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ISSN 1560-3547, Regular and Chaotic Dynamics, 2011, Vol. 16, No. 5, pp. 536­549. c Pleiades Publishing, Ltd., 2011.

Statistical Irreversibility of the Kac Reversible Circular Mo del
Valery V. Kozlov*
V.A. Steklov Mathematical Institute Russian Academy of Sciences ul. Gubkina 8, Moscow, 119991 Russia
Received Octob er 29, 2010; accepted Decemb er 4, 2010

Abstract--The Kac circular model is a discrete dynamical system which has the property of recurrence and reversibility. Within the framework of this model M. Kac formulated necessary conditions for irreversibility over "short" time intervals to take place and demonstrated Boltzmann's most important exploration methods and ideas, outlining their advantages and limitations. We study the circular model within the realm of the theory of Gibbs ensembles and offer a new approach to a rigorous proof of the "zeroth" law of thermodynamics based on the analysis of weak convergence of probability distributions. MSC2010 numbers: 37A60 DOI: 10.1134/S1560354711050091 Keywords: reversibility, stochastic equilibrium, weak convergence

INTRODUCTION One of the key problems in namely to establish that an is the time goes infinitely. To do and give a rigorous definition statistical mechanics is to prove the "zeroth" law of thermodynamics, olated system irreversibly tends to a state of thermal equilibrium as so, we need to define the term "thermal equilibrium" more carefully of convergence to this state.

It is b elieved that convergence to equilibrium takes place as t +; to b e more precise, the "arrow of time" is determined by the direction of the evolution of a system. However, this p oint of view contradicts the initial prop erty of reversibility of the equations of motion for mechanical systems. Therefore, the classical Boltzmann theory and its subsequent versions do not seem to b e well founded. These approaches are based on an explicit or implicit assumption of the Markov property of the process of evolution. As a matter of fact, the Markov chains were used by T. and P. Ehrenfest in their well-known probabilistic models for the proof of irreversibility (see [1, 2]). Mention should b e made here of the earlier works of A. Markov himself, namely those dealing with the urn models, which were not related to problems of statistical mechanics, though (see, e.g., [3]). The Kac circular model, which has the prop erty of recurrence and reversibility, is a more advanced version. In particular, these prop erties enable one to investigate the classical paradoxes of Loschmidt and Zermelo from a fundamental p oint of view (see [4, 5]). On the other hand, the Kac circular model provides new approaches to the proof of irreversibility [6, 7] to develop the forgotten ideas of H. Poincar´ [8] and allows the problem of the increase of entropy of an isolated system to e b e clarified.
*

E-mail: kozlov@pran.ru

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1. THE KAC MODEL Consider a circumference and a regular n-gon inscrib ed into it. Choose a set S consisting of m (m < n) fixed vertices. There is a white or black ball at each of n vertices. The dynamics of such a discrete system are determined as follows. The circumference is turned through an angle of 2 /n in a unit time counter-clockwise, and each ball moves to a neighb oring p osition provided that the ball leaving the set S changes its color, but if the vertex does not b elong to S , the ball leaving it does not change its color. This is the Kac circular model. We present this model in more common terms used in statistical mechanics. Let us numb er the vertices of the n-gon by integers p mod n and assign to each of them the numb er xp = +1, if the ball at p oint p is black, -1, if the ball at p oint p is white.

The sets of numb ers x = (x1 , . . . , xn ) are states of the system. Next it is convenient to represent them as column vectors. It is easy to understand that the numb er of different states is equal to N = 2n . The set of all states is the phase space of the system. Denote it by . On there are a variety of additive measures. We choose one of them and take the measure of each state to b e equal to 1. Denote this measure by µ and call it the Liouvil le measure. It is clear that µ() = N . Now we define the phase flow of the Kac system. This is a set of integer degrees {T t }+ t=- where the transformation T : corresp onds to the turn of the circumference through an angle of 2 /n. The transformation is linear and is given by the matrix x Tx 0 0 1 0 T = 0 2 . . . . . . where p = -1, if p S, +1, if p S. / ··· ··· ··· .. . 0 0 0 . . .
-1

(1.1) n 0 0 , . . . 0

(1.2)

0 0 · · · n

Obviously, the transformation (1.1) preserves the Liouville measure. This simple observation is a discrete variant of the Liouvil le theorem from statistical mechanics on the preservation of phase volume. Thus, we have a discrete dynamical system (, T , µ) with an invariant measure. The transformation (1.1) is obviously p eriodic with p eriod 2n: T
2n

