3.2.3 Quantum statistical reduced distributions

In the quantum case, the distribution function f <sup>N</sup> is replaced by the statistical operator ρ that describes the state of the system, and the equation of motion is the von Neumann equation (8). The quantum statistical reduced density matrix is defined as average over creation and annihilation operators,

$$\rho\_{\boldsymbol{\varsigma}}(\boldsymbol{r}\_{1},...,\boldsymbol{r}\_{\boldsymbol{\varsigma}}^{\prime},\boldsymbol{t}) = \operatorname{Tr}\left\{\rho(\boldsymbol{t})\boldsymbol{\upmu}^{\dagger}(\boldsymbol{r}\_{1})...\boldsymbol{\upmu}^{\dagger}(\boldsymbol{r}\_{\boldsymbol{\varsigma}})\boldsymbol{\upmu}\left(\boldsymbol{r}\_{\boldsymbol{\varsigma}}^{\prime}\right)...\boldsymbol{\upmu}\left(\boldsymbol{r}\_{1}^{\prime}\right)\right\}.\tag{99}$$

It is related to correlation functions, the Wigner function, Green functions, dynamical structure factor, and others.

We consider the equations of motion for reduced distribution functions. For the single-particle density matrix in momentum representation, we have

$$\rho\_1(\mathbf{p}, \mathbf{p}', t) = \text{Tr}\{\rho(t)\boldsymbol{\uprho}^\dagger(\mathbf{p})\boldsymbol{\uprho}(\mathbf{p}')\}. \tag{100}$$

Derivation with respect to time gives

$$\frac{\partial}{\partial t}\rho\_1(\mathbf{p},\mathbf{p}',t) = \frac{1}{\mathbf{i}\hbar}\text{Tr}\left\{[H,\rho]\psi^\dagger(\mathbf{p})\psi(\mathbf{p}')\right\} = \frac{1}{\mathbf{i}\hbar}\text{Tr}\left\{\rho\left[\psi^\dagger(\mathbf{p})\psi(\mathbf{p}'),H\right]\right\}.\tag{101}$$

Similar as for the BBGKY hierarchy, we obtain in general a hierarchy of equations of the form

$$\frac{d\rho\_s(t)}{dt} = \text{function of } \{\rho\_s(t), \rho\_{s+1}(t)\}. \tag{102}$$

Like in the classical case, we have to truncate this chain of equations. For example, in the Boltzmann equation for f <sup>1</sup>ð Þt , the higher order distribution function f <sup>2</sup>ð Þt is replaced by a product of single-particle distribution functions f <sup>1</sup>ð Þt .

### 3.2.4 Stoßzahlansatz and Boltzmann equation

To evaluate the averages of single-particle properties such as particle current or kinetic energy, only the single-particle distribution must be known. Then, the single-particle distribution contains the relevant information, the higher distributions are irrelevant and will be integrated over.

We are looking for an equation of motion for the single-particle distribution function f <sup>1</sup>ð Þ r; p; t , taking into account short range interactions and binary collisions. For the total derivative with respect to time we find, see Eq. (95),

$$\frac{\partial f\_1}{\partial t} = \frac{\partial}{\partial t} f\_1 + \dot{r} \frac{\partial}{\partial r} f\_1 + \dot{\mathbf{p}} \frac{\partial}{\partial \mathbf{p}} f\_1 = \frac{\partial}{\partial t} f\_1 + \nu \frac{\partial}{\partial r} f\_1 + \mathbf{F} \frac{\partial}{\partial \mathbf{p}} f\_1 = 0.1$$

The crucial point in this equation is the force F. It is the sum of external forces Fext acting on the system under consideration and all forces resulting from the interaction Vij ri; r<sup>j</sup> � � between the constituents of the system.

Before discussing the derivation of kinetic equations using the method of the nonequilibrium statistical operator, we give a phenomenological approach using empirical arguments. To describe the change in the distribution function f <sup>1</sup> due to collisions among particles, we write

$$\frac{\partial}{\partial t}f\_1 = \left(\frac{\partial}{\partial t}f\_1\right)\_{\text{D}} + \left(\frac{\partial}{\partial t}f\_1\right)\_{\text{St}},\tag{103}$$

We can write the single-particle distribution as an average (93) of a microscopic

The self-consistency conditions (18) are realized with the Lagrange

N i¼1

ð

f <sup>1</sup> rj; p<sup>j</sup>

; t � �, Zrel <sup>¼</sup>

ð

np � �<sup>t</sup>

Below, we show that it increases with time for nonequilibrium distributions. The relevant distribution can be used to derive the collision term (107), for details see [3]. We will switch over to the quantum case where the presentation is

In the quantum case, we consider the single-particle density matrix. In the case of a homogeneous system (n1ð Þ¼ r n), ρ<sup>1</sup> p; p<sup>0</sup> ð Þ is diagonal. The set of relevant observ-

Considering these mean values as given, we construct the relevant statistical

� �,

<sup>ρ</sup>relðÞ¼ <sup>t</sup> <sup>e</sup>�Φð Þ�<sup>t</sup> <sup>∑</sup>pF1ð Þ <sup>p</sup>;<sup>t</sup> <sup>n</sup><sup>p</sup> , <sup>Φ</sup>ðÞ¼ <sup>t</sup> ln Tre�∑pF1ð Þ <sup>p</sup>;<sup>t</sup> <sup>n</sup><sup>p</sup> : (115)

parameter F1ð Þ r; p; t . The relevant distribution Frel reads (see (19) and replace ∑<sup>n</sup>

F<sup>1</sup> ri; p<sup>i</sup> ; <sup>t</sup> � � � �, <sup>Φ</sup>ðÞ¼ <sup>t</sup> ln <sup>ð</sup>

e�F1ð Þ <sup>r</sup>;p;<sup>t</sup> d3

� ��<sup>1</sup>

This means, we can eliminate the Lagrange parameters F1ð Þ r; p; t that are expressed in terms of the given distribution function f <sup>1</sup>ð Þ r; p; t . The relevant

r d<sup>3</sup> p

, n<sup>1</sup> <sup>r</sup>1; <sup>p</sup>1; …; <sup>r</sup>N; <sup>p</sup>N; <sup>r</sup>; <sup>p</sup> � � <sup>¼</sup> <sup>ℏ</sup><sup>3</sup> <sup>∑</sup>

N i¼1 δ3

exp � ∑ N i¼1

, F1ð Þ¼� r; p; t ln f <sup>1</sup>ð Þ r; p; t :

