**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*, satisfying the following properties:


$$|u\_{n,k}| < \varepsilon\_n + \nu \, |u\_{n,k-1}| \text{ and } \varepsilon\_n \to 0 \text{ if } n \to \infty. \tag{65}$$

*P I*ð Þ *<sup>K</sup>*, *TK* <sup>≈</sup> <sup>X</sup>

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

notations of 2.8

*iKn*, *iKn*þ1, … *i*ð Þ *<sup>K</sup>*þ<sup>1</sup> *<sup>n</sup>*�<sup>1</sup>

mesoscopic system.

we have

*P I*ð Þ¼ *<sup>K</sup>*, *TK An*,*K*ð Þþ *IK*, *Tk*

Master Eq. (72) exactly. So, writing

*Un*,*<sup>K</sup>*ð Þ¼ *IK*, *TK An*,*<sup>K</sup>*ð Þþ *IK*, *Tk*

infinite if, furthermore, the following.

property (*c*) holds for typical actual systems.

eigenstate with eigenvalue 1.

tion (*c*) is satisfied:

**101**

have an upper bound <1.

*I K*

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

X *I K*

This approximate Master Equation can now be written more precisely, in the

where *An*,*K*ð Þ *IK*, *Tk* is just the *an*,*<sup>K</sup>* term of (71) expressed in the notations of 2.8 where *TK* is the group of *n* successive times: *kn*, *kn* þ 1, *::*,ð Þ *k* þ 1 *n* � 1, and *IK* ¼

� � <sup>∈</sup> <sup>M</sup>*<sup>n</sup>* describes the corresponding partial history of the

On the other hand, we know that the stationary distribution *P*<sup>0</sup> satisfies the

X *IK*

From (72)–(75) it results that *Un*,*<sup>K</sup>*ð Þ *IK*, *TK* tend to 0 when *n* and *K* tend to

condition (*c*) holds. In fact, the (*M<sup>n</sup>*- dim) vector *Un,K* is orthogonal to the left-

(*c)* When *n* increases, the absolute values of the nonstationary eigenvalues of *W*

This property is likely to hold if the actual stationary mesoscopic process is not too different from an exact Markov process. So, it is reasonable to conjecture that

of matrix *W*. All the eigenvalues of the projection of *W* in the corresponding subspace have an absolute value smaller than 1. Thus the lemma 1 applies if condi-

Note that *W* depends of *n,* but is independent of *K*.

*W IK*, *TK IK*�1, *TK*�<sup>1</sup> �

*Un*,*<sup>K</sup>*ð Þ� *IK*, *TK P I*ð Þ� *<sup>K</sup>*, *TK <sup>P</sup>*<sup>0</sup>ð Þ *IK*, *TK :* (74)

*W I*ð Þ *<sup>K</sup>* j*IK*�<sup>1</sup> *P I*ð Þ *<sup>K</sup>*�1, *TK*�<sup>1</sup> *:* (72)

� � � *P I*ð Þ *<sup>K</sup>*�1, *TK*�<sup>1</sup> *:* (73)

*W I*ð Þ *<sup>K</sup>* j*IK*�<sup>1</sup> *Un*,*K*�<sup>1</sup>ð Þ *IK*�1, *TK*�<sup>1</sup> *:* (75)

Then u*n*,*<sup>k</sup>* ! 0 if *n* ! ∞ and *k* ! ∞.

In fact, for any positive *ε*, there is an integer *n*<sup>0</sup> such that *ε<sup>n</sup>* < *ε* if *n* > *n*0, and

$$|u\_{n,k}| < \varepsilon \left(1 + \nu + \ldots + \nu^{k-1}\right) + \nu^k u\_{n,0} < \frac{\varepsilon}{1 - \nu} + \nu^k |u\_{n,0}| \text{ if } n > n\_{0.} \tag{66}$$

So, |*un,k*| can be made as small as desired by chosing *n* and *k* large enough.

