Finite and infinite dynamical systems of identical interacting particles

Physica A 387 (2008) 2719–2735 www.elsevier.com/locate/physa

Finite and infinite dynamical systems of identical interacting particles Miriam Lemanska Soreq NRC, Yavne 81800, Israel Received 17 January 2008 Available online 28 January 2008

Abstract A dynamical system of infinite volume and of infinite number of identical interacting particles occupying energy levels ei (i = 1, 2, . . . , I ) has been constructed as the limit of an infinite sequence of finite, equivalent systems of increasing size and ∗ = lim S ∗ , respectively, particle number. Systems both in equilibrium and in non-equilibrium state (designated S∞ = lim Sk , S∞ k k = 1, 2, . . .) were investigated. The main results are: (i) The values in the T -limit (thermodynamic limit) of the physical quantities characterizing these systems are determined. ∗ systems is governed by the non-linear rate equations p (t)/dt = (ii) The time evolution process both in Sk∗ and in S∞ i − ln pi (t) + a(t) + ei b(t) (i = 1, 2, . . . , I ) with common initial conditions pi (t0 ), where pi (t) = n i (t)/N are the occupation ∗ systems is the same. The asymptotic approach to the equilibrium probabilities at time t. The time evolution process in the Sk∗ and S∞ state is proved. (iii) For the case of the equilibrium state, the Boltzmann probability distribution pi is given by the equation − ln pi +a+ei b = 0 common to Sk and S∞ systems with the same value of a and b. The term a = β −1 ae , where ae is the free energy per particle, and b = −β (=−1/k B T ). (iv) The conditions for the equivalence of the systems being in equilibrium and also of the ones in non-equilibrium are stated. c 2008 Elsevier B.V. All rights reserved.

Keywords: Finite and infinite dynamical systems; Thermodynamic limit; Equivalent systems; Time evolution process; Approach to equilibrium

1. Introduction In this work we consider a physical dynamical system, which extends throughout the space R k , k < ∞ (usually k = 3) and contains an infinite number of identical, interacting particles occupying the basic energy levels 0 ≤ ei ≤ 1 (i = 1, 2, . . . , I ). Such system will be called in what follows an infinite one. By a finite system we understand an isolated system, which consists of a large but finite number of identical, interacting particles occupying the energy levels defined above. It should be noted that the terms, the infinite dimensional system (met in the literature) and the infinite one (used here), designate the same system of infinite volume and of infinite particle number. The first name is due to the infinite dimension of the phase space. The aim of this work is to construct and to investigate the infinite system and its connection with the finite ones. Both equilibrium and non-equilibrium states are considered. In

E-mail address: [email protected]. c 2008 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2008.01.089

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order to study the time evolution process in a perturbed infinite system the non-linear model for relaxation of Ref. [1] is applied (see also Eq. (A.1) of Appendix). Several authors have studied the dynamical behavior in large time of the finite and infinite systems generated by various differential equations, for example Refs. [3–8] and the references therein. Hale [8] gives an extensive theory of flow defined by dynamical systems, which are generated by continuous semigroups. Most of the investigations are concentrated on the following subjects (see Introduction to Refs. [5,6], Teman in Ref. [6]): (i) Determining the spatial discretization in order that the behavior of the discrete system and the initial (infinite) one will be the same in large time physical processes. (ii) The existence of a compact global attractor for infinite dynamical systems. The global attractor attracts all trajectories formed during the time evolution process in the phase space. The global attractor is an invariant compact set. (iii) The existence of the inertial manifold and the inertial system. This subject is related to (ii) and is studied in many recent works. The important property of an invariant system is the asymptotic behavior in the approach to the equilibrium. The bifurcation problem and the connection between the finite and infinite system are related to the above subjects. Problems (i)–(iii) are not relevant to our mathematical model shown in Eq. (A.1) (Appendix). Eq. (A.1) has been derived in Ref. [1] in order to study the time evolution process in finite systems defined at the beginning of this section. The treatment of an infinite system by Eq. (A.6), (which is the integrated form of Eq. (A.1)) is suggested by its significant properties recalled from Appendix A.1 of the Appendix: (1) The occupation numbers n i∗ (t) and the total particle number N ∗ are not contained explicitly in Eq. (A.6). The terms of our equation include only the ratios pi∗ (t) = n i∗ (t)/N ∗ . (2) I -the number of Eq. (A.6) is finite and depends only on the number of the assumed energy levels, 0 ≤ ei ≤ 1 (i = 1, . . . , I ). The idea that an infinite system is isolated along with the properties (1), (2) allows to apply Eq. (A.6) to the study of the time evolution process both in the finite systems and the infinite one. From property (1) we conclude that problem (i) is not relevant to Eq. (A.6). It has been proved in the Appendix that the finite and the infinite systems approach to the equilibrium state, the Boltzmann distribution, in a monotonic and asymptotic manner. Hence, in the place of a global attractor we have a point attractor (fixed point), the Boltzmann distribution. To attain our purpose noted at the beginning of this section, the following questions have been examined. (1) The construction of an infinite system and associated questions. We denote by Sk and Sk∗ (k = 1, 2 . . .) finite systems with increasing size and increasing particle number, being in the equilibrium and in the non-equilibrium state respectively. The sequence {Sk , } is constructed in Section 2.1 by increasing the volume, the total particle number and the occupation numbers by the same factor greater than one (see Eqs. (2.1) and (2.2)). The sequence {Sk∗ } is obtained in Section 3.1 by introducing into the systems Sk the same perturbation (repopulation of a particular energy level e j ) as shown in Eq. (3.1). The infinite systems are regarded as ∗ = lim S ∗ k → ∞ (Sections 2.2 and 3.2). limit of the above sequences: S∞ = lim Sk and S∞ k It is shown that the members of the sequence {Sk } and S∞ are equivalent to each other. The same is valid for the ∗ systems. Of course, the concept of the equivalence of systems being in non-equilibrium state includes {Sk∗ } and S∞ the notion of an identical time evolution process (see Section 4). The sufficient and necessary conditions for the equivalence are stated in Sections 2.4 and 4.3. The concept of equivalent systems allows us to understand the relations between finite and infinite systems, Sections 2.4 and 4.3. Thus we have obtained the thermodynamic (T )-limit by a simple limiting process (Sections 2.2, 2.3 and 3.2). In Refs. [3,4,7] the pressure and the free energy were determined in the T -limit, by a limiting process which requires several constraints concerning the particle interaction and the convergence procedure. (2) The time evolution process. The relations between the initial, the new equilibrium state and the intermediate non-equilibrium one. The equivalence of the finite and infinite systems in non-equilibrium state. The topological treatment of the time evolution process, the proof of the monotonic behavior of the occupation probabilities and of the asymptotic approach to the equilibrium state are given in the Appendix. The approach to the equilibrium state is a fundamental problem of the non-equilibrium statistical mechanics and is to date topical for the case of isolated systems [2]. ∗ systems are the same, i.e. the probabilities We have shown that the time evolution processes in the Sk∗ and S∞ ∗ pi (t) are common and approach to the same distribution pi (Section 4). The influence of the induced perturbation,