<sup>F</sup>rel <sup>r</sup>1; …; <sup>p</sup>N; <sup>t</sup> � � <sup>n</sup><sup>1</sup> <sup>r</sup>1; …; <sup>p</sup>N; <sup>r</sup>; <sup>p</sup> � �d<sup>Γ</sup> are solved

ð Y j

<sup>f</sup> <sup>1</sup>ð Þ <sup>r</sup>; <sup>p</sup>; <sup>t</sup> ln <sup>f</sup> <sup>1</sup>ð Þ <sup>r</sup>; <sup>p</sup>; <sup>t</sup>

e

f <sup>1</sup> rj; p<sup>j</sup>

d3 r d<sup>3</sup> p

¼ f <sup>1</sup>ð Þ p; t : (114)

; t � �dΓ<sup>N</sup> <sup>¼</sup> <sup>N</sup><sup>N</sup>

ð Þ <sup>r</sup> � <sup>r</sup><sup>i</sup> <sup>δ</sup><sup>3</sup> <sup>p</sup> � <sup>p</sup><sup>i</sup>

F<sup>1</sup> ri; p<sup>i</sup> ; <sup>t</sup> � � � �dΓ:

� �:

(109)

(110)

(111)

<sup>N</sup>! <sup>≈</sup>e<sup>N</sup>:

(112)

<sup>h</sup><sup>3</sup> : (113)

(dynamic) variable, the single-particle density

DOI: http://dx.doi.org/10.5772/intechopen.84707

<sup>f</sup> <sup>1</sup>ð Þ¼ <sup>r</sup>; <sup>p</sup>; <sup>t</sup> <sup>n</sup><sup>1</sup> <sup>r</sup>1; …; <sup>p</sup>N; <sup>r</sup>; <sup>p</sup> � � � � <sup>t</sup>

Nonequilibrium Statistical Operator

<sup>F</sup>rel <sup>r</sup>1; …; <sup>p</sup>N; <sup>t</sup> � � <sup>¼</sup> exp �ΦðÞ�<sup>t</sup> <sup>∑</sup>

The constraints <sup>f</sup> <sup>1</sup>ð Þ� <sup>r</sup>; <sup>p</sup>; <sup>t</sup> <sup>Ð</sup>

N e�F1ð Þ <sup>r</sup>;p;<sup>t</sup>

Zrel Y j

<sup>S</sup>relðÞ¼� <sup>t</sup> <sup>k</sup>Bh i ln <sup>F</sup>rel <sup>t</sup> ¼ �k<sup>B</sup>

ables are the occupation number operators n<sup>p</sup>

The Boltzmann entropy is then

by Ð d3 rd<sup>3</sup> p=h<sup>3</sup> )

according to

<sup>f</sup> <sup>1</sup>ð Þ¼ <sup>r</sup>; <sup>p</sup>; <sup>t</sup> <sup>h</sup><sup>3</sup>

distribution is

<sup>F</sup>rel <sup>r</sup>1; …; <sup>p</sup>N; <sup>t</sup> � � <sup>¼</sup> <sup>1</sup>

more transparent.

operator as

27

where the drift term contains the external force,

$$\left(\frac{\partial}{\partial t}f\_1\right)\_{\rm D} = -\nu \frac{\partial}{\partial \mathbf{r}} f\_1 - \mathbf{F^{ext}} \frac{\partial}{\partial \mathbf{p}} f\_1 \tag{104}$$

and the internal interactions are contained in the collision term <sup>∂</sup> <sup>∂</sup><sup>t</sup> f <sup>1</sup> � � St for which, from the BBGKY hierarchy (98), an exact expression has already been given:

$$
\left(\frac{\partial}{\partial t}f\_1\right)\_{\text{St}} = \int \frac{\text{d}^3 r' \text{d}^3 \mathbf{p}'}{h^3} \frac{\partial V(r, r')}{\partial r} \frac{\partial}{\partial \mathbf{p}'} f\_2(r \mathbf{p}, r' \mathbf{p}', t). \tag{105}
$$

Collisions or interactions among particles occur due to the interaction potential V r; r<sup>0</sup> ð Þ, which depends on the coordinates of the two colliding partners. For every particle, one has to sum over collision with all partners in the system. In this way, we have an equation for the single-particle distribution function, but it is not closed because the right-hand side contains the two-particle distribution function f <sup>2</sup> rp; r<sup>0</sup> p<sup>0</sup> ð Þ ; t .

As an approximation, similar to the master equation, we assume a balance between gain and loss:

$$\left(\frac{\partial f\_1}{\partial t}\right)\_{\text{St}} = G - L.\tag{106}$$

With some phenomenological considerations [5], we can find the collision term as

$$
\left(\frac{\partial f\_1}{\partial t}\right)\_{\text{St}} = \int \mathbf{d}^3 \boldsymbol{\nu}\_2 \left[ d\Omega \frac{\mathbf{d}\sigma}{d\Omega} \, |\boldsymbol{\nu}\_1 - \boldsymbol{\nu}\_2| \{f\_1(\boldsymbol{r}, \boldsymbol{\nu}\_1', t) f\_1(\boldsymbol{r}, \boldsymbol{\nu}\_2', t) - f\_1(\boldsymbol{r}, \boldsymbol{\nu}\_1, t) f\_1(\boldsymbol{r}, \boldsymbol{\nu}\_2, t)\} \right], \tag{107}
$$

where we have introduced the differential cross section

$$\frac{\mathbf{d}\sigma}{\mathbf{d}\mathfrak{Q}} = \frac{b(\mathfrak{d})}{\sin\mathfrak{G}} \left| \frac{\mathbf{d}b(\mathfrak{d})}{\mathbf{d}\mathfrak{G}} \right|. \tag{108}$$

Inserting expression (108) into Eq. (103), we obtain a kinetic equation only for the single-particle distribution, the Boltzmann equation.

### 3.2.5 Derivation of the Boltzmann equation from the nonequilibrium statistical operator

The relevant observable to describe the nonequilibrium state of the system is the single-particle distribution function. First, we consider classical mechanics where the single-particle distribution function is f <sup>1</sup>ð Þ r; p; t .

