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V. V. KOZLOV
Steklov Mathematical Institute Russian Academy of Sciences 117966 Moscow, Russia E-mail: kozlov@pran.ru

BILLIARDS, INVARIANT MEASURES, AND EQUILIBRIUM THERMODYNAMICS. II
Received November 03, 2003

DOI: 10.1070/RD2004v009n02ABEH000268

The kinetics of collisionless continuous medium is studied in a bounded region on a curved manifold. We have assumed that in statistical equilibrium, the probability distribution density depends only on the total energy. It is shown that in this case, all the fundamental relations for a multi-dimensional ideal gas in thermal equilibrium hold true.

According to Gibbs, the basic ob ject of statistical mechanics is an ensemble of identical Hamiltonian systems. The systems do not interact with each other and the assembly of them makes essentially a collisionless continuous medium. From the viewp oint of kinetics, the Hamiltonian systems with elastic impacts, i. e. bil liards , are esp ecially interesting. These are systems where particles move inertially inside a b ounded region and b ounce elastically against the b oundaries of the region. As it is shown in Refs. [1, 2], in a billiards, the probability distribution density as a function of time t (this function satisfies the classic Liouville equation) necessarily has the weak limit as t ±. This result justifies the Zeroth Law of Thermo dynamics in the Gibbs theory. The weak limit is a first integral of the Hamilton equations and dep ends, in the ergo dic case, only on the system's energy. It is noted in Ref. [3] that this is very often justified even without any ergo dic hyp othesis : it is imp ortant here to keep in mind the function class, to which the probability distribution density function b elongs. In Ref. [3], we develop ed the thermo dynamics of billiards in the Euclidean space. It turns out to b e p ossible to extend these observations to the general case of a curved configurational space. Let M n b e a compact configurational space of a natural mechanical system with n degrees of freedom, x = (x1 , . . . , xn ) b e lo cal co ordinates on M , and y = (y 1 , . . . , yn ) b e the conjugate canonical momenta. The motion to b e considered is inertial. So the Hamiltonian is a p ositively defined quadratic form with resp ect to the momenta : H=1 2
n

aij (x)yi yj .
i,j =1

(1)

If we denote the matrix of co efficients ||a i,j || as A, then H = (Ay , y )/2. Let f (·) b e a nonnegative summable function of one variable. By analogy with the Gibbs canonical distribution, we intro duce the density of the stationary probability distribution in the phase space = = T M : f ( H ) (x, y ) = . (2) f ( H ) dn y dn x
n


M

Here,the factor is intro duced to non-dimensionalize the argument of f . It is customary to take = = 1/k , where k is the Boltzmann constant, and is the absolute temp erature.
Mathematics Sub ject Classification 82C22, 70F07, 70F45

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The denominator in (2) is referred to in Ref. [3] as the generalized statistic integral. It can b e easily expressed in terms of , the only external thermo dynamical parameter here b eing M , Riemann's volume of the manifold. To this effect, we p erform the linear change of variables y p: y = C (x)p, Thus, F=
n


C T AC = E .

f ( H ) dn y dn x =


f
M
n

2

p

2 i

(det A)

-

1 2

M

bv dn x d n p = n , ( )

where v=
M

(det A

1 -1 2 )

dn x

is the volume M with resp ect to Riemannian metric (1),
n 2 b = 2 n n-1

r
0

f

r2 2

dr = const,

2

where is the Euler gamma function. Now we calculate the average kinetic energy : E= 1 F
n


1 (Ay , y )f 2
M

(Ay , y ) 2 p
2 j

dn x d n y =
- 1 2

=1 F


1 2
n

p2 f j

2

(det A)

dn x d n p =

M

= a, b where a=
n 2

n
2

r
0

n+1

f

r2 2

dr = const.

It is interesting to note that the average internal energy of a collisionless medium do es not dep end on the volume, which correlates with Joule's law for ideal gas. As it is shown in Ref. [4], if a Hamiltonian is a homogeneous function with resp ect to momenta, then the quantities calculated using the general routines of statistical mechanics and density (2) satisfy the First and the Second Laws of Thermo dynamics. Let us calculate, for example, the thermo dynamic entropy. For this, we should first (according to Ref. [3]) write down the following relation: E = F , which gives the co efficient . According to the general theory, this co efficient must b e a function of the statistical integral F . In the case in question, = - 2a 1 . bn F
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Let (F ) b e the antiderivative of (F ). Then, as shown in Ref. [3], the thermo dynamic entropy is given by the equation S = - . Hence, S = a + 2a ln F = const + 2a b bn bn ln v + n ln 2 . (3)

In the case of the Gibbs canonical distribution ( f (z ) = e -z ), one can easily show, up on integration by parts, that 2a = nb. Usually, the entropy of ideal gas in a three-dimensional vessel with volume w is N ln w + 3N ln + const, 2 (4)

where N is the numb er of gas particles (this expression is sometimes multiplied by the Boltzmann constant k , but we do without it). To compare (3) and (4), let us consider the Boltzmann­Gibbs gas consisting of N identical small balls, moving in a vessel . The balls collide elastically with each other and with the walls of the vessel. Then, obviously, n = 3N , while the volume v is approximately equal to wN (for, in the case of non-interacting balls, the configurational space M of the system is the direct pro duct of N copies of ). Having made these remarks, we see that (3) and (4) b ecome identical up to the insignificant constant factor 2a/nb that dep ends on the typ e of the function f (·) and on the numb er of degrees of freedom in the system. On the other hand, the entropy in statistical mechanics is given by the integral S=-


ln dn x dn y .

(5)

In the case of the canonical distribution, this integral coincides with the thermo dynamical entropy. Of course, for more general distributions of the form (2), this remarkable Gibbs' result is not valid. However, the Gibbs entropy (5) lo oks as follows: + ln F , b where
n 2 = - 2 n

(6)

r
0

n-1

f

r2 2

ln f

r2 2

dr = const.

2

We see that (3) and (6) coincide up to an insignificant additive constant and a somewhat less insignificant constant p ositive factor. The latter remark is very imp ortant for the kinetics of a collisionless medium, esp ecially for the validation of the Second Law in the case of irreversible pro cesses. The matter is that (as proved in Refs. [1] and [5]) if we replace the density in (5) with its weak limit, then the Gibbs entropy gets a nonnegative increment. If the entropies from (3) and (5) were not so closely related, this general result would not allow a natural thermo dynamical interpretation. In the case of ergodic billiards, we can make not only general conclusions on increase in entropy in irreversible pro cesses, but we can also calculate these increments. As a simple example, let us consider the case where a collisionless medium is initially enclosed in the p ortion M - M (regions M- and M \M- are separated with a wall), b eing in statistical equilibrium. After removal of the wall, the medium expands irreversibly, tending to fill the whole region M . During this, its internal energy (and, consequently, its temp erature) do es not change. According to (3), the entropy gets a p ositive
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increment, prop ortional to logarithm of the ratio of volumes M - /M+ , where M+ = M . This result is in a go o d agreement with the predictions of the phenomenological thermo dynamics. We should, probably, also mention that ergo dicity of the Boltzmann­Gibbs gas for a vessel shap ed as a rectangular parallelepip ed was ascertained by Ya. G. Sinai [6]. According to Ref. [3], the thermo dynamical variable P , conjugate to the volume v , is given as P = - 1 . v Hence, P = 2a k . nb v (8) (7)

This is the equation of state for the considered system in statistical equilibrium. This equation is identical in form with the classical Clap eyron equation. If f (z ) = e -z , then 2a = nb, and (8) exactly fits the Clap eyron equation for a mole of ideal gas. The physical meaning of the variable P is pressure. Let us return to the Boltzmann­Gibbs gas of N small balls in a three-dimensional vessel with volume w. Then n = 3N , and we can assume that v = w N . Substituting this expression into (8), we obtain an equation, which is different from the Clap eyron equation. However, there is no contradiction here, for P stands for the pressure of 3N -dimensional gas. The pressure p in ordinary gas, as a thermo dynamical quantity, conjugate to the volume w, is given by (7), only should first b e presented as a function of and w : p = - 1 = - 1 dv = 2a k . w v dw 3b w For the Maxwell distribution (where f (z ) = e -z ), 2a/3b = N , and we obtain the classical ideal gas equations : E = 3 N k , pw = N k . (9) 2 For non-Maxwellian distributions, the value of 2a/3b, surely, differs from N . However, within a wide range of distributions, for large N , this value is approximately equal to N : lim 2a(N ) = 1. 3N b(N ) (10)

N

For example, this range includes distributions with densities
2 f ( r ) = g (r )e 2 -r 2 /2

,

(11)

where g (r ) is an arbitrary non-negative p olynomial in r . Indeed,


r
0

n+-1 -r 2 /2

e

dz = 1 n+
0

r

n++1 -r 2 /2

e

dz .

n Since n + 1 when is fixed, this results in the limit relation (10). Recall that functions of the form (11) are referred to as partial sums of Gram-Charlier series , and are commonly used to approximate the distribution densities of arbitrary random variables. In the case in question, such an approximation is p ossible due to the well-known observation (traced as far back as to Boltzmann) that in the ma jor p ortion of a high-dimensional space, any distribution is close to normal (strict formulations and discussion can b e found, for example, in Ref. [7]). Since N , as a rule, is extremely large (of the order of 1023 ) and not precisely known, we can as well use the classical equations (9) instead of E = (a/b)k and pw = (2a/3b)k . 94 REGULAR AND CHAOTIC DYNAMICS, V. 9,
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BILLIARDS, INVARIANT MEASURES, AND EQUILIBRIUM THERMODYNAMICS. II

Thus, we have built a complete (nonequilibrium) theory of ideal gas within the framework of Gibbs' general approach, using the concept of weak limits of probability distributions and the result concerning the ergo dic b ehaviour of the Boltzmann­Gibbs gas. As distinct from Boltzmann's approach, we do not use any additional assumptions (like the condition of statistical indep endence of double collisions). The substantial difference from Boltzmann's approach is that in our theory the gas reaches statistical (thermal) equilibrium b oth as t + and as t -, these equilibriums b eing identical. This fact is fully consistent with the invertibility prop erty of the equations of motion. Note also that by virtue of (10), the entropy equation (3) yields the classical formula (4) for monatomic ideal gas. Besides, the statistical entropy (5) coincides (up to an additive constant) with the thermo dynamical entropy (3) as t .

A supplement. Particle distribution functions
Let N (x1 , . . . , xN , t) (12) b e the distribution density of the Boltzmann­Gibbs gas, which is a system of N small identical balls enclosed in a rectangular b ox; xj denotes the co ordinates and momenta of the j -th ball. The function (12) satisfies the Liouville equation and the initial condition N (x, 0) at t = 0. According to Bogolyub ov (see Ref. [8]), it is useful to intro duce s-particle distribution functions s (x1 , . . . , xs , t), averaging density (12) over xs+1 , . . . , xN . The particle distribution functions satisfy the infinite chain of "ho oked" equations, a so-called BBGKY (Bogolyub ov, Born, Green, Kirkwo o d and Yvon) chain. Under some additional assumptions (sp ecifically, that of molecular chaos "in the past"), in the case of rarefied Boltzmann­Gibbs gas, one derives the kinetic Boltzmann equation for one-particle distribution function 1 . These assumptions are not self-evident, do not follow from the principles of Gibbs' statistical mechanics, and are to certain extent similar to Boltzmann's assumption of statistical indep endence of the balls' velo cities b efore a double collision. Two imp ortant facts follow from the Boltzmann equation : 1) Boltzmann's entropy - monotonously increases with time, and 2) as t +, the distribution 1 tends to the Maxwell distribution. However, these conclusions (at least, the former) cannot b e directly verified by exp eriment. The matter is that the thermo dynamical entropy is intro duced only for equilibrium states. The ideas of determination of entropy for nonequilibrium states (like those given, for example, in Refs. [9,10]) are of metho dical nature, prop osing to intro duce an infinite numb er of additional internal thermo dynamical parameters. Using these ideas, the reader can find, for example, the entropy of ideal gas as a function of time as the gas is adiabatically expanding into vacuum (Joule's classical exp eriment). We develop a different approach in the nonequilibrium statistical mechanics of the Boltzmann­ Gibbs gas. It has nothing to do with the analysis of additional assumptions that can b e used to close Bogolyub ov's chain of equations. We evolve Gibbs' classical principles and try to avoid entirely additional assumptions of conceptual nature. The crucial idea of our approach is : transition to thermo dynamical (statistical) equilibrium is equal to replacement of the distribution (12) with its weak limit. This idea arises very naturally when one pro ceeds from microscopic to macroscopic description of a dynamical system. The weak limit (as t + and t -) of density (12) (if it exists) coincides with Birkhoff 's average (x1 , . . . , xN ). ¯
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Let (x1 ) b e a test function. Then 1 d6 x1 = - where 1 (x1 ) = N d6 x2 . . . d 6 xN .
N

N d6 x1 . . . d6 x
N

N

-- -

t

N d6 x1 . . . d6 x

=

1 d6 x1 ,

(13) is a summable function that

If the initial system with 3N degrees of freedom is ergo dic, then dep ends only on the total energy :
N

= f(

m 2 2 (v + . . . + vN ))/v 21
3N


f d 3 v1 . . . d 3 vN .

(14)

2 Here m is the mass of the p oints, vj is the squared velo city of the j -th ball, the parameter has the dimension of the inverse of energy (it is intro duced to non-dimensionilize the argument of f ) and v is the volume of the 3N -dimensional configurational space of the system of N balls. The denominator in (14) makes the integral of N over the whole phase space equal to unity. Formula (14) shows that every p ossible p osition of the N balls is equiprobable. This fact, noted earlier in Ref. [2], means that a homogeneous distribution is established in the state of thermal equilibrium. In fact, a similar conclusion follows Boltzmann's theory : the density of gas particles gets equalized and at the same time the Maxwell velo city distribution is established. Hence, the limit one-particle distribution 1 do es not dep end on the co ordinates, and therefore, the averaging in (13) can b e replaced with merely averaging over the velo cities v 2 , . . . , vN . As a result, the following simple expression is obtained :

f 1 (u) =


3N -3


1 (u2 + v 2 + . . . + v 2 ) d3 v . . . d3 v 2 2 N 2 1 (v 2 + v 2 + . . . + v 2 ) d3 v . . . d3 v 1 2 N 2 1

N

(15)
N

f
3N

where = /m, u R3 . It turns out that when certain additional constraints (of analytical nature, not statistical) are imp osed on the function f , the limit one-particle distribution function 1 tends to the Maxwell distribution as N . That is, for nearly any initial distribution N (x1 , . . . , xN , 0) (even without assuming that N is symmetrical relative to x1 , . . . , xN ), the balls' velo city distribution in the state of thermal equilibrium is, to all practical purp ose, normal (if, as usual, N is sufficiently large). It should b e underlined that, in such an approach, it is meaningless to sp eak of the rate of convergence of 1 (as a function of time) to the limit distribution 1 , since the 1 itself do es not tend anywhere at all. One can only sp eak of the rate of convergence of the average values of the dynamic quantities. If, for example, one takes the characteristic function of certain region inside the vessel as a test function , then it will b e just reasonable to consider the rate of equalization of the numb er of balls in this region. To derive an expression for the limit distribution (as N ), we put 3N = m + 2 and transform (15) : 1+ m 1 (u) =
3/2

2

r
0

m-2

f

u2 + u 2 + u 2 + r 1 2 3 2


2

dr . dr (16)

1+ m-3 2

r
0

m+1

f

r2 2

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BILLIARDS, INVARIANT MEASURES, AND EQUILIBRIUM THERMODYNAMICS. II

Here u = (u1 , u2 , u3 ) and is the gamma function. The density, actually, dep ends on the velo city |u|. Therefore, its k -th moment 1 (|u|)|u|k du1 du2 du ¯
3


3

is equal to


4
0

1 (x)x ¯
k+3 2

k +2

dx =
m+k +1

2 1 + m =
2




0

f
2 2

2 2

d . (17)



k+m +1 2


0

m+1

f

d

When deriving this expression, we used (16) and the elementary prop erties of the gamma function. Assume that the variable x takes all real values; then, it is natural to consider the density 1 (x) ¯ an even function. To simplify the notation, we put = 1 (or replace the function f (z ) with f ( z )). Our goal is to show that, as m , a distribution with the density 2 x2 1 (x), x R ¯ tends to the normal one. To this end, we accept two assumptions : (a) the limit (as m ) density (18) has a finite p ositive variance (the second moment, k = 2), and (b) the function f has a summable derivative. Condition (a) is the condition of non-degeneracy of the limit distribution, while condition (b) is of technical nature and can probably b e weakened. Besides, f should decay at infinity faster than any p ower function : otherwise the integrals in (16) and (17) are not defined for every m. Putting k = 2 in (17), integrating by parts and using assumption (a), we obtain :
m -

(18)

lim

2 x4 1 (x) dx = ¯
m+3

3 = lim m m + 2
0

f f

2 2 2 2

d = d


0

m+1

= -3
0 lim m 0

m+3

f

2 2 2 2

d = 3c > 0. d (19)



m+3

f

Here 3c is the variance, which exists due to (a).
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Let us put f + cf = g . Then, according to (19),



0

m+3

2 g d 2
0



m+3

f

2 d 0 2

(20)

as m . Now, we calculate the limit of the fourth moments (k = 4):


2 1 + m 7
m

lim

2 2 m +3 2


0

m+5

f

2 2

d . d (21)


0

m+1

2 f 2

According to (20), the integral in the numerator of (21) can b e replaced with the integral


-c
0



m+5

f

2 2

d = c(m + 4)
0



m+3

f

2 2

d .

Using (19), it is easy to calculate the limit (21). It is equal to 1 · 3 · 5c 5 . In the similar way, we prove that when k = 2n, the limit (17), as m , is (2n + 1)!!cn . (22)

All the o dd moments are, obviously, equal to zero. Now, let 1 b e the density of the normal distribution in the three-dimensional Euclidean space : ^ 1 e ( 2 )3 Then 2 x2 1 (x) = 1 x2 e ^ 2 3 Let us calculate the variance of this distribution :
- - u2 +u2 +u 1 2 2
2 3

.

x2 2

.

-

x4 e 2 3

-

x2 2

dx = 3 2 .

The fourth moment is equal to 1 · 3 · 5 4 ; more generally, the 2n-th moment is equal to (2n + 1)!! 2n . (23)

Formulas (22) and (23) coincide if we put c = 2 . Hence, according to the Chebyshev­Markov moment theorem (see Ref. [11]), lim 1 (u) = (u), u R3 ; = c ¯ ^ (24)
m

Up to now, we have b een using the assumption that the function f do es not dep end on the numb er of particles. In general, of course, this is not the case, and instead of a single function f , we have a sequence of functions, fm+2 . Nevertheless, we can again put gm+2 = fm+2 + cfm+2 (provided that the
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limit distribution has a finite p ositive variance). It is easy to show that (24) remains valid if the limit expression (20) is replaced with a more general one :



0

m+2k +1

g

m+2

2 d 2
0



m+2k +1

f

m+2

2 d 0 2

as m for any integer k 1. Recall that, for a fixed value of the total energy, the distribution (14) is called microcanonical . According to Maxwell and Borel, it can b e transformed to the canonical Gibbs distribution if one assumes that the total energy is equal to N E , the average energy E of a single particle b eing indep endent of N (see Ref. [12]). Of course, this is an imp ortant assumption. A more general implementation of this idea, applied to an ensemble of weakly interacting identical subsystems, can b e found, for example, in Ref. [13]. In essence, the author sp ecifies the conditions, under which the micro canonical distribution weakly converges to the canonical distribution as the numb er of subsystems increases indefinitely. Besides, as test functions, the author uses so called adders , symmetrical functions of some sp ecially chosen canonical variables. Our construction of the normal distribution is based on different ideas. One should b ear in mind that Boltzmann's and Bogolyub ov's theories are not free of all these problems, either. Supp ose that, at the initial time t = 0, the distribution N coincides with the distribution (14). The distribution (14) is stationary, and corresp onds to the state of thermo dynamical equilibrium (in Gibbs' approach). In Boltzmann's theory, however, density 1 (u) (which is given by (15)) corresp onds, in the general case, to the initial nonstationary distribution and should tend, in the course of time, to the Maxwell distribution. In Bogolyub ov's theory, we have a similar case : not every summable function can b e readily used as the density of the initial distribution. It was supp osed that the velo cities in a particle system should in some remote past (when the particles were far from each other) b e indep endent (so that every s-particle distribution function was reduced to a pro duct of one-particle functions). It should b e underlined that this remote past cannot b e replaced with the remote future (for discussion, see [8]). However, in our approach, the tendency to statistical equilibrium is invariant under the time reversal. The work was supp orted by the Russian Foundation for Basic Research (grant 01-01-22004) and the Foundation for Leading Scientific Scho ols (grant 136.2003.1).

References
[1] V. V. Kozlov, D. V. Treshchev. Weak convergence of solutions of the Liouville equation for nonlinear Hamiltonian systems. Teor. Mat. Fiz. 2003. V. 134. 3. P. 388­400 . English transl.: Theor. Math. Phys. 2003. V. 134. 3. P. 339­350 . [2] V. V. Kozlov, D. V. Treshchev. Evolution of measures in the phase space of nonlinear Hamiltonian systems. Teor. Mat. Fiz. 2003. V. 136. 3. P. 496­506 . English transl.: Theor. Math. Phys. 2003. V. 136. 3. P. 1325­1335 . [3] V. V. Kozlov. Billiards, Invariant Measures, and Equilibrium Thermodynamics. Reg. & Chaot. Dyn. 2000. V. 5. 2. P. 129­138 .


[4] V. V. Kozlov. Thermodynamics of Hamiltonian Systems and Gibbs Distribution. Dokl. Akad. Nauk. 2000. V. 370. 3. P. 325­327. English transl.: Doklady Mathematics. 2000. V. 61. 1. P. 123­125 . [5] V. V. Kozlov. Kinetics of Collisionless Continuous Medium. Reg. & Chaot. Dyn. 2001. V. 6. 3. P. 235­251 . [6] Y. G. Sinai. Dynamical Systems with Elastic Reflections. Ergodic Properties of Dispersing Billiards. Usp. Mat. Nauk. 1970. V. 125. 2. P. 141­192. English transl.: Russ. Math. Surv. 1970. V. 25. P. 137­189 . [7] V. V. Ten. On normal Distribution in Velocities. Reg. & Chaot. Dynamics 2002. V. 7. 1. P. 11­20.


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[8] G. E. Uhlenbeck, G. W. Ford. Lectures in Statistical Mechanics. Amer. Math. Soc. Providence. 1963. [9] E. Fermi. Thermodynamics. Hall. 1937. New York: Prentice-

[11] N. I. Akhiezer. The Classical Moment Problem. Moscow: Nauka. 1961. (In Russian) [12] M. Kac. Probability and Related Topics in Physical Sciences. Interscience Publichers. 1958. [13] F. A. Berezin. Lectures on Statistical Physics. Moscow-Izhevsk: Institute of Computer Science. 2002. (In Russian)

[10] M. A. Leontovich. Introduction into Thermodynamics. Moscow-Leningrad: Gostekhizdat. 1952. (In Russian)

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