(1.3)

= I,

where I is an identity p ermutation. This yields a straightforward but imp ortant prop osition. Prop osition 1. For n > 2 the dynamical system (1.3) is not ergodic. Moreover, it will not even b e transitive: the orbit
{T k x}+=- k

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does not coincide with the entire for any x . Indeed, each orbit contains no more than 2n different various p oints, and consists of 2n elements. It only remains to note that 2n > 2n for n > 2. The systems of interacting particles with the Lennard­Jones p otential, which are common in statistical mechanics, are obviously not ergodic on the surfaces of the level of energy integral either. All invariant measures on are constructed as follows. A measure is given on each ergodic comp onent (orbit of transformation T ) such that the measure of each p oint of the orbit is the same. The prop erty of p eriodicity of mapping T can b e sp ecified. The following algebraic lemma holds Lemma. T n = 1 . . . n I . M. Kac did not p oint out or used this prop erty. The proof follows immediately from the kind of successive degrees of the matrix (1.2): 0 0 · · · n n-1 0 0 0 1 n 0 ··· T 2 = 2 1 0 · · · 0 0 , 0 3 2 · · · 0 0 ··· ··· ··· ··· ··· (1.4) 0 · · · n n-1 n-2 0 0 0 0 0 ··· 0 1 n n-1 0 0 0 ··· 0 0 2 1 n 3 , . . . T = 3 2 1 0 ··· 0 0 0 0 4 3 2 · · · 0 0 0 ··· ··· ··· ··· ··· ··· The lemma yields a series of imp ortant corollaries. Corollary 1. If m is an even number, T n = I . Corollary 2. If m is an odd number, T n = -I . The latter implies that for an odd numb er of elements of the chosen set S after n successive turns all balls change their color. This, in its turn, yields the following statement, which we shall use in § 3. Prop osition 2. If m is odd for each orbit of transformation T , the numbers of black and white bal ls coincide. Indeed, when n t < 2n the system evolves in just the same way as when 0 that all balls change color to the opp osite. t < n, except

For even m this prop erty is not achieved. This can b e verified by sp ecific examples. Furthermore, the following prop osition holds. Prop osition 3. If m is even and n is odd for each orbit of transformation T , the total number of black bal ls never coincides with that of white bal ls.
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Indeed, according to Corollary 1, if m is even each orbit is p eriodic with p eriod n. Altogether there are n2 balls on this orbit. If n is odd this numb er is odd. In some cases the total numb ers of white and black balls differ only by one. The figure b elow gives two examples for n = 5 and n = 7 and shows the arrangement of the set S (m = 2). The Kac system is reversible in the sense that there exists a backward transformation T -1 with the matrix 0 1 0 · · · 0 0 0 2 · · · 0 . . . .. . . . . . . .. . . . 0 0 0 · · · n-1 n 0 0 · · · 0 As is seen from (1.4), the matrix T
n-1

has such a diagonal form.

2. SPACE AVERAGING Following M. Kac, we consider a problem of distribution of the numb er of black and white balls in the process of evolution of the system. Let us fix the initial distribution x(0); Kac considered the case where at the initial moment of time t = 0 all balls are black: x(0) = (1, 1, . . . , 1). Let Nc (t), (Nb (t)) b e the numb er of black (white) balls at the current moment of time t. We are interested in the relative difference N (t) - Nb (t) . (2.1) (t) = c n This function of time essentially dep ends on the arrangement of the set S . We demonstrate it for m = 2 and even n in two cases: when S consists of two neighb oring p oints and when S is comp osed of two diametrically opp osite p oints. In the first of them we obviously have Nc (0) = n, Nc (t) = n - 2 at 1 t n - 1. Consequently, in this interval of time n-4 (t) = 1, n if n is large. In the second case we have Nc (t) = n - 2t in the interval 0 t n/2. At t = n/2 all balls b ecome white. Then the numb er of black balls b egins to increase linearly: Nc (t) = 2(t - n/2) at n/2 t n. The graph of function is shown b elow. In order to avoid such a strong dep endence on the arrangement of the p oints of the set S , M. Kac suggested averaging the relative difference (2.1) over all p ossible p ositions of this set as b eing equivalent. This averaging should b e regarded as an alternative to "averaging over space"; we denote the result of averaging by angle brackets.
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As a result Kac arrived at the formula (t) =
1 z 1-z 1+z
2 2

t (1+z 2 )n z 2m
2 )n

dz . (2.2)