The crucial point in this equation is the force F. It is the sum of external forces Fext acting on the system under consideration and all forces resulting from the

Before discussing the derivation of kinetic equations using the method of the nonequilibrium statistical operator, we give a phenomenological approach using empirical arguments. To describe the change in the distribution function f <sup>1</sup> due to

> D þ

∂

which, from the BBGKY hierarchy (98), an exact expression has already been given:

Collisions or interactions among particles occur due to the interaction potential V r; r<sup>0</sup> ð Þ, which depends on the coordinates of the two colliding partners. For every particle, one has to sum over collision with all partners in the system. In this way, we have an equation for the single-particle distribution function, but it is not closed because

As an approximation, similar to the master equation, we assume a balance

St

With some phenomenological considerations [5], we can find the collision term as

<sup>1</sup>; <sup>t</sup> � �<sup>f</sup> <sup>1</sup> <sup>r</sup>; <sup>v</sup><sup>0</sup>

dbð Þ ϑ dϑ

� � � �

� � � �

Inserting expression (108) into Eq. (103), we obtain a kinetic equation only for

3.2.5 Derivation of the Boltzmann equation from the nonequilibrium statistical operator

The relevant observable to describe the nonequilibrium state of the system is the single-particle distribution function. First, we consider classical mechanics where the

<sup>∂</sup><sup>V</sup> <sup>r</sup>; <sup>r</sup><sup>0</sup> ð Þ ∂r

∂

<sup>∂</sup><sup>p</sup> <sup>f</sup> <sup>2</sup> rp; <sup>r</sup><sup>0</sup>

∂ ∂t f 1 � �

<sup>∂</sup><sup>r</sup> <sup>f</sup> <sup>1</sup> � <sup>F</sup>ext <sup>∂</sup>

St

, (103)

<sup>∂</sup><sup>p</sup> <sup>f</sup> <sup>1</sup> (104)

<sup>∂</sup><sup>t</sup> f <sup>1</sup> � �

p<sup>0</sup> ð Þ ; t : (105)

p<sup>0</sup> ð Þ ; t .

(107)

¼ G � L: (106)

<sup>2</sup>; <sup>t</sup> � � � <sup>f</sup> <sup>1</sup>ð Þ <sup>r</sup>; <sup>v</sup>1; <sup>t</sup> <sup>f</sup> <sup>1</sup>ð Þ <sup>r</sup>; <sup>v</sup>2; <sup>t</sup> � �,

: (108)

St for

� � between the constituents of the system.

interaction Vij ri; r<sup>j</sup>

collisions among particles, we write

Non-Equilibrium Particle Dynamics

∂ <sup>∂</sup><sup>t</sup> <sup>f</sup> <sup>1</sup> � �

between gain and loss:

∂f 1 ∂t � �

26

St ¼ ð d3 v2 ð <sup>d</sup><sup>Ω</sup> <sup>d</sup><sup>σ</sup>

∂ ∂t

where the drift term contains the external force, ∂ <sup>∂</sup><sup>t</sup> <sup>f</sup> <sup>1</sup> � �

> St ¼ ð d3 r0 d3 p0 h3

<sup>f</sup> <sup>1</sup> <sup>¼</sup> <sup>∂</sup> ∂t f 1 � �

> D ¼ �v

and the internal interactions are contained in the collision term <sup>∂</sup>

the right-hand side contains the two-particle distribution function f <sup>2</sup> rp; r<sup>0</sup>

∂f 1 ∂t � �

<sup>d</sup><sup>Ω</sup> <sup>∣</sup>v<sup>1</sup> � <sup>v</sup>2<sup>∣</sup> <sup>f</sup> <sup>1</sup> <sup>r</sup>; <sup>v</sup><sup>0</sup>

where we have introduced the differential cross section

the single-particle distribution, the Boltzmann equation.

single-particle distribution function is f <sup>1</sup>ð Þ r; p; t .

dσ <sup>d</sup><sup>Ω</sup> <sup>¼</sup> <sup>b</sup>ð Þ <sup>ϑ</sup> sin ϑ

We can write the single-particle distribution as an average (93) of a microscopic (dynamic) variable, the single-particle density

$$f\_1(r, p, t) = \left< n\_1(r\_1, \dots, p\_N, r, p) \right>^t, \quad n\_1(r\_1, p\_1, \dots, r\_N, p\_N; r, p) = \hbar^3 \sum\_{i=1}^N \delta^3(r - r\_i) \delta^3(p - p\_i). \tag{109}$$

The self-consistency conditions (18) are realized with the Lagrange parameter F1ð Þ r; p; t . The relevant distribution Frel reads (see (19) and replace ∑<sup>n</sup> by Ð d3 rd<sup>3</sup> p=h<sup>3</sup> )

$$F\_{\rm rel}(\mathbf{r}\_1, \dots, \mathbf{p}\_N, t) = \exp\left\{-\Phi(t) - \sum\_{i=1}^N F\_1(\mathbf{r}\_i, \mathbf{p}\_i, t)\right\}, \quad \Phi(t) = \ln\int \exp\left\{-\sum\_{i=1}^N F\_1(\mathbf{r}\_i, \mathbf{p}\_i, t)\right\} d\Gamma. \tag{110}$$

The constraints <sup>f</sup> <sup>1</sup>ð Þ� <sup>r</sup>; <sup>p</sup>; <sup>t</sup> <sup>Ð</sup> <sup>F</sup>rel <sup>r</sup>1; …; <sup>p</sup>N; <sup>t</sup> � � <sup>n</sup><sup>1</sup> <sup>r</sup>1; …; <sup>p</sup>N; <sup>r</sup>; <sup>p</sup> � �d<sup>Γ</sup> are solved according to

$$f\_1(r, \mathbf{p}, t) = h^3 N \operatorname{e}^{-F\_1(r, \mathbf{p}, t)} \left\{ \int \operatorname{e}^{-F\_1(r, \mathbf{p}, t)} \operatorname{d}^3 r \, \mathrm{d}^3 \mathbf{p} \right\}^{-1}, \qquad F\_1(r, \mathbf{p}, t) = -\ln f\_1(r, \mathbf{p}, t). \tag{111}$$

This means, we can eliminate the Lagrange parameters F1ð Þ r; p; t that are expressed in terms of the given distribution function f <sup>1</sup>ð Þ r; p; t . The relevant distribution is

$$F\_{\rm rel}(\mathbf{r}\_1,...,\mathbf{p}\_N,t) = \frac{1}{Z\_{\rm rel}} \prod\_j f\_1(\mathbf{r}\_j,\mathbf{p}\_j,t), \qquad Z\_{\rm rel} = \int \prod\_j f\_1(\mathbf{r}\_j,\mathbf{p}\_j,t) \, \mathrm{d}\Gamma\_N = \frac{N^N}{N!} \approx \mathbf{e}^N. \tag{112}$$

The Boltzmann entropy is then

$$S\_{\rm rel}(t) = -k\_{\rm B} \langle \ln F\_{\rm rel} \rangle^{t} = -k\_{\rm B} \left[ f\_{1}(r, p, t) \ln \frac{f\_{1}(r, p, t)}{\mathbf{e}} \frac{\mathbf{d}^{3} r \, \mathbf{d}^{3} \mathbf{p}}{h^{3}}.\right] \tag{113}$$

Below, we show that it increases with time for nonequilibrium distributions.