**B.2.** For given integers *n* and *K* larger than 1, and states *ik* ∈ M, *k* = 0, 1, … (*K* + 1)*n*-1, we will write.

*p i*0, 0; *<sup>i</sup>*1, 1; *::*; *<sup>i</sup>*ð Þ *<sup>K</sup>*þ<sup>1</sup> *<sup>n</sup>*�1,ð Þ *<sup>K</sup>* <sup>þ</sup> <sup>1</sup> *<sup>n</sup>* � <sup>1</sup> � � � *<sup>p</sup>*ð Þ 0, 1, … ,ð Þ *<sup>K</sup>* <sup>þ</sup> <sup>1</sup> *<sup>n</sup>* � <sup>1</sup> for the sake of simplicity. In these abbreviated notations, we have

$$p(Kn, Kn+1; \ldots, (K+1)n - 1) \quad = \sum\_{i\_0, i\_1, \ldots, i\_{K-1}} p((K+1)n, -1 \ldots Kn | Kn - 1, \ldots 0)$$

$$p(0; \ 1; \ldots, Kn - 1). \tag{67}$$

We know that

$$p\left((K+1)n, -1\ldots Kn | Kn-1, \ldots 0\right) \\ = p^0((K+1)n, -1\ldots Kn | Kn-1, \ldots 0) \\ \tag{68}$$

*p*<sup>0</sup> being the stationary probabilities, and that, for large *n*

$$p^0 \left( (K+1)n, \ -1 \dots Kn \vert Kn-1, \dots 0 \right) \approx p^0 ((K+1)n, \ -1 \dots Kn \vert Kn-1, \dots K(n-1)). \tag{69}$$

More precisely, in the conditions discussed previously, for any given positive *ε*, there is an integer *n*(*ε*) such that

$$\begin{pmatrix} p^0 \left( (K+1)n, -1 \dots Kn \vert Kn-1, \dots 0 \right) - p^0 \left( (K+1)n, -1 \dots Kn \vert Kn-1, \dots K(n-1) \right) \mid < \varepsilon \quad \text{if } n \ge n(\varepsilon) \tag{70}$$

So, Eq. (67) becomes

$$\begin{aligned} p\left(\mathrm{Kn}, \mathrm{Kn} + 1; \ldots, (K+1)n - 1\right) &= \\ \sum\_{i\_0, \ldots, i\_{k-1}} \left[ p\left( (K+1)n, -1 \ldots Kn \middle| \mathrm{Kn} - 1, \ldots 0 \right) - p^0((K+1)n, -1 \ldots Kn \middle| \mathrm{Kn} - 1, \ldots 0) \right] p\left( 0; \ 1; \ldots, \mathrm{Kn} - 1 \right) \\ + \sum\_{i\_0, \ldots, i\_{k-1}} p^0\left( (K+1)n, -1 \ldots Kn \middle| \mathrm{Kn} - 1, \ldots (K-1)n \middle| \, p(K-1)n; \ldots, \mathrm{Kn} - 1 \right) \\ \equiv a\_{n, K} + \sum\_{i\_0, \ldots, i\_{k-1}} p^0\left( (K+1)n, -1 \ldots Kn \middle| \mathrm{Kn} - 1, \ldots (K-1)n \right) p\left( K-1 \middle| n; \ldots, \mathrm{Kn} - 1 \right). \end{aligned} \tag{71}$$

where the 1st term of the last line satisfies j j *a n*ð Þ , *K* < *ε* if *n* ≥ *n*(*e*).

The second term in the last line of (71) is, in other notations, the righth hand side term of the approximate Master Eq. (47) of Section 2.8

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

$$P(I\_K, T\_K) \approx \sum\_{I\_K} \, \_{\mathcal{U}} \mathcal{W}(I\_K \, | \, I\_{K-1}) \, \, P(I\_{K-1}, T\_{K-1}) . \tag{72}$$

This approximate Master Equation can now be written more precisely, in the notations of 2.8

$$P(I\_K, T\_K) = A\_{\mathfrak{n}, \mathcal{K}}(I\_K, T\_k) + \sum\_{L\_k} W(I\_K, T\_K \mid I\_{K-1}, T\_{K-1}) \; P(I\_{K-1}, T\_{K-1}) \; . \tag{73}$$

where *An*,*K*ð Þ *IK*, *Tk* is just the *an*,*<sup>K</sup>* term of (71) expressed in the notations of 2.8 where *TK* is the group of *n* successive times: *kn*, *kn* þ 1, *::*,ð Þ *k* þ 1 *n* � 1, and *IK* ¼ *iKn*, *iKn*þ1, … *i*ð Þ *<sup>K</sup>*þ<sup>1</sup> *<sup>n</sup>*�<sup>1</sup> � � <sup>∈</sup> <sup>M</sup>*<sup>n</sup>* describes the corresponding partial history of the mesoscopic system.