1 (1+z z z 2m

dz

Here the integration extends over a small circumference in a complex plane including the p oint n z = 0. As a matter of fact, the denominator contains the binomial coefficient m . Formula (2.2) has b een obtained on the assumption that at the initial moment of time all balls are black. Besides, in deriving (2.2) it was assumed that t n. All details of the derivation are contained in [5]. By the way, for t = n the average (2.2) is equal to +1 or -1 dep ending on whether m is even or odd. This observation is in good agreement with the derivations of corollaries 1 and 2 from the algebraic lemma. Next M. Kac fixes the time t and lets m and n tend to infinity such that m/n µ as n . Moreover, it is assumed that 0 < 2µ < 1. In these assumptions the asymptotic formula (t) (1 - 2µ)t . (2.4) is derived by the saddle-p oint-technique from (2.2). In the assumption (2.3) this average tends to zero exp onentially fast as t increases. Of course, one should not forget that t must remain much less that n. This result implies equalization of the numb er of black and white balls after averaging over all p ossible p ositions of the set S . M. Kac interprets this fact as an analog of the so-called H -Botzmann theorem. More generally, this is a proof of the fact that the system irreversibly tends to the state of equilibrium in a limited finite range of time (after an appropriate additional averaging over space). And how does the quantity (2.2) b ehave on larger time intervals comparable to n? We supplement the result of M. Kac with a new formula showing that the average numb er of white and black balls also b ecomes equal but at a different sp eed. Let n b e even. Set t = n/2 = k. Then the integral in the numerator (2.2) is equal to 1 (1 - z 4 )k dz . z z 2m (2.5) (2.3)

If m is odd, this integral is obviously equal to zero. Then at this moment of time the average numb er of white and black balls b ecomes equal. Let m = 2l and l b e an integer. As ab ove, assume that l m = µ n k as n . It is straightforward to see that the integral (2.5) is equal to the binomial coefficient (-1)k k . l
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Consequently, in the case at hand the average (2.2) is equal to (-1)k k l 2k . 2l

Using Stirling's formula, it is easy to obtain the asymptotic form of this expression for k (n ): (2.6) (-1)t 2[µµ (1 - µ)1-µ ]t .

Since 0 < µ < 1, the expression in square brackets is less than one. Therefore, as t the relative difference of the numb er of black and white balls oscillates and tends to zero as exp onentially fast. Note that for µ = 1/2 the asymptotic formula of Kac (2.4) degenerates and, vice versa, the expression in square brackets in formula (2.6) b ecomes maximal. In [4, 5] a simplified circular model is considered where in the formula (1.2) j are regarded as random quantities assuming the values ±1. This is a simple example of the Markov chain. To refine this model, transition probabilities should b e given. As a result the asymptotic formula (2.4) also holds with overwhelming p ossibility (see [4, 5] for details). In fact it is precisely this simplified probabilistic model that may serve as an analog of the Boltzmann theory. However, the following significant circumstance should b e kept in mind here. The simplified probabilistic model as opp osed to the original deterministic model is conceptually irreversible: the state of the system at previous moments of time can only b e sp ecified with a certain probability. Therefore, as soon as we include the probability into the definition of the dynamics of the system, we imp ose the direction of evolution from the very outset, which makes the system irreversible. Incidentally, by introducing the averaging over all p ossible p ositions of the set S , we also lose information on the previous states of the system. Thus, the introduction of such an averaging renders our system irreversible as well. 3. GIBBS' THEORY AND AVERAGING OVER TIME

In addition to the Boltzmann approach to the validation of thermodynamics (the approach is based on the time-irreversible classical kinetic Boltzmann equation, which contradicts the reversibility of initial dynamical equations), there is the general Gibbs approach related to the introduction of the dynamical system of a probability measure (transferred by the flow) on the phase space (see [6, 9]). The idea of introducing two matched mathematical structures (the phase flow and probabilistic measure) on the phase space is undoubtedly very b eautiful and useful. However, some authors consider it to b e too general and insist that the probability in statistical mechanics should manifest itself in many different ways (see [4] for more discussions of this). Using the Kac circular model as an example, we will show that within the framework of Gibbs' theory it is also p ossible to solve the problem of a reversible system irreversibly tending to the state of statistical equilibrium. According to Gibbs, a probability measure , which accounts for the inaccuracy in the determination of the initial state, is p ostulated at the initial time t = 0 on the phase space . In our case the measure is given by a sequence of nonnegative numb ers (x), which add up to one: x , (3.1)

(x) = 1.