The relevant distribution can be used to derive the collision term (107), for details see [3]. We will switch over to the quantum case where the presentation is more transparent.

In the quantum case, we consider the single-particle density matrix. In the case of a homogeneous system (n1ð Þ¼ r n), ρ<sup>1</sup> p; p<sup>0</sup> ð Þ is diagonal. The set of relevant observables are the occupation number operators n<sup>p</sup> � �,

$$\left< n\_{\mathbf{p}} \right>^{t} = f\_{1}(\mathbf{p}, t). \tag{114}$$

Considering these mean values as given, we construct the relevant statistical operator as

$$\rho\_{\rm rel}(t) = \mathbf{e}^{-\Phi(t) - \sum\_{p} F\_1(p, t) n\_p}, \qquad \Phi(t) = \ln \operatorname{Tr} \mathbf{e}^{-\sum\_{p} F\_1(p, t) n\_p}. \tag{115}$$

The Lagrange parameters F1ð Þ p; t are obtained from the self-consistency conditions (114) similar to Eq. (111);

$$f\_1(\mathbf{p}, t) = \frac{\mathrm{Tr}\left\{\mathbf{e}^{-\sum\_{\mathbf{p}'} F\_1(\mathbf{p}', t) \mathbf{n}\_{\mathbf{p}'}} \boldsymbol{n}\_{\mathbf{p}}\right\}}{\mathrm{Tr}\left\{\mathbf{e}^{-\sum\_{\mathbf{p}'} F\_1(\mathbf{p}', t) \mathbf{n}\_{\mathbf{p}'}}\right\}} = \frac{\prod\_i \sum\_{\mathbf{n}\_i} \mathrm{e}^{-F\_1(\mathbf{p}\_i, t) \mathbf{n}\_i} \left(\mathbf{1} + \delta\_{\mathbf{p}\_i, \mathbf{p}}(\mathbf{n}\_i - \mathbf{1})\right)}{\prod\_i \sum\_{\mathbf{n}\_i} \mathrm{e}^{-F\_1(\mathbf{p}\_i, t) \mathbf{n}\_i}} \tag{116}$$

∂f 1 <sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>π</sup> ℏ ð

Nonequilibrium Statistical Operator

DOI: http://dx.doi.org/10.5772/intechopen.84707

T ≈ ∑ <sup>p</sup>1, <sup>p</sup>2, <sup>p</sup>10p2<sup>0</sup>

<sup>∂</sup><sup>f</sup> <sup>1</sup> <sup>p</sup><sup>1</sup> � � ∂t � �

w p<sup>1</sup> p<sup>2</sup> p<sup>0</sup>

<sup>1</sup> p<sup>0</sup> 2 � � <sup>¼</sup> <sup>2</sup><sup>π</sup>

quantization, the T matrix is then

collision term (time t is dropped)

St ¼ ∑ p2p<sup>0</sup> 1p0 2

a† p1 a† p2

�f <sup>1</sup> p<sup>1</sup>

with the transition probability rate

<sup>ℏ</sup> <sup>t</sup> <sup>p</sup><sup>1</sup> <sup>p</sup><sup>2</sup> <sup>p</sup><sup>0</sup>

3.2.6 Properties of the Boltzmann equation

single-particle distribution functions.

Srel ¼ k<sup>B</sup> ∑

w p1p2p<sup>0</sup> 1p0 2 � �ln <sup>1</sup>

> � �<sup>f</sup> <sup>1</sup> <sup>p</sup><sup>0</sup> 2 � �:

dSrel

<sup>d</sup><sup>t</sup> ¼ �k<sup>B</sup> <sup>∑</sup> p ∂f 1 ∂t

<sup>¼</sup> <sup>k</sup><sup>B</sup> <sup>∑</sup><sup>p</sup>1p2p<sup>0</sup> 1p0 2

�f <sup>1</sup> p<sup>1</sup> � �<sup>f</sup> <sup>1</sup> <sup>p</sup><sup>0</sup> 1 � �<sup>f</sup> <sup>1</sup> <sup>p</sup><sup>2</sup>

29

relevant statistical operator, the entropy is

p

The change with time follows from

ln f <sup>1</sup> � k<sup>B</sup> ∑ p ∂f 1 ∂t þ k<sup>B</sup> ∑ p ∂f 1 ∂t

> f <sup>1</sup> p<sup>1</sup> � � <sup>∓</sup> <sup>1</sup> ! <sup>1</sup>

with the two-particle T matrix t p1; p2; p<sup>0</sup>

w p<sup>1</sup> p<sup>2</sup> p<sup>0</sup>

<sup>1</sup> p<sup>0</sup> 2 � � <sup>∓</sup><sup>t</sup> <sup>p</sup><sup>1</sup> <sup>p</sup><sup>2</sup> <sup>p</sup><sup>0</sup>

� � � � �

which leads to the quantum statistical Boltzmann equation.

can use the density matrix or the Wigner function to characterize the

considered. The assumption of molecular chaos means that correlations are neglected, the two-particle distribution function is replaced by the product of

� �<sup>f</sup> <sup>1</sup> <sup>p</sup><sup>2</sup>

dETr T; n<sup>p</sup>

t p1; p2; p<sup>0</sup>

<sup>1</sup> p<sup>2</sup> � � <sup>f</sup> <sup>1</sup> <sup>p</sup><sup>0</sup>

� � <sup>1</sup>∓<sup>f</sup> <sup>1</sup> <sup>p</sup><sup>0</sup>

� �<sup>δ</sup> <sup>E</sup> � <sup>H</sup><sup>0</sup> � � <sup>T</sup>; <sup>ρ</sup>rel ½ �<sup>δ</sup> <sup>E</sup> � <sup>H</sup><sup>0</sup> � � � � , (123)

δ p<sup>1</sup> þ p<sup>2</sup> � p<sup>0</sup>

� � � � �

� �. With this T matrix, we find the

� � � � <sup>1</sup>∓<sup>f</sup> <sup>1</sup> <sup>p</sup><sup>2</sup>

<sup>1</sup> � E<sup>p</sup><sup>0</sup> 2 � �<sup>δ</sup> <sup>p</sup><sup>1</sup> <sup>þ</sup> <sup>p</sup><sup>2</sup> � <sup>p</sup><sup>0</sup>

(125)

<sup>1</sup> � p<sup>0</sup> 2

(126)

� �,

<sup>1</sup> � p<sup>0</sup> 2 � �, (124)

