**3.1 Uniformization time in a mesoscopic cell: An elementary estimation for Hamiltonian systems**

Using oversimplified, but reasonable arguments [17], we now coarsely estimate the uniformization time *T* in a mesoscopic cell. As an example, we consider *n* identical particles initially located in this cell, among *N* identical particles in an isolated vessel. The complete system obeys Hamilton mechanics.

Assume that the particles constitute a gas under normal conditions, with density *ρ* ≈ 3. 10<sup>25</sup> molecules.m�<sup>3</sup> . A mesocopic state can be reasonably represented by a cube of size *l* ≈ 10�<sup>6</sup> m (as an order of magnitude), which contains *n* ≈ 3. 10<sup>7</sup> molecules. We now divide the mesoscopic cell into *m* "microscopic" cells whose size *λ* is comparable to the size of a molecule: each of these microscopic cells, however, should contain a sufficient number particles for allowing them to interact from time to time. We can take *λ* ≈ 10�<sup>8</sup> m, so each microscopic cell approximately contains 30 molecules, and there are *m* ≈ 106 microscopic cells in a mesocopic cell. The particles have an average absolute value *v* ≈ 500 m.s�<sup>1</sup> in typical conditions. They can jump between the various microcells of the same mesocopic cell. They can also jump out of their initial mesoscopic cell, but they are replaced by molecules proceeding from other cells, and we assume that these contrary effects coarsely compensate themselves, except in the first stage of the evolution if the initial mesocopic distribution is strongly inhomogeneous.

Because all particles are identical, an almost microscopic configuration of a mesoscopic cell can be defined by specifying the number of particles in each of its microscopic cells. Focusing on a given mesoscopic cell, we compute the number of its possible configurations, and we estimate the average time *θ* necessary for the system to visit all these configurations. Note that the uniformization time *T* is obviously much larger than *θ*: *T* > > *θ*. So, *θ* is a lower bound of *T.*

The number of ways of partitioning the *n* identical particles into the *m* microscopic cells is

$$C = \frac{(m+n-1)!}{n!(m-1)!} \approx \exp\left[\left(m+n\right)\rho(\mathbf{x})\right] \text{ with } \rho(\mathbf{x})$$

$$= -\varkappa \ln \pi - (\mathbf{1} - \boldsymbol{\varkappa}) \ln \left(\mathbf{1} - \boldsymbol{\varkappa}\right) \text{ and } \boldsymbol{\varkappa} = n/(m+n). \tag{56}$$

The system jumps from one of these configurations to another one each time one the present particles jumps to another microscopic cell. The order of magnitude of the time needed for a particle to cross a micro-cell is *λ/v*, and the time between two configurations changes is *τ* ≈ (1/*n*) *λ/v*.. In order that all configurations are visited during time *θ* we should have at least *θ* ≈ *C τ* (in fact, *θ* should be much larger than *Cτ* because of the multiple visits during *θ*). So we conclude from (46) and relevant approximations that a lower bound of *Θ* satisfies

$$n\frac{\rho(\mathbf{x})}{\mathbf{x}} \approx \ln \frac{v}{\lambda} \text{ with } \mathbf{x} = n/(m+n) \approx 1. \tag{57}$$

With the previous numerical values

$$\theta \approx \frac{\lambda}{v} \left(\frac{n}{m}\right)^m \approx 2.10^{-11} \text{.} (30)^{10^6} \text{ s.} \tag{58}$$

*Stochastic Theory of Coarse-Grained Deterministic Systems: Martingales and Markov… DOI: http://dx.doi.org/10.5772/intechopen.95903*

which is far larger than the age of universe (now estimated to be about <sup>14</sup> � <sup>10</sup><sup>9</sup> years, or 4.4 � <sup>10</sup><sup>17</sup> s)!

Although these calculations are very rudimentary, it is clear that, in the framework of purely Hamiltonian systems, the microscopic distribution within a mesocopic cell remains far from uniformity during any realistic time if it is initially fairly inhomogeneous.

More generally, it is clear that the uniformization time *T* should be of the order of Poincaré time [32–36] in a mesoscopic cell, which is known to be extraordinarily long [9, 37].

### **3.2 An elementary, empirical approach of mesoscopic systems**

The practical relevance of Markov processes to model a large class of physical systems is supported by a vast literature. We have seen that the progressive erasure of its memory over time allows one to justify the use of a Markov process to represents the evolution of the coarse-grained system. However, such representation can also stem from random disturbances due to the measurements or other sources of stochasticity: then, one has to renounce to a purely deterministic microscopic dynamics, as formerly proposed by many authors, even without adopting the formalism of Quantum Mechanics. It is interesting to compare the time scales of the relaxation to equilibrium in both approaches with an elementary example.

