marrero & Gani, 2001; Brown et al., 2010; Monago & Otobrise, 2010; Sales-Cruz et al., 2010; Manohar & Udaya Sankar, 2011; Garcia et al., 2012), parameters of state equations (Pereda et al., 2010; Schmid & Gmehling, 2012), acentric factor (Constantinou & Gani, 1994; Brown et al., 2010; Monago & Otobrise, 2010), activity coefficients (Tochigi et al., 2005; Tochigi & Gmehling, 2011), vapour pressure (Poling et al., 2001; Miller, 1964), liquid viscosity (Joback & Reid, 1987; Conte et al., 2008; Sales-Cruz et al., 2010), gas viscosity (Reichenberg, 1975), heat capacity (Joback & Reid, 1987; Ruzicka & Domalski, 1993a; Ruzicka & Domalski, 1993b; Kolská et al., 2008), enthalpy of vaporization (e.g. Chickos et al., 1995; Chickos & Wilson, 1997; Marrero & Gani, 2001; Kolská et al. 2005, etc.), entropy of vaporization (Chiskos et al., 1998; Kolská et al. 2005), normal boiling temperature (Joback & Reid, 1987; Gani & Constantinou, 1996; Marrero & Gani, 2001), liquid thermal conductivity (Nagvekar & Daubert, 1987), gas thermal conductivity (Chung et al., 1984), gas permeability and diffusion coefficients (Yampolskii et al., 1998), liquid density (Campbell & Thodos, 1985; Sales-Cruz et al., 2010; Shahbaz et al., 2012), surface tension (Brock, 1955; Conte et al., 2008; Awasthi et al., 2010), solubility parameters of fatty acid methyl esters (Lu et al., 2011), flash temperatures (Liaw & Chiu, 2006; Liaw et al., 2011). Large surveys of group contribution methods for enthalpy of vaporization and liquid heat capacity have been presented in references (Zábranský et al., 2003; Kolská, 2004; Kolská et al., 2005; Kolská et. al, 2008; Zábranský et al, 2010a). Group-contribution-based property estimation methods ca be also used to predict the missing UNIFAC group-interaction parameters for the calculation of vapor-

Group contribution methods can be used for pure compounds, even inorganic compounds (e.g. Williams, 1997; Briard et al., 2003), organometallic compounds (e.g. Nikitin et al., 2010) and also for mixtures (e.g. Awasthi et al., 2010; Papaioannou et al., 2010; Teixeira et al., 2011; Garcia et al., 2012). Also e.g. estimation of thermodynamic properties of polysacharides was presented (Lobanova et al., 2011). Discussion about determination of properties of polymers has been also published (Satyanarayana et al., 2007; Bogdanic, 2009; Oh & Bae, 2009). Property models based on the group contribution approach for lipid technology have been

During last years also models for ionic liquids and their variable properties were developed, e.g. for density, thermal expansion and viscosity of cholinium-derived ionic liquids (Costa et al., 2011; Costa et al., 2012), viscosity (Adamová et al., 2011), the glass-transition temperature and fragility (Gacino et al., 2011), experimental data of mixture with ionic liquid were compared with group contribution methods (Cehreli & Gmehling, 2010) or thermophysical

Some of these group contribution methods were developed for only limited number of compounds, for some family of compounds, e.g. for fluorinated olefins (Brown et al., 2010), hydrocarbons (Chickos et al., 1995), fatty acid methyl esters (Lu et al., 2011), etc., most of

approaches were established for a wide range of organic compounds.

In this chapter for most of estimations the modified group contribution method by Marrero and Gani (Marrero & Gani, 2001; Kolská et al., 2005; Kolská et. al. 2008) was applied, which has been originally developed for estimation of different thermodynamic properties at one temperature only (Constantinou & Gani, 1996; Marrero & Gani, 2001). Determination of group contribution parameters is performed in three levels, primary, secondary and third. At first, all compounds are divided into the primary (first) order group contributions. This primary level uses contributions from simple groups that allow description of a wide variety of organic compounds. Criteria for their creation and calculation have been described (Marrero & Gani, 2001; Kolská et al., 2005; Kolská et al., 2008). The primary level groups, however, are insufficient to capture a proximity effect (they do not implicate an influence of their surroundings) and differences between isomers. Using primary level groups enables to estimate correctly properties of only simple and monofunctional compounds, but the estimation errors for more complex substances are higher. The primary level contributions provide an initial approximation that is improved at the second level and further refined at the third level, if that is possible and necessary. The higher levels (second and third) involve polyfunctional and structural groups that provide more information about a molecular structure of more complex compounds. These higher levels are able to describe more correctly polyfunctional compounds with at least one ring in a molecule, or non-ring chains including more than four carbon atoms in a molecule, and multi-ring compounds with a fused or non-fused aromatic or non-aromatic rings. The differences between some isomers are also able to distinguish by these higher levels. Complex polycyclic compounds or systems of fused aromatic or nonaromatic rings are described by the third order contributions. They are still bigger and more complex than the first, even the second order ones. The multilevel scheme enhances the accuracy, reliability and the range of application of group contribution method for an almost all classes of organic compounds.

After these all three levels the total value of predicted property *X* is obtained by the summing up of all group contributions, which occur in the molecule. First order groups, second and third order ones, if they are in.

$$X = x\_0 + \sum\_{\substack{\mathbf{I} \\ \mathbf{j}}}^{\mathcal{M}} N\_{\mathbf{I}} \mathbf{C}\_{\mathbf{X}\mathbf{j}} + \sigma \sum\_{\substack{\mathbf{I} \\ \mathbf{j}}}^{\mathcal{M}} M\_{\mathbf{j}} D\_{\mathbf{X}\mathbf{j}} + z \sum\_{\substack{\mathbf{I} \\ \mathbf{k}}}^{\mathcal{O}} O\_{\mathbf{k}} E\_{\mathbf{x}\mathbf{k}} \tag{1}$$

where *X* stands the estimated property, *x*0 is an adjustable parameter for the relevant property, *Cxi* is the first-order group contribution of type *i*, *Dxj* is the second-order group contribution of type *j*, *Exk* is the third-order group contributions of the type *k* and *Ni*, *Mj*, *Ok* denote the number of occurrences of individual group contributions. The more detail description of parameters calculation is mentioned in original papers (Marrero & Gani, 2001; Kolská et al., 2005; Kolská et al., 2008)

To develop reliable and accurate group contribution model three important steps should be realized: (i) to collect input database, rather of critically assessed experimental data, from which parameters, group contributions, would be calculated; (ii) to design structural

fragments for description of all chemical structures for compounds of input database; (iii) to divide all chemical structures into defined structural fragments correctly. It can be realized either manually, when databases of chemical structures and structural fragments inclusive several members only, either via computer program, when databases contain hundreds of compounds and structural fragments are more complex. To calculate group contribution parameters for thermophysical properties the ProPred program has been used (Marrero, 2002). Description and division chemical structures for other estimations have been made handy. Molecular structures for electronical splitting of all compounds from the basic data set were input in the Simplified Molecular Input Line Entry Specification, so-called the SMILES format (Weininger et al., 1986; Weininger, 1988; Weininger et al., 1989; Weininger, 1990).

Group Contribution Methods for Estimation of Selected Physico-Chemical Properties of Organic Compounds 141

for their universality and simplicity. More rich survey of estimation methods for enthalpy of

Large databases of critically assessed data have been used for group contribution calculations: data for 831 compounds have been used for estimations at 298.15 K, and data for 589 compounds have been used for estimations at the normal boiling temperature. Organic compounds were divided into several classes (aliphatic and acyclic saturated and unsaturated hydrocarbons, aromatic hydrocarbons, halogenated hydrocarbons, compounds containing oxygen, nitrogen or sulphur atoms and miscellaneous compounds). Especially calorimetrically measured experimental data from the compilation (Majer & Svoboda, 1985) and data from some other sources mentioned in original paper (Kolská et al., 2005) were

Results for estimations of these three properties are presented in the following Tables, Table 1 for enthalpy of vaporization at 298.15 K, Table 2 for enthalpy of vaporization at normal boiling temperature and Table 3 for entropy of vaporization at normal boiling temperature, *NC* means a number of compounds used for development of model and contributions calculation, *NG* is number of applied structural fragments (groups), *AAE* is absolute average

Table 1 shows that values for 831 compounds were used for estimation of enthalpy of vaporization at 298.15 K. When only first level groups were used, the prediction was performed with the *AAE* and the *ARE* of 1.3 kJ/mol and 2.8%, resp. Values of 116 group contributions were calculated at this step. Then, 486 compounds were described by the second order groups. Prediction of these compounds improved after the use of these contributions from the value of 1.3 kJ/mol to 0.8 kJ/mol (from 2.8% to 1.8%) in comparison when using only the first level groups. At the end only 55 compounds were suitable for refining by the third order groups. The results were refined from the values of 1.4 kJ/mol to 1.1 kJ/mol (from 2.5% to 2.1%). The total prediction error was cut down from the value of 1.3 kJ/mol to 1.0 kJ/mol for *AAE* and from 2.8% to 2.2 % for *ARE* after usage of all three-level groups, as it is obvious from this table. A similar pattern of results for other predicted

Estimation level *NC NG AAE* / kJ/mol *ARE* / % **FIRST 831 116 1.3 2.8 SECOND 486 91 0.8 1.8 486 compounds after only the FIRST (1.3) (2.8) THIRD 55 15 1.1 2.1 55 compounds After FIRST + SECOND (1.4) (2.5) ALL LEVELS 831 222 1.0 2.2** 

**Table 1.** Results for estimation of enthalpy of vaporization at 298.15 K (Kolská et al., 2005)

vaporization is presented in papers (Kolská, 2004; Kolská et al., 2005).

error and *ARE* is average relative error (Kolská et al., 2005).

properties are presented in Tables 2 and 3.

employed.

For more universal usage of computer fragmentation a suitable computer program has been developed (Kolská & Petrus, 2010). The main goal of the newly developed program is to provide a powerful tool for authors using group contribution methods for automatic fragmentation of chemical structures.

## **2.2. Estimation of selected physico-chemical properties of compounds**

The models for estimation of several physical or physico-chemical properties of pure organic compounds, such as enthalpy of vaporization, entropy of vaporization (Kolská et al., 2005), liquid heat capacity (Kolská et al., 2008) and a Nafion swelling (Randová et al, 2009) is presented below. Most of them are developed to estimate property at constant temperature 298.15 K and at normal boiling temperature (Kolská et al., 2005; Kolská et al., 2008; Randová et al., 2009), liquid heat capacity as a temperature dependent (Kolská et al., 2008). Hitherto unpublished results for estimation of a flash temperature or organic compounds and for determination of reactivation abbility of reactivators of acetylcholinesterase inhibited by inhibitor are presented in this chapter.

## *2.2.1. Enthalpy of vaporization and entropy of vaporization*

Enthalpy of vaporization, *H*V, entropy of vaporization, *S*V are important thermodynamic quantities of a pure compound, necessary for chemical engineers for modelling of many technological processes with evaporation, for extrapolation and prediction of vapour pressure data, or for estimation of the other thermodynamic properties, e.g. solubility parameters. It can be also used for extrapolation and prediction of vapour pressure data.

There are several methods to determine these properties, experiment-based and modelbased. Experiment-based methods, such as calorimetry or gas chromatography, provide generally reliable data of good accuracy. In the case of model-based methods, we can distinguish several groups of methods on the basis of the input information they require. Methods based on the Clausius-Clapeyron equation and vapour pressure data, variable empirical correlations, methods based on the tools of statistical thermodynamics or quantum mechanics. During last decades the group contribution methods are widely used for their universality and simplicity. More rich survey of estimation methods for enthalpy of vaporization is presented in papers (Kolská, 2004; Kolská et al., 2005).

140 Thermodynamics – Fundamentals and Its Application in Science

fragmentation of chemical structures.

inhibitor are presented in this chapter.

*2.2.1. Enthalpy of vaporization and entropy of vaporization* 

1990).

fragments for description of all chemical structures for compounds of input database; (iii) to divide all chemical structures into defined structural fragments correctly. It can be realized either manually, when databases of chemical structures and structural fragments inclusive several members only, either via computer program, when databases contain hundreds of compounds and structural fragments are more complex. To calculate group contribution parameters for thermophysical properties the ProPred program has been used (Marrero, 2002). Description and division chemical structures for other estimations have been made handy. Molecular structures for electronical splitting of all compounds from the basic data set were input in the Simplified Molecular Input Line Entry Specification, so-called the SMILES format (Weininger et al., 1986; Weininger, 1988; Weininger et al., 1989; Weininger,

For more universal usage of computer fragmentation a suitable computer program has been developed (Kolská & Petrus, 2010). The main goal of the newly developed program is to provide a powerful tool for authors using group contribution methods for automatic

The models for estimation of several physical or physico-chemical properties of pure organic compounds, such as enthalpy of vaporization, entropy of vaporization (Kolská et al., 2005), liquid heat capacity (Kolská et al., 2008) and a Nafion swelling (Randová et al, 2009) is presented below. Most of them are developed to estimate property at constant temperature 298.15 K and at normal boiling temperature (Kolská et al., 2005; Kolská et al., 2008; Randová et al., 2009), liquid heat capacity as a temperature dependent (Kolská et al., 2008). Hitherto unpublished results for estimation of a flash temperature or organic compounds and for determination of reactivation abbility of reactivators of acetylcholinesterase inhibited by

Enthalpy of vaporization, *H*V, entropy of vaporization, *S*V are important thermodynamic quantities of a pure compound, necessary for chemical engineers for modelling of many technological processes with evaporation, for extrapolation and prediction of vapour pressure data, or for estimation of the other thermodynamic properties, e.g. solubility parameters. It can be also used for extrapolation and prediction of vapour pressure data.

There are several methods to determine these properties, experiment-based and modelbased. Experiment-based methods, such as calorimetry or gas chromatography, provide generally reliable data of good accuracy. In the case of model-based methods, we can distinguish several groups of methods on the basis of the input information they require. Methods based on the Clausius-Clapeyron equation and vapour pressure data, variable empirical correlations, methods based on the tools of statistical thermodynamics or quantum mechanics. During last decades the group contribution methods are widely used

**2.2. Estimation of selected physico-chemical properties of compounds** 

Large databases of critically assessed data have been used for group contribution calculations: data for 831 compounds have been used for estimations at 298.15 K, and data for 589 compounds have been used for estimations at the normal boiling temperature. Organic compounds were divided into several classes (aliphatic and acyclic saturated and unsaturated hydrocarbons, aromatic hydrocarbons, halogenated hydrocarbons, compounds containing oxygen, nitrogen or sulphur atoms and miscellaneous compounds). Especially calorimetrically measured experimental data from the compilation (Majer & Svoboda, 1985) and data from some other sources mentioned in original paper (Kolská et al., 2005) were employed.

Results for estimations of these three properties are presented in the following Tables, Table 1 for enthalpy of vaporization at 298.15 K, Table 2 for enthalpy of vaporization at normal boiling temperature and Table 3 for entropy of vaporization at normal boiling temperature, *NC* means a number of compounds used for development of model and contributions calculation, *NG* is number of applied structural fragments (groups), *AAE* is absolute average error and *ARE* is average relative error (Kolská et al., 2005).

Table 1 shows that values for 831 compounds were used for estimation of enthalpy of vaporization at 298.15 K. When only first level groups were used, the prediction was performed with the *AAE* and the *ARE* of 1.3 kJ/mol and 2.8%, resp. Values of 116 group contributions were calculated at this step. Then, 486 compounds were described by the second order groups. Prediction of these compounds improved after the use of these contributions from the value of 1.3 kJ/mol to 0.8 kJ/mol (from 2.8% to 1.8%) in comparison when using only the first level groups. At the end only 55 compounds were suitable for refining by the third order groups. The results were refined from the values of 1.4 kJ/mol to 1.1 kJ/mol (from 2.5% to 2.1%). The total prediction error was cut down from the value of 1.3 kJ/mol to 1.0 kJ/mol for *AAE* and from 2.8% to 2.2 % for *ARE* after usage of all three-level groups, as it is obvious from this table. A similar pattern of results for other predicted properties are presented in Tables 2 and 3.


**Table 1.** Results for estimation of enthalpy of vaporization at 298.15 K (Kolská et al., 2005)


Group Contribution Methods for Estimation of Selected Physico-Chemical Properties of Organic Compounds 143

1 **CH3** 2 2.266 13 **aCH** 2 4.297

18 **aC-CH3** 2 8.121 107 **CH2 (cyclic)** 2 4.013 109 **C (cyclic)** 1 3.667

frequency

**subring** 2 6.190

**in 0..1) 2** 0.279

*<sup>p</sup>* is an important thermodynamic quantity of a pure

Group contribution value for Δ*H*V at 298.15 K / kJ/mol

fragment no. Group fragment definition Its

<sup>15</sup>**aC fused with nonaromatic** 

**THIRD 6 aC-CHncyc (fused rings) (n** 

presented in original paper (Kolská et al., 2005).

*2.2.2. Liquid heat capacity* 

Isobaric heat capacity of liquid *Cl*

**FIRST** *x***<sup>0</sup> Adjustable parameter 1 9.672** 

**value** 64.48 **SECOND 55 Ccyc-CH3 2** -1.355

**value** 63.59

 **19 AROM.FUSED[2]s1s4 1** -0.615

**Table 4.** Results for estimation of enthalpy of vaporization at 298.15 K for 1,1,4,7-tetramethylindane, aC

Group contribution methods by Ducros (Ducros et al., 1980; Ducros et al., 1981; Ducros & Sannier, 1982; Ducros & Sannier, 1984), by Chickos (Chickos et al., 1996), the empirical method, equations nos. 6 and 7 by Vetere (Vetere, 1995) and method by Ma and Zhao (Ma & Zhao, 1993) were used for comparison of results obtained in this work for estimation at 298.15 K and at normal boiling temperature, resp. While the new approach (Kolská et al., 2005) was applied for enthalpy of vaporization at 298.15 K for 831 organic compounds with the *ARE* of 2.2 %, the Ducros´s method could be applied to only 526 substances with the *ARE* of 3.1 % and the Chickos´s one for 800 compounds with the *ARE* of 4.7 %. For comparison of the results of estimation at the normal boiling temperature the new model provided for 589 compounds, the *ARE* was 2.6 % for enthalpy of vaporization and 1.8 % for entropy of vaporization (Kolská et al., 2005), the Vetere´s method was capable of estimating the values of for the same number of compounds with the following results: 4.6 % (Eq. 6, Vetere, 1995) and 3.4 % (Eq. 7, Vetere, 1995), model by Ma and Zhao (Ma & Zhao, 1993) for 549 compounds with the *ARE* of 2.5 %. The error for the enthalpy of vaporization, based on an independent set of various 74 compounds not used for correlation, has been determined to be 2.5%. Group contribution description and values for next usage of readers are

compound. Its value must be known for the calculation of an enthalpy difference required

**value** 62.13

means carbon atom in aromatic ring, abbreviation cyc is used for cycle (Kolská et al., 2005)

Estimation level

**Estimated** 

**Estimated** 

**Estimated** 

Group

**Table 2.** Results for estimation of enthalpy of vaporization at normal boiling temperature (Kolská et al., 2005)


**Table 3.** Results for estimation of entropy of vaporization at normal boiling temperature (Kolská et al., 2005)

As an example of the use of all three levels we have chosen the molecule of 1,1,4,7 tetramethylindane. Its chemical structure is shown in Fig. 3 and its division into individual first, second and third order groups with the result for vaporization enthalpy at 298.15 K is presented in Table 4. When we sum up all group contribution of the first level, we have got value of 64.48 kJ/mol. The first level provides an initial approximation with the relative error of estimated value exceeding 5 % in comparison with experimental value 61.37 kJ/mol. Estimated value of vaporization enthalpy at 298.15 K is then improved at the second level and further refined at the third level, after those the relative error reduced to 1.2 %.

**Figure 3.** Chemical structure of 1,1,4,7-tetramethylindane


**Table 4.** Results for estimation of enthalpy of vaporization at 298.15 K for 1,1,4,7-tetramethylindane, aC means carbon atom in aromatic ring, abbreviation cyc is used for cycle (Kolská et al., 2005)

Group contribution methods by Ducros (Ducros et al., 1980; Ducros et al., 1981; Ducros & Sannier, 1982; Ducros & Sannier, 1984), by Chickos (Chickos et al., 1996), the empirical method, equations nos. 6 and 7 by Vetere (Vetere, 1995) and method by Ma and Zhao (Ma & Zhao, 1993) were used for comparison of results obtained in this work for estimation at 298.15 K and at normal boiling temperature, resp. While the new approach (Kolská et al., 2005) was applied for enthalpy of vaporization at 298.15 K for 831 organic compounds with the *ARE* of 2.2 %, the Ducros´s method could be applied to only 526 substances with the *ARE* of 3.1 % and the Chickos´s one for 800 compounds with the *ARE* of 4.7 %. For comparison of the results of estimation at the normal boiling temperature the new model provided for 589 compounds, the *ARE* was 2.6 % for enthalpy of vaporization and 1.8 % for entropy of vaporization (Kolská et al., 2005), the Vetere´s method was capable of estimating the values of for the same number of compounds with the following results: 4.6 % (Eq. 6, Vetere, 1995) and 3.4 % (Eq. 7, Vetere, 1995), model by Ma and Zhao (Ma & Zhao, 1993) for 549 compounds with the *ARE* of 2.5 %. The error for the enthalpy of vaporization, based on an independent set of various 74 compounds not used for correlation, has been determined to be 2.5%. Group contribution description and values for next usage of readers are presented in original paper (Kolská et al., 2005).

### *2.2.2. Liquid heat capacity*

142 Thermodynamics – Fundamentals and Its Application in Science

2005)

2005)

Estimation level *NC NG AAE* / kJ/mol *ARE* / % **FIRST 589 111 1.2 3.2 SECOND 377 100 0.9 2.5 377 compounds after only the FIRST (1.2) (3.4) THIRD 23 14 1.1 2.1 23 compounds After FIRST + SECOND (1.3) (2.7) ALL LEVELS 589 225 0.9 2.6 Table 2.** Results for estimation of enthalpy of vaporization at normal boiling temperature (Kolská et al.,

Estimation level *NC NG AAE* / J/(K·mol) *ARE* / % **FIRST 589 111 2.1 2.2 SECOND 377 100 1.8 1.9 377 compounds after only the FIRST (2.3) (2.4) THIRD 23 14 1.9 1.9 23 compounds After FIRST + SECOND (2.5) (2.5) ALL LEVELS 589 225 1.7 1.8 Table 3.** Results for estimation of entropy of vaporization at normal boiling temperature (Kolská et al.,

As an example of the use of all three levels we have chosen the molecule of 1,1,4,7 tetramethylindane. Its chemical structure is shown in Fig. 3 and its division into individual first, second and third order groups with the result for vaporization enthalpy at 298.15 K is presented in Table 4. When we sum up all group contribution of the first level, we have got value of 64.48 kJ/mol. The first level provides an initial approximation with the relative error of estimated value exceeding 5 % in comparison with experimental value 61.37 kJ/mol. Estimated value of vaporization enthalpy at 298.15 K is then improved at the second level

and further refined at the third level, after those the relative error reduced to 1.2 %.

**Figure 3.** Chemical structure of 1,1,4,7-tetramethylindane

Isobaric heat capacity of liquid *Cl <sup>p</sup>* is an important thermodynamic quantity of a pure compound. Its value must be known for the calculation of an enthalpy difference required

for the evaluation of heating and cooling duties. Liquid heat capacity also serves as an input parameter for example in the calculation of temperature dependence of enthalpy of vaporization, for extrapolation of vapour pressure and the related thermal data by their simultaneous correlation, etc.

Group Contribution Methods for Estimation of Selected Physico-Chemical Properties of Organic Compounds 145

into the database. Compounds were divided into several families, such as hydrocarbons (saturated, cyclic, unsaturated, aromatic), halogenated hydrocarbons containing atoms of fluorine, chlorine, iodine, bromine, compounds containing oxygen (alcohols, phenols, ethers, ketones, aldehydes, acids, esters, heterocycles, other miscellaneous compounds), compounds containing nitrogen (amines, nitriles, heterocycles, other miscellaneous compounds), compounds containing sulphur (thioles, sulphides, heterocycles) and compounds containing silicon. Also data of organometallic compounds, compounds containing atoms of phosphorus and boron as well as some inorganic compounds were included. Also the list of families of compounds has been extended by a new group denoted as ionic liquids due to an increased interest in physical-chemical properties of these compounds in recent years. Data for approximately 40 ionic liquids were included. Altogether new data for almost 500 compounds, out of them about 250 compounds were not covered the in previous works (Zábranský et al., 1996; Zábranský et al., 2001), were

Prediction of the physical and chemical properties of pure substances and mixtures is a serious problem in the chemical process industries. One of the possibilities for prediction of the properties is the group contribution method. The anisotropic swelling of Nafion 112 membrane in pure organic liquids (solvents) was monitored by an optical method. Nafion is a poly(tetrafluoroethylene) (PTFE) polymer with perfluorovinyl pendant side chains ended by sulfonic acid groups. The PTFE backbone guarantees a great chemical stability in both reducing and oxidizing environments. Nafion membrane is important in chemical industry. It is used in fuel cells, membrane reactors, gas dryers, production of NaOH, etc. (Randová et al., 2009). In many applications Nafion is immersed in liquid, which significantly affects the membrane properties, namely swelling and transport properties of permeates (Randová et al., 2009). The change in the size of the membrane sample is taken as a measure of swelling. All experimental data were presented (Randová et al., 2009) and these results were used as a basis for application of the group contribution method to the relative expansion in equilibrium. From a total of 38 organic liquids under study, 26 were selected as an evaluational set from which the group and structural group contributions were assigned.