For further treatment, we choose the approximation of binary collisions, that means that only two particles change their momentums during a collision. In second

> 2 ap0 1

1 � �<sup>f</sup> <sup>1</sup> <sup>p</sup><sup>0</sup>

1 � � � � <sup>1</sup>∓<sup>f</sup> <sup>1</sup> <sup>p</sup><sup>0</sup>

> <sup>2</sup> p<sup>0</sup> 1

The Boltzmann equation is a nonlinear integro-differential equation for the single-particle distribution function in the classical case. In the quantum case, we

nonequilibrium state of the system. The Boltzmann equation is valid in low-density limit (only binary collisions). At higher densities also three-body collisions, etc., must be taken into account. Further density effects such as the formation of quasi particles and bound states have to be considered. The collision term is approximated to be local in space and time, no gradients in the density and no memory in time is

The increase of entropy (Boltzmann H theorem) can be proven. In terms of the

ln 1∓ f <sup>1</sup> � � <sup>þ</sup> <sup>k</sup><sup>B</sup> <sup>∑</sup>

f <sup>1</sup> p<sup>0</sup> 1 � � <sup>∓</sup><sup>1</sup> ! <sup>1</sup>

<sup>∓</sup><sup>1</sup> <sup>þ</sup> <sup>f</sup> <sup>1</sup>ð Þ <sup>p</sup> � �ln 1<sup>∓</sup> <sup>f</sup> <sup>1</sup>ð Þ <sup>p</sup> � � � <sup>f</sup> <sup>1</sup>ð Þ <sup>p</sup> ln <sup>f</sup> <sup>1</sup>ð Þ <sup>p</sup> � �: (127)

p ∂f 1 ∂t

> � <sup>1</sup> f <sup>1</sup> p<sup>1</sup> � � <sup>∓</sup><sup>1</sup> ! <sup>1</sup>

( ) !

f <sup>1</sup> p<sup>2</sup> � � <sup>∓</sup><sup>1</sup>

(128)

f <sup>1</sup> p<sup>0</sup> 2 � � <sup>∓</sup><sup>1</sup> !

� 2

<sup>1</sup>; p<sup>0</sup> 2

> 2 � � <sup>1</sup>∓<sup>f</sup> <sup>1</sup> <sup>p</sup><sup>1</sup>

2 � �� � �

δ E<sup>p</sup><sup>1</sup> þ E<sup>p</sup><sup>2</sup> � E<sup>p</sup><sup>0</sup>

<sup>1</sup>; p<sup>0</sup> 2 � �a<sup>p</sup><sup>0</sup>

so that

$$f\_1(\mathbf{p}, t) = \begin{cases} \mathbf{1} & + \\ \mathbf{e}^{F\_1(\mathbf{p}, t)} \pm \mathbf{1} \ - & \text{: Bosons} \end{cases} \}, \qquad F\_1(\mathbf{p}, t) = \ln\left[\mathbf{1} \mp f\_1(\mathbf{p}, t)\right] - \ln f\_1(\mathbf{p}, t). \tag{117}$$

As in the classical case, also in the quantum case, the Lagrange parameters can be eliminated explicitly.

We now derive the Boltzmann equation for the quantum case, see [3]. With the statistical operator (Eq. (25) after integration by parts)

$$\rho(t) = \rho\_{\rm rel}(t) - \int\_{-\infty}^{t} \mathbf{e}^{\epsilon(t\_1 - t)} \frac{\mathbf{d}}{\mathbf{d}t\_1} \left\{ \mathbf{e}^{-\frac{i}{\hbar}H(t - t\_1)} \rho\_{\rm rel}(t\_1) \mathbf{e}^{\frac{i}{\hbar}H(t - t\_1)} \right\} \mathbf{d}t\_1 \tag{118}$$

With <sup>n</sup>\_ <sup>p</sup> <sup>¼</sup> <sup>i</sup> <sup>ℏ</sup> H; n<sup>p</sup> � �, we get the time derivative of the single-particle distribution function

$$\frac{\partial}{\partial t} f\_1(\mathbf{p}, t) = \text{Tr}\left\{\rho\_{\text{rel}}(t)\dot{n}\_{\text{p}}\right\} - \int\_{-\infty}^{0} \mathbf{e}^{\epsilon t'} \text{Tr}\left\{\dot{n}\_{\text{p}} \frac{\mathbf{d}}{\mathbf{d}t'} \left[\mathbf{e}^{\dot{\mathbf{d}} \cdot \mathbf{H}^{\prime}} \rho\_{\text{rel}}(t + t')\mathbf{e}^{-\dot{\mathbf{d}} \cdot \mathbf{H}^{\prime}}\right]\right\} \mathbf{d}t'. \tag{119}$$

Because the trace is invariant with respect to cyclic permutations and ρrelð Þt commutes with np, see (115),

$$\operatorname{Tr}\left\{\rho\_{\rm rel}(t)\dot{n}\_{\rm p}\right\} = \frac{\operatorname{i}}{\hbar}\operatorname{Tr}\left\{\rho\_{\rm rel}\left[H,n\_{\rm p}\right]\right\} = \frac{\operatorname{i}}{\hbar}\operatorname{Tr}\left\{H\left[n\_{\rm p},\rho\_{\rm rel}\right]\right\} = 0,\tag{120}$$

and Eq. (119) can be written as

$$\frac{\partial \mathbf{f}\_1}{\partial t} = \frac{1}{\hbar^2} \int\_{-\infty}^0 \mathbf{d}t' \,\mathbf{e}^{\mathbf{c}t'} \text{Tr}\left\{ [H, n\_p] \,\mathbf{e}^{\dot{\mathbf{g}}Ht'} [H, \rho\_{\text{rel}}] \,\mathbf{e}^{-\dot{\mathbf{g}}Ht'} \right\},\tag{121}$$

if we neglect the explicit time dependence of ρrelð Þt (no memory effects, the collision term is local in space and time). Next, we introduce two more integrations via delta functions to get rid of the time dependence in the trace:

$$\frac{\partial \mathbf{f}\_1}{\partial t} = \frac{1}{\hbar^2} \int\_{-\infty}^{\infty} \mathrm{d}E \int\_{-\infty}^{\infty} \mathrm{d}E' \int\_{-\infty}^{0} \mathrm{d}t' \,\mathrm{e}^{\left[c + \frac{i}{\hbar}(E - E')\right] \mathrm{f}'} \mathrm{Tr}\left\{ \left[V, n\_\mathrm{p}\right] \delta(E - H) [V, \rho\_\mathrm{rel}] \delta(E' - H) \right\}.\tag{122}$$