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the repopulation, for example, of energy level e j (see Ref. [1]) in a system is considered and discussed in Section 4.2. On the basis of our numerical work we have shown how the occupation probabilities pi∗ (t) change during the evolution process. The relationship between the perturbed system S ∗ (finite or infinite), the earlier S f and the new S n equilibrium states is examined in Section 4.2. (3) The equation for the equilibrium distribution. Its properties and solution. The approach to the equilibrium allows us to derive the Eq. (2.14) for the equilibrium distribution pi from Eq. (A.1) as t → ∞ (Section 2.3). The terms of Eq. (2.14) are expressed by the important thermodynamic quantities, β and the free energy per particle ae . The solution is the Boltzmann distribution. The results of Section 2.3 show that the equilibrium state both of finite and infinite systems is completely determined by the particle density and the temperature. The results of our work are discussed in Section 5. 2. Finite and infinite systems in an equilibrium state Let Sk be a finite system in an equilibrium state, containing a large number Nk of identical, interacting particles in a volume Vk . We denote by dk the particle density, by n ik the number of the particles occupying the energy level ei , by P P pik = n ik /Nk the occupation probability. E k = i ei pik is the mean energy, Se,k = − i pik ln pik , i = 1, 2, 3, . . . , I is the entropy of the occupation probabilities. In Section 2,1 an infinite sequence of finite systems Sk (k = 1, 2, . . .) will be constructed by increasing the size of S1 . Then the infinite system S∞ will be obtained as limk Sk . k → ∞. 2.1. The infinite sequence {Sk } We assume that the volume Vk , the particle number Nk and the occupation numbers n ik increase in the same ratio ak (>1), i.e. Vk = ak Vk−1 , n ik

=

Nk = ak Nk−1

ak · n ik−1 .

(2.1) (2.2)

Proposition 2.1. The particle density dl , and the probabilities pil are common to the systems Sk (k = 1, 2, . . .). Proof. From Conditions (2.1) and (2.2) the following relations are obtained: dk = Nk /Vk = ak N1 /ak V1 = N1 /V1 = d1

(2.3)

pik = n ik /Nk = ak n i1 /ak N1 = pi1 .

(2.4)

Note: (i) The probabilities pi are intensive quantities. P (ii) The total energy E ktot = i ei n ik increases by the factor ak like the particle number Nk , but the energy density u k = u 1 is common to all systems Sk .  Corollary 2.1. The normalization condition (A.3), the mean energy E 1 and the entropy Se,l are maintained in all systems Sk . Proof. As a direct consequence of Eq. (2.4) we have for every k = 1, 2, 3, . . . X pik = 1

(2.5)

i

Ek =

X

ei pik = E 1

(2.6)

I

Se,k =

X i

− pik ln pik = Se,l .

(2.7)

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Note that Eqs. (2.6) and (2.7) are not in contradiction to the extensive character of the energy and the entropy, because E k and Se,k are quantities averaged per system. The additivity property of Se,k (called also the mean entropy [4,9]) is evident if pi = f i1 f i2 , and f are probabilities of independent events [9].  2.2. The infinite system S∞ To determine the infinite system we find the limit of the sequence Sk, i.e. the value of the quantities of Section 2.1 as k → ∞. Proposition 2.2. The particle density dl , the probabilities pil are maintained in the T -limit. Proof. From Eqs. (2.3) and (2.4) it follows that lim dk = dl ,

as k → ∞

(2.8)

lim u k = u l ,

as k → ∞

(2.9)

lim

pik

=

pil ,

as k → ∞. 

(2.10)

Corollary 2.2. The normalization condition (2.5), the mean energy E 1 and the mean entropy Se,1 are preserved in the thermodynamic limit. Proof. Following Eq. (2.10) we have X lim pik = 1, as k → ∞

(2.11)

i

lim

X

ei pik = E 1 ,

as k → ∞

(2.12)

i

lim Se,k = Se,l

as k → ∞.

(2.13)

To complete the determination of the T -limit, the free energy and pressure will be found in Section 2.3. The infinite system will be designated by S∞ . Owing to the common property of the physical quantities considered above the indices l and k will be omitted in what follows.  2.3. Equation for pi , free energy and pressure First, the equations for the equilibrium distribution pi are formulated and solved. Secondly, the free energy per particle and the pressure of the Sk and S∞ systems are determined. That completes the T -limit of Section 2.2. Notes: (i) Eq. (A.1) is derived for isolated system. (ii) An infinite system is isolated because it is extended throughout the whole space, universe. (iii) It is shown in Ref. [1] and in the Appendix, that the time evolution process governed by Eq. (A.1) approaches to an equilibrium state. Theorem 2.1. The particle distribution of finite and infinite systems (defined in Section 1) being in equilibrium state is given by the equations − ln pi + a + ei b = 0 (i = 1, 2, . . . , I )

(2.14)

where b = −β and a = βa0 , a0 is the free energy per particle. The Boltzmann probability distribution is the unique solution. Proof. From the above Note Eq. (2.14) is obtained by putting d pi (t)/dt = 0 in Eq. (A.1). The solution of Eq. (2.14) is pi = exp(a) exp(ei b).

(2.15)

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The term exp(a) is eliminated using the condition

P

i

pi = 1.

!−1 X

exp(a) =

.

exp(bei )

(2.16a)

i

Substituting (2.16b)

b = −β into Eqs. (2.15) and (2.16a) we obtain X exp(−βei ), pi = exp(−βei )/

i = 1, 2, . . . , I.