Due to limited number of compounds the more complex and known group contribution methods could not been taken. Authors have to develop new group structural fragments. The proposed method utilizes the four kinds of the structural units: constants, C-backbone, functional groups, and molecular geometry (Randová et al., 2009). Constants were presented as alcohols, ketones, ethers, esters, carboxylic acids. As C-backbone were taken groups CH3, -CH2- and >CH-. Functional groups as hydroxyl OH-, carbonyl -C=O and ether –O- and fragments for molecular geometry for cycles and branched chains were taken. The relative expansions *A*exp (for the drawing direction) and/or *B*exp (for the perpendicular direction) were calculated from the side lengths of the dry membrane sample (*a*10, *a*20, *b*10, *b*20) and the side lengths of the swelled membrane sample in equilibrium (*a*1, *a*2, *b*1, *b*2) according to the

compiled and critically evaluated.

The remaining 12 compounds were used as the testing set.

eq. (3). Description of mentioned sizes is presented in Fig. 4.

*2.2.3. Nafion swelling* 

In work (Kolská et al., 2008) the three-level group contribution method by Marrero and Gani (Marrero & Gani, 2001) mentioned above, which is able to calculate liquid heat capacity at only one temperature 298.15 K, was applied, and this approach has been extended to estimate heat capacity of liquids as a function of temperature. Authors have employed the combination of equation for the temperature dependence of heat capacity and the model by Marrero and Gani to develop new model (Kolská et al., 2008).

For parameter calculation 549 organic compounds of variable families of compounds were taken. In Table 5 are presented results of this estimation. *NG* means number of applied structural groups and *ARE* is the average relative error. More detailed results are presented in original paper (Kolská et al., 2008).


**Table 5.** Results for estimation of liquid heat capacity in temperature range of pure organic compounds (Kolská et al., 2008)

Also these estimated values were compared with results obtained by other estimation methods (Zábranský & Růžička, 2004; Chickos et al., 1993) for the basic dataset (compounds applied for parameter calculation) and also for 149 additional compounds not used in the parameter calculation (independent set). The first method (Zábranský & Růžička, 2004) was applied for all temperature range, the method proposed by Chickos (Chickos et al., 1993) was only used for temperature 298.15 K with the following results: new model was applied for 404 compounds with *ARE* of 1.5 %, the older method by Zábranský (Zábranský & Růžička, 2004) for the same number of compounds with the *ARE* of 1.8 % and the Chickos´s one for 399 compounds with the *ARE* of 3.9 %.

For the heat capacity of liquids authors used recommended data from the compilations by (Zábranský et al., 1996; Zábranský et al., 2001). Because the experimental data are presented permanently, it is necessary to update database of critically assessed and recommended data. Therefore authors´s work has been also aimed at updating and extending two publications prepared earlier within the framework of the IUPAC projects (Zábranský et al., 1996; Zábranský et al., 2001). These publications contain recommended data on liquid heat capacities for almost 2000 mostly organic compounds expressed in terms of parameters of correlating equations for temperature dependence of heat capacity. In new work (Zábranský et al., 2010b) authors collected experimental data on heat capacities of pure liquid organic and inorganic compounds that have melting temperature below 573 K published in the primary literature between 1999 and 2006. Data from more than 200 articles are included into the database. Compounds were divided into several families, such as hydrocarbons (saturated, cyclic, unsaturated, aromatic), halogenated hydrocarbons containing atoms of fluorine, chlorine, iodine, bromine, compounds containing oxygen (alcohols, phenols, ethers, ketones, aldehydes, acids, esters, heterocycles, other miscellaneous compounds), compounds containing nitrogen (amines, nitriles, heterocycles, other miscellaneous compounds), compounds containing sulphur (thioles, sulphides, heterocycles) and compounds containing silicon. Also data of organometallic compounds, compounds containing atoms of phosphorus and boron as well as some inorganic compounds were included. Also the list of families of compounds has been extended by a new group denoted as ionic liquids due to an increased interest in physical-chemical properties of these compounds in recent years. Data for approximately 40 ionic liquids were included. Altogether new data for almost 500 compounds, out of them about 250 compounds were not covered the in previous works (Zábranský et al., 1996; Zábranský et al., 2001), were compiled and critically evaluated.

### *2.2.3. Nafion swelling*

144 Thermodynamics – Fundamentals and Its Application in Science

Marrero and Gani to develop new model (Kolská et al., 2008).

simultaneous correlation, etc.

in original paper (Kolská et al., 2008).

one for 399 compounds with the *ARE* of 3.9 %.

(Kolská et al., 2008)

for the evaluation of heating and cooling duties. Liquid heat capacity also serves as an input parameter for example in the calculation of temperature dependence of enthalpy of vaporization, for extrapolation of vapour pressure and the related thermal data by their

In work (Kolská et al., 2008) the three-level group contribution method by Marrero and Gani (Marrero & Gani, 2001) mentioned above, which is able to calculate liquid heat capacity at only one temperature 298.15 K, was applied, and this approach has been extended to estimate heat capacity of liquids as a function of temperature. Authors have employed the combination of equation for the temperature dependence of heat capacity and the model by

For parameter calculation 549 organic compounds of variable families of compounds were taken. In Table 5 are presented results of this estimation. *NG* means number of applied structural groups and *ARE* is the average relative error. More detailed results are presented

**Table 5.** Results for estimation of liquid heat capacity in temperature range of pure organic compounds

Also these estimated values were compared with results obtained by other estimation methods (Zábranský & Růžička, 2004; Chickos et al., 1993) for the basic dataset (compounds applied for parameter calculation) and also for 149 additional compounds not used in the parameter calculation (independent set). The first method (Zábranský & Růžička, 2004) was applied for all temperature range, the method proposed by Chickos (Chickos et al., 1993) was only used for temperature 298.15 K with the following results: new model was applied for 404 compounds with *ARE* of 1.5 %, the older method by Zábranský (Zábranský & Růžička, 2004) for the same number of compounds with the *ARE* of 1.8 % and the Chickos´s

For the heat capacity of liquids authors used recommended data from the compilations by (Zábranský et al., 1996; Zábranský et al., 2001). Because the experimental data are presented permanently, it is necessary to update database of critically assessed and recommended data. Therefore authors´s work has been also aimed at updating and extending two publications prepared earlier within the framework of the IUPAC projects (Zábranský et al., 1996; Zábranský et al., 2001). These publications contain recommended data on liquid heat capacities for almost 2000 mostly organic compounds expressed in terms of parameters of correlating equations for temperature dependence of heat capacity. In new work (Zábranský et al., 2010b) authors collected experimental data on heat capacities of pure liquid organic and inorganic compounds that have melting temperature below 573 K published in the primary literature between 1999 and 2006. Data from more than 200 articles are included

Estimation level *NG ARE* / % First 111 1.9 Second 88 1.6 Third 25 1.5

Prediction of the physical and chemical properties of pure substances and mixtures is a serious problem in the chemical process industries. One of the possibilities for prediction of the properties is the group contribution method. The anisotropic swelling of Nafion 112 membrane in pure organic liquids (solvents) was monitored by an optical method. Nafion is a poly(tetrafluoroethylene) (PTFE) polymer with perfluorovinyl pendant side chains ended by sulfonic acid groups. The PTFE backbone guarantees a great chemical stability in both reducing and oxidizing environments. Nafion membrane is important in chemical industry. It is used in fuel cells, membrane reactors, gas dryers, production of NaOH, etc. (Randová et al., 2009). In many applications Nafion is immersed in liquid, which significantly affects the membrane properties, namely swelling and transport properties of permeates (Randová et al., 2009). The change in the size of the membrane sample is taken as a measure of swelling. All experimental data were presented (Randová et al., 2009) and these results were used as a basis for application of the group contribution method to the relative expansion in equilibrium. From a total of 38 organic liquids under study, 26 were selected as an evaluational set from which the group and structural group contributions were assigned. The remaining 12 compounds were used as the testing set.

Due to limited number of compounds the more complex and known group contribution methods could not been taken. Authors have to develop new group structural fragments. The proposed method utilizes the four kinds of the structural units: constants, C-backbone, functional groups, and molecular geometry (Randová et al., 2009). Constants were presented as alcohols, ketones, ethers, esters, carboxylic acids. As C-backbone were taken groups CH3, -CH2- and >CH-. Functional groups as hydroxyl OH-, carbonyl -C=O and ether –O- and fragments for molecular geometry for cycles and branched chains were taken. The relative expansions *A*exp (for the drawing direction) and/or *B*exp (for the perpendicular direction) were calculated from the side lengths of the dry membrane sample (*a*10, *a*20, *b*10, *b*20) and the side lengths of the swelled membrane sample in equilibrium (*a*1, *a*2, *b*1, *b*2) according to the eq. (3). Description of mentioned sizes is presented in Fig. 4.

**Figure 4.** Description of membrane dimensions *a*10, *a*20, *b*10, *b*20 side lengths of the dry membrane and *a*1, *a*2, *b*1, *b*2 side lengths of the swelled membrane in equilibrium (Randová et al., 2009)

$$u\_{a1} = \frac{a\_1 - a\_{10}}{a\_{10}} \quad , \quad u\_{a2} = \frac{a\_2 - a\_{20}}{a\_{20}} \quad , \quad u\_{b1} = \frac{b\_1 - b\_{10}}{b\_{10}} \quad , \quad u\_{b2} = \frac{b\_2 - b\_{20}}{b\_{20}} \tag{3}$$

11 1

*ijk T T NC MD z O E* 

where *Tf*° is an adjustable parameter, *Ci* is the first-order group contribution of type *i*, *Dj* is the second-order group contribution of type j, *Ek* is the third-order group contribution of the type *k* and *Ni*, *Mj*, *Ok* denote the number of occurrences of individual group contributions. Determination of contributions and of adjustable parameters was performed by a three-step regression procedure (Marrero & Gani, 2001). To evaluate the method error the following statistical quantities for each compound, absolute error *AE* (eq. 5) and relative error *ARE* (eq.

> f f exp

*T T*

*T* 

where subscripts "exp" and "est" mean experimental and estimated value of the flash temperature. 186 compounds from the basic data set were described by the first level group contributions (Kolská et al., 2005). From this large database only 114 compounds could be selected to be described by the original second level groups as defined earlier (Kolská et al., 2005). The total absolute and the relative average errors for all 186 compounds were equal to

6.3 K and 2.0 %. Results for individual estimation levels are presented in Table 6.

**Table 6.** Results for Estimation of flash temperature, *NC* is number of compounds

presented in Tables 7-9, resp.

f exp

**Estimation level** *NC AAE* **/ K** *ARE* **/ %**  FIRST 186 7.9 2.4 SECOND 105 5.7 1.8 THIRD 11 2.9 0.8 ALL LEVELS 186 6.5 2.0

Individual calculated structural fragments of the first, second and third estimation levels are

*ii j j kk*

(4)

ff f exp *est AE T T T* (5)

(6)

<sup>100</sup> *est*

*nm o*

hydrocarbons, aromatic hydrocarbons, alcohols, halogenated hydrocarbons, compounds containing oxygen, nitrogen or sulphur atoms and miscellaneous compounds. To collect more data for development of reliable method was not able due to that all databases collect some values obtained via closed cup type measuring method and others measured by open

Flash temperature was calculated by relationship (4) similar to eq. (1):

f

*RE T*

f f

cup one and data both of methods vary.

6) were used:

Calculation approach is presented in original paper (Randová et al., 2009). Value of ±1.5% in relative expansions was determined to be the experimental error. Maximum differences between the experimental and calculated relative expansions in both sets did not exceed the value of ±3% (Randová et al., 2009).

The values of 13 contributions for individual membrane relative expansions were determined on the basis of experimental data on relative expansion of Nafion membrane. Obtained results are in good agreement with experimental data. Maximum differences between experimental and calculated values are nearly the same, only twice greater than the experimental error.

#### *2.2.4. Flash temperature of organic compounds*

The flash temperature *Tf* and lower flammability limit (LFL) are one of the most important variables to consider when designing chemical processes involving flammable substances. These characteristics are not fundamental physical points. Flash temperature is one of the most important variables used to characterize fire and explosion hazard of liquids. The flash temperature is defined as the lowest temperature at which vapour above liquid forms flammable mixture with air at a pressure 101 325 Pa. Usual approach for flash temperature estimation is linear relationship between flash temperature *Tf* and normal boiling temperature *Tb* (Dvořák, 1993). Some models for flash temperature were presented earlier (Liaw & Chiu, 2006; Liaw et al., 2011).

In this work to estimate flash temperature of organic compound authors applied the modified group contribution method (Kolská et al., 2005) and calculate group contribution values data for 186 compounds (Steinleitner, 1980) were used. The database for calculation of parameters contains data for aliphatic and acyclic saturated and unsaturated hydrocarbons, aromatic hydrocarbons, alcohols, halogenated hydrocarbons, compounds containing oxygen, nitrogen or sulphur atoms and miscellaneous compounds. To collect more data for development of reliable method was not able due to that all databases collect some values obtained via closed cup type measuring method and others measured by open cup one and data both of methods vary.