#### *3.2.1 Uniformization induced by randomization*

Suppose now that the measure process does not induce any significant change in the average molecules energy - so, their average velocity remains unchanged – but that it causes a random reorientation of their velocity. A rudimentary, one dimensional model of such a randomization could be to assume that each time a molecule is about to pass to a neighboring cell, it will go indifferently to one of the neighboring microscopic cell. In a one dimensional version of the model, a molecule perform a random walk on the *η* = *l*/*λ* = 10<sup>2</sup> points representing the microscopic cells contained in the mesoscopic cell, and we adopt periodic conditions at the boundaries of the mesocopic cell. The *η* � *η* transition matrix of the process is a circulant matrix which, in its simplest version, has transition probabilities ½ to jump from any state to one of its neighbors, and it is known that its eigenvalues *λ<sup>k</sup>* are *λ<sup>k</sup>* = cos (2π*k*/*η*), *k* = 0, 1, … [*η*/2]. The number of jumps necessary for relaxing to the uniform, asymptotic distribution is of the order

$$\mathbf{1}/(-\ln \lambda\_1) \mathbf{s} \mathbf{2} (2\pi/\mathfrak{n})^{-2} \approx \mathbf{500} \text{ s.t.}$$

which correspond to a relaxation time of 500. *λ*/*v* ≈ 10�<sup>8</sup> s, which is very short for current measurements, but comparable with (or even larger than) the time scale of fast modern experiments. Considering a 3-dim model would not change this time scale significantly. It is conceivable that he molecules are not necessarily reoriented each time they leave a microscopic cell. Even if the proportion of reoriented molecules is as low as 10�<sup>6</sup> , the relaxation time is of order 10�<sup>2</sup> s, which is insignificant in many simple measures. In this case the Markov representation can be justified.

#### *3.2.2 Semi-classical Hamiltonian systems*

In analogy with the previous randomized system, we can introduce a new source of stochasticity in the coarse-grained deterministc systems considered in Sections 2 and 3. This could be done by assuming that a particle cannot be described by a

needed to reach uniformity in a mesoscopic scale, starting from strong inhomogeneity. In a first approximation it is reasonable to use this uniformization time *T* to characterize the time scale over which a Markov evolution can describe the system.

Using oversimplified, but reasonable arguments [17], we now coarsely estimate

Assume that the particles constitute a gas under normal conditions, with density

. A mesocopic state can be reasonably represented by a

**3.1 Uniformization time in a mesoscopic cell: An elementary estimation**

*Advances in Dynamical Systems Theory, Models, Algorithms and Applications*

the uniformization time *T* in a mesoscopic cell. As an example, we consider *n* identical particles initially located in this cell, among *N* identical particles in an

cube of size *l* ≈ 10�<sup>6</sup> m (as an order of magnitude), which contains *n* ≈ 3. 10<sup>7</sup> molecules. We now divide the mesoscopic cell into *m* "microscopic" cells whose size *λ* is comparable to the size of a molecule: each of these microscopic cells, however, should contain a sufficient number particles for allowing them to interact from time to time. We can take *λ* ≈ 10�<sup>8</sup> m, so each microscopic cell approximately contains 30 molecules, and there are *m* ≈ 106 microscopic cells in a mesocopic cell. The particles have an average absolute value *v* ≈ 500 m.s�<sup>1</sup> in typical conditions. They can jump between the various microcells of the same mesocopic cell. They can also jump out of their initial mesoscopic cell, but they are replaced by molecules proceeding from other cells, and we assume that these contrary effects coarsely compensate themselves, except in the first stage of the evolution if the initial mesocopic

Because all particles are identical, an almost microscopic configuration of a mesoscopic cell can be defined by specifying the number of particles in each of its microscopic cells. Focusing on a given mesoscopic cell, we compute the number of its possible configurations, and we estimate the average time *θ* necessary for the system to visit all these configurations. Note that the uniformization time *T* is

The number of ways of partitioning the *n* identical particles into the *m*

*<sup>n</sup>*!ð Þ *<sup>m</sup>* � <sup>1</sup> ! <sup>≈</sup> exp ½ � ð Þ *<sup>m</sup>* <sup>þ</sup> *<sup>n</sup> <sup>φ</sup>*ð Þ *<sup>x</sup>* with *<sup>φ</sup>*ð Þ *<sup>x</sup>*