(2.17)

i

This is the Boltzmann distribution. From Eq. (2.16) we have, X a = − ln exp(−βei ) = β −1 ae

(2.18)

i

ae = −β −1 ln

X

exp(−βei )

(2.19)

i

is the free energy per particle. Like [3, p. 124–6] the relations below show that ae has the known property of being a thermodynamic potential. X X dβae /dβ = ei exp(−βek )/ exp(−βei ) = E (2.20) i

I

d(ae /T )/d(1/T ) = E −(dae /dT ) = k B Se

(2.21) (2.22)

ae = E − k B T Se .

(2.23)

Obviously, the mean energy E and the mean entropy Se are expressed in the terms of the Boltzmann distribution.



We next return to the systems Sk and S∞ . From Sections 2.1 and 2.2 we have Corollary 2.3. The Boltzmann distribution, the value of the free energy per particle ae are common to the Sk and S∞ systems. Next, the pressure P(d, T ) will be derived from the free energy ae . Theorem 2.2. The pressure of any finite and infinite systems (Section 1) is X exp(−βei ) P(d, T ) = −dae = dT ln

(2.24)

i

β, d, T and ae have their usual meaning. Proof. We apply the formula P = (∂ A/∂ V )T (2.24*) of Table 4.4.1 [3], where A is the free energy of the system of interest. Substituting A = dV ae into Eq. (2.24*) we obtain Eq. (2.24).  Proposition 2.3. The pressure P(d, β) = −dae = dβ −1 ln

P

exp(−βei ) is common to the Sk and S∞ systems.

Proof. The validity follows from the Corollary 2.3 and Theorem 2.2.



It is worth noting that the derivations of the quantities ae and P are quite general and do not require any limitative conditions met in the literature, see Refs. [4,7]. Corollary 2.4. The pressure of an arbitrary finite and infinite systems in equilibrium state is determined by the temperature T and the density d (Section 1).

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2.4. The equivalence of finite and infinite systems in equilibrium The results of the above sections elucidate the relations between the infinite and finite systems. Some remarks and conclusions are summarized as follows. The connection between the finite and infinite systems is related to the concept of their equivalence expressed by the common property of the intensive quantities [3]. The common values of the average quantities pi , E, Se, , ae , d, T, P enable us to state that the infinite S∞ system and the finite Sk ones are equivalent. As it is manifested in the above sections, this equivalence is the consequence of the common particle density d and the temperature T . Therefore the propositions and corollaries of Sections 2.1–2.3 prove the following statement. Theorem 2.3. Finite and infinite systems (defined in Section 1) being in the equilibrium state are equivalent (one to the other) iff they have a common particle density d and a common temperature T . 3. Infinite system in a non-equilibrium state As in Section 2.2, an infinite system in non-equilibrium state is regarded as a limit of an infinite sequence of finite systems Sk∗ k = 1, 2, . . . being in a non-equilibrium state. The systems Sk∗ are obtained by a repopulation of a particular energy level e j by the same factor f in all systems Sk . Clearly, the so perturbed systems are out of equilibrium. We designate the physical quantities characterizing Sk∗ systems by the asterisk *. 3.1. The infinite sequence Sk∗ We assume that the probabilities p kj are changed by the same factor f ( f > 1 or f < 1) at the time t0 in all Sk systems. That means k p kj → p ∗,k j (t0 ) = f p j

k = 1, 2, 3 . . .

(3.1)

k n ∗,k j = fnj

(3.2a)

Nk∗ = Nk + ( f − 1)n kj .

(3.2b)

Note that Vk is kept fixed, but the particle number Nk has been changed. According to the condition (A.3), the probabilities are renormalized and denoted by pi∗,n (t0 ). ! X ∗,k k k k p j (t0 ) = f p j / pi. + f p j

(3.3a)

i6= j

! pi∗,n (t0 )

=

pin /

X

pik

+

f p kj

.

(3.3b)

i6= j

Proposition 3.1. The density d1∗ and the probabilities pi∗,1 (t0 ) are common to all systems Sk∗ k = 1, 2, . . . . Proof. dk∗ = (Nk + ( f − 1)n kj )/Vk = (ak N1 + ( f − 1)ak nlj )/ak V1 = (N1 + ( f − 1)nlj )/V1 = d1∗ k = 1, 2, . . . .

(3.4)

From Eqs. (2.4) and (3.3) it follows that pi∗,k (t0 ) = pi∗,l (t0 )

k = 1, 2, . . . , i = 1, 2, 3, . . . I.



(3.5)

∗ (t ), are maintained in all members of the sequence S ∗ . Corollary 3.1. The mean energy E 1∗ , the mean entropy, Se,l 0 k

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M. Lemanska / Physica A 387 (2008) 2719–2735

Proof. Following Eq. (3.5) we obtain for k = 1, 2, . . . X X ei pi∗,k (t0 ) = ei pi∗,l (t0 ) = E 1∗ E k∗ = i ∗ Se,k (t0 ) =

(3.6)

i

X

pi∗kn (t0 ) ln pi∗,k (t0 ) = −

X

∗ pi∗ , 1(t0) ln pi∗,l (t0 ) = Se,l (t0 ).

(3.7)

i

The new β ∗ (or T ∗ ) is defined by the mean energy E ∗ = Eq. (3.6) we conclude : 

i ei pi (t0 )

P

=

P

I ei

exp(−β ∗ ei )/

P

I

exp(−β ∗ ei ). From

Corollary 3.2. The value of β ∗ (=1/T ∗ ) is common to all systems Sk∗ . ∗ 3.2. The infinite system S∞

Proposition 3.2. The values of the particle density d1∗ and the initial distribution pi∗1 (t0 ) are preserved in the T -limit. Proof. As in Section 2.2, the limiting process gives lim dk∗ = d1∗

as k → ∞

lim pi∗,k (t0 ) = pi∗,1 (t0 )

(3.8) as k → ∞.



(3.9)

∗ (t ) are preserved Corollary 3.3. The normalization condition (A.3), the mean energy E 1∗ and the mean entropy Se,l 0 in the thermodynamic limit.

Proof. From Eq. (3.9) it follows, X ∗,k lim pi (t0 ) = 1 as k → ∞

(3.10)

i ∗ E∞ = lim

X

ei pi∗,k = E 1∗

as k → ∞

(3.11)

i ∗ Se,∞ (t0 ) = lim −

X

∗ pi∗,k (t0 ) ln pi∗,k (t0 ) = Se,1 (t0 )

as k → ∞.