(We take into account that the kinetic energy in H commutes with n<sup>p</sup> so that only the potential energy V remains.) This equation can be expressed by so-called T matrices, <sup>T</sup> <sup>¼</sup> <sup>V</sup> <sup>þ</sup> <sup>V</sup> <sup>1</sup> <sup>E</sup>�<sup>H</sup> <sup>T</sup>,

Nonequilibrium Statistical Operator DOI: http://dx.doi.org/10.5772/intechopen.84707

The Lagrange parameters F1ð Þ p; t are obtained from the self-consistency

Q

<sup>i</sup> <sup>∑</sup>ni <sup>e</sup>�F<sup>1</sup> <sup>p</sup><sup>i</sup> ð Þ;<sup>t</sup> ni <sup>1</sup> <sup>þ</sup> <sup>δ</sup><sup>p</sup><sup>i</sup>

<sup>i</sup> <sup>∑</sup>ni <sup>e</sup>�F<sup>1</sup> <sup>p</sup><sup>i</sup> ð Þ;<sup>t</sup> ni

Q

As in the classical case, also in the quantum case, the Lagrange parameters can be

We now derive the Boltzmann equation for the quantum case, see [3]. With the

e�i

Because the trace is invariant with respect to cyclic permutations and ρrelð Þt

Tr H; n<sup>p</sup> � �e i ℏHt<sup>0</sup>

if we neglect the explicit time dependence of ρrelð Þt (no memory effects, the collision term is local in space and time). Next, we introduce two more integrations

0

(We take into account that the kinetic energy in H commutes with n<sup>p</sup> so that only the potential energy V remains.) This equation can be expressed by so-called T

Tr V; n<sup>p</sup>

<sup>ℏ</sup>Hðt�t1ρrelð Þ <sup>t</sup><sup>1</sup> <sup>e</sup>

� �, we get the time derivative of the single-particle distribution

d <sup>d</sup>t<sup>0</sup> <sup>e</sup> i ℏHt<sup>0</sup>

ℏ

n o

i <sup>ℏ</sup>Hðt�t<sup>1</sup>

ρrel t þ t <sup>0</sup> ð Þe�<sup>i</sup>

� � h i

Tr H np; ρrel

<sup>H</sup>; <sup>ρ</sup>rel ½ �e�<sup>i</sup> ℏHt<sup>0</sup> n o

ℏHt<sup>0</sup>

� � � � <sup>¼</sup> <sup>0</sup>, (120)

� �δð Þ <sup>E</sup> � <sup>H</sup> <sup>V</sup>; <sup>ρ</sup>rel ½ �<sup>δ</sup> <sup>E</sup><sup>0</sup> ð Þ � <sup>H</sup> � �:

,pð Þ ni � <sup>1</sup> � �

, F1ð Þ¼ <sup>p</sup>; <sup>t</sup> ln 1∓<sup>f</sup> <sup>1</sup>ð Þ <sup>p</sup>; <sup>t</sup> � � � ln <sup>f</sup> <sup>1</sup>ð Þ <sup>p</sup>; <sup>t</sup> :

(116)

(117)

dt1, (118)

dt 0 : (119)

, (121)

(122)

np

þ : Fermions � : Bosons

conditions (114) similar to Eq. (111);

Non-Equilibrium Particle Dynamics

1 eF1ð Þ <sup>p</sup>;<sup>t</sup> � 1

ρðÞ¼ t ρrelðÞ�t

f <sup>1</sup>ð Þ¼ p; t Tr ρrelð Þt n\_ <sup>p</sup>

commutes with np, see (115),

Tr ρrelð Þt n\_ <sup>p</sup> � � <sup>¼</sup> <sup>i</sup>

∞ð

dE ∞ð

�∞

dE<sup>0</sup> ð 0

<sup>E</sup>�<sup>H</sup> <sup>T</sup>,

�∞

dt <sup>0</sup> e <sup>ϵ</sup>þ<sup>i</sup> <sup>ℏ</sup> <sup>E</sup>�E<sup>0</sup> ½ � ð Þ <sup>t</sup>

�∞

matrices, <sup>T</sup> <sup>¼</sup> <sup>V</sup> <sup>þ</sup> <sup>V</sup> <sup>1</sup>

and Eq. (119) can be written as

∂f 1 <sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>1</sup> ℏ2 ð 0

<sup>ℏ</sup> H; n<sup>p</sup>

Tr e� <sup>∑</sup>p<sup>0</sup> <sup>F</sup><sup>1</sup> <sup>p</sup><sup>0</sup> ð Þ ;<sup>t</sup> <sup>n</sup>p<sup>0</sup>

Tr e� <sup>∑</sup>p<sup>0</sup> <sup>F</sup><sup>1</sup> <sup>p</sup><sup>0</sup> ð Þ ;<sup>t</sup> <sup>n</sup>p<sup>0</sup> n o ¼

;

� �

statistical operator (Eq. (25) after integration by parts)

<sup>e</sup><sup>ϵ</sup>ð Þ <sup>t</sup>1�<sup>t</sup> <sup>d</sup> dt<sup>1</sup>

> ð 0

eϵt 0 Tr n\_ <sup>p</sup>

�∞

Tr ρrel H; n<sup>p</sup> � � � � <sup>¼</sup> <sup>i</sup>

�∞

ðt

� � �

ℏ

�∞

dt <sup>0</sup> e<sup>ϵ</sup><sup>t</sup> 0

via delta functions to get rid of the time dependence in the trace:

n o

f <sup>1</sup>ð Þ¼ p; t

eliminated explicitly.

With <sup>n</sup>\_ <sup>p</sup> <sup>¼</sup> <sup>i</sup>

function

∂ ∂t

> ∂f 1 <sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>1</sup> ℏ2

28

so that

f <sup>1</sup>ð Þ¼ p; t

$$\frac{\partial \mathcal{f}\_1}{\partial t} = \frac{\pi}{\hbar} \Big[ \text{dETr} \left\{ [T, n\_p] \delta(E - H^0) [T, \rho\_{\text{rel}}] \delta(E - H^0) \right\},\tag{123}$$