(3.2)

The latter means that if the system is in some state it is a sure event. The function : R+ can b e thought of as the density of measure relative to the Liouville measure: = µ. Then this measure, which is transferred by the flow {T t }, b egins to evolve in time. Denote it by t at time t and let t (x) b e its "density" in the sense of (3.1) and (3.2). By the Liouvil le equation, t (x) = (T
-t

x).

(3.3)

To b e more precise, this is a solution of the classical Liouville equation, which constitutes the basis of Gibbs' theory. For t = 0 we obtain the initial measure (3.1).
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Let us derive formula (3.3). Supp ose t (x) is the probability of the system b eing in a state x at time t. It is clear that 0 (x) = (x) and t-1 (x) = t (T x) for all x . The latter formula is a condition for the measure to b e transferred by the phase flow. Hence we obtain successively t (x) =
t-1

(T

-1

x) =

t-2

(T

-2

x) = . . . = 0 (T

-t

x).

As was to b e shown. Gibbs hop ed to show that in a sense the measure t converges to which is invariant relative to the phase flow {T t }. Besides, one would b e "microcanonical". In our case it would lead to = const = 2-n or, with the Liouville measure µ (up to the multiplier 2-n ). Then all equivalent and the distributions of black (and white) balls binomial: 2-
n

the stationary distribution , like the limit distribution to which is the same, coincidence states in the limit would b e

n , k

0

k

n.

However, these hop es are not destined to come true (for Gibbs' arguments in the continuous case see [9, Chapter 12]). The equation = 2-n µ would mean ergodicity of the Kac system. This, however, is not the case according to Prop osition 1. On the other hand, the sequence t (x) (t = 0, 1, 2, . . .) is p eriodic (with p eriod 2n) and is generally nonconstant. Consequently, it cannot have an ordinary limit as t . However, these difficulties can b e overcome if we use a somewhat different approach to solving the problem. First, we need to give an exact definition of the system's "tendency" to statistical equilibrium. The idea is to replace the ordinary convergence with a stronger method of summing M. We require this method to b e linear, regular and natural. This implies the following. To some sequences a0 , a1 , a2 , . . . we assign the numb ers a and call them limits of these sequences (we write them as follows: The following conditions are satisfied: an a (M) or lim an = a (M)). (3.4)

a) if an a (M) and bn b (M), then for any real and ; an + bn a + b (M)

b) if an a in the ordinary sense, then an a (M); c) if an a (M), the sequence a1 - a0 , a2 - a0 , . . . converges to a - a0 (in the sense of definition M). The simplest (except for the ordinary convergence) example of the linear and regular summation method is probably the C`saro method (C ) : an a (C ), if e a0 + a1 + . . . + an a n+1

as n .
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Prop osition 4. Let the sequence (3.4) be periodical with period p and M be a linear, regular and natural summation method. If an a (M), then a= a0 + . . . + ap p
-1

.

(3.5)

This statement actually b elongs to D. Bernulli, except that he dealt with p eriodic divergent series rather than sequences [10]. Of course, any p eriodic sequence converges in the C`saro sense e to (3.5). Let us return to the Kac model. In our case as t . Note that the limiting function is obtained by averaging over the orbit of each p oint x of mapping T . This function has apparently the following prop erties: 1) of invariant relative to the action of phase flow {T t }, 2) 3)
x

t (x) (x) (C )

(3.6)

0, (x) = 1.

Consequently, the limiting function is the stationary density of some probability distribution on . The state of the system with the probability measure = µ will b e a state of statistical equilibrium and the system irreversibly tends to this state in the sense of the C`saro convergence (3.6). It must b e emphasized that the limiting relation (3.6) takes place e b oth for t + and for t -. Thus the states of statistical equilibrium in the distant past and future coincide. This fact entirely corresp onds to the reversible nature of the initial Kac system. Convergence (3.6) with prop erties 1)­3) is, of course, a quite particular case of the Birkhoff­ Khinchin ergodic theorem for the dynamical system (, T , µ). Recall that, in fact, the ergodic theory arose in an attempt to substantiate statistical mechanics. The definition of statistical equilibrium (3.6) may seem too formal. However, we can give it a more concrete definition. Actually our interest is not in the evolution of probability distribution as such but in the way the average values of dynamical quantities (functions) vary with time on the phase space. Let : R b e a discrete dynamical quantity. Its average value at time t relative to probability measure t is equal to dt =


t dµ =
x

(T

-t

x)(x).