(3.12)

i

Corollary 3.4. The value of β ∗ (or T ∗ ) is preserved in the thermodynamic limit. ∗ . In what follows the indices k and 1 will be omitted.) (The T -limit is designated by S∞ Note that the physical quantities considered in this section at the time t0 establish the initial conditions for the time evolution process described by Eq. (A.6). ∗ . Corollary 3.5. The initial conditions pi∗ (t0 ), (i = 1, 2, . . . , I ) to Eq. (A.6) are the same for the systems Sk∗ and S∞

4. Time evolution in finite and infinite systems ∗ and S ∗ systems is discussed. It is shown in what follows that this In this section the time evolution process in S∞ k process is the same. ∗ system 4.1. The equation for the time evolution process in S∞

As in Section 2.3, we have Corollary 4.1. The time evolution process both in infinite and finite systems (Section 1) being in a non-equilibrium state is governed by Eq. (A.6).

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4.2. The time evolution process Some results from Ref. [1] and Appendix concerning the evolution process are recalled below. (1) The solution of Eq. (A.6), pi∗ (t) (i = 1, 2, . . . I ) (t ∈< t0 , ∞) is determined uniquely by the initial conditions pi∗ (t0 ). (2) The mapping (A.6) is one-to-one. (3) The finite and infinite systems, defined in Section 1, being in a non-equilibrium state approach asymptotically to the equilibrium. I.e. lim pi∗ (t) = pi∗ , pi∗

as t → ∞

(4.1) β∗

where is the Boltzmann distribution of the new equilibrium state related to and given by Eq. (2.17). (4) The probabilities pi∗ (t) are monotonic functions of t. The difference pi∗ − pi∗ (t0 ) shows whether pi∗ (t) increases or decreases during the time Pevolution process. (5) The entropy Se∗ (t) = − i pi∗ (t) ln pi∗ (t) is a positive, monotonic, increasing function of the time t, and ∗ is common to S ∗ and S ∗ systems. approaches as t → ∞ to the equilibrium. The maximum entropy Se,B ∞ k The items (1)–(3), the Corollaries 3.5 and 4.1 and the above items prove the following statements. ∗ approach to the equilibrium states Proposition 4.1. The finite systems Sk∗ and the corresponding infinite one S∞ having the same Boltzmann distribution. ∗ is identical. Proposition 4.2. The course of the time evolution process in the systems Sk∗ and S∞

The significant statements given in items (1)–(5) are independent of the perturbation induced to systems Sk . From the item (4) arise the following questions: (i) Which of the pi∗ (t) increase and which decrease. (ii) What is the relation between the distribution pi∗ in the new equilibrium and pi in the earlier one. To answer this, the influence of the induced perturbation E ∗ − E should be examined. As in Section 3 the index j denotes the perturbed energy level and f —the perturbation factor. From Eq. (3.3) and from the results of our numerical work Ref. [1], we can formulate the following items. (I) f > 1. (Ia) Eq. (3.3) give p ∗j (t0 ) > p j and pi∗ (t0 ) < pi for i 6= j. During the evolution process p ∗j (t) decreases and pi∗ (t) (i 6= j) increase. P (Ib) From Eq. (3.3) and the formula E ∗ = ei pi∗ (t0 ) we have: if the perturbed energy level e j > E, then E ∗ > E. The relation between pi∗ and pi (new and earlier equilibrium) is: pi∗ > pi for i ≥ j, pi∗ < pi for i < j. (Ic) As in (Ib), when e j < E, then E ∗ < E. The relation between pi∗ and pi is : pi∗ > pi for i ≤ j and pi∗ < pi for i > j. (II) f < 1. (Iia) From Eq. (3.3) we obtain pj ∗ (t0 ) < pj and the remainder pi ∗ (t0 ) > pi for i 6= j. pj ∗ (t) increases and pi∗ (t) for i 6= j decrease during the evolution process. (Iib) If ej > E, or ej < E, the relations are like the ones of items (Ib), (Ic) with > changed to <. The above relations are illustrated by the numerical results of the evolution process of the system given in Table 1, taken over from Ref. [1]. Some of the pi∗ (t) are graphed on Fig. 1. Let S ∗ denote a finite or infinite system in non-equilibrium state and S f , S n its former (earlier) and new equilibrium states, respectively. We will find below the connection between the former S f , the new S n and the intermediate S ∗ states. Note that systems S f , S ∗ , S n have the same volume V . The S ∗ , S n have also the same density d ∗ and the same temperature T ∗ (= the same mean energy E ∗ ). Assume that the states S f and S n are known. The aim is to find the intermediate S ∗ . In other words, the quantities d ∗ , T ∗ , pi∗ (t0 ) should be found. First, it follows from the above Note that the system S ∗ and S n have

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M. Lemanska / Physica A 387 (2008) 2719–2735 Table 1 I = 9, β = 2.0, the energy level e3 = 0.25 is perturbed by the factor f = 4, E = 0.30755, E ∗ = 0.289697 i

pi

pi∗ (t0 )

pi∗

1 2 3 4 5 6 7 8 9

.2472 .1925 .1499 .1168 .9096−1 .7084−1 .5517−1 .4297−1 .3346−1

.1705 .1328 .4137 .8855−1 .6273−1 .4885−1 .3805−1 .2963−1 .2308−1

.2643 .2002 .1516 .1148 .8696−1 .6585−1 .4988−1 .3777−1 .2963−1

Fig. 1. Time behavior of p2 , p3 , p6 , p9 of the system I = 9, β = 2. The energy level e3 was perturbed.

common density d ∗ , temperature T ∗ and the mean energy E ∗ . Also only the probabilities pi∗ (t0 ) should be found. First we will determine the perturbed energy level. As follows from items (I), (II), the differences E ∗ − E and pi∗ − pi , i = 1, 2, . . . , I , should be examined. Without loss of the generality, as an example, the following case will be considered. Let E ∗ < E, pi∗ > pi for i ≤ j and pi∗ < pi for i > j. Therefore we say (see item(I)) that the energy level e j has been perturbed. Then we find the perturbation factor f and the distribution pi∗ (t0 ). Using Eq. (3.3), the mean energy E ∗ can be written as follows. ! ! X X ∗ E = ei pi + ( f − 1)e j p j pi + ( f − 1) p j . (4.2) i

Since

P

i ei pi

= E and

i

P

i

pi = 1, Eq. (4.2) gives

f − 1 = (E ∗ − E)/[ p j (e j − E ∗ )].