Flash temperature was calculated by relationship (4) similar to eq. (1):

146 Thermodynamics – Fundamentals and Its Application in Science

1 10

 , 2 20 2

*a*

*u*

*a a*

*a*

10

*2.2.4. Flash temperature of organic compounds* 

(Liaw & Chiu, 2006; Liaw et al., 2011).

1

value of ±3% (Randová et al., 2009).

experimental error.

*a*

*u*

**Figure 4.** Description of membrane dimensions *a*10, *a*20, *b*10, *b*20 side lengths of the dry membrane and *a*1,

 , 1 10 1

Calculation approach is presented in original paper (Randová et al., 2009). Value of ±1.5% in relative expansions was determined to be the experimental error. Maximum differences between the experimental and calculated relative expansions in both sets did not exceed the

The values of 13 contributions for individual membrane relative expansions were determined on the basis of experimental data on relative expansion of Nafion membrane. Obtained results are in good agreement with experimental data. Maximum differences between experimental and calculated values are nearly the same, only twice greater than the

The flash temperature *Tf* and lower flammability limit (LFL) are one of the most important variables to consider when designing chemical processes involving flammable substances. These characteristics are not fundamental physical points. Flash temperature is one of the most important variables used to characterize fire and explosion hazard of liquids. The flash temperature is defined as the lowest temperature at which vapour above liquid forms flammable mixture with air at a pressure 101 325 Pa. Usual approach for flash temperature estimation is linear relationship between flash temperature *Tf* and normal boiling temperature *Tb* (Dvořák, 1993). Some models for flash temperature were presented earlier

In this work to estimate flash temperature of organic compound authors applied the modified group contribution method (Kolská et al., 2005) and calculate group contribution values data for 186 compounds (Steinleitner, 1980) were used. The database for calculation of parameters contains data for aliphatic and acyclic saturated and unsaturated

*b*

*u*

10

 , 2 20 2

*b*

*u*

20

(3)

*b b*

*b*

*b b*

*b*

*a*2, *b*1, *b*2 side lengths of the swelled membrane in equilibrium (Randová et al., 2009)

20

*a a*

*a*

$$T\_{\mathbf{f}} = T\_{\mathbf{f}} \, ^\circ + \sum\_{i=1}^n \mathbf{N}\_i \mathbf{C}\_i + \sigma \sigma \sum\_{j=1}^m \mathbf{M}\_j \mathbf{D}\_j + \mathbf{z} \sum\_{k=1}^o \mathbf{O}\_k \mathbf{E}\_k \tag{4}$$

where *Tf*° is an adjustable parameter, *Ci* is the first-order group contribution of type *i*, *Dj* is the second-order group contribution of type j, *Ek* is the third-order group contribution of the type *k* and *Ni*, *Mj*, *Ok* denote the number of occurrences of individual group contributions. Determination of contributions and of adjustable parameters was performed by a three-step regression procedure (Marrero & Gani, 2001). To evaluate the method error the following statistical quantities for each compound, absolute error *AE* (eq. 5) and relative error *ARE* (eq. 6) were used:

$$AE\left[\left.T\_{\mathbf{f}}\right]\right] = \left| \left(T\_{\mathbf{f}}\right)\_{\exp} - \left(T\_{\mathbf{f}}\right)\_{\mathrm{est}} \right| \tag{5}$$

$$RE\left[\left.T\_{\rm f}\right] = \left(\frac{\left|\left(T\_{\rm f}\right)\_{\rm exp} - \left(T\_{\rm f}\right)\_{est}\right|}{\left(\left.T\_{\rm f}\right)\_{\rm exp}}\right|\right) \cdot 100\tag{6}$$

where subscripts "exp" and "est" mean experimental and estimated value of the flash temperature. 186 compounds from the basic data set were described by the first level group contributions (Kolská et al., 2005). From this large database only 114 compounds could be selected to be described by the original second level groups as defined earlier (Kolská et al., 2005). The total absolute and the relative average errors for all 186 compounds were equal to 6.3 K and 2.0 %. Results for individual estimation levels are presented in Table 6.


**Table 6.** Results for Estimation of flash temperature, *NC* is number of compounds

Individual calculated structural fragments of the first, second and third estimation levels are presented in Tables 7-9, resp.


rings) -6.04 CH multiring 0.98 AROM.FUSED[4a] -26.07

In the last years regarding to valid legislation on dangerous compounds it is necessary to know many of important characteristics of chemical compounds. Due to this new models for their estimation were developed. New models for estimation of reactivation ability of reactivators for acetylcholinesterase inhibited by (i) chloropyrifos (*O*,*O*-diethyl *O*-3,5,6 trichloropyridin-2-yl phosphorothioate) as a representative of organophosphate insecticide and by (ii) sarin ((*RS*)-propan-2-yl methylphosphonofluoridate) as a representative of nerve agent is now presented. Both of these family compounds, organophosphate pesticide and nerve agent, are highly toxic and have the same effect to living organisms, which is based on an inhibition of acetylcholinesterase (AChE). New compounds able to reactivate the inhibited AChE, so-called reactivators of AChE, are synthesized. Reactivation ability of these reactivators is studied using standard reactivation *in vitro* test (Kuča & Kassa, 2003). Reactivation ability of reactivators means the percentage of original activity of AChE (Kuča & Patočka, 2004). New models for determination values of reactivation ability of reactivators AChE inhibited by (i) chloropyrifos and (ii) sarin have been developed. Concentration of reactivators was *c*=110-3 moldm-3. In comparison with previous cases (estimations of thermophysical properties) authors have only less experimental data for development of model (about 20 for each of cases). Due to their long names and complex chemical structures these compounds in this chapter only are presented as their codes taken from original papers (Kuča & Kassa, 2003; Kuča et al., 2003a; Kuča et al., 2003b; Kuča et al., 2003c; Kuča & Patočka, 2004; Kuča & Cabal, 2004a; Kuča & Cabal, 2004b; Kuča et. al., 2006. Data of reactivation ability for these reactivators were given by the mentioned author team (Kuča et al.). Classical group contribution method includes groups describing some central atom, central atom with its bonds, or central atom with its nearest surrounding. However these models commonly used experimental data of hundreds or thousands compounds for parameters calculation. Due to for much small database in these cases it was necessary to design new fragments depending on the molecular structures available compounds. Structural fragments in this work cover larger and more complex part of molecules in comparison with other papers focused to group contribution methods. Reactivation potency

**/ K** 

12.92

(different rings) -1.87 AROM.FUSED[2] 8.69

**Structural fragment** 

**Contribution / K** 

**/ K Structural fragment Contribution**

aC-CHm-aC (different rings) (m in 0..2)

**Table 9.** Group contribution of the third level for estimation of flash temperature

*2.2.5. Reactivation ability of some reactivators of acetylcholinesterase* 

is given in the group contribution method by the following relation, eq. 7:

p p 1

*i i*

(7)

*n*

*i R xR* 

**Structural fragment** 

OH-(CHn)m-OH

aC-aC (different

aC-CHncyc (fused

rings) (n in 0..1) -4.34

**Contribution**

(m>2, n in 0..2) -33.36 CHcyc-CHcyc



**Table 8.** Group contribution of the second level for estimation of flash temperature


**Table 9.** Group contribution of the third level for estimation of flash temperature

#### *2.2.5. Reactivation ability of some reactivators of acetylcholinesterase*

148 Thermodynamics – Fundamentals and Its Application in Science

**Contribution / K** 

**Structural fragment** 

**Table 7.** Group contribution of the first level for estimation of flash temperature

**Structural fragment** 

CHm=CHn-Cl

aC-CHn-X (n in

aC-CHn-OH (n

CH(CH3)C(CH3)2 22.69 aC-CH(CH3)2 1.80 AROMRINGs1s2 -2.24

(k,m,n,p in 0..2) 0.53 aC-CF3 0.13 AROMRINGs1s3 1.72

0..2) 2.23 CHcyc-CH3 -5.70 AROMRINGs1s2s4 -2.84

0..2; p in 0..1) 3.78 CHcyc-CH2 17.80 AROMRINGs1s2s4s5 6.22

(m,n in 0..2) 13.41 CHcyc-Cl 1.59 (3 F) -0.13

8.52

CHcyc-CH=CHn (n in 1..2)

CHcyc-C=CHn

**Table 8.** Group contribution of the second level for estimation of flash temperature

(CHn=C)cyc-

**Contribution / K** 

**Contribution / K** 

(m,n in 0..2) -0.50 CHcyc-OH -1.31

1..2) X: Halogen 1.04 Ccyc-CH3 -0.23

CH3 (n in 0..2) 0.46 AROMRINGs1s4 -0.84

(n in 1..2) -1.20 (CH=CHOCH=CH)cyc -4.24

(perFlouro) 2.66E-17

in 1..2) 5.51 >Ncyc-CH3 -1.11E-17

*Tf*° 194.35 aCH 12.39 aC-OH 85.26 CH3 5.38 aC 21.15 CH2Cl 45.56 CH2 13.28 aC 26.84 CHCl 42.83 CH 15.77 aC 31.53 CCl 37.51 C 13.59 aN 25.51 CHCl2 67.42 CH2=CH 11.51 aC-CH3 28.32 CCl3 100.38 CH=CH 34.57 aC-CH2 37.11 aC-Cl 50.70 CH2=C 19.63 aC-CH 37.38 aC-F 53.47 CH=C 27.52 aC-C 19.88 aC-Br 64.97 C=C 29.27 aC-CH=CH2 50.66 -I 78.85 CH#C 14.94 OH 64.22 -Br 59.25 C#C 15.04 -SH 55.62 CH=CH 18.21 -F 2.99 CH2 10.89 CH=C 37.59 -Cl 29.32 CH 22.89 N 52.99 CH2SH 56.33 C -9.50 O -3.39

**Structural fragment** 

**Structural fragment Contribution** 

PYRIDINEs3s5 9.98E-18

 **/ K** 

**Contribution / K** 

**Structural fragment** 

**Structural fragment Contribution**

(CH3)2CH -1.45

(CH3)3C -3.98

CH(CH3)CH(CH3) 7.68

CHn=CHm-CHp=CHk

CH3-CHm=CHn (m,n in 0..2) -2.01

CH2-CHm=CHn (m,n in

CHp-CHm=CHn (m,n in

CHOH -3.92

COH -4.98

CHm(OH)CHn(OH)

 **/ K** 

In the last years regarding to valid legislation on dangerous compounds it is necessary to know many of important characteristics of chemical compounds. Due to this new models for their estimation were developed. New models for estimation of reactivation ability of reactivators for acetylcholinesterase inhibited by (i) chloropyrifos (*O*,*O*-diethyl *O*-3,5,6 trichloropyridin-2-yl phosphorothioate) as a representative of organophosphate insecticide and by (ii) sarin ((*RS*)-propan-2-yl methylphosphonofluoridate) as a representative of nerve agent is now presented. Both of these family compounds, organophosphate pesticide and nerve agent, are highly toxic and have the same effect to living organisms, which is based on an inhibition of acetylcholinesterase (AChE). New compounds able to reactivate the inhibited AChE, so-called reactivators of AChE, are synthesized. Reactivation ability of these reactivators is studied using standard reactivation *in vitro* test (Kuča & Kassa, 2003). Reactivation ability of reactivators means the percentage of original activity of AChE (Kuča & Patočka, 2004). New models for determination values of reactivation ability of reactivators AChE inhibited by (i) chloropyrifos and (ii) sarin have been developed. Concentration of reactivators was *c*=110-3 moldm-3. In comparison with previous cases (estimations of thermophysical properties) authors have only less experimental data for development of model (about 20 for each of cases). Due to their long names and complex chemical structures these compounds in this chapter only are presented as their codes taken from original papers (Kuča & Kassa, 2003; Kuča et al., 2003a; Kuča et al., 2003b; Kuča et al., 2003c; Kuča & Patočka, 2004; Kuča & Cabal, 2004a; Kuča & Cabal, 2004b; Kuča et. al., 2006. Data of reactivation ability for these reactivators were given by the mentioned author team (Kuča et al.). Classical group contribution method includes groups describing some central atom, central atom with its bonds, or central atom with its nearest surrounding. However these models commonly used experimental data of hundreds or thousands compounds for parameters calculation. Due to for much small database in these cases it was necessary to design new fragments depending on the molecular structures available compounds. Structural fragments in this work cover larger and more complex part of molecules in comparison with other papers focused to group contribution methods. Reactivation potency is given in the group contribution method by the following relation, eq. 7:

$$\mathcal{R}\_{\mathbf{p}} = \sum\_{i=1}^{n} \mathbf{x}\_{i} \cdot \mathcal{R}\_{\mathbf{p}^{i}} \tag{7}$$

where *R*p*<sup>i</sup>* is value of individual fragment *i* presented in molecule by which it contributes to total value of *R*p, *x* is number of frequency of this fragment *i* in molecule. Parameters *R*p*<sup>i</sup>* were obtained by minimization function *SR*p, eq. 8:

$$\mathcal{S}\_{\text{Rp}} = \sum\_{i=1}^{m} \left( \mathcal{R}\_{\text{pi}, \text{calc}} - \mathcal{R}\_{\text{pi}, \text{exp}} \right)^{2} \tag{8}$$