For further treatment, we choose the approximation of binary collisions, that means that only two particles change their momentums during a collision. In second quantization, the T matrix is then

$$T \approx \sum\_{\mathbf{p}\_{\mathcal{V}}, \mathbf{p}\_{\mathcal{D}}, \mathbf{p}\_{\mathcal{V}} \mathbf{p}\_{\mathcal{V}}} a\_{\mathbf{p}\_1}^{\dagger} a\_{\mathbf{p}\_1}^{\dagger} t (\mathbf{p}\_1, \mathbf{p}\_2, \mathbf{p}\_1', \mathbf{p}\_2') a\_{\mathbf{p}\_1'} a\_{\mathbf{p}\_1'} \delta(\mathbf{p}\_1 + \mathbf{p}\_2 - \mathbf{p}\_1' - \mathbf{p}\_2'), \tag{124}$$

with the two-particle T matrix t p1; p2; p<sup>0</sup> <sup>1</sup>; p<sup>0</sup> 2 � �. With this T matrix, we find the collision term (time t is dropped)

$$\begin{aligned} \left(\frac{\partial f\_1(\mathbf{p}\_1)}{\partial t}\right)\_{\mathrm{St}} &= \sum\_{\mathbf{p}\_2 \mathbf{p}\_1' \mathbf{p}\_2'} w(\mathbf{p}\_1 \mathbf{p}\_2 \mathbf{p}\_1' \mathbf{p}\_2) \{f\_1(\mathbf{p}\_1') f\_1(\mathbf{p}\_2') (\mathbb{1} \mp f\_1(\mathbf{p}\_1)) (\mathbb{1} \mp f\_1(\mathbf{p}\_2)) \\ &- f\_1(\mathbf{p}\_1) f\_1(\mathbf{p}\_2) (\mathbb{1} \mp f\_1(\mathbf{p}\_1')) (\mathbb{1} \mp f\_1(\mathbf{p}\_2')) \} \end{aligned} \tag{125}$$

with the transition probability rate

$$w\left(\mathbf{p}\_1\,\mathbf{p}\_2\,\mathbf{p}\_1'\,\mathbf{p}\_2'\right) = \frac{2\pi}{\hbar} \left| t\left(\mathbf{p}\_1\,\mathbf{p}\_2\,\mathbf{p}\_1'\,\mathbf{p}\_2'\right) \mp t\left(\mathbf{p}\_1\,\mathbf{p}\_2\,\mathbf{p}\_2'\,\mathbf{p}\_1'\right) \right|^2 \delta\left(\mathbf{E}\_{\mathbf{p}\_1} + \mathbf{E}\_{\mathbf{p}\_2} - \mathbf{E}\_{\mathbf{p}\_1'} - \mathbf{E}\_{\mathbf{p}\_1'}\right) \delta\left(\mathbf{p}\_1 + \mathbf{p}\_2 - \mathbf{p}\_1' - \mathbf{p}\_2'\right), \tag{12.6}$$

which leads to the quantum statistical Boltzmann equation.

### 3.2.6 Properties of the Boltzmann equation

The Boltzmann equation is a nonlinear integro-differential equation for the single-particle distribution function in the classical case. In the quantum case, we can use the density matrix or the Wigner function to characterize the nonequilibrium state of the system. The Boltzmann equation is valid in low-density limit (only binary collisions). At higher densities also three-body collisions, etc., must be taken into account. Further density effects such as the formation of quasi particles and bound states have to be considered. The collision term is approximated to be local in space and time, no gradients in the density and no memory in time is considered. The assumption of molecular chaos means that correlations are neglected, the two-particle distribution function is replaced by the product of single-particle distribution functions.

The increase of entropy (Boltzmann H theorem) can be proven. In terms of the relevant statistical operator, the entropy is

$$S\_{\rm rel} = k\_{\rm B} \sum\_{\mathbf{p}} \left\{ \left( \mp \mathbf{1} + f\_1(\mathbf{p}) \right) \ln \left( \mathbf{1} \mp f\_1(\mathbf{p}) \right) - f\_1(\mathbf{p}) \ln f\_1(\mathbf{p}) \right\}. \tag{127}$$

The change with time follows from

$$\begin{split} \frac{dS\_{\mathrm{rel}}}{dt} &= -k\_{\mathrm{B}} \sum\_{p} \frac{\partial f\_{1}}{\partial t} \ln f\_{1} - k\_{\mathrm{B}} \sum\_{p} \frac{\partial f\_{1}}{\partial t} + k\_{\mathrm{B}} \sum\_{p} \frac{\partial f\_{1}}{\partial t} \ln \left( \mathbbm{1} \mp f\_{1} \right) + k\_{\mathrm{B}} \sum\_{p} \frac{\partial f\_{1}}{\partial t} \\ &= k\_{\mathrm{B}} \sum\_{p, p\_{1}p\_{1}p\_{1}} w \left( \mathfrak{p}\_{1} \mathfrak{p}\_{2} \mathfrak{p}\_{2}^{\prime} p\_{2}^{\prime} \right) \ln \left( \frac{1}{f\_{1}(\mathfrak{p}\_{1})} \mp 1 \right) \left\{ \left( \frac{1}{f\_{1}(\mathfrak{p}\_{1}^{\prime})} \mp 1 \right) \left( \frac{1}{f\_{1}(\mathfrak{p}\_{2}^{\prime})} \mp 1 \right) - \left( \frac{1}{f\_{1}(\mathfrak{p}\_{1})} \mp 1 \right) \left( \frac{1}{f\_{1}(\mathfrak{p}\_{2})} \mp 1 \right) \right\}. \end{split} \tag{128}$$

We interchange indices 1 \$ 2, 1<sup>0</sup> \$ 2<sup>0</sup> ; furthermore 1 \$ 1<sup>0</sup> , 2 \$ 2<sup>0</sup> ; and 1 \$ 2<sup>0</sup> , 2 \$ 1<sup>0</sup> , use the symmetries of w p1p2p<sup>0</sup> 1p0 2 � � and ð Þ <sup>x</sup><sup>1</sup> � <sup>x</sup><sup>2</sup> ð Þ ln <sup>x</sup><sup>1</sup> � ln <sup>x</sup><sup>2</sup> <sup>≥</sup><sup>0</sup> because ln x is a monotonous function of x. We obtain 4 <sup>d</sup>Srel <sup>d</sup><sup>t</sup> ≥0, the Boltzmann (relevant) entropy can increase.