(3.7)

We shall say that the measure t weakly converges to a measure = µ as t if for any "trial" function : R t dµ


dµ.

The time function on the left will again b e p eriodical in t and, therefore, there is actually no ordinary convergence here. The difficulty can b e resolved by introducing an additional averaging over time, thereby replacing the ordinary convergence with the C`saro convergence: e 1 lim
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t dµ =
t=0

dµ.

(3.8)

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It is straightforward to show that the weak convergence in the C`saro sense in the Kac model e always takes place, moreover, the limiting function is determined again by the passage to the limit (3.6). Relation (3.8) is a discrete version of von Neumann's ergodic theorem, which historically preceded the Birkhoff­Khinchin theorem. In our case relations (3.6) and (3.8) are, of course, equivalent. However, the discussion of the velocity of convergence of probability distribution to the state of statistical equilibrium lies b eyond the scop e of this approach. But this approach enables us to consider the velocity of the convergence of C`saro means of various dynamical quantities. e Let now x + . . . + xn . (3.9) = 1 n Then integral (3.7) is the exp ected value of the relative difference of the numb er of black and white balls (2.1) at time t. Theorem 1. If m is odd, then regard less of the initial distribution the average value of quantity (3.9) in statistical equilibrium is equal to zero. Thus, the average numb ers of black and white balls irreversibly equalize. Generally sp eaking, for an odd value m Theorem 1 does not hold, as the example with m = 2 from § 2 shows. The proof of the theorem is straightforward. On each orbit of the phase flow {T t } the function assumes constant values. Further, if m is odd, the p oints of the orbit are divided into pairs x and x such that (x ) + (x ) = 0 (Prop osition 2). But then the integral on the right in (3.8) along this orbit is equal to zero. Summing over all orbits, we obtain the required result. We demonstrate the ab ove-said using a simple example where n = 6 and the set S consists of only one vertex (m = 1). In this case the phase space (consisting of 26 = 64 elements) is divided into six disjoint orbits of the phase flow {T t }; we denote them by a, b, c, d, e and f . The numb er of p oints of in the orbits a­e is equal to 12 and the orbit f consists of four p oints (5 · 12 + 4 = 64). Recall that 12 = 6 · 2 is a p eriod of the mapping T . The representatives of the orbits are shown in this figure.

The mapping T : is not ergodic and, therefore, the study of the dynamics reduces to the study of individual closed orbits. The limit distributions of the numb er of white balls on each of the orbits are shown in the following figure. All of them differ from the binomial distribution. In general the limit distribution of white balls is obtained by "mixing" these particular distributions. It is easy to see that, regardless of the initial distribution, the exp ected numb er of white (and black) balls in statistical equilibrium is equal to 3 = 6/2. 4. THE GROWTH OF ENTROPY First, we make some general observations. Let f : R b e a function on the phase space. f (T t x) dµ =
x

Prop osition 5. The integral

f (T t x)

(4.1)

does not change with time.
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This statement is almost obvious. It also holds for smooth dynamical systems under an additional assumption of summability of functions f . In the most general case Prop osition 5 follows from two prop erties of the transformation T : it is reversible and conserves the Liouville measure µ. In (4.1), t can, of course, b e replaced with -t. Usually Prop osition 5 is formulated in a different form: if f is a function of density , then integral (4.1) is a constant as time function (see [8, 9]). In particular, the entropy s(t) = -


(T t x) ln (T t x) dµ

(4.2)

does not change with time. Integral (4.2) is sometimes called informational entropy or Gibbs' entropy. As Gibbs showed, for the so-called canonical distribution integral (4.2) expresses the entropy from thermodynamics. On the face of it, the constancy of entropy (4.2) seems inconsistent with the law of entropy growth in an isolated system. For this reason, some authors (e.g., I. Prigogine [11]) even think it incorrect to identify integral (4.2) with the entropy. However, all these questions are solved in a natural way in the theory of weak limits of probability distributions. Let f b e the density function f = F (). Set (t) =


F ((T

-t

x)) dµ.

(4.3)

As has b een said ab ove, this integral does not actually dep end on t. Now we replace the density t (x) = (T with its weak limit (x) = lim t (x) (C ).
t -t

x)

Let =


F ((x)) dµ.