On the other hand, we know that the stationary distribution *P*<sup>0</sup> satisfies the Master Eq. (72) exactly. So, writing

$$U\_{\mathfrak{n},\mathcal{K}}(I\_K, T\_K) \equiv P(I\_K, T\_K) - P^0(I\_K, T\_K). \tag{74}$$

we have

i. it is absolutely bounded: there is a positive real number *M* such that |*un,k*|

j j *un*,*<sup>k</sup>* < *ε<sup>n</sup>* þ *ν* j j *un*,*k*�<sup>1</sup> and *ε<sup>n</sup>* ! 0 if *n* ! ∞*:* (65)

1 � *ν*

<sup>þ</sup> *<sup>ν</sup><sup>k</sup>* j j *un*,0 if *<sup>n</sup>*<sup>&</sup>gt; *<sup>n</sup>*0*:* (66)

*p K*ðð Þ þ 1 *n*, � 1 … *Kn Kn* j � 1, *:::*0Þ

(68)

(70)

(71)

*p*ð Þ 0; 1; *::*,*Kn* � 1 *:* (67)

ii. for all *n*, *k,* there are positive numbers *ε<sup>n</sup>* (independent of *n*) and *ν*

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

In fact, for any positive *ε*, there is an integer *n*<sup>0</sup> such that *ε<sup>n</sup>* < *ε* if *n* > *n*0, and

So, |*un,k*| can be made as small as desired by chosing *n* and *k* large enough. **B.2.** For given integers *n* and *K* larger than 1, and states *ik* ∈ M, *k* = 0, 1, …

*i*

*p*<sup>0</sup> being the stationary probabilities, and that, for large *n*

*p i*0, 0; *<sup>i</sup>*1, 1; *::*; *<sup>i</sup>*ð Þ *<sup>K</sup>*þ<sup>1</sup> *<sup>n</sup>*�1,ð Þ *<sup>K</sup>* <sup>þ</sup> <sup>1</sup> *<sup>n</sup>* � <sup>1</sup> � � � *<sup>p</sup>*ð Þ 0, 1, … ,ð Þ *<sup>K</sup>* <sup>þ</sup> <sup>1</sup> *<sup>n</sup>* � <sup>1</sup> for the sake of

<sup>0</sup>, *i*1, … *iKn*�<sup>1</sup>

*p K*ð Þ <sup>þ</sup> <sup>1</sup> *<sup>n</sup>*, � <sup>1</sup> … *Kn Kn* <sup>j</sup> � 1, *:::*0<sup>Þ</sup> <sup>¼</sup> *<sup>p</sup>*<sup>0</sup>ðð Þ *<sup>K</sup>* <sup>þ</sup> <sup>1</sup> *<sup>n</sup>*, � <sup>1</sup> … *Kn Kn* <sup>j</sup> � 1, *:::*0<sup>Þ</sup> *:* �

*<sup>p</sup>*<sup>0</sup> ð Þ *<sup>K</sup>* <sup>þ</sup> <sup>1</sup> *<sup>n</sup>*, � <sup>1</sup> … *Kn Kn* <sup>j</sup> � 1, *:::*0Þ<sup>≈</sup> *<sup>p</sup>*<sup>0</sup>ðð Þ *<sup>K</sup>* <sup>þ</sup> <sup>1</sup> *<sup>n</sup>*, � <sup>1</sup> … *Kn Kn* <sup>j</sup> � 1, … *K n*ð ÞÞ � <sup>1</sup> *:* � (69)

More precisely, in the conditions discussed previously, for any given positive *ε*,

*<sup>p</sup>*<sup>0</sup> ð Þ *<sup>K</sup>* <sup>þ</sup> <sup>1</sup> *<sup>n</sup>*, �<sup>1</sup> … *Kn Kn* <sup>j</sup> � 1, *:::*0Þ�*p*<sup>0</sup>ðð Þ *<sup>K</sup>* <sup>þ</sup> <sup>1</sup> *<sup>n</sup>*, �<sup>1</sup> … *Kn Kn* <sup>j</sup> � 1, … *K n*ð ÞÞ � <sup>1</sup> <sup>j</sup> <sup>&</sup>lt; *<sup>ε</sup>* if *<sup>n</sup>*≥*n*ð Þ*<sup>ε</sup>* � � �

*p K*ð Þ <sup>þ</sup> <sup>1</sup> *<sup>n</sup>*, �<sup>1</sup> … *Kn Kn* � 1, *:::*0Þ � *<sup>p</sup>*<sup>0</sup>ðð Þ *<sup>K</sup>* <sup>þ</sup> <sup>1</sup> *<sup>n</sup>*, �<sup>1</sup> … *Kn Kn* <sup>j</sup> � 1, *:::*0<sup>Þ</sup> �

*<sup>p</sup>*<sup>0</sup>ð Þ ð Þ *<sup>K</sup>* <sup>þ</sup> <sup>1</sup> *<sup>n</sup>*, �<sup>1</sup> … *Kn Kn* <sup>j</sup> � 1, … ð Þ *<sup>K</sup>* � <sup>1</sup> *<sup>n</sup>*Þ*p K*ð Þ � <sup>1</sup> *<sup>n</sup>*; *::*, *Kn* � <sup>1</sup>

where the 1st term of the last line satisfies j j *a n*ð Þ , *K* < *ε* if *n* ≥ *n*(*e*).

term of the approximate Master Eq. (47) of Section 2.8

The second term in the last line of (71) is, in other notations, the righth hand side

� � *<sup>p</sup>*ð Þ 0; 1; *::*, *Kn* � <sup>1</sup> � �

*<sup>p</sup>*<sup>0</sup>ð Þ ð Þ *<sup>K</sup>* <sup>þ</sup> <sup>1</sup> *<sup>n</sup>*, �<sup>1</sup> … *Kn Kn* <sup>j</sup> � 1, … ð Þ *<sup>K</sup>* � <sup>1</sup> *<sup>n</sup>*<sup>Þ</sup> *p K*ð Þ � <sup>1</sup> *<sup>n</sup>*; *::*,*Kn* � <sup>1</sup> *:*

*un*, 0 <sup>&</sup>lt; *<sup>ε</sup>*

< *M* for all integers *n, k.*

Then u*n*,*<sup>k</sup>* ! 0 if *n* ! ∞ and *k* ! ∞.

j j *un*,*<sup>k</sup>* <sup>&</sup>lt; *<sup>ε</sup>* <sup>1</sup> <sup>þ</sup> *<sup>ν</sup>* <sup>þ</sup> … <sup>þ</sup> *<sup>ν</sup>k*�<sup>1</sup> � � <sup>þ</sup> *<sup>ν</sup><sup>k</sup>*

simplicity. In these abbreviated notations, we have

*p Kn* <sup>ð</sup> ,*Kn* <sup>þ</sup> 1; *::*,ð Þ *<sup>K</sup>* <sup>þ</sup> <sup>1</sup> *<sup>n</sup>* � <sup>1</sup>Þ ¼ <sup>X</sup>

there is an integer *n*(*ε*) such that

So, Eq. (67) becomes

*p Kn* ð , *Kn* þ 1; *::*,ð Þ *K* þ 1 *n* � 1Þ ¼

*i* <sup>0</sup>, … *iKn*�<sup>1</sup>

� *an*,*<sup>K</sup>* <sup>þ</sup><sup>X</sup>

X *i* <sup>0</sup>, … *iKn*�<sup>1</sup>

þ X *i* <sup>0</sup>, … *iKn*�<sup>1</sup>

**100**

(*K* + 1)*n*-1, we will write.

We know that

(independent of *n* and *k*) such that

$$U\_{n, \mathcal{K}}(I\_{\mathcal{K}}, T\_{\mathcal{K}}) = \ \ A\_{n, \mathcal{K}}(I\_{\mathcal{K}}, T\_{\mathcal{k}}) + \sum\_{\mathcal{I}\_{\mathcal{K}}} W(I\_{\mathcal{K}} | I\_{\mathcal{K}-1}) \ \ U\_{n, \mathcal{K}-1}(I\_{\mathcal{K}-1}, T\_{\mathcal{K}-1}). \tag{75}$$

Note that *W* depends of *n,* but is independent of *K*.