(4.3)

Substituting (4.3) into Eq. (3.3) we obtain the distribution pi∗ (t0 ). The numerical results of the Table 1 are in good agreement with Eq. (4.3). The proofs for other cases given in items (I) and (II) are similar. Hence the following theorem is valid. Theorem 4.1a. The system S ∗ is completely determined, if S f and S n are known. The statements below are given without the proof. Theorem 4.1b. The S n system is completely determined, if S ∗ is known.

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Corollary 4.2. If S f and the induced perturbation are known, then S ∗ and S n are completely determined. But knowledge of the system S f only (i.e. with the perturbation unknown), does not give any information (besides the volume) about the S ∗ and S n . Corollary 4.3. Knowledge of the system S n gives only the particle density d ∗ , the temperature T ∗ and the mean energy E ∗ of S ∗ . System S f is forgotten. 4.3. Equivalence of finite and infinite systems in non-equilibrium state ∗ results in a common value of the characteristic physical quantities, like the particle Construction of Sk∗ and S∞ ∗ density d , the temperature T ∗ , the mean energy E ∗ , the distribution pi∗ (t0 ) and the mean entropy Se∗ (t0. ) Also ∗ are equivalent. This fact suggests what attending to the Proposition 4.2 we can say that the systems Sk∗ and S∞ follows.

Definition 4.1. The finite systems and the infinite ones being in non-equilibrium state (Section 1) are said to be equivalent (one to the other), if the particle density d ∗ , the temperature T ∗ (=the mean energy E ∗ ) are common to them and the time evolution process is the same. By the same (identical) evolution process we understand that the systems pass through the same states, p(t) = { p1 (t), p2 (t), . . . , p I (t)}, t0 ≤ t < ∞, during the approach to the equilibrium state having the same Boltzmann distribution (common β). Note that the same evolution process produces the same entropy Se∗ (t). To simplify the formulation of the below statements we denote the collection of finite and infinite systems (Section 1) being in a non-equilibrium state by {Sl∗ } = {S1∗ , S2∗ , . . . , SL∗ }, where the natural number L ≥ 2 (and may be also infinite). Similarly the collections of the corresponding former and new equilibrium states are denoted by f f f {Sl } = {S1 , . . . , SL } and {Sln } = {S1n , . . . , SLn }, respectively. Theorem 4.2. The systems {Sl∗ }, (l = 1, 2, 3, . . . , L) are equivalent iff they have the same particle density d ∗ and the same temperature T ∗ . Proof. Necessary condition: The equivalence of Sl∗ systems is assumed. Then it follows from the Definition 4.1 that they have common d ∗ and T ∗ . Sufficient condition: Let the {Sl∗ } systems have the same density d ∗ and the same temperature T ∗ . Also d ∗ , T ∗ are common to {Sln }. It follows from Theorem 2.3 that the {Sln } systems are equivalent. Therefore the members of {Sl∗ } have the same mean energy E ∗ . Then, attending to Definition 4.1, it should be shown that the probabilities pi∗ (t0 ) are common to the systems {Sl∗ }. Let Sk∗ , S ∗j ∈ {Sl∗ } be chosen arbitrary. The common d ∗ and E ∗ give Vk /V j = Nk∗ /N ∗j = ak X X ∗ E∗ = ei n i ,∗j (t0 )/N ∗j = ei n i,k (t0 )/ak N ∗j . i

i

Hence ∗ n i,k (t0 ) = ak n i,∗ j (t0 )

for S ∗j , Sk∗ ∈ {Sl∗ }.

Attending the items (1)–(3) of Section 4.2 we say that the time evolution process in the systems {Sl∗ } is the same.



The below theorems are given without the proof. Theorem 4.3a. Let the systems {Sl∗ } be equivalent, then the {Sln } ones are equivalent. Theorem 4.3b. Let the systems {Sln } be equivalent, then the {Sl∗ } ones are equivalent. f

Theorem 4.3c. Let the systems {Sl } be equivalent and perturbed in the same manner; then {Sl∗ } and {Sln } both are equivalent.

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5. Remarks and conclusions We note the following points: (1) The construction of the {Sk } and {Sk∗ } sequences is based on the assumption Vl /Vk = Nl /Nk = nli /n ik (see Eqs. (2.1) and (2.2)). As a result, the equivalence of the systems belonging to the same sequence has been obtained. ∗ have been determined. The method of Hence, by a simple limiting process, k → ∞, the infinite systems S∞ and S∞ Sections 2.1 and 3.1 is quite general because the choice of the system S1 and the induced perturbation is arbitrary. (2) The conditions for the equivalence of the finite and infinite systems and their properties are given for the case of the equilibrium and non-equilibrium states. As far as we know, the equivalence of the systems (defined in Section 1) being in non-equilibrium state is considered in this work for the first time. The Theorems 2.3 and 4.2 show that the conditions (common density and temperature) for the equivalence are the same for equilibrium and non-equilibrium f states. Theorems 4.3a–4.3c show how the equivalence property is preserved in the systems {Sl }, {Sl∗ }, {Sln }. Note that these statements are not trivial. The proofs require the knowledge of items (1)–(3) and of the relation between the states S f , S ∗ , S n, (Section 4.2). On the ground of the results of Sections 2–4, we can say that the connection between the finite systems and the infinite one is in fact the equivalence of them. (3) The derivation of Eq. (A.6) is based on two P assumptions. The first one is the time dependent Onsager–Machlup phenomenological formulation d pi (t)/dt = j L i, j ∂ S/∂ p j [1, in the Appendix]. By reason of this, neither a Hamiltonian nor a dissipative term are included in our model Eq. (A.6). Physical processes (e.g., particle interactions) are reflected only in the entropy Se∗ (t) and the time evolution process is expressed by the P changes of the Poccupation probabilities, pi (t), with the time. The second assumption is that the conservation laws i pi = 1 and i ei pi = E [Eqs. (A.3) and (A.4)] are satisfied at all times. They play a significant role in the considerations of Appendix A.2. First, the conservation laws define the time independent properties of the dynamical system. Second, with the help of Eqs. (A.3) and (A.4) the domain and the range of the mapping (A.6) are found. The intersection of two polygons, see (A.13), belonging to the hyperplanes, defined by Eqs. (A.3) and (A.4) is a closed segment P in Euclidean R I space. The coordinates of every point p ∈ P satisfy the two above constrains. Then the domain and the range of the transformation (A.6) are found, the closed segment P ∗ ⊂ P. The coordinates of every point p ∈ P ∗ represent the occupation probability distribution { p1,... p I }, state of the system at the specified time (Appendix A.2). It is shown in Section A.3.2 that the evolution process approaches to the equilibrium having the Boltzmann distribution p B ∈ P ∗ . Also we may say that the two above assumptions included in Eq. (A.6) create a dynamical system, in which the time evolution processes confirm the Boltzmann hypothesis. (4) The statements below, given intuitively by Kac [10], follow from the asymptotic approach to the equilibrium state and from the Boltzmann distribution as the fixed point of the mapping (A.6). The finite and infinite systems in the equilibrium state in the absence of a perturbation event will never go out of it. But finite and infinite systems in non-equilibrium state will always go towards it. (5) Another very important question in statistical mechanics is how the pi∗ (t0 ) change into the new equilibrium distribution pi∗ . In order to answer, the influence of the induced perturbation should be known. The items (I) and (II) of Section 4.2 establish a suitable tool for solving this question. The numerical results of Table 1 are in good agreement with the items (I) and (II), see Section 4.2. (6) The connection between the states S f , S n and S ∗ (the former, the new equilibrium states and the intermediate non-equilibrium one) is an interesting problem treated by Theorems 4.1a and 4.1b. It is worth noting that the known S n system gives only the quantities d ∗ , T ∗ and E ∗ of S ∗ . To determine pi∗ (t0 ) the mean energy E of the system S f should be known. The new equilibrium S n “has forgotten” the earlier S f . (7) The remarkable relation ae = E − k B T Se (Eq. (2.23)) suggests to define the function ae (t) = E − k B T Se (t).