Theorem 2. If the function of one variable F is convex (concave ), then
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For integral (4.2) F (z ) = -z ln z , z > 0. This function is concave, therefore, the entropy in equilibrium s=-


ln dµ

is not less than that at the initial time instant. Thus, the entropy remains constant over time, and at the p oints t = ± it admits a nonnegative jump s - s(0). Of course, this result also holds for smooth dynamical systems and the magnitude of the jump is entirely in agreement with the predictions of phenomenological thermodynamics (see [6]). It should also b e emphasized that the entropy makes the ab ove-mentioned jump in b oth the future and the past. This observation corresp onds to the reversible nature of the Kac circular model. Given that is obtained from t by the simple op eration of taking the arithmetic mean over the orbits of the action of the mapping T , inequalities (4.4) are merely the consequences of Jensen's classical inequality for convex functions. Note that if F > 0 (F < 0) and t as a time function is non-constant, these inequalities will b e strict. There is another method of proving the law of increase in the entropy of an isolated system. This method was already suggested by Gibbs and substantiated in [12] for smooth dynamical systems. It involves replacing (4.2) with the so-called coarse-grained entropy. The phase space is split into a finite or countable set of measurable pieces, on each of which at each instant of time the averaging of density t occurs. Let t b e the result of such an averaging, which, of course, dep ends on the breakdown of . The integral -


t ln t dµ,

(4.5)

is called a coarse-grained entropy. Generally sp eaking, this integral changes with time. The conditions for increase of the integral (4.5) as t ± are describ ed in [12]. Of considerable imp ortance is the assumption of a small decomp osition diameter, which is difficult to formulate for discrete dynamical systems. 5. THE FOURIER­RADEMACHER ANALYSIS AND LIMIT DISTRIBUTIONS Following M. Kac, let us consider the following functions on the phase space : 1; xp ; xp xq ; xp xq xr ; . . . ; x1 x2 . . . xn (p < q < r < . . . < n).

(5.1)

It is easy to understand that the numb er of such functions is exactly 2n , as is the numb er of various p oints in . Indeed, the numb er of linear functions xp is exactly n, that of quadratic functions xp xq (p < q ) is exactly n etc. Next the classical binomial identity should b e used. Thus, functions (5.1) 2 form a basis in the linear space of all functions on . The key prop erty of the set of functions (5.1) is their orthogonality: the integral of the product of any two of them over the entire with resp ect to the Liouville measure is equal to zero. Their construction resembles the construction of the Rademacher functions on a unit segment. The prop erties of completeness and orthogonality allow us to simply expand any function on into a Fourier series according to (5.1): (x) = c0 + cp xp +
p
cpq xp xq + . . . + c12...n x1 x2 . . . xn .

(5.2)

The coefficients in this formula are easily calculated from the values of the function itself. For example, 1 1 (x)xp xq . cpq = n (x)xp xq dµ = n 2 2


According to this formula the coefficient c0 in (5.1) is obviously equal to 2-n .
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If (5.2) is taken as the initial distribution density, then its "Fourier­Rademacher series" can b e found at each time instant. The corresp onding formula has b een obtained by Kac: t (x) = 1 + 2n +
p
cp x
p

p+t p p+1

. . . p

+t-1

cpq xp+t xq+t (p . . . p

+t-1

)(q . . . q

+t-1

) + ....

(5.3)

In order to find the state of statistical equilibrium, we have to determine the C`saro mean of each e term. This is a separate conceptual combinatorial problem. We approach this problem from a different angle. We ascertain the structure of functions of the form (5.2), which are invariant relative to the transformation T . According to § 3, such and only such functions are weak C`saro limits of functions (5.3) as t ±. e In series x1 , . . if the view of the , it suffices . , xn . Let u coefficients single-valuedness of expansion of the functions on into Fourier­Rademacher to find conditions for invariance of homogeneous forms relative to variables s start with linear terms. The form cp xp is invariant relative to T if and only satisfy the relations c1 = c2 1 , c2 = c3 2 , ..., cn
-1

= cn n

-1

,

cn = c1 n .

(5.4)

The condition for its solvability reduces to the equality 1 2 . . . n = 1. (5.5)

It is only satisfied if m (the numb er of elements in the set S ) is even. In this case system (5.4) has a one-parameter family of solutions c1 = , c2 = 1 , c3 = 1 2 , ...,

where is an arbitrary real numb er. If m is odd, all cj are equal to zero. Contrary to the linear case, invariant quadratic forms always exist regardless of whether m is even. Their coefficients are determined by equalities of the form (5.4), which we will not present here. Nontrivial invariant forms of the third degree exist again only if (5.5) is satisfied. And so on. In particular, the following theorem holds. Theorem 3. If m is odd, is an even function. In this case, therefore, the average of the odd function over the stationary probability measure µ is zero. Applying this observation to the linear function (3.9), we obtain again the conclusion of Theorem 1.