Group Contribution Methods for Estimation of Selected Physico-Chemical Properties of Organic Compounds 151

26.365 P*<sup>9</sup>*

15.365 P*<sup>10</sup>*

46.792 P*<sup>11</sup>*





P*<sup>17</sup>*

**Table 10.** List of structural fragments and their values for estimation of reactivation ability of

P*<sup>14</sup>*

**/ % no. Fragment description** *R***p***<sup>i</sup>*

two oxime groups in positions *o*- due to a quarternary nitrogen atom in aromatic ring

two oxime groups in positions *m*- due to a quarternary nitrogen atom N+ in aromatic ring

two oxime groups in positions *p*- due to a quarternary nitrogen atom N+ in aromatic ring

two oxime groups, one in position *o*-, other in position *m*- due to a quarternary nitrogen atom N+ in aromatic ring

two oxime groups, one in position *o*-, other in position *p*- due to a quarternary nitrogen atom N+ in aromatic ring

two oxime groups, one in position *m*-, other in position *p*- due to a quarternary nitrogen atom N+ in aromatic ring

Oxygen atom O bonded in aliphatic ring between two aromatic rings

Cycle between two aromatic rings

Double bond between two aromatic rings

**/ %** 

26.580

15.737

47.105

52.105

25.842

56.105




**no. Fragment description** *R***p***<sup>i</sup>*

Oxime group (=NOH) in position *o*due to a quarternary nitrogen N+ atom in aromatic ring

Oxime group (=NOH) in position *p*due to a quarternary nitrogen N+ atom in aromatic ring

Other quarternary nitrogen atom N+ with 4 CHx- groups in molecule, in aliphatic ring bonded to nitrogen atom N in aromatic ring

Number of members bonded in aliphatic ring after the group P*<sup>3</sup>*

Cycle ring bonded to nitrogen atom N in aromatic ring

Oxygen atom O bonded in aliphatic ring bonded to one aromatic ring

Presence of other aliphatic ring

Number of members bonded in aliphatic ring following group N-CHx- (nitrogen atom N is a part of aromatic ring), (which are not included in other groups)

reactivators for acetylcholinesterase inhibited by chloropyrifos

bonded to aromatic one 88.073 P*<sup>15</sup>*

P*1*

P*2*

P*3*

P*4*

P*5*

P*6*

P*7*

P*8*

where suffix exp presents experimental data and suffix calc the calculated values of *R*p, *m* is number of compounds in dataset. The results obtained by this new approach were compared with experimental data using the following statistical quantities - an absolute error of individual compounds *AE* (eq. 9) and the average absolute error of dataset *AAE* (eq. 10):

$$AE\_i = R\_{\text{pi}, \text{calc}} - R\_{\text{pi}, \text{exp}} \tag{9}$$

$$AAE = \sum\_{i=1}^{m} \left( \frac{\left| R\_{\text{pi},\text{calc}} - R\_{\text{pi},\text{exp}} \right|}{m} \right) \tag{10}$$

Parameters of new model were calculated from the experimental data of the basic dataset. For model for reactivators AChE inhibited by chloropyrifos the input database included data of reactivation ability *R*p for 24 reactivators (K 135, K 078, TO 096, TO 100, K 076, TO 094, TO 063, TO 097, TO 098, K 347, TO 231, K 117, K 074, K 033, K 106, K 107, K 110, K 114, HI-6, K 282, K 283, K 285, K 129, K 099) of concentration *c*=110-3 moldm-3. Values for 17 groups with the *AAE* of 1.85 % of *R*p were calculated. Designed groups with their calculated values of *R*p*<sup>i</sup>* are presented in Table 10. These calculated parameters were tested on the test set of 5 independent compounds (TO 238, K 111, K 113, Methoxime, K 280) of which experimental data were not applied to group contributions determination. The *AAE* of *R*<sup>p</sup> prediction for this test-set was 1.45 %. Table 11 presents experimental data and predicted values for these 5 independent compounds. Also illustration of usage of this method for two compounds from this test set is added below.

As it is clear from Table 10 the highest values of contributions are given for fragments P*3*, P*<sup>7</sup>* for monoaromatic reactivators and P*11*, P*12* and P*14* for two aromatic rings in reactivator molecule. On the other hand the smallest contribution (the negative ones) to total value of reactivation ability yields fragments P*5* a P*6* for monoaromatic compounds and P*16* and P*<sup>17</sup>* for two aromatic ring reactivators. These values resulted in fact that reactivation ability of new reactivators for reactivation AChE inhibited by chloropyrifos should be increased by presence of the following functional groups in molecules: another quarternary nitrogen atom in aliphatic ring bonded to aromatic quarternary nitrogen atom, the oxime groups in *para*- or *meta*- positions and presence of other aliphatic rings bonded to aromatic ring in other position than quarternary nitrogen and oxime groups. In all cases it is clear that reactivation ability decreases with presence of cycle ring, double bond and also in a less range with the presence of oxygen atoms presented in molecules. Also *ortho*- position of oxime group does not contribute positively.

were obtained by minimization function *SR*p, eq. 8:

compounds from this test set is added below.

oxime group does not contribute positively.

10):

where *R*p*<sup>i</sup>* is value of individual fragment *i* presented in molecule by which it contributes to total value of *R*p, *x* is number of frequency of this fragment *i* in molecule. Parameters *R*p*<sup>i</sup>*

p p ,calc p ,exp

where suffix exp presents experimental data and suffix calc the calculated values of *R*p, *m* is number of compounds in dataset. The results obtained by this new approach were compared with experimental data using the following statistical quantities - an absolute error of individual compounds *AE* (eq. 9) and the average absolute error of dataset *AAE* (eq.

*R ii*

1

1

*i*

*AAE*

*i S RR* 

*m*

p ,calc p ,exp

*<sup>m</sup> i i*

 *m* 

Parameters of new model were calculated from the experimental data of the basic dataset. For model for reactivators AChE inhibited by chloropyrifos the input database included data of reactivation ability *R*p for 24 reactivators (K 135, K 078, TO 096, TO 100, K 076, TO 094, TO 063, TO 097, TO 098, K 347, TO 231, K 117, K 074, K 033, K 106, K 107, K 110, K 114, HI-6, K 282, K 283, K 285, K 129, K 099) of concentration *c*=110-3 moldm-3. Values for 17 groups with the *AAE* of 1.85 % of *R*p were calculated. Designed groups with their calculated values of *R*p*<sup>i</sup>* are presented in Table 10. These calculated parameters were tested on the test set of 5 independent compounds (TO 238, K 111, K 113, Methoxime, K 280) of which experimental data were not applied to group contributions determination. The *AAE* of *R*<sup>p</sup> prediction for this test-set was 1.45 %. Table 11 presents experimental data and predicted values for these 5 independent compounds. Also illustration of usage of this method for two

As it is clear from Table 10 the highest values of contributions are given for fragments P*3*, P*<sup>7</sup>* for monoaromatic reactivators and P*11*, P*12* and P*14* for two aromatic rings in reactivator molecule. On the other hand the smallest contribution (the negative ones) to total value of reactivation ability yields fragments P*5* a P*6* for monoaromatic compounds and P*16* and P*<sup>17</sup>* for two aromatic ring reactivators. These values resulted in fact that reactivation ability of new reactivators for reactivation AChE inhibited by chloropyrifos should be increased by presence of the following functional groups in molecules: another quarternary nitrogen atom in aliphatic ring bonded to aromatic quarternary nitrogen atom, the oxime groups in *para*- or *meta*- positions and presence of other aliphatic rings bonded to aromatic ring in other position than quarternary nitrogen and oxime groups. In all cases it is clear that reactivation ability decreases with presence of cycle ring, double bond and also in a less range with the presence of oxygen atoms presented in molecules. Also *ortho*- position of

*R R*

2

(8)

*AE R R ii i* p ,calc p ,exp (9)

(10)


**Table 10.** List of structural fragments and their values for estimation of reactivation ability of reactivators for acetylcholinesterase inhibited by chloropyrifos



**/ % no. Fragment description** *R***p***<sup>i</sup>*

Other member of ring between two quarternary nitrogen atoms N+ or/and bonded at the last quarternary nitrogen atom N+ of molecule

Presence of oxygen atom O in molecule other than mentioned in the following group 2.16

Presence of group -NH*x*(*x* = 0, .., 2) in

Presence of a double bond between two carbon atoms in a ring between two quarternary nitrogen atoms N+ in molecule

molecule -12.20

46.03 P*<sup>9</sup>* Presence of group >C=O in molecule 7.88

**/ %** 


1.66

**no. Fragment description** *R***p***<sup>i</sup>*

Quarternary nitrogen atom N inclusive in aromatic ring

Presence of oxime group

*ortho*- position of substituent on aromatic ring

*meta*- position of substituent on aromatic ring

*para*- position of substituent on aromatic ring

Presence of cycle in a

acetylcholinesterase inhibited by sarin

molecule -10.03

reactivators for acetylcholinesterase inhibited by sarin

22.50 P*<sup>7</sup>*


14.49 P*<sup>10</sup>*

40.01 P*<sup>11</sup>*

**Table 12.** List of structural fragments and their values for estimation of reactivation ability of

As it is shown in Table 12, the highest and the positive values of group contributions are given for fragments P*1*, P*3*-P*5*, P8 and P9. On the other hand the smallest contribution (the negative ones) to the total value of reactivation ability yield fragments P*6*, P*7* and P*10*. Also the value of fragment P*2* for oxime group seems to have a negative effect to the total value but it should be said, that the oxime group has to be summed up with some group for its position on aromatic ring. It results in a fact that the oxime group in *meta*- position has the negative influence to the total value of reactivation ability, on the other hand the total value of *Rp* increases with oxime group in positions of *ortho*- or *para*-. These values resulted in fact that reactivation ability of new reactivators for reactivation AChE inhibited by sarin should be increased by the presence of the following function groups in molecules: another quarternary nitrogen atom in aromatic ring, the oxime groups in *ortho*- or *para*– positions, presence of oxygen atom or group >C=O in molecule. It is clear that reactivation ability decreases with presence of cycle ring and also with presence of the group NH*x* (*x* = 0, .., 2) in molecules. Also *meta*- position of oxime group, as same as the longer ring (CH*x*)*n* (*x* = 0, .., 2) bonded at quarternary nitrogen atoms, that means group P*7*, do not contribute positively.

**Reactivator** *R***p,exp / %** *R***p,calc / % Deviation / %**  TO 055 30.00 32.38 2.38 TO 058 25.00 27.63 2.63 K 197 4.00 4.08 0.08 Obidoxime 41.00 44.70 3.70

**Table 13.** Results for estimation of reactivation ability of the test dataset of 4 reactivators of

P*1*

P*2*

P*3*

P*4*

P*5*

P*6*

**Table 11.** Results for estimation of reactivation ability of the test dataset of 5 reactivators of acetylcholinesterase inhibited by chloropyrifos

Illustration of new method for reactivation ability prediction of two reactivators (TO 238 and K 280) of which experimental data were not used for parameters calculation follows.

**Figure 5.** Chemical structure of two reactivators of acetylcholinesterase signed as TO 238 and K 280

Example of usage of the new model for reactivation ability prediction for TO 280 reactivator:

*Rp*,calc(TO 238) = P*1* + 2P*6* + P*7* = 26.365 + 2(-32.437) + 88.073 = 49.546 % *Rp*,exp(TO 238) = 48.00 % *AE* = *Rp*,calc(TO 238) - *Rp*,exp(TO 238) = 1.55 %.

Example of usage of the new model for reactivation ability prediction for K 280 reactivator:

*Rp*,calc(K 280) = P*9* + P*17* = 26.580 + (-22.105) = 4.475 % *Rp*,exp(K 280) = 4.00 % *AE* = *Rp*,calc(K 280) - *Rp*,exp(K 280) = 0.48 %.

For model development for reactivators AChE inhibited by sarin the input database included data of reactivation ability *Rp* for 18 reactivators (K 127, K 128, K 141, K 276, K 311, K 277, K 077, K 142, K 131, K 100, K 233, K 194, K 191, K 067, K 119, K 053, Pralidoxime, HI-6) of concentration *c*=110-3 moldm-3 were taken. Due to the smaller database in comparison with the chloropyrifos-inhibited case it was not possible to apply the same structural fragments. Values for 11 new structural different groups with the *AAE* of 3.39 % of *R*p have been calculated. Designed groups with their calculated values of *R*p*<sup>i</sup>* are presented in Table 12. These calculated parameters were tested on the test set of 4 independent compounds (TO 055, TO 058, K 197, Obidoxime) of which experimental data were not applied to group contributions determination. The *AAE* of *R*p prediction for this test-set was 2.18 %. Table 13 presents experimental data and predicted values for 4 independent compounds.

152 Thermodynamics – Fundamentals and Its Application in Science

acetylcholinesterase inhibited by chloropyrifos

*Rp*,exp(TO 238) = 48.00 %

*Rp*,exp(K 280) = 4.00 %

*AE* = *Rp*,calc(TO 238) - *Rp*,exp(TO 238) = 1.55 %.

*AE* = *Rp*,calc(K 280) - *Rp*,exp(K 280) = 0.48 %.

*Rp*,calc(K 280) = P*9* + P*17* = 26.580 + (-22.105) = 4.475 %

**Reactivator** *R***p,exp / %** *R***p,calc / % Deviation / %**  TO 238 48.00 49.55 1.55 K 111 8.00 5.26 -2.74 K 113 37.00 36.63 -0.37 Methoxime 45.00 47.11 2.11 K 280 4.00 4.48 0.48

Illustration of new method for reactivation ability prediction of two reactivators (TO 238 and K 280) of which experimental data were not used for parameters calculation follows.

**Figure 5.** Chemical structure of two reactivators of acetylcholinesterase signed as TO 238 and K 280

TO 238 K 280

*Rp*,calc(TO 238) = P*1* + 2P*6* + P*7* = 26.365 + 2(-32.437) + 88.073 = 49.546 %

Example of usage of the new model for reactivation ability prediction for TO 280 reactivator:

Example of usage of the new model for reactivation ability prediction for K 280 reactivator:

For model development for reactivators AChE inhibited by sarin the input database included data of reactivation ability *Rp* for 18 reactivators (K 127, K 128, K 141, K 276, K 311, K 277, K 077, K 142, K 131, K 100, K 233, K 194, K 191, K 067, K 119, K 053, Pralidoxime, HI-6) of concentration *c*=110-3 moldm-3 were taken. Due to the smaller database in comparison with the chloropyrifos-inhibited case it was not possible to apply the same structural fragments. Values for 11 new structural different groups with the *AAE* of 3.39 % of *R*p have been calculated. Designed groups with their calculated values of *R*p*<sup>i</sup>* are presented in Table 12. These calculated parameters were tested on the test set of 4 independent compounds (TO 055, TO 058, K 197, Obidoxime) of which experimental data were not applied to group contributions determination. The *AAE* of *R*p prediction for this test-set was 2.18 %. Table 13

presents experimental data and predicted values for 4 independent compounds.

**Table 11.** Results for estimation of reactivation ability of the test dataset of 5 reactivators of


**Table 12.** List of structural fragments and their values for estimation of reactivation ability of reactivators for acetylcholinesterase inhibited by sarin

As it is shown in Table 12, the highest and the positive values of group contributions are given for fragments P*1*, P*3*-P*5*, P8 and P9. On the other hand the smallest contribution (the negative ones) to the total value of reactivation ability yield fragments P*6*, P*7* and P*10*. Also the value of fragment P*2* for oxime group seems to have a negative effect to the total value but it should be said, that the oxime group has to be summed up with some group for its position on aromatic ring. It results in a fact that the oxime group in *meta*- position has the negative influence to the total value of reactivation ability, on the other hand the total value of *Rp* increases with oxime group in positions of *ortho*- or *para*-. These values resulted in fact that reactivation ability of new reactivators for reactivation AChE inhibited by sarin should be increased by the presence of the following function groups in molecules: another quarternary nitrogen atom in aromatic ring, the oxime groups in *ortho*- or *para*– positions, presence of oxygen atom or group >C=O in molecule. It is clear that reactivation ability decreases with presence of cycle ring and also with presence of the group NH*x* (*x* = 0, .., 2) in molecules. Also *meta*- position of oxime group, as same as the longer ring (CH*x*)*n* (*x* = 0, .., 2) bonded at quarternary nitrogen atoms, that means group P*7*, do not contribute positively.


**Table 13.** Results for estimation of reactivation ability of the test dataset of 4 reactivators of acetylcholinesterase inhibited by sarin

Illustration of new method for reactivation ability prediction of two reactivators (TO 055 and TO 058) of which experimental data were not used for parameters calculation follows.

Group Contribution Methods for Estimation of Selected Physico-Chemical Properties of Organic Compounds 155

without any other input information. The presented models have been developed for estimation of many variable properties, enthalpy of vaporization, entropy of vaporization, liquid heat capacity, swelling of Nafion, flash temperature and reactivation ability of reactivators of acetylcholinesterase inhibited by organophosphate compounds. Proposed models and their structural fragments, accuracy and reliability depend mainly on frequency of input data and their accuracy, correctness and reliability. The most of presented models of group contribution methods, not only in the cases presented in this chapter, can be applied for the wide variety of organic compounds, when groups describing these molecules are presented. Some of models can be applied from only limited families of compounds due to their parameters were calculated only for limited database of compounds. Group contribution methods can be applied either for estimation or prediction of properties at one temperature or as a temperature function depending on their development. The accuracy of developed models is the higher, the input database is more

*Faculty of Science, J. E. Purkyně University, Usti nad Labem, Czech Republic* 

This work was supported by the GA CR under the project P108/12/G108. Authors also thank to Ing. Michal Karlík from ICT Prague, Czech Republic, for data for flash temperature estimation and Prof. Kamil Kuča from Department of Toxicology, Faculty of Millitary

Baum, E. J. (1997). *Chemical Property Estimation: Theory and Application*, CRC Press LLC, ISBN

Poling, B. E.; Prausnitz, J. M. & O´Connell, J. P. (2001). *The Properties of Gases and Liquids*, fifth

González, H. E.; Abildskov, J.; & Gani, R. (2007). Computer-aided framework for pure component properties and phase equilibria prediction for organic systems. *Fluid Phase* 

Majer, V.; Svoboda, V. & Pick, J. (1989). *Heats of Vaporization of Fluids*. Elsevier, ISBN 0-444-

Majer, V. & Svoboda V. (1985). *Enthalpies of Vaporization of Organic Compounds, Critical Review and Data Compilation*. IUPAC. Chemical Data Series No. 32, Blackwell, Oxford, ISBN 0-

Health Science Hradec Kralove, Czech Republic, for data on reactivation ability.

edition. McGraw-Hill, ISBN 0-07-011682-2, New York, USA

*Equilibria*, Vol.261, No.1-2, pp. 199–204, ISSN 0378-3812

reliable.

**Author details** 

**Acknowledgement** 

**4. References** 

632-01529-2

Milan Zábranský and Alena Randová

978-0873719384, Boca Raton, USA

98920-X. Amsterdam, Netherlands

*Institute of Chemical Technology, Prague, Czech Republic* 

Zdeňka Kolská

**Figure 6.** Chemical structure of two reactivators of acetylcholinesterase signed as TO 055 and TO 058

Example of usage of the new model for reactivation ability prediction for TO 055 reactivator: *Rp*,calc(TO 055) = 3P*1* + 3P*2* + 3P*5* + 3P*6* + 3P*7* + P*10* = 3(22.50) + 3(-31.21) + 3(40.01) + 3(-10.03) + 3(-6.41) + (-12.20) = 32.38 %; *Rp*,exp(TO 055) = 30.00 %

*AE* = *Rp*,calc(TO 055) - *Rp*,exp(TO 055) = 2.38 %.

Example of usage of the new model for reactivation ability prediction for TO 055 reactivator: *Rp*,calc(TO 058) = 2P*1* + 2P*2* + 2P*5* + 2P*6* + 3P*7* + 2P*8* = 2(22.50) + 2(-31.21) + 2(40.01) + 2(- 10.03) + 3(-6.41) + 2(2.16) = 27.63 %; *Rp*,exp(TO 058) = 25.00 %

$$AE = R\_{\text{p,calc}}(\text{TO } 058) - R\_{\text{p,exp}}(\text{TO } 058) = 2.63 \text{ } \%.$$

As it is clear, in comparison with the previous cases, these models are applicable only for the same inhibitors but for new reactivators of ACHE inhibited by the same inhibitors (the first for chloropyrifos, the second one for sarin). But on the other hand, it can be also used as a tool for easy prediction of reactivation potency of some newly synthesized reactivators without any other *in vitro* standard tests.

## **3. Conclusion**

Most of the industrial applications and products contain a mixture of many components and for the production it is important to know the properties of individual substance and the properties of aggregates. The accomplishments of all of these experiments are too expensive and time-consuming, so the calculation or estimation methods are good way to solve this problem. The group contribution methods are the important and favourible estimation method, because they permit to determine value of property of extant or hypothetic compound. Group contribution methods are the suitable tool for estimation of many physico-chemical quantities of pure compounds and mixtures too as it was showed and confirmed above for some cases. It can be used for estimation of pure compounds, as well as mixtures, for one temperature estimation, as well as for temperature range, etc. The biggest advantage of these methods is they need knowledge only chemical structure of compounds without any other input information. The presented models have been developed for estimation of many variable properties, enthalpy of vaporization, entropy of vaporization, liquid heat capacity, swelling of Nafion, flash temperature and reactivation ability of reactivators of acetylcholinesterase inhibited by organophosphate compounds. Proposed models and their structural fragments, accuracy and reliability depend mainly on frequency of input data and their accuracy, correctness and reliability. The most of presented models of group contribution methods, not only in the cases presented in this chapter, can be applied for the wide variety of organic compounds, when groups describing these molecules are presented. Some of models can be applied from only limited families of compounds due to their parameters were calculated only for limited database of compounds. Group contribution methods can be applied either for estimation or prediction of properties at one temperature or as a temperature function depending on their development. The accuracy of developed models is the higher, the input database is more reliable.

## **Author details**

154 Thermodynamics – Fundamentals and Its Application in Science

+ 3(-6.41) + (-12.20) = 32.38 %; *Rp*,exp(TO 055) = 30.00 %

10.03) + 3(-6.41) + 2(2.16) = 27.63 %; *Rp*,exp(TO 058) = 25.00 %

*AE* = *Rp*,calc(TO 055) - *Rp*,exp(TO 055) = 2.38 %.

*AE* = *Rp*,calc(TO 058) - *Rp*,exp(TO 058) = 2.63 %.

without any other *in vitro* standard tests.

**3. Conclusion** 

Illustration of new method for reactivation ability prediction of two reactivators (TO 055 and TO 058) of which experimental data were not used for parameters calculation follows.

**Figure 6.** Chemical structure of two reactivators of acetylcholinesterase signed as TO 055 and TO 058

Example of usage of the new model for reactivation ability prediction for TO 055 reactivator: *Rp*,calc(TO 055) = 3P*1* + 3P*2* + 3P*5* + 3P*6* + 3P*7* + P*10* = 3(22.50) + 3(-31.21) + 3(40.01) + 3(-10.03)

TO 055 TO 058

Example of usage of the new model for reactivation ability prediction for TO 055 reactivator: *Rp*,calc(TO 058) = 2P*1* + 2P*2* + 2P*5* + 2P*6* + 3P*7* + 2P*8* = 2(22.50) + 2(-31.21) + 2(40.01) + 2(-

As it is clear, in comparison with the previous cases, these models are applicable only for the same inhibitors but for new reactivators of ACHE inhibited by the same inhibitors (the first for chloropyrifos, the second one for sarin). But on the other hand, it can be also used as a tool for easy prediction of reactivation potency of some newly synthesized reactivators

Most of the industrial applications and products contain a mixture of many components and for the production it is important to know the properties of individual substance and the properties of aggregates. The accomplishments of all of these experiments are too expensive and time-consuming, so the calculation or estimation methods are good way to solve this problem. The group contribution methods are the important and favourible estimation method, because they permit to determine value of property of extant or hypothetic compound. Group contribution methods are the suitable tool for estimation of many physico-chemical quantities of pure compounds and mixtures too as it was showed and confirmed above for some cases. It can be used for estimation of pure compounds, as well as mixtures, for one temperature estimation, as well as for temperature range, etc. The biggest advantage of these methods is they need knowledge only chemical structure of compounds Zdeňka Kolská *Faculty of Science, J. E. Purkyně University, Usti nad Labem, Czech Republic* 

Milan Zábranský and Alena Randová *Institute of Chemical Technology, Prague, Czech Republic* 

## **Acknowledgement**

This work was supported by the GA CR under the project P108/12/G108. Authors also thank to Ing. Michal Karlík from ICT Prague, Czech Republic, for data for flash temperature estimation and Prof. Kamil Kuča from Department of Toxicology, Faculty of Millitary Health Science Hradec Kralove, Czech Republic, for data on reactivation ability.

## **4. References**


NIST database. http://webbook.nist.gov/chemistry/

Chickos, J. S.; Acree, W. E. Jr. & Liebman J. F. (1998). Phase Change Enthalpies and Entropies. In: *Computational Thermochemistry: Prediction and Estimation of Molecular* Thermodynamics. D. Frurip and K. Irikura, (Eds.), ACS Symp. Ser. 677, pp. 63-91, Washington, D. C.

Group Contribution Methods for Estimation of Selected Physico-Chemical Properties of Organic Compounds 157

Constantinou, L. & Gani, R. (1994). New Group Contribution Method for Estimating Properties of Pure Compounds. *AIChE Journal*, Vol.40, No.10, pp. 1697-1710, ISSN ISSN

Tochigi, K.; Kurita, S.; Okitsu, Y.; Kurihara, K. & Ochi, K. (2005). Measurement and Prediction of Activity Coefficients of Solvents in Polymer Solutions Using Gas Chromatography and a Cubic-Perturbed Equation of State with Group Contribution.