(5.1)

Eqs. (2.23) and (5.1) contain the same physical quantities and are of the same form. E and T are fixed and Se (t) → Se . We have ae (t) → ae as t → ∞. Since Se is the maximum entropy, ae (t) approaches to its minimum value ae in the equilibrium state. It is worth noting that the known relations between the free energy, the internal energy and the entropy for the case of an equilibrium (see Ref. [3] and Eqs. (2.21) and (2.22) of Section 2.3) hold for a nonequilibrium state. Namely, for any specified time t we have d(ae (t)/T )/dt (1/T ) = E

(5.2)

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d(ae (t))/dT = k B Se (t).

(5.3)

The function ae (t) defined by (5.1) may be regarded as the extension of the free energy notion to the case of nonequilibrium state. Appendix. Mathematical model A.1. The equation for time evolution process We recall here the definitions and the equations of Ref. [1] related to our mathematical model for the time evolution process. For more details see Ref. [1]. The equations of interest are of the form d pi (t)/dt = ν[− ln pi (t) + a(t) + ei b(t)] i = 1, 2, 3, . . . I

(A.1)

pi (t0 ), i = 1, 2, . . . , I are initial condition for the time t = t0 . The terms a(t) and b(t) are linear combinations of ln pi (t). Their explicit form is given below. Eq. (A.1) describe a state at the time t of a thermally isolated system, consisting of identical particles, which occupy the energy levels 0 ≤ ei ≤ 1 with the time dependent probabilities pi (t) = n i (t)/N . n i is the occupation number and N the total number of particles. 0 < pi (t) < 1,

t ∈ hto , ∞).

(A.2)

The following constraints are included in Eq. (A.1): X pi (t) = 1

(A.3)

i

X

ei pi (t) = E.

(A.4)

i

E is the mean energy. Combining Eqs. (A.1) and (A.4), we obtain Eqs. (A.5), which determine the functions a(t) and b(t). X ln pi (t) − I a(t) − αb(t) = 0. (A.5.1) i

X

ei ln pi (t) − αa(t) − βb(t) = 0.

(A.5.2)

i

where α =

β=

P

i ei

2 i ei

P

! a(t) = α

X

ei ln pi (t) − β

X

i

ln pi (t) /(α 2 − Iβ)

(A.5.3)

i

! b(t) = α

X i

ln pi (t) − I

X

ei ln pi (t) /(α 2 − Iβ).

(A.5.4)

i

In this way the conservation laws (A.3), (A.4) are included in Eq. (A.1) and preserved during the dynamical process. As pi (t) are continuous functions of t, an equivalent system of non-linear integral equations is obtained: Z t pi (t) = ν[− ln pi (t 0 ) + a(t 0 ) + ei b(t 0 )]dt 0 + pi (t0 ), i = 1, 2 . . . , I. (A.6) t0

We recall here some important properties of Eq. (A.6) [1]: (i) Eq. (A.6) do not include any information about the relaxation mechanism. They do not explicitly contain either a Hamiltonian or a dissipative term. (ii) To solve Eq. (A.6) it is sufficient to know only the number of the energy levels I , the basic energies ei and the initial (or starting) distribution pi (t0 ).

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(iii) The occupation number n i and the total number of particles N are not of interest. Only the ratios pi = n i /N play a significant role in our model. In applications N may be of the order of the Avogadro number for a finite system. For the case of an infinite one, N is infinite. (iv) The discretization of the basic energy interval 0 ≤ e ≤ 1 ensures a finite number of Eq. (A.6). (v) Eq. (A.6) give a statistical interpretation of the relaxation process. A.2. Properties of the transformation (A.6) The right-hand side of Eq. (A.6) defines a collection of the transformations Tt = (Tt,1 , Tt,2 , . . . , Tt,I )

(A.7)

defined by pi (t) = Tt,i p(t) = T2 [νT1,i p(t)] + po,i

(A.6.1)

where T1,i p(t) = ν[− ln pi + a(t) + ei b(t)] Z t T2 = (·)dt.

(A.8) (A.9)

t0

In what follows the mapping Tt will be considered. A.2.1. The function p(t) Let us consider the solution of Eq. (A.6) for each t ∈ ht0 , ∞) as a finite sequence p(t) = { p1 (t), p2 (t), . . . , p I (t)}.

(A.10)

Since pi = pi (t) are real numbers, we have in fact for each t ∈ ht0 , ∞) a point p = ( p1 , p2 , . . . , p I ) ∈ P ⊂ R I

(A.11)

RI

where is the I -dimensional Euclidean space. The coordinates pi are the occupation probabilities of the system of interest at the specified time t. Proposition A.1. The set P belongs to the intersection of the hyperplanes determined by the conservation laws (A.3) and (A.4). Proof. From Eqs. (A.3) and (A.4) it follows that the points p (see Eq. (A.11)) belong to two hyperplanes, ( ) X I: H= p∈R : pi = 1

(A.12a)

i

( ∗

H =

) I

p∈R :

X

ei pi = E .