6. THE KAC MODEL AND THE BOGOLIUBOV CHAIN OF EQUATIONS All these observations allow us to discuss some crucial problems that arise in the theory of the Bogoliub ov chain of equations [13, 14]. This theory plays a fundamental role in the statistical mechanics of interacting particle systems. The starting p oint of N. N. Bogoliub ov's theory is the classical Liouville equation for the system consisting of n equal interacting particles. The initial distribution 0 (x1 , . . . , xn ) is assumed to b e symmetric ab out the states of individual particles x1 , . . . , xn . That is a reflection of the "identity principle" of particles.
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548

KOZLOV

In the Kac model, symmetric distributions can also b e used as a starting p oint. However, in the process of evolution the symmetry is, as a rule, destroyed. Therefore M. Kac suggested symmetrizing the distribution at each instant of time in addition to averaging over all p ositions of the set S . But this additional averaging does not allow a strict substantiation of the system's tendency to a uniform distribution either. See [5] for more discussions of this. Next, N. N. Bogoliub ov derives a chain of engaging equations for the so-called s-particle distribution functions t (x1 , . . . , xs ), by averaging the "full" distribution function over the states of n - s particles xs+1 , . . . , xn . Of sp ecial interest is the one-particle function (when s = 1), which describ es the evolution of particle distribution by coordinates and velocities (as in Boltzmann's theory). Bogoliub ov's chain of equations is, of course, equivalent to the initial Liouville equation and its strict analysis is not at all simpler than the study of the initial Liouville equation. In order to simplify this analysis, N. N. Bogoliub ov makes two additional essential assumptions which do not follow immediately from the principles of dynamics. 1) After a relatively short interval (of the order of the "time of interaction" of particles) the time evolution of all s-particle distribution functions is entirely determined by the dep endence on the time of the first distribution function: t
(s) (s)

= fs (x1 , . . . , xs | t ).
(1)

(1)

(6.1)

The vertical slash means that the dep endence on t is a functional dep endence. 2) In the distant past (when t -) the states of individual particles are statistically indep endent. As a result, a Boltzmann-typ e equation (the so-called kinetic equation), which is irreversible in time, is derived for the first distribution function. We shall not formulate and discuss precisely the latter assumption. We shall only note that if the condition of statistical indep endence in the past is replaced with the indep endence in the future (when t +), then, according to Bogoliub ov, the direction of evolution will change direction. In fact, the former assumption is crucial in Bogoliub ov's theory. However, as far as the author knows, no serious attempts have b een made to validate it. Of course, it cannot b e universal. For example, this assumption does not obviously hold for the system of non-interacting particles: the equations for (s) separate completely. N. N. Bogoliub ov himself assumed that there exist only repulsive forces b etween particles. If such a system of repulsive particles is placed in a vessel of b ounded volume, one would exp ect mixing at the levels of constant full energy. But this conclusion is not obvious either and requires validation. A great deal dep ends here on the form of the vessel. Most probably there is no mixing in regions with a nearly spherical b oundary. Let us discuss the p ossibility of representing (6.1) in the Kac system. First we consider a simplified probabilistic model already mentioned in § 2: when turning in a unit time the ball changes (preserves) its color with probability µ (accordingly, 1 - µ). Taking the changes of color in various p ositions to b e indep endent, we transform the Kac system into a simple Markov chain. Using the Fourier­Rademacher representation, M. Kac obtained an explicit formula for distribution density at the current time instant [5]: t (x) = 1 + (1 - 2µ)t 2n ck xk
+t

+ (1 - 2µ)2t

ckl xk+t xl
k
+t

+ ... (6.2)

+ (1 - 2µ)nt c1...n x1+t x2+t . . . xn+t 1 ck-t xk + (1 - 2µ)2t = n + (1 - 2µ)t 2 + (1 - 2µ)nt c1...n x1 x2 . . . xn .

ck
kl

-t, l-t

xk xl + . . .