Tochigi, K. & Gmehling, J. (2011). Determination of ASOG Parameters-Extension and Revision. *Journal of Chemical Engineering of Japan*, Vol.44, No.4, pp. 304-306, ISSN 0021-

Miller, D. G. (1964). Estimating Vapor Pressures-Comparison of Equations. *Industrial and* 

Conte, E.; Martinho, A.; Matos, H. A.; & Gani, R. (2008). Combined Group-Contribution and Atom Connectivity Index-Based Methods for Estimation of Surface Tension and Viscosity. *Industrial & Engineering Chemistry Research*, Vol.47, No.20, pp. 7940–7954,

Reichenberg, D. (1975). New Methods for Estimation of Viscosity Coefficients of Pure Gases at Moderate Pressures (With Particular Reference To Organic Vapors). *AIChE Journal*,

Ruzicka, V. & Domalski, E. S. (1993a). Estimation of the Heat-Capacities of Organic Liquids as a Function of Temperature Using Group Additivity. 1. Hydrocarbon Compounds. *Journal of Physical and Chemical Reference Data*, Vol.22, No.3, pp. 597-618, ISSN 0047-2689. Ruzicka, V. & Domalski, E. S. (1993b). Estimation of the Heat-Capacities of Organic Liquids as a Function of Temperature Using Group Additivity. 2. Compounds of Carbon, Hydrogen, Halogens, Nitrogen, Oxygen, and Sulfur. *Journal of Physical and Chemical* 

Kolská, Z.; Kukal, J.; Zábranský, M. & Růžička, V. (2008). Estimation of the Heat Capacity of Organic Liquids as a Function of Temperature by a Three-Level Group Contribution Method. *Industrial & Engineering Chemistry Research*, Vol.47, No.6, pp. 2075-2085, ISSN

Chickos, J. S.; Hesse, D. G.; Hosseini, S.; Liebman, J. F.; Mendenhall, G. D.; Verevkin, S. P.; Rakus, K.; Beckhaus, H.-D. & Ruechardt, C. (1995). Enthalpies of vaporization of some highly branched hydrocarbons. *Journal of Chemical Thermodynamics*, Vol.27, No.6, pp.

Chickos, J. S. & Wilson, J. A. (1997). Vaporization Enthalpies at 298.15K of the n-Alkanes from C21-C28 and C30. *Journal of Chemical and Engineering Data*, Vol.42, No.1, pp. 190-

Kolská, Z.; Růžička, V. & Gani, R. (2005). Estimation of the Enthalpy of Vaporization and the Entropy of Vaporization for Pure Organic Compounds at 298.15 K and at Normal Boiling Temperature by a Group Contribution Method. *Industrial & Engineering* 

*Chemistry Research,* Vol.44, No.22, pp. 8436-8454, ISSN 0888-5885

*Fluid Phase Equilibria*, Vol.228, No. Special Issue, pp. 527-533, ISSN 0378-3812

*Engineering Chemistry*, Vol.56, No.3, pp. 46-&, ISSN 0019-7866

*Reference Data*, Vol.22, No.3, pp. 619-657, ISSN 0047-2689

0001-1541

9592

ISSN 0888-5885

0888-5885

693-705, ISSN 0021-9614

197, ISSN 0021-9568

Vol.21, No.1, 181-183, ISSN 0001-1541


Constantinou, L. & Gani, R. (1994). New Group Contribution Method for Estimating Properties of Pure Compounds. *AIChE Journal*, Vol.40, No.10, pp. 1697-1710, ISSN ISSN 0001-1541

156 Thermodynamics – Fundamentals and Its Application in Science

Washington, D. C.

Rep. Chem. 92

0098-6445

NPL Rep. Chem. 98

No.2, pp. 142-148

1768

UK

3812

NIST database. http://webbook.nist.gov/chemistry/

Chickos, J. S.; Acree, W. E. Jr. & Liebman J. F. (1998). Phase Change Enthalpies and Entropies. In: *Computational Thermochemistry: Prediction and Estimation of Molecular* Thermodynamics. D. Frurip and K. Irikura, (Eds.), ACS Symp. Ser. 677, pp. 63-91,

Lydersen, A. L. (1955). *Estimation of Critical Properties of Organic Compounds*. Eng. Exp. Stn.

Ambrose, D. (1978). *Correlation and Estimation of Vapour-Liquid Critical Properties. I. Critical Temperatures of Organic Compounds*. National Physical Laboratory, Teddington: NPL

Ambrose, D. (1979). *Correlation and Estimation of Vapour-Liquid Critical Properties. II. Critical Pressures and Volumes of Organic Compounds*. National Physical Laboratory, Teddington:

Joback, K. G. & Reid, R. C. (1987). Estimation of pure-component properties from groupcontributions. *Chemical Engineering Communications*, Vol.57, No.1-6, pp. 233-243, ISSN

Gani, R. & Constantinou, L. (1996). Molecular structure based estimation of properties for process design. *Fluid Phase Equilibria*, Vol.116, No.1-2, pp. 75-86, ISSN 0378-3812 Marrero J. & Gani R. (2001). Group-contribution based estimation of pure component properties. *Fluid Phase Equilibria*, Vol.183, Special Issue, pp. 183-208, ISSN 0378-3812 Brown, J. S.; Zilio, C. & Cavallini, A. (2010). Thermodynamic properties of eight fluorinated olefins. *International Journal of Refrigeration*, Vol.33, No.2, pp. 235-241, ISSN 0140-7007 Monago, K. O. & Otobrise, C. (2010). Estimation of pure-component properties of fatty acid sand esters from group contributions. *Journal of Chemical Society of Nigeria*, Vol.35,

Sales-Cruz, M.; Aca-Aca, G.; Sanchez-Daza, O. & Lopez-Arenas, T. (2010). Predicting critical properties, density and viscosity of fatty acids, triacylglycerols and methyl esters by group contribution methods. *Computer-Aided Chemical Engineering*, Vol.28, pp. 1763-

Manohar, B. & Udaya Sankar, K. (2011). Prediction of solubility of Psoralea corylifolia L. Seed extract in supercritical carbon dioxide by equation of state models. *Theoretical* 

Pereda, S.; Brignole, E. & Bottini, S. (2010). Equations of state in chemical reacting systems. In: *Applied Thermodynamics of Fluids*, Goodwin, A. R. H.; Sengers, J. V. & Peters, C. J. (Eds.), 433-459, Royal Society of Chemistry; 1st Ed., ISBN 978-1847558060, Cambridge,

Schmid, B. & Gmehling, J. (2012). Revised parameters and typical results of the VTPR group contribution equation of state. *Fluid Phase Equilibria*, Vol.317, pp. 110-126, ISSN 0378-

*Foundations of Chemical Engineering*, Vol.45, No.4, pp. 409-419, ISSN 0040-5795 Garcia, M.; Alba, J.; Gonzalo, A.; Sanchez, J. L. & Arauzo, J. (2012). Comparison of Methods for Estimating Critical Properties of Alkyl Esters and Its Mixtures. *Journal of Chemical &* 

*Engineering Data*, Vol.57, No.1, pp. 208-218, ISSN 0021-9568

rept. 3; University of Wisconsin College of Engineering: Madison, WI

	- Nagvekar, M. & Daubert, T. E. (1987). A Group Contribution Method for Liquid Thermal-Conductivity. *Industrial & Engineering Chemistry Research*, Vol.26, No.7, pp. 1362-1365, ISSN 0888-5885

Williams, J. D. (1997). *Prediction of melting and heat capacity of inorganic liquids by the method of* 

Briard, A. J.; Bouroukba, M.; Petitjean, D. & Dirand, M. (2003). Models for estimation of pure n-alkanes' thermodynamic properties as a function of carbon chain length. *Journal of* 

Nikitin, E. D.; Popov, A. P.; Yatluk, Y. G. & Simakina, V. A. (2010). Critical Temperatures and Pressures of Some Tetraalkoxytitaniums. *Journal of Chemical and Engineering* 

Papaioannou, V.; Adjiman, C. S.; Jackson, G. & Galindo, A. (2010). Group Contribution Methodologies for the Prediction of Thermodynamic Properties and Phase Behavior in Mixtures. In: *Molecular Systems Engineering*. Pistikopoulos E. N.; Georgiadis, M. C.; Due, V.; Adjiman, C. S.; Galindo, A. (Eds.), 135-172, Wiley-VCH, ISBN 978-3-527-31695-3,

Teixeira, M. A.; Rodriguez, O.; Mota, F. L.; Macedo, E. A. & Rodrigues, A. E. (2010). Evaluation of Group-Contribution Methods To Predict VLE and Odor Intensity of Fragrances. *Industrial & Engineering Chemistry Research*, Vol.50, No.15, pp. 9390-9402,

Lobanova, O.; Mueller, K.; Mokrushina, L. & Arlt, W. (2011). Estimation of Thermodynamic Properties of Polysaccharides. *Chemical Engineering & Technology*, Vol.34, No.6, pp. 867-

Satyanarayana, K. C.; Gani, R.; & Abildskov, J. (2007). Polymer property modeling using grid technology for design of structured products. *Fluid Phase Equilibria*, Vol.261, No.1-

Bogdanic, G. (2009). Additive Group Contribution Methods for Predicting the Properties of Polymer Systems. In: *Polymeric Materials*. Nastasovic, A. B. & Jovanovic, S. M.

Díaz-Tovar, C.; Gani, R. & Sarup, B. (2011). Lipid technology: Property prediction and process design/analysis in the edible oil and biodiesel industries. *Fluid Phase Equilibria*,

Costa, A. J. L.; Esperanca, J. M. S. S.; Marrucho, I. M. & Rebelo, L. P. N. (2011). Densities and Viscosities of 1-Ethyl-3-methylimidazolium n-Alkyl Sulfates. *Journal of Chemical &* 

Costa, A. J. L.; Soromenho, M. R. C.; Shimizu, K.; Marrucho, I. M.; Esperanca, J. M. S. S.; Lopes, J. N. C. & Rebelo, L. P. N. (2012). Density, Thermal Expansion and Viscosity of Cholinium-Derived Ionic Liquids. *Chemphyschem: a European journal of chemical physics* 

Adamova, G.; Gardas, Ramesh L.; Rebelo, L. P. N.; Robertson, A. J. & Seddon, K. R. (2011). Alkyltrioctylphosphonium Chloride Ionic Liquids: Synthesis and Physicochemical

Properties. *Dalton Transactions*, Vol.40, No.47, pp. 12750-12764, ISSN 1477-9226

(Eds.), 155-197, Transworld Research Network, ISSN 978-81-7895-398-4, Kerala Oh, S. Y. & Bae, Y. C. (2009). Group contribution method for group contribution method for estimation of vapor liquid equilibria in polymer solutions. *Macromolecular Research*,

*group contributions*. Thesis, New Mexico State Univ., Las Cruces, NM, USA

*Chemical and Engineering Data*, Vol.48, No.6, pp. 1508-1516, ISSN 0021-9568

*Data*, Vol.55, No.1, pp. 178-183, ISSN 0021-9568

Weinheim

ISSN 0888-5885

876, ISSN 0930-7516

2, pp. 58–63, ISSN 0378-3812

Vol.17, No.11, pp. 829-841, ISSN 1598-5032

Vol.302, No.1-2, pp. 284–293, ISSN 0378-3812

*Engineering Data*, Vol.56, No.8, pp. 3433-3441, ISSN 0021-9568

*and physical chemistry*, Ahead of Print, ISSN 1439-7641


Williams, J. D. (1997). *Prediction of melting and heat capacity of inorganic liquids by the method of group contributions*. Thesis, New Mexico State Univ., Las Cruces, NM, USA

158 Thermodynamics – Fundamentals and Its Application in Science

Data, Vol.30, No.1, pp. 102-111, ISSN 0021-9568

*AIChE Journal*, Vol.1, No.2, pp. 174-177, ISSN 0001-1541

ISSN 0888-5885

ISSN 0196-4313

ISSN 0040-6031

ISSN 0378-3812

London

328-334, ISSN 0009-2770

1620-32, ISSN 0001-1541

7388

3812

Nagvekar, M. & Daubert, T. E. (1987). A Group Contribution Method for Liquid Thermal-Conductivity. *Industrial & Engineering Chemistry Research*, Vol.26, No.7, pp. 1362-1365,

Chung, T. H.; Lee, L. L. & Starling, K. E. (1984). Applications of Kinetic Gas Theories and Multiparameter Correlation for Prediction of Dilute Gas Viscosity and Thermal-Conductivity. *Industrial & Engineering Chemistry Fundamentals*, Vol.23, No.1, pp. 8-13,

Yampolskii, Y.; Shishatskii, S.; Alentiev, A. & Loza, K. (1998). Group Contribution Method for Transport Property Predictions of Glassy Polymers: Focus on Polyimides and Polynorbornenes. *Journal of Membrane Science*, Vol.149, No.2, pp. 203-220, ISSN 0376-

Campbell, S. W. & Thodos, G. (1985). Prediction of Saturated-Liquid Densities and Critical Volumes for Polar and Nonpolar Substances. Journal of Chemical and Engineering

Shahbaz, K.; Baroutian, S.; Mjalli, F. S.; Hashim, M. A. & AlNashef, I. M. (2012). Densities of ammonium and phosphonium based deep eutectic solvents: Prediction using artificial intelligence and group contribution techniques. *Thermochimica Acta*, Vol.527, pp. 59-66,

Brock, J. R. & Bird, R. B. (1955). Surface Tension and the Principle of Corresponding States.