(A.12b)

i

Since 0 ≤ pi ≤ 1, one obtains two hyperpolygons of interest, H1 = { p ∈ H : 0 ≤ pi ≤ 1}

(A.13a)

∗

(A.13b)

H2 = { p ∈ H : 0 ≤ pi ≤ E/ei }. E/ei is replaced by 1, if ei = 0. In this manner the set P is defined by P = H1 ∩ H2 .

(A.14)

The sets P, H1 and H2 are compact. The set P plays the role of the “phase space” in the dynamical model (A.6). To illustrate the above, we specify the sets H1 , H2 and P for the case of I = 3 (i.e. in R 3 space, see Fig. A.1). To find the geometrical meaning of p(t), let us denote by P(t) a slice P at t of the Cartesian product K = ht0 , ∞)x P, p(t) ∈ P(t). p(t) is continuous function of t in the space K . 

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Fig. A.1. Domains. H1 is the triangle with the vertices A(0, 1, 0), B(0, 0, 1), C(1, 0, 0). H2 is the quadrilateral D(0, E/e2 , 0), E(0, 0, E/e3 ), F(1, 0, E/e3 ), G(1, E/e2 , 0). The segment ab = P.

A.2.2. The transformation Tt The transformations (A.6.1) and (A.7)–(A.10) are considered below. From Eq. (A.6.1) it follows that the domain and the range of Tt belong to the set P. The mapping T1,i is time independent and transforms a finite, real tuple into a real number. We assume a closed time interval ht0 , tk i. The mapping T1,i p includes the terms ln pi . Therefore it is not defined for the points p ∈ P having a coordinate equals to 0 or to 1. Also, we should find the domain of T1,i as a subset of the set P. Proposition A.2. The solution of Eq. (A.6), p(t) = { pi (t)} (i = 1, 2 . . . I ) on the closed time interval ht0 , tk i k = 1, 2, . . . , forms a compact set, closed segment, Pk∗ ⊂ P. Proof. From the continuity of the functions pi (t) on the interval ht0 , tk i it follows m i ≤ pi (t) ≤ Mi

i = 1, 2 . . . , I

(A.15)

where Mi and m i are its maximum and minimum. The segment hm i , Mi i ⊂ (0, 1). Thus we obtain Pk∗ = { p ∈ P : m i ≤ pi ≤ Mi } ⊂ P. P ∗ It is shown in Section A.3.2 that ∞ k=1 Pk ⊂ P.

(A.16) 

In virtue of the appropriate theorems of Refs. [11–13] we may conclude the following corollaries. Corollary A.1. (1) The continuous mapping T1,i ( p) is defined on the set Pk∗ and its range is a closed segment of the space R— a real line. (2) T1,i ( p) is uniformly continuous on a set Pk∗ . Thus T1,i (Pk∗ ) = hm i∗ , Mi∗ i, where m i∗ Mi∗ are the minimum and maximum. (3) The mapping T1,i (p) is one-to-one. Corollary A.2. From Eq. (A.1) it follows, that if T1,i p is positive (negative), then pi (t) increases (decreases). The transformation Tt,i . The operator T2 is an integral over hto , t K i. We remember that T1,i (PK∗ ) = hm i∗ , Mi∗ i 6= hm i , Mi i. It has been observed in our numerical work that the operation T2 [νT1,i ( p)] + po,i may yield pi 6∈ hm i , Mi i. For this reason it will be shown that there exists ∆t = tk − tk−1 tk−1 , tk ∈ ht0 , t K i such that the mapping Tt,i results in pi (t) ∈ hm i , Mi i for i = 1, 2, . . . , I . Let us denote Mi∗∗ = maxi [|Mi∗ |, |m i∗ |]. Proposition A.3. The mapping T2 T1,i ( p) results in pi ∈ hm i , Mi i, if ∆t ≤ (Mi − m i )/ν Mi∗∗ . Proof. Let tk > tk−1 and tk−1 , tk ∈ ht0 , t K i. Let p(tk−1 ) be a known solution of Eq. (A.6). Then Z tk pi (tk ) = νT1,i p(t)dt + pi (tk−1 ) tk−1

(A.6*)

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Remember, m i∗ ≤ T1,i p ≤ Mi∗ . From Eq. (A.6*) we get Z tk ν|T1,i p(t)|dt ≤ ∆tν Mˆ i . | pi (tk ) − pi (tk−1 )| ≤

(A.6**)

tk−1

Putting in Eq. (A.6**) ∆t = tk − tk−1 ≤ (Mi − m i )/ν Mi∗∗

(A.17.1)

we obtain | pk (t j ) − pi (tk−1 )| < (Mi − m i ), i.e. pi (tk ) is correct. The common ∆t for i = 1, 2, 3, . . . , I is: ∆t ≤ min[(Mi − m i )/(max ν Mi∗∗ )]. i

i



(A.17.2)

Note that Eq. (A.6) should be solved with many time steps. Thus we have obtained R(Tt ) = Pk∗ . It should be mentioned that the value of an appropriate ∆t we have found very easily by computational testing. A.3. The approach to the equilibrium state A.3.1. The monotonicity and the convergence of the probabilities pi (t) The following properties of the transformation (A.6) are used to prove the next proposition. (i) The probabilities pi (t), i = 1, 2, . . . , I , are continuous functions of the time t. (ii) The mapping Tt,i ( p) is one–one. (iii) In Euclidean space the coordinates with regard to the same axis of the points on a segment behave monotonically (i.e. the coordinates either increase or decrease or are constant). Proposition A.4. The probabilities pi (t) are bounded, monotonic and convergent functions. Lim pi (t) = p B,i as t → ∞ is unique. Proof. From the items (i)–(iii) it follows that the probabilities pi (t) are monotonic functions on the time interval ht0 , ∞). The pi (t) are bounded, because P is a closed segment. From the monotonicity and the boundness of pi (t) it follows that lim pi (t) = pi,B

as t → ∞.

(A.18)

The uniqueness of this limit follows from the metric property of the set P. By the coordinate convergence and the compactness of P we obtain lim p(t) = p B ∈ P

as t → ∞

(A.19)

where p B = p B,1 . . . , p B,I .