The collision integral guarantees conservation of total momentum, particle number, and kinetic energy. However, the total energy including the interaction part is not conserved. The equilibrium solution f 0 <sup>1</sup> ð Þ <sup>p</sup> follows from <sup>d</sup>Srel <sup>d</sup><sup>t</sup> ¼ 0:

$$\left(\frac{1}{f\_1^0(\mathbf{p})} \mp 1\right) \left(\frac{1}{f\_1^0(\mathbf{p}\_1)} \mp 1\right) - \left(\frac{1}{f\_1^0(\mathbf{p'})} \mp 1\right) \left(\frac{1}{f\_1^0(\mathbf{p'\_1})} \mp 1\right) = 0. \tag{129}$$

If f 0 <sup>1</sup> ð Þ p depends only on energy, we find the well-known result for ideal quantum gases,

$$\frac{1}{f\_1^0(\mathfrak{p})} \mp \mathbf{1} = \mathbf{e}^{\beta \left(E\_p - \mu\right)}, \qquad f\_1^0(\mathfrak{p}) = \left[\mathbf{e}^{\beta \left(E\_p - \mu\right)} \pm \mathbf{1}\right]^{-1}. \tag{130}$$

3.2.8 The linearized Boltzmann equation

DOI: http://dx.doi.org/10.5772/intechopen.84707

Nonequilibrium Statistical Operator

tion of electrical conductivity in plasmas.

ior of the plasma following Ohm's Law:

¼ ∑ s es ð d3

> p m ∂

3.2.9 Example: conductivity of the Lorentz plasma

collisions), the interaction part of the Hamiltonian reads

H<sup>0</sup> ¼ ∑ i

<sup>∂</sup><sup>r</sup> <sup>f</sup> <sup>1</sup> <sup>þ</sup> <sup>e</sup><sup>E</sup> <sup>∂</sup>

We recall the Boltzmann equation

the polarization field εP]. j

<sup>j</sup>e<sup>l</sup> <sup>¼</sup> <sup>1</sup>

distribution, we have

St ¼ ð d<sup>3</sup> p0 Ω

factors 1 � <sup>f</sup> <sup>1</sup>ð Þ <sup>p</sup> � �.

∂ ∂t f 1 � �

31

particle distribution function f <sup>1</sup>

<sup>Ω</sup> <sup>∑</sup> N i eiv<sup>i</sup> � �

Different approximations are known to obtain solutions of the Boltzmann equation, see [4, 5]. A serious problem in solving the Boltzmann equation is its nonlinearity as we have terms of the form <sup>f</sup> <sup>1</sup> <sup>p</sup>1; <sup>t</sup> � �<sup>f</sup> <sup>1</sup> <sup>p</sup>2; <sup>t</sup> � �. Special cases that allow for linearization are two-component systems with a large difference in the masses or concentration. Linearization is also possible in the case where the deviation from some equilibrium distribution is small. As an application, we consider the calcula-

We investigate a plasma of ions and electrons under the influence of an external electric field Eext. For simplicity, we assume Eext to be homogeneous and independent of time (statical conductivity σ). For moderate fields, we await a linear behav-

[Note that in Eq. (132) E is not the external field, but the effective electric field in the medium (the plasma), being the superposition of the external field Eext and

vvf <sup>1</sup>ð Þ¼ v; s ∑

Here, we have kept the index s for the different sorts. In the following, we will skip this index as we only consider electrons being responsible for the electric current.

<sup>∂</sup><sup>p</sup> <sup>f</sup> <sup>1</sup> <sup>þ</sup>

ð Þ <sup>2</sup>π<sup>ℏ</sup> <sup>3</sup> <sup>f</sup> <sup>1</sup> <sup>p</sup><sup>0</sup> ð Þwpp<sup>0</sup> <sup>1</sup> � <sup>f</sup> <sup>1</sup>ð Þ <sup>p</sup> � � � <sup>f</sup> <sup>1</sup>ð Þ <sup>p</sup> <sup>w</sup><sup>p</sup><sup>0</sup>

where wpp<sup>0</sup> is the transition rate from the momentum state p to the state p<sup>0</sup>

In the Lorentz plasma model, the electron-electron collisions are neglected, and only electron-ion collisions are considered, interaction potential Veið Þr . In the adiabatic approximation where the ions are regarded as fixed at positions R<sup>i</sup> (elastic

quantum behavior of the collisions is taken into account via the Pauli blocking

m is the electron mass and �e the electron charge. The first term in this equation vanishes because of homogeneity of the system. For the collision term, we take the expression Eq. (125) in the generalized form for quantum systems. After the distribution function of the collision partner has been replaced by the equilibrium

s es ms

∂ ∂t f 1 � �

el ¼ σE: (132)

ð Þ <sup>2</sup>π<sup>ℏ</sup> <sup>3</sup> <sup>p</sup><sup>f</sup> <sup>1</sup>ð Þ <sup>p</sup>; <sup>s</sup> : (133)

¼ 0, (134)

. The

el is the average electric current defined via the single-

ð d<sup>3</sup> p

St

<sup>p</sup> <sup>1</sup> � <sup>f</sup> <sup>1</sup> <sup>p</sup><sup>0</sup> ð Þ � � � � , (135)

Veið Þ r � R<sup>i</sup> : (136)

j

In the classical limit, we have f 0 <sup>1</sup> ð Þ¼ <sup>p</sup> <sup>e</sup>�<sup>β</sup>ð Þ <sup>E</sup>p�<sup>μ</sup> with eβμ <sup>¼</sup> <sup>N</sup> Ω 2πℏ<sup>2</sup> mkBT � �<sup>3</sup>=<sup>2</sup> 1 ð Þ <sup>2</sup>sþ<sup>1</sup> , where s denotes the spin of the particle.

## 3.2.7 Beyond the Boltzmann kinetic equation

In deriving the Boltzmann equation, different approximations have been performed: only binary collisions are considered, three-particle, and higher order collisions are neglected. Memory effects and spatial inhomogeneities have been neglected. The single-particle distribution was considered as relevant observable in the Markov approximation. These approximations can be compared with the Born-Markov approximation discussed in context with the quantum master equation. Instead of the Born approximation that is possible for weak interactions, the binary collision approximation is possible in the low-density limit, where three- and higher order collisions are improbable.

In the case of thermal equilibrium, the Boltzmann entropy Srel (127) coincides with the entropy of the ideal (classical or quantum) gas. The equilibrium solution of the Boltzmann equation leads to the entropy of the ideal gas and gives not the correct equation of state for an interacting system that are derived from the Gibbs entropy (Φ ¼ ln Z is the Matthieu-Planck function)

$$\mathcal{S}\_{\rm eq} = -k\_{\rm B} \int d\Gamma(\Phi + \beta H) \, \exp\left[-\Phi - \beta H\right],\tag{131}$$

see Eq. (13). This deficit of the Boltzmann equation arises because binary collisions are considered where the kinetic energy of the asymptotic states is conserved. Only the single-particle distribution is a relevant observable and is correctly reproduced. It can be improved if the total energy, which is conserved, is considered as a relevant observable. Alternatively, we can also include the two-particle distribution function in the set of relevant observables. An important example is the formation of bound states as a signature of strong correlations in the system. Then, the momentum distribution of bound states has to be included in the set of relevant observables.