From (72)–(75) it results that *Un*,*<sup>K</sup>*ð Þ *IK*, *TK* tend to 0 when *n* and *K* tend to infinite if, furthermore, the following.

condition (*c*) holds. In fact, the (*M<sup>n</sup>*- dim) vector *Un,K* is orthogonal to the lefteigenstate with eigenvalue 1.

of matrix *W*. All the eigenvalues of the projection of *W* in the corresponding subspace have an absolute value smaller than 1. Thus the lemma 1 applies if condition (*c*) is satisfied:

(*c)* When *n* increases, the absolute values of the nonstationary eigenvalues of *W* have an upper bound <1.

This property is likely to hold if the actual stationary mesoscopic process is not too different from an exact Markov process. So, it is reasonable to conjecture that property (*c*) holds for typical actual systems.

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

**References**

Barth; 1898

[1] Boltzmann L. Vorlesungen über Gastheorie. In: *part*. Vol. *2*. Leipzig: J. A.

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

[14] Gallavotti G, Cohen EGD. Phys.

[15] Yvon J. *La Théorie Statistique des Liquides*. Paris: Herman; 1935

[17] B. Gaveau and L.S. Schulman, EPJST

[18] Gaveau B, Moreau M. Chaos. 2020;

[19] Arnold VI, Avez A. *Ergodic probems of Classical Mechanics*. Mathematical Physics Monographs: Benjamin; 1968

[20] Doob J. *Stochastic Processes*. N.Y:

[21] Levy P. *Théorie de l'addition des variables aléatoires*. Paris: Gauthier-

[22] Feller W. *An Introduction to Probability Theory and Its Applications*,

[23] McKean HP. *Stochastic Integrals*. London: Academic Press, NY; 1968

*a particle in a very rarefied gas*,

[24] Gaveau B, Gaveau MA. *Diffusion of*

Symposium on rarefied gas dynamics, Muntz, Weaver, Campell eds. Progress in Astronomics and Aeronautics. 1988;

[25] Jaynes ET. Phys. Rev. II **108**, 171 (1957). Phys. Rev. II. 1957;**106**:620

*Foundations of Statistical Mechanics*. N.Y:

[27] C. E. Shannon, *A mathematical theory of communication*, Bell System

[26] Khinchin AI. *Mathemtical*

vol II. N.Y: Wiley; 1971

[16] Forster D. *Hydrodynamic Fluctuations*. Benjamen, Reading, Masschusetts: *Broken Symmetry and*

*Correlation Function*s; 1975

**224**, 8, p 891 (2015).

**30**:083104

Wiley; 1953

Villars; 1937

**118**(61)

Dover; 1949

Rev. Lett. 1995;**74**:2694

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

[2] L. Boltzmann. Reply to Zermelo's Remarks on the theory of heat.In: History ofmodern physical sciences: The kinetic theory of gases, ed. S. Brush, ImperialCollege Press, **57**, 567 (1896).

[3] Ehrenfest P, Ehrenfest T. *The conceptual foundations of the*

York: Dover; 1990

1980

*statisticalapproach in Mechanics*. New

[4] Uhlenbeck GE. *Anoutline of Statistical Mechanics, in Fundamental problemsin Statistical Mechanics*. II. ed. Cohen, North Holland, Amsterdam: E. G. D; 1968

[5] Landau LD, Lifshitz EM. *Statistical Physics*. 3rd ed. Oxford: Pergamon Press;

[6] Landau LD, Pitaevskii LP. *Physical Kinetics*. Oxford: Pergamon Press; 1981

[7] H.B. Callen,*Thermodynamics and an Introduction to Thermostatistics*( John Wiley and Sons, New York, (1985).

[8] Reif F. *Fundamentals of Statistical and Thermal Physics*. NY: Mc Graw Hill; 1965

[9] J.P. Eckmann and D. Ruelle, *Ergodic theory of chaos and strange attractors*, Rev

[10] J.R. Dorfman, A, Introduction to Chaos I, Nonequilibrium Statistical Mechanics (Cambridge, New York,

[11] Gallavotti G. Statistical Mechanics: a Short Treatise. Berlin: Springer; 1999

[12] Gallavotti G. Journal of Statistical

[13] Evans DJ, Cohen EGD, Morries GP.

Mod Phys**57**, 617 (1985).

Physics. 1995;**78**:1571

Phs. Rev. Lett**.** 1993;**71**:15

(1999).

**103**