(A.20)

Following Farquhar [15, p. 87], and [14] we conclude from Eq. (A.21) that the evolution process governed by Eq. (A.6) is strongly mixing and approaches to the equilibrium. The point p B is a fixed point (point attractor) [16] of the collection of Eq. (A.6). This means that the system is attracted toward the point p B and the evolution process is dead in equilibrium. Now, we shall show that the point p B is the Boltzmann distribution. Proposition A.5. A point p ∈ P is invariant (fixed point) under the transformation T t ( p) iff T1,i ( p) = 0 for i = 1, 2, . . . , I . Proof. The validity of this statement follows from the form of Eq. (A.6).



From the Proposition A.5, Theorem 2.1 and Eq. (2.17) it follows Corollary A.3. The Boltzmann distribution p B = { p B,1 , . . . , p B,I } ∈ P is the fixed point (point attractor) of the collection of Eq. (A.6).

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The above results allow to formulate the following conclusion. Corollary A.4. The evolution process takes place on the segment P ∗ = h p(t0 ), p B i ⊂ P. Beginning at the point p(t0 ) the points p created by the evolution process fill the segment P ∗ towards the point p B . A.3.2. The sequence {Pk∗ } and its limit as k → ∞ We consider here certain important properties of the sets Pk∗ as a topological illustration of the results of Appendix A.3.1. The appropriate theorems used in this section are given in Refs. [11–13]. Let be given a sequence of time intervals {ht0 , tk i},

k = 1, 2 . . .

(A.21)

where t0 < t1 , . . . < tk−1 < tk and tk → ∞ as k → ∞. ∗ } (k = 1, 2, . . .). From Appendix A.2 it follows that the sequence (A.21) generates two sequences {Pk∗ } and {Pc,k ∗ ∗ ∗ The set Pc,k is the complement of Pk on the segment P . It follows from the construction of sets Pk∗ and from the previous section that ∗ P1∗ ⊂ P2∗ ⊂ · · · ⊂ Pk∗ ⊂ Pk+1..

Pk∗ Pk∗

∩

∪ Pl∗ = Pl∗ for k < Pl∗ = Pk∗ for k < l

l

(A.22) (A.23) (A.24)

{Pk∗ } is a monotone sequence of increasing closed subsets Pk∗ ⊂ P ∗ . Also we have ∞

∗ ∗ ∗ = lim Pk∗ = ∪∞ P∞ k=1 Pk ⊂ P . k=1

(A.25)

∗ = P ∗ − P ∗ . Thus we have the sequence of open subsets {P ∗ }, k = 1, 2 . . . . The complement of Pk∗ is an open set Pc,k k c,k From the relation (A.22) we have ∗ ∗ ∗ Pc,1 ⊃ Pc,2 · · · ⊃ Pc,k ⊃ ···

(A.26)

∗ ∗ ∗ Pc,k ∪ Pc,l = Pc,k

for k > l

(A.27)

∗ Pc,k

for k > l.

(A.28)

∩

∗ Pc,l

=

∗ Pc,l

∗ } is a monotonic sequence of open decreasing sets. d(P ∗ ) → 0, as k → ∞ (d(A) is a diameter of a set A). {Pc,k c,k ∗ → p as k → ∞. The point p is common to all sets P ∗ k = 1, 2, . . . . Thus the countable Hence the set Pc,k B B c,k intersection ∗ ∗ Pc,∞ = ∩∞ n=1 Pc,k = p B

(A.29)

∗ is of the type G . Hence P ∗ is an open set of the type F . Both belong to is a one point closed set. Also the set Pc,∞ δ σ ∞ ∗ the set P . From the above considerations we deduce what follows.

Corollary A.5. The open segment ∗ P∞ = h p(t0 ), p B ) ⊂ P ∗

(A.30)

∗ P ∗ = P∞ ∪ p B = h(t0 ), p B i.

(A.31)

and

The above relations show that the time evolution process is closed under the equilibrium state. The approach to the equilibrium state is asymptotic.

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References [1] M. Lemanska, Z. Jaeger, A nonlinear model for relaxation in excited closed physical systems, Physica D 170 (2002) 72. [2] A.R. Vasconcellos, J.G. Ramos, R. Luzzi, Ensemble formalism for nonequilibrium systems and an associated irreversible statistical thermodynamics, Bras. J. Phys. 35 (2005) 689. [3] R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics, Wiley Interscience, New York, 1975. [4] D. Ruelle, Statistical Mechanics Rigorous Results, W.A. Benjamin, INC, Amsterdam, 1969. [5] J.C. Robinson, P.A. Glendinning (Eds.), From Finite to Infinite Dimensional Dynamical Systems, Kluwer Academic Publishers, 2001. [6] B. Nicalenko, C. Foias, R. Temam (Eds.), The Connection Between Infinite Dimensional and Finite Dimensional Dynamical Systems, Contemporary Mathematics, vol. 99, American Mathematical Society, Providence, RI, 1989. [7] H. Spohn, Large Scale Dynamics of Interacting Particles, Springer Verlag, New York, 1991. [8] Jack K. Hale, Louis T. Magalhaes, Valddyr M. Oliva, Dynamics in Infinite Dimension, Springer Verlag, New York, 2002. [9] M. Le Bellac, F. Mortessagne, G.G. Batroni, Equilibrium and Non-Equilibrium Statistical Thermodynamics, Cambridge University Press, Cambridge, 2004. [10] M. Kac, Probability and Related Topics in Physical Sciences, Interscience, London, 1959. [11] Sikorski Roman, Real Functions, vol. I, Panstwowe Wydawnictwo Naukowe, Warszawa, 1958 (in Polish). [12] Arch W. Naylor, George R. Sell, Linear Operator Theory in Engineering and Science, Holt, Rinehart and Wiston, Inc., New York, 1971. [13] P.R. Halmos, Measure Theory, D. Van Nostrand Company, 1950. [14] I.P. Cornfeld, S.V. Fomin, Ya.G. Sinai, Ergodic Theory, Springer Verlag, New York, 1982. [15] I.E. Farquhar, Ergodic Theory in Statistical Mechanics, John Wiley and Sons, London, 1964. [16] David Campbell, Jim Crutchfield, Doyne Farmer, Erica Jen, Experimental mathematics: The role of computation in nonlinear science, Communication of the ACM 28 (1985) 374–383.