In writing these formulae (as in (5.3)) the natural convention xp = xq was used if p q (mod n). The Fourier­Rademacher coefficients are the products of the exp onential function by the p eriodic time functions of p eriod n.
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STATISTICAL IRREVERSIBILITY OF THE KAC REVERSIBLE CIRCULAR MODEL

549

Consider an imp ortant particular case where the initial distribution (5.2) is symmetric (invariant relative to the p ermutations of variables x1 , . . . , xn ). Then all ck are equal, all ckl are also equal etc. In this case the p eriodic multipliers are constant and all coefficients in (6.2) are simply expressed in terms of the same function (1 - 2µ)t . After averaging (6.2) over variables xs+1 , . . . , xn we obtain a function with the same structure of coefficients, which makes the representation of (6.1) p ossible. If there is no initial symmetry, then the Bogoliub ov hyp othesis does not hold in a typical situation. As is seen from (6.2), the probabilistic simplification of the original Kac model involves the loss of its key prop erties of recurrence and reversibility. Without the simplifying assumptions the functional representation (6.1) is imp ossible. To show this, we consider a particular initial distribution 1 + x1 x2 . . . xn . (6.3) 2n It is symmetric, and for small values of this function assumes everywhere p ositive values. It is easy to understand that 0 (x) = 1 + (1 2 . . . n )t x1 x2 . . . xn . (6.4) 2n If m, the numb er of elements of a given set S , is odd, then 1 2 . . . n = -1. In this case t is a nonconstant function of time. Averaging over variables x2 , . . . , xn gives a "one-particle" distribution function, which assumes at p oints x1 = ±1 the equal value 1/2. Therefore, when m is odd, formula (6.1) does not obviously hold for s = n. From the ab ove observations it may b e concluded that the first hyp othesis of N. N. Bogoliub ov evidently holds under additional simplifying assumptions related to the replacement of the initial dynamical system with the corresp onding Markov process. As for determinate dynamical systems, the conditions for this hyp othesis to hold need to b e ascertained. t (x) = REFERENCES
1. Ehrenfest, P. and Ehrenfest, T., Bemerkung zur Theorie der Entropiezunahme in der "Statistischen Mechanik" von W. Gibbs, Sitzungsberichte Akad. Wiss. Wien, 1906, vol. 115, pt. IIa, pp. 89­98. ¨ 2. Ehrenfest, P. and Ehrenfest, T., Uber zwei bekannte Eniwande gegen das Boltzmannsche H -Theorem, ¨ Phys. Zschr., Jg. 9, vol. 8, pp. 311­314. 3. Markov, A. A., Generalization of the Problem of the Consecutive Exchange of Balls, Izv. Akad. Nauk, 4 ser., 1918, vol. 12, pp. 261­266 (Russian). 4. Kac, M., Probability and Related Topics in Physical Sciences, New York­London: Intersci. Publ., 1958. 5. Kac, M., Some Stochastic Problems in Physics and Mathematics, Colloquium Lectures in Pure and Applied Science, Dallas, TX: Magnolia Petroleum Co., 1956. 6. Kozlov, V. V., Thermal Equilibrium in the Sense of Gibbs and Poincar´, Moscow­Izhevsk: Institute of e Computer Science, 2002 (Russian). 7. Kozlov, V. V., Gibbs Ensembles and Non-equilibrium Statistical Mechanics, Moscow­Izhevsk: Institute of Computer Science, 2008 (Russian). 8. Poincar´, H., R´fflexions sur la th´orie cin´tique des gaz, J. Phys. th´oret. et appl., 4-e s´r., 1906, vol. 5, e e e e e e pp. 369­403. 9. Gibbs, W., Elementary Principles in Statistical Mechanics, Developed with Especial Reference to the Rational Foundation of Thermodynamics, New York: Schribner, 1902. 0. Hardy, G. H., Divergent series, Oxford: Oxford Univ. Press, 1949. 1. Prigogine, I. and Stengers, I., Order out of Chaos: Man's New Dialogue with Nature, London: Heinemann, 1984. 2. Kozlov, V. V. and Treshchev, D. V., Fine-Grained and Coarse-Grained Entropy in Problems of Statistical Mechanics, Teoret. Mat. Fiz., 2007, vol. 151, no. 1, pp. 120­137 [Theoret. and Math. Phys., 2007, vol. 151, no. 1, pp. 539­555]. 3. Bogolyubov, N. N., Problems of Dynamical Theory in Statistical Physics, Moscow: Gostekhizdat, 1946 (Russian). 4. Uhlenbeck, G. E. and Ford, G. W., Lectures in Statistical Mechanics, Providence, RI: AMS, 1963.

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2011