Awasthi, A.; Tripathi, B. S. & Awasthi, A. (2010). Applicability of corresponding-states group-contribution methods for the estimation of surface tension of multicomponent liquid mixtures at 298.15 K. *Fluid Phase Equilibria*, Vol.287, No.2, pp. 151-154, ISSN 0378-

Lu, X.; Yang, Y. & Ji, J. (2011). Application of solubility parameter in solubility study of fatty

Liaw, H. & Chiu, Y. (2006). A general model for predicting the flash point of miscible mixtures. *Journal of Hazardous Materials*, Vol.137, No.1, pp. 38-46, ISSN 0304-3894 Liaw, H.; Gerbaud, V. & Li, Y. (2011). Prediction of miscible mixtures flash-point from UNIFAC group contribution methods. *Fluid Phase Equilibria*, Vol.300, No.1-2, pp. 70-82,

Zábranský, M.; Růžička, V. & Malijevský, A. (2003). Odhadové metody tepelných kapacit

Kolská, Z. (2004). Odhadové metody pro výparnou entalpii. *Chemické Listy*, Vol.98, No.6, pp.

Zábranský, M.; Kolská, Z.; Růžička, V. & Malijevský, A. (2010a). The Estimation of Heat Capacities of Pure Liquids. In *Heat capacities: liquids, solutions and vapours*. T.M. Letcher & E. Wilhelm (Eds.), 421-435, The Royal Society of Chemistry, ISBN 978-0-85404-176-3,

Gonzalez, H. E.; Abildskov, J.; Gani, R.; Rousseaux, P. & Le Bert, B. (2007). A Method for Prediction of UNIFAC Group Interaction Parameters. *AIChE Journal*, Vol. 53, No.6, pp.

acid methyl esters . *Zhongguo Liangyou Xuebao*, Vol.26, No.6, pp. 60-65

čistých kapalin. *Chemické Listy*, Vol.97, No.1, pp. 3-8, ISSN 0009-2770

	- Gacino, F. M.; Regueira, T.; Lugo, L.; Comunas, M. J. P. & Fernandez, J. (2011). Influence of Molecular Structure on Densities and Viscosities of Several Ionic Liquids. *Journal of Chemical & Engineering Data*, Vol.56, No.12, pp. 4984-4999, ISSN 0021-9568

Vetere A. (1995). Methods to predict the vaporization enthalpies at the normal boiling temperature of pure compounds revisited. *Fluid Phase Equilibria*, Vol.106, No.1-2, pp. 1-

Ma, P. & Zhao, X. (1993). Modified Group Contribution Method for Predicting the Entropy of Vaporization at the Normal Boiling Point. *Industrial & Engineering Chemistry Research*,

Zábranský, M. & Růžička, V. (2004). Estimation of the Heat Capacities of Organic Liquids as a Function of Temperature Using Group Additivity. An Amendment. *J. Phys. Chem. Ref.* 

Chickos, J. S.; Hesse, D. G. & Liebman, J. F. (1993). A Group Additivity Approach for the Estimation of Heat-Capacities of Organic Liquids and Solids at 298 K. *Structural* 

Zábranský, M.; Růžička, V.; Majer, V. & Domalski E. S. (1996). Heat capacity of liquids: Volume II. Critical review and recommended values. *Journal of Physical and Chemical Reference Data*, Monograph, 815-1596, American Chemical Society: Washington, D.C.,

Zábranský, M.; Růžička, V. & Domalski E. S. (2001). Heat Capacity of Liquids: Critical Review and Recommended Values. Supplement I. *Journal of Physical and Chemical* 

Zábranský, M.; Kolská, Z.; Růžička, V. & Domalski, E. S. (2010b). Heat Capacity of Liquids: Critical Review and Recommended Values. Supplement II. *Journal of Physical and* 

Steinleitner, H. G. (1980). *Tabulky hořlavých a nebezpečných látek*, Transl. Novotný, V.; Benda,

Kuča, K. & Kassa, J. (2003). A Comparison of the Ability of a New Bisperidinium Oxime-1,4- (hydroxyiminomethylpyridinium)-4-(4-carbamoylpyridinium)butane Dibromide and Currently used Oximes to Reactivate Nerve Agent-Inhibited Rat Brain Acetylcholinesterase by In Vitro Methods. *Journal of Enzyme Inhibition*, Vol.18, No.6, pp.

Kuča, K.; Bielavský, J.; Cabal, J. & Kassa, J. (2003a). Synthesis of a New Reactivator of Tabun-Inhibited Acetylcholnesterase. *Bioorganic & Medicinal Chemistry Letters*, Vol.13, No.20,

Kuča, K.; Bielavský, J.; Cabal, J. & Bielavská, M. (2003b). Synthesis of a Potential Reactivator of Acetylcholinesterase-(1-(4-hydroxyiminomethylpyridinium)-3-(carbamoylpyridinium) propane dibromide. Tetrahedron Letters, Vol.44, No.15, pp. 3123-3125, ISSN 0040-4039 Kuča, K.; Patočka, J. & Cabal, J. (2003c). Reactivation of Organophosphate Inhibited


in vitro. *Journal of Applied Biomedicine*, Vol.1, No.4, pp. 207-211. ISSN 1214-0287 Kuča K. & Patočka J. (2004). Reactivation of Cyclosarine-Inhibited Rad Brain Acetylcholinesterase by Pyridinium-Oximes. *Journal of Enzyme Inhibition and Medicinal* 

*Chemical Reference Data*, Vol.39, No.1, pp. 013103/1-013103/404, ISSN 0047-2689 Dvořák, O. (1993). Estimation of the flash point of flammable liquids. *Chemický průmysl*,

10 , ISSN 0378-3812

ISSN 1063-0651

Vol.32, No.12, pp. 3180-3183, ISSN 0888-5885

*Data*, Vol.33, No.4, pp. 1071-1081, ISSN 0047-2689

*Chemistry*, Vol.4, No.4, pp. 261-269, ISSN 1040-0400

*Reference Data*, Vol.30, No.5, pp. 1199-1689

Vol.43, No.5, pp. 157-158., ISSN 0009-2789

E. Svaz požární ochrany ČSSR , 1st ed. Praha

529-533, ISSN 8755-5093

pp. 3545-3547, ISSN 0960-894X

Acetyylcholinesterase Activity by

*Chemistry*. Vol.19, No.1, pp. 39-43, ISSN 1475-6366


Vetere A. (1995). Methods to predict the vaporization enthalpies at the normal boiling temperature of pure compounds revisited. *Fluid Phase Equilibria*, Vol.106, No.1-2, pp. 1- 10 , ISSN 0378-3812

160 Thermodynamics – Fundamentals and Its Application in Science

0886-6716

*Equilibria*, Vol.295, No.1, pp. 125-129, ISSN 0378-3812

*Engineering Data*, Vol.55, No.4, pp. 1505-1515, ISSN 0021-9568

Engineering, DTU Denmark, released date: May 15, 2002

*Computer Sciences*, Vol.28, No.1, pp. 31-36, ISSN 0095-2338

*Polymer Science*, Vol.111, No.4, pp. 1745-1750, ISSN 0021-8995

*Termochimica Acta*, Vol.44, No.2, pp. 131-140, ISSN 0040-6031

Vol.54, No.1-2, pp. 153-157, ISSN 0040-6031

Vol.75, No.3, pp. 329-340, ISSN 0040-6031

No.2, pp. 97-101, ISSN 0095-2338

404, ISSN 0010-0765

Gacino, F. M.; Regueira, T.; Lugo, L.; Comunas, M. J. P. & Fernandez, J. (2011). Influence of Molecular Structure on Densities and Viscosities of Several Ionic Liquids. *Journal of* 

Cehreli, S. & Gmehling, J. (2010). Phase Equilibria for Benzene-Cyclohexene and Activity Coefficients at Infinite Dilution for the Ternary Systems with Ionic Liquids. *Fluid Phase* 

Gardas, R. L.; Ge, R.; Goodrich, P.; Hardacre, C.; Hussain, A. & Rooney, D. W. (2010). Thermophysical Properties of Amino Acid-Based Ionic Liquids. *Journal of Chemical &* 

Marrero J. (2002). Programm *ProPred*, Version 3.5, Jorge Marrero, Department of Chemical

Weininger, D.; Weininger, A. & Weininger, J. (1986). Smiles. A Modern Chemical Language and Information System. *Chemical Design Automation News*, Vol.1, No.8, pp. 2-15, ISSN

Weininger, D. (1988). Smiles. A Chemical Language and Information-System. 1. Introduction to Metodology and Encoding Rules. *Journal of Chemical Information and* 

Weininger, D.; Weininger, A. & Weininger, J. (1989). Smiles. 2. Algorithm for Generation of Unique Smiles Notation. *Journal of Chemical Information and Computer Sciences*, Vol.29,

Weininger, D. (1990). Smiles. 3. Depict – Graphical Depiction of Chemical Structures. *Journal of Chemical Information and Computer Sciences*, Vol.30, No.3, pp. 237-243, ISSN 0095-2338 Kolská, Z. & Petrus P. (2010). Tool for group contribution methods – computational Fragmentation *Collection of Czechoslovak Chemical Communications*, Vol.75, No.4, pp. 393–

Randová, A.; Bartovská, L.; Hovorka, Š.; Poloncarzová, M.; Kolská, Z. & Izák, P. (2009). Application of the Group Contribution Approach to Nafion Swelling. *Journal of Applied* 

Ducros, M.; Gruson, J. F. & Sannier H. (1980). Estimation of enthalpies of vaporization for liquid organic compounds. Part 1. Applications to alkanes, cycloalkanes, alkenes, benzene hydrocarbons, alkanethiols, chloro- and bromoalkanes, nitriles, esters, acids,

Ducros, M.; Gruson, J. F. & Sannier H. (1981). Estimation of the enthalpies of vaporization of liquid organic compounds. Part 2. Ethers, thioalkanes, ketones and amines.

Ducros, M. & Sannier, H. (1982). Estimation of the enthalpies of vaporization of liquid compounds. Part 3. Application to unsaturated hydrocarbons. *Termochimica Acta*,

Ducros, M. & Sannier, H. (1984). Determination of vaporization enthalpies of liquid organic compounds. Part 4. Application to organometallic compounds. *Termochimica Acta*,

and aldehydes. *Termochimica Acta*, Vol.36, No.1, pp. 39-65, ISSN 0040-6031

*Chemical & Engineering Data*, Vol.56, No.12, pp. 4984-4999, ISSN 0021-9568

	- Kuča K. & Cabal, J. (2004a). In Vitro Reaktivace Acetylcholinesterázy Inhibované O-Isopropylmethylfluorofofonátem užitím biskvarterního oximu HS-6. *Česká a Slovenská Farmacie*, Vol.53, No.2, pp. 93-95, ISSN 1210-7816

**Chapter 7** 

© 2012 Bakker, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

*pTV x* , *<sup>m</sup>* , (1)

(2)

distribution, and reproduction in any medium, provided the original work is properly cited.

, *V nT*

**Thermodynamic Properties and Applications** 

Ronald J. Bakker

**1. Introduction** 

http://dx.doi.org/10.5772/50315

Additional information is available at the end of the chapter

temperature (*T*), composition (*x*), and molar volume (*V*m).

according to the Maxwell's relations [5].

**of Modified van-der-Waals Equations of State** 

Physical and chemical properties of natural fluids are used to understand geological processes in crustal and mantel rock. The fluid phase plays an important role in processes in diagenesis, metamorphism, deformation, magmatism, and ore formation. The environment of these processes reaches depths of maximally 5 km in oceanic crusts, and 65 km in continental crusts, e.g. [1, 2], which corresponds to pressures and temperatures up to 2 GPa and 1000 C, respectively. Although in deep environments the low porosity in solid rock does not allow the presence of large amounts of fluid phases, fluids may be entrapped in crystals as fluid inclusions, i.e. nm to µm sized cavities, e.g. [3], and fluid components may be present within the crystal lattice, e.g. [4]. The properties of the fluid phase can be approximated with equations of state (Eq. 1), which are mathematical formula that describe the relation between intensive properties of the fluid phase, such as pressure (*p*),

This pressure equation can be transformed according to thermodynamic principles [5], to calculate a variety of extensive properties, such as entropy, internal energy, enthalpy, Helmholtz energy, Gibbs energy, et al., as well as liquid-vapour equilibria and homogenization conditions of fluid inclusions, i.e. dew point curve, bubble point curve, and critical points, e.g. [6]. The partial derivative of Eq. 1 with respect to temperature is used to calculate total entropy change (*dS* in Eq. 2) and total internal energy change (*dU* in Eq. 3),

> *<sup>p</sup> dS dV T*

and reproduction in any medium, provided the original work is properly cited.