The system jumps from one of these configurations to another one each time one the present particles jumps to another microscopic cell. The order of magnitude of the time needed for a particle to cross a micro-cell is *λ/v*, and the time between two configurations changes is *τ* ≈ (1/*n*) *λ/v*.. In order that all configurations are visited during time *θ* we should have at least *θ* ≈ *C τ* (in fact, *θ* should be much larger than *Cτ* because of the multiple visits during *θ*). So we conclude from (46) and relevant

<sup>≈</sup> <sup>2</sup>*:*10�<sup>11</sup>*:*ð Þ <sup>30</sup> <sup>10</sup><sup>6</sup>

¼ �*x* ln *x* � ð Þ 1 � *x* ln 1ð Þ � *x* and *x* ¼ *n=*ð Þ *m* þ *n :* (56)

*<sup>λ</sup>* with *<sup>x</sup>* <sup>¼</sup> *<sup>n</sup>=*ð Þ *<sup>m</sup>* <sup>þ</sup> *<sup>n</sup>* <sup>≈</sup>1*:* (57)

s*:* (58)

obviously much larger than *θ*: *T* > > *θ*. So, *θ* is a lower bound of *T.*

isolated vessel. The complete system obeys Hamilton mechanics.

**for Hamiltonian systems**

*ρ* ≈ 3. 10<sup>25</sup> molecules.m�<sup>3</sup>

microscopic cells is

**96**

distribution is strongly inhomogeneous.

*<sup>C</sup>* <sup>¼</sup> ð Þ *<sup>m</sup>* <sup>þ</sup> *<sup>n</sup>* � <sup>1</sup> !

approximations that a lower bound of *Θ* satisfies

*<sup>x</sup>* <sup>≈</sup> ln *<sup>v</sup><sup>Θ</sup>*

*n m <sup>m</sup>*

*<sup>θ</sup>* <sup>≈</sup> *<sup>λ</sup> v*

*n φ*ð Þ *x*

With the previous numerical values

point, but by a probability density centered on the point that would represent it classically: such a description borrows one, but not all, of the axioms of wave mechanics, and it can be qualified as a "semi-quantical" description. A similar assumption can be introduced without referring to quantum mechanics, by noticing that a particle cannot be localized in a given mesoscopic cell with complete certainty, because of its finite size: if it is mainly attributed to a given cell, there exists a small probability that it also belongs to a neighboring cell. Even without formalizing these possibilities, one can presume that such random effects shorten drastically the memory of the mesoscopic process, and make it short with respect to ordinary measure times: then the Markov approximation described in Section 2 can correctly represent the evolution of the observed coarse-grained process.

*p iN*þ*n*�1, *<sup>N</sup>* <sup>þ</sup> *<sup>n</sup>* � 1; … ; *iN*, *N iN*�1, *<sup>N</sup>* � 1; … ; *iN*�*<sup>n</sup>*, *<sup>N</sup>* � *<sup>n</sup>*; … ; *<sup>i</sup>*0, 0<sup>Þ</sup> � � <sup>¼</sup> �

*Stochastic Theory of Coarse-Grained Deterministic Systems: Martingales and Markov…*

The last line of (49) is *<sup>p</sup>*ð Þ *<sup>n</sup> iN*þ*n*�1, *<sup>N</sup>* <sup>þ</sup> *<sup>n</sup>* � 1; … ; *iN*, *N i*<sup>j</sup> *<sup>N</sup>*�1, *<sup>N</sup>* � 1; … ; *<sup>i</sup>*0, 0<sup>Þ</sup> � .

*p iN*þ*n*�1, *<sup>N</sup>* <sup>þ</sup> *<sup>n</sup>* � 1; … ; *iN*, *N iN*�1, *<sup>N</sup>* � 1; … ; *iN*�*<sup>n</sup>*, *<sup>N</sup>* � *<sup>n</sup>*; … ; *<sup>i</sup>*0, 0<sup>Þ</sup> � � � �

*<sup>p</sup>*ð Þ *<sup>n</sup> iN*þ*n*�1, *<sup>N</sup>* <sup>þ</sup> *<sup>n</sup>* � 1; … ; *iN*, *N iN*�1, *<sup>N</sup>* � 1; … ; *iN*�*<sup>n</sup>*, *<sup>N</sup>* � *<sup>n</sup>*; … ; *<sup>i</sup>*0, 0<sup>Þ</sup> �

*lk*ð Þ� *<sup>i</sup>*0, … *ik*�<sup>1</sup> ln *<sup>Π</sup>k*ð Þ *ik*

*pN*ð Þ *<sup>i</sup>*0, 0; … ; *ik*, *<sup>k</sup>* ln *<sup>Π</sup>k*ð Þ *ik*

(Note that *σ<sup>k</sup>* is positive, although this not necessarily true for *lk*). By (24) for

D E � � <sup>≤</sup> h i *<sup>σ</sup>k*ð Þ *<sup>i</sup>*0, … *ik*�<sup>1</sup> <sup>&</sup>lt; *<sup>ε</sup>* if *<sup>n</sup>* is large enough*:* (63)

*δs n*ð Þ� *sn* ð Þ� *p s*∞ð Þ *p* can be interpreted as an entropy fluctuation with respect to its equilibrium thermodynamic value. If such a fluctuation relaxes exponentially to 0 with time, as usual, the last term of (54) tends to 0 when *n* ! ∞. Then, the *n*-times Markov approximations 4.2 and 5.1 are justified. Although exponential relaxation can be considered as a characteristic of "normal" physical systems, slower relaxations can occur: in this case the Markov approximation may be invalid.

**Appendix B: Tendency to the stationary mesoscopic distribution**

This tendency can be reasonably expected from the approximation of the exact mesocopic process by Markov processes, but it can only be affirmed by adding additional assumptions to the basic assumptions. We first prove a simple, useful lemma. **B.1. Lemma.** Consider a *d-*dim sequence *un*,k with 2 positive, integer indices *n*, *k*,

*Π*ð Þ *<sup>n</sup> <sup>k</sup>* ð Þ *ik*

> *Π*ð Þ *<sup>n</sup> <sup>k</sup>* ð Þ *ik*

½ � *sn*ð Þ� *p sN*þ*<sup>k</sup>*ð Þ *p* ∝ *n s*½ �¼ *<sup>n</sup>*ð Þ� *p s*∞ð Þ *p nδs n*ð Þ*:*

� � �*::* … *p iN*, *N i*<sup>j</sup> *<sup>N</sup>*�1, *<sup>N</sup>* � 1; … ; *<sup>i</sup>*0, 0<sup>Þ</sup> �

� � � … *<sup>p</sup>*ð Þ *<sup>n</sup> iN*, *N i*<sup>j</sup> *<sup>N</sup>*�1, *<sup>N</sup>* � 1; … ; *iN*�*<sup>n</sup>*, *<sup>N</sup>* � *<sup>n</sup>*Þ*:* �

� *<sup>Q</sup>*ð Þ *<sup>n</sup>*

*:* (61)

¼ h i *lk*ð Þ *i*0, … *ik*�<sup>1</sup> � *σk*ð Þ *n :*

(59)

*<sup>N</sup> :*

(60)

(62)

(64)

<sup>¼</sup> *p iN*þ*n*�1, *<sup>N</sup>* <sup>þ</sup> *<sup>n</sup>* � <sup>1</sup> *iN*þ*n*�2, *<sup>N</sup>* <sup>þ</sup> *<sup>n</sup>* � 2; … ; *:i*0, 0 �

and for *k* > *n* we define *lk*, using the abbreviations (32)

*i*0, … *ik*

Averaging the logarithm of Eq. (60) we have

*k*¼0

<sup>≈</sup>*p*ð Þ *<sup>n</sup> iN*þ*n*�1, *<sup>N</sup>* <sup>þ</sup> *<sup>n</sup>* � <sup>1</sup> *iN*þ*n*�2; … *iN*�1, *<sup>N</sup>* � <sup>1</sup> �

*DOI: http://dx.doi.org/10.5772/intechopen.95903*

We write

�

We have by (24)

*sn* ð Þ� *<sup>p</sup> sk*ð Þ¼ *<sup>p</sup>* <sup>X</sup>

any positive *ε*

*L*ð Þ *<sup>n</sup> N*

**99**

2 *d*<sup>2</sup>

D E � ln *<sup>Q</sup>*ð Þ *<sup>n</sup>*

*pk*, *<sup>p</sup>*ð Þ *<sup>n</sup> k*

*N* D E <sup>¼</sup> <sup>X</sup>*<sup>n</sup>*�<sup>1</sup>

satisfying the following properties:
