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V. V. KOZLOV
Department of Mechanics and Mathematics Moscow State University, Vorob'ievy Gory 119899, Moscow, Russia

ON JUSTIFICATION OF GIBBS DISTRIBUTION
Received January 10, 2001

DOI: 10.1070/RD2002v007n01ABEH000190

The paper develop a new approach to the justification of Gibbs canonical distribution for Hamiltonian systems with finite number of degrees of freedom. It uses the condition of nonintegrability of the ensemble of weak interacting Hamiltonian systems.

Gibbs distribution. system with the density

We consider the probability distribution in the phase space of Hamiltonian = ce
- H k

,

(1)

where H is a Hamiltonian, is an absolute temp erature, k is the Boltzmann constant, c is a normalized factor. It plays the key role in the equilibrium statistical mechanics. Gibbs show in [1] that the averaging with resp ect to probability measure with density (1.1) give rise to the fundamental relations of equilibrium thermo dynamics. To deduce the canonical Gibbs distribution one usually consider the ensemble of Hamiltonian systems with Hamiltonian function of the following form
0


=


(P , Q) +
n

1

(P , Q),
(s)

(2)

where
0

=
s=1 (s) m

H0 (p ),

(s)

,q

),
(s) 1

p

(s)

= (p

(s) 1

, ..., p

Thus at = 0 the system with Hamiltonian (1.2) is decomp osed on n identical systems with m degrees of freedom and Hamiltonian H 0 . The canonical variables P , Q are the momenta p (1) , . . . , p(n) and co ordinates q (1) , . . . , q (n) of separate subsystems. The p erturbing function 1 is the energy of interaction of n subsystems; it usually dep ends on their co ordinates Q. Small parameter is the characteristic of intensity of subsystems' interaction. We consider the case, when the Hamiltonian is sufficiently smo oth with resp ect to variables P , Q. However, in application we often see cases with singular interaction. The classical example is the Boltzmann-Gibbs gas, the assembly of rigid balls in cub e that elastically collide with each other (see [1, 2, 3]). The traditional approach to the deduction of Giibs distribution suggested by Fowler and Darwin ([4], the rigorous exp osition see in [5, 6]) essentially uses the ergo dic hyp othesis: for all small > 0 the Hamiltonian system with Hamiltonian (1.2) is ergo dic on fixed energy manifolds = const. With some additional conditions some systems with Hamiltonian H 0 are distributed in accordance with formula (1.1) as 0 and n .
Mathematics Sub ject Classification 37H10, 70F45


¡

REGULAR AND CHAOTIC DYNAMICS, V. 7,



q

(s)

= (q

, ..., q

(s) m

).

(3)

1, 2002

1


V. V. KOZLOV

But the pro of of ergo dic hyp othesis for sp ecific Hamiltonian systems is usually pretty difficult problem. Moreover the ergo dic hyp othesis often contradicts with results of KAM theory. In particular if the Hamiltonian system with Hamiltonian H 0 is completely integrable and the energy surfaces H 0 = = const are compact, then the ergo dic prop erty is not present with certainty. In view of this remark we can set the following interesting problem: prove that in analytical (or even in infinitely differentiable) case if the energy surfaces H 0 = const are compact, then the Hamiltonian system with Hamiltonian (1.2) never satisfy ergo dic hyp othesis. We can try to mo dify the Fowler­Darwin metho d assuming that = 0 and n dep end on in such way that n() as 0. We can assume that for some functions n() the system with Hamiltonian (1.2) is ergo dic on the surfaces = const as a result of huge numb er of its degrees of freedom. This somewhat weakened version of ergo dic hyp othesis is closely related to the unsolved problem of estimation of small parameter in KAM theory, when the "last" Kolmogorov torus disapp ears. The weaker conjecture on transitivity of system with Hamiltonian (1.2) on energy surfaces = const for large values of n and small is not proved yet. If this problem has the p ositive answer, then we can assert the presence of diffusion in Hamiltonian systems with many degrees of freedom (see [7­9]). Probability density as an integral of Hamiltonian equations. A different approach to the deduction of canonical Gibbs distribution was prop osed in the pap er [10]. This approach is based on the fact that the stationary density of probability distribution is the integral of Hamiltonian differential equations uniquely defined in the whole phase space. In this case the numb er of interacting subsystem n 2 is fixed. More exactly in [10] we consider the case, when the subsystems have one degree of freedom: m = 1. Under some natural condition we can pro ceed to the angle­action variables in each subsystem, and using the well-known Poincar´ metho d we can obtain the constructive condition of nonexistence of e new integrals (see [11, 12]). The results of pap er [10] can b e easily converted to the more general case, when H0 is a Hamiltonian of completely integrable system. We are going to find out the sufficient conditions (constructive if p ossible) of nonintegrability of systems (1.2)­(1.3). We study the conditions of existence of an integral (P , Q, ) of the canonical differential equations P = - , Q= (4) Q P


where 0 and 1 are functions of class C 2 and C 1 with resp ect to P and Q corresp ondingly. Probably we can weaken the requirements to the class of smo othness of integral , and the following arguments still will b e correct. But this is a separate problem and we are not going to discuss it here. Non-p erturb ed problem. Supp ose = 0. Then we have a system of n indep endent subsystems. It is strongly nonergo dic: at = 0 system of differential equations (2.1) has n indep endent first integrals Hs = H0 (p(s) , q (s) ), 1 s n. (6) It is clear that the function 0 from expansion (2.2) is a first integral of this unjointed system. Let's show that under the sp ecific conditions the function 0 dep ends only on H1 , . . . , Hn . In particular these conditions will imply that any separate subsystem with m degrees of freedom do es not nave
2 REGULAR AND CHAOTIC DYNAMICS, V. 7,
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=


0

(P , Q) +

1

(P , Q) + o(),

(5)

1, 2002





with a Hamiltonian parameter . Poincar´ e as a p ower series with the variables P , Q and


of the form (1.2)­(1.3). We emphasize that the integral dep ends on the considered the analytical case; in particular we can construct the integral resp ect to . We supp ose that is a function of class C 2 with resp ect to all . Hence we can supp ose








ON JUSTIFICATION OF GIBBS DISTRIBUTION

first integrals indep endent on the integral of energy. The ideas of our arguments follows Poincar´ e metho d [11]. Let M b e a phase space of Hamiltonian system with Hamiltonian function (3.1). Certainly for all s these spaces are identical. A phase space of new system is the direct pro duct


= M â . . . â M,


dim

= 2nm.

Let hs b e a value of total energy of system with numb er s, and (hs ) = {p
(s)

,q

(s)

: Hs (p

(s)

,q

(s)

) = hs }

is the a energy surface. If the value of h s is uncritical, then is a smo oth 2m - 1-dimensional manifold. At fixed values of h = (h1 , . . . , hn ) and = 0 unjoined Hamiltonian system (2.1) is reduced to the direct pro duct of dynamical systems defined on


S (h) =

(h1 ) â . . . â

(hn )

.

The ergo dic prop erty of system with Hamiltonian H s on (hs ) do es not necessarily imply the constancy of integral 0 on the invariant set S (h). Consider the following simple example Example. Supp ose the following dynamical system xi = i , y j = j ; i, j = 1, . . . , k (7)


with constant incommensurable systems = ( 1 , . . . , k ) is defined on direct pro duct of k -dimensional tori Tk {x mo d 2 } â Tk {y mo d 2 }. According to the Weyl theorem, each separate subsystem is ergo dic on Tk . But equations (3.2) have single-valued nonconstant integrals sin(x i - yi ) (1 i k ).
¡

Remark. However, if Hamiltonian systems are weakly mixing (mixing) systems on (hs ), then their direct product also has weakly mixing (mixing) property on S (h). In particular in these cases the function 0 is constant on any connected component of manifold S (h).

Let T1 b e a p erio d, and s = of this tra jectory new variables the

nondegenerate p erio dic tra jectory of system with numb er s with energy h s , Ts its 2 /Ts its frequency. According to Flo quet­Lyapunov theorem, in a neighb orho o d (s) (s) on (hs ) we can express the co ordinates s mo d 2 , z1 , . . . , z2m-2 , so that in the motion equation obtain the following form: s = s + fs (s , z
(s)

),

z

(s)

= s z

(s)

+ gs (s , z

(s)

).

(8)

Here fs = O (|z (s) |), gs = o(|z (s) |), and constant square matrix s of order 2m - 2 is nondegenerate. Assuming in (3.3) z (s) = 0 we obtain the equation of p erio dic tra jectory: s =
s

(1

s

n).

(9)

According to the assumption on non degeneracy of p erio dic tra jectory T 1 , nondegenerate p erio dic tra jectories with similar p erio d are situated on close energy surfaces (hs ); p erio ds and frequencies continuously dep end on energy hs . It is clear that the direct pro duct T 1 â . . . â T1 = Tn is n-dimensional invariant torus of canonical system of differential equations (2.1) at = 0 situated on S (h). In the neighb orho o d of this torus the equation of motion have form (3.3). Hence, such torus is reducible and nondegenerate (see, for example, [12]). On the torus the equation are reduced to a conditionally-p erio dic form (3.4). As usually, we call an invariant torus nonresonance if the frequencies 1 , . . . , n are indep endent on the ring of integers. In the further analysis the following condition is essential A) For almost all admissible values of h R n nonresonance tori are everywhere dense on the manifold S (h).
REGULAR AND CHAOTIC DYNAMICS, V. 7,
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3


V. V. KOZLOV

Example. Let separate subsystems describ e the inertial motion on the manifold of negative curvature. Energy h is non-negative. All p erio dic tra jectories with p ositive energy are hyp erb olical; hence they are not degenerate. Perio dic tra jectories are the motions on closed geo detics on with l . Hence the frequency different sp eed. If l is a length of closed geo detic, then the p erio d is equal to is defined by the formula 2h . l Since the lengths of n geo detics l1 , . . . , ln are fixed, then for almost all p ositive values of energy h1 , . . . , hn the frequencies 1 , . . . , n are incommensurable. It is p ossible to show (and this is a separate problem) that in a considered situation condition A is fulfilled. 2
0




2h

Prop osition 1. If condition A is fulfil led, then for al l h R n the function any connected component of S (h).


is constant on

Proof. Since 0 is an integral of system of equations (2.1) at = 0, then (according to the Kronecker theorem) 0 is constant on any nonresonance torus T n . Since this torus is reducible and nondegenerate, then d 0 = 0 in p oints T (see [12], ch. IV). According to condition A, for almost all values of h Rn nonresonance tori are everywhere dense on S (h). Hence, d 0 = 0 on such manifolds S (h). Therefore 0 is constant on their connected comp onents. For other values of h the conclusion of prop osition 1 follows by continuity.
Remark. The proof shows that in condition A instead of "for almost all admissible values of h R n " we can say "for everywhere dense set of values of h Rn ". However, such weakening of condition A practically does not give anything new.


In further analysis we will use the prop osition on everywhere density of the set of maximum resonance tori (when all frequencies are rationally expressed through one frequency). This condition together with condition A pro duces "an alternation" of resonance and nonresonance invariant tori and replaces the condition of nondegeneracy of non-p erturb ed completely integrable system in the Poincar´ theory. e Energy surfaces. Let us consider critical values a1 < a2 < . . . < ar , and a1 the total energy h0 passes through the (h0 ) on h0 is lost. In that moment its a case, when function the H 0 : M R has a finite numb er of = min H0 . Such situation is common in applications. When critical value, the continuous dep endence of energy surface top ology generally changes.

Fig. 1

In fig. 1 we present the plot of Hamiltonian H 0 with three critical values. The p oints a 1 and a3 are stationary values of H0 , and the critical p oint a2 is not a stationary value. The presence of
4 REGULAR AND CHAOTIC DYNAMICS, V. 7,
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ON JUSTIFICATION OF GIBBS DISTRIBUTION

the nonstationary critical p oints is the characteristic prop erty of p otentials describing gravitational or Coulomb interaction. Let's denote as Ki1 i2 ... in an op en parallelepip ed in Rn = {h1 , . . . , hn }. This parallelepip ed is a direct pro duct of the intervals a i1 < h 1 < a If numb er are shown manifold, S (h) do es is + 1 is larger than r , then for n = 2 and r = 3. Each p which may consist of several not dep end on a p oint h K
i1 +1

, ..., a

i

n

< hn < a

in +1

.

(10)

we replace a is +1 with a symb ol . In fig. 2 these domains oint h K i , i = (i1 , . . . , in ) corresp onds to a smo oth regular connected parts. The quantity of connected comp onents of i ; we denote this numb er by symb ol i .

Fig. 2


Let us intro duce in the phase space ai1 < H1 (p
(1)

op en areas
i1 +1

i

defined by the inequalities similar to (4.1):
(n)

,q

(1)

)
, ..., a

i

n

< Hn (p

,q

(n)

)

in +1

.
i


It is clear that the closure of these domains in the whole covers all i connected comp onents.


. Also each

has exactly

such that the fol lowing equality is fulfil led in this domain
0


Prop osition 2. For any connected component of domain ferentiable function fi : Ki R, = fi (H1 , H2 , . . . , Hn ).

i

there exists the continuously dif-

(11)

Remark. Actually function fi belongs to the class of smoothness C 2 . However it is not essential for the further analysis.

Proof. It is clear that the domain i is foliated to the regular surfaces S . The function 0 is constant on these surfaces (more exactly on their connected comp onents) (the prop osition 1). Hence, on any connected comp onent i the function 0 has natural representation (4.2). By definition in any p oint i functions H1 , . . . , Hn are indep endent. Therefore we can intro duce lo cally new co ordinates so that H1 , . . . , Hn will app ear as n of new variables. The transition to such co ordinates is carried out with certainty with the help of continuously differentiable reversible transformation. In new variables the function 0 is continuously differentiable and dep ends only on n variables H 1 , . . . , Hn . The prop osition is proved.
REGULAR AND CHAOTIC DYNAMICS, V. 7,
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1, 2002











5


V. V. KOZLOV

Resonances. Let us consider again an invariant torus T of non-p erturb ed system. The equations of motion on the torus are reduced to form (3.4). This torus we call completely resonance if there exist n - 1 linearly indep endent integer vectors u = (u1 , . . . , un ), such that (u, ) = u1 1 + . . . + un n = 0, (u, ) = 0, . . . , (w, ) = 0. (12) In other words all frequencies s are rationally expressed through the prop osition that all solutions of differential equations (3.4) on same p erio d. Let b e a restriction of p erturbing function : M R on that is 2 -p erio dic function of (1 , . . . , n ) = . We asso ciate =


v = (v1 , . . . , vn ), . . . ,

w = (w1 , . . . , wn ),

one of them. This is equivalent to the torus T n are p erio dic with the the invariant torus T n . It is clear it with multiple Fourier series (13)

k exp i(k , ).
k
n

A nonresonance torus we call the Poincar´ torus if the factors of Fourier decomp osition (5.2) with e numb ers u, v , . . . , w are nonzero. Since the Poincar´ tori consist of the separate closed tra jectories, e they have no "rigidity" prop erty and collapse after the addition of p erturbation. We can not exclude the p ossibility that the family of degenerate p erio dic tra jectories, comp onents of the Poincar´ torus, at e p erturbation give rise to the finite numb er of nondegenerate p erio dic tra jectories with close transition. Let us intro duce finally the Poincar´ set P R n . It is a set of p oints from Rn = {h1 , . . . , hn }, e which are the images of the Poincar´ tori under the "energy" mapping e R n : a p oint with co ordinates (P , Q) = (p(1) , . . . , p(n) , q (n) , . . . , q (n) ) passes to a p oint with co ordinates h1 = H1 (p
¡

(1)

,q

(1)

), . . . ,

Remark. In the Poincar´ theory [11, 12] the usually supposition is that the non-perturbed system with e the Hamiltonian 0 is completely integrable and nondegenerate. Therefore the set of its first integrals is a set of action variables "numerating" the invariant tori. The Poincar´ set (introduced in [12]) is defined here e as a set of points in space of action variables corresponding to the completely resonance tori collapsing after the addition of perturbation. In our case functions H1 , . . . , Hn are the set of integrals of the non-perturbed problem and the Poincar´ set is a set of points in space of values of these integrals. e

Prop osition 3. In points of the Poincar´ set P functions e
n s=1


0

=

and

are dependent. Proof. Let { } b e a Poisson bracket connected with symplectic structure on . Since function is the first integral of initial system (2.1), then { , } = 0 for all values of . Since { 0 , 0 } = 0, then {,} lim = 0. (14) 0 Using decomp osition (1.2) and (2.2) we receive from (5.3) the equality


0

1

0

¡

6

REGULAR AND CHAOTIC DYNAMICS, V. 7,





{

,

}={



hn = Hn (p

(n)

,q

(n)

).

0

= f (H1 , . . . , Hn )



,

1

}.
1, 2002

(15)


ON JUSTIFICATION OF GIBBS DISTRIBUTION

n 0


1

s=1

Now let us restrict equality (5.4) on the invariant torus T n . It is clear that { 0 , 1 } is equal to the derivative of restriction of function 1 on Tn by virtue of system of differential equations (3.4). Let : Tn R b e a restriction of 1 on Tn and =


k exp i(k , )
k
n

its Fourier series. In p oints Zn the left part of (5.4) b ecomes i(k , )k exp i(k , ).


k

n

In view of formulas (5.2) and (5.5) the right part of relation (5.4) in p oints of the invariant torus is equal to f f i k + ... + k k exp i(k , ). (19) H1 1 1 Hn n n n Comparing (5.7) and (5.8) we obtain a chain of equalities (k , )k = ( k1 1 + . . . + n kn n )k , k Zn . (20)


k

Now let Tn b e the Poincar´ torus. Setting k equal to u, v , . . . , w we obtain that as a vector e of n-dimensional space is orthogonal to the hyp erplane generated by linearly indep endent vectors u, v , . . . , w. Since (u, ) = . . . = (w, ) = 0, and u = 0, . . . , w = 0, then we obtain from (5.9) n - 1 linear relations: u1 (1 1 ) + . . . + un (n n ) = 0, ............................... w1 (1 1 ) + . . . + wn (n n ) = 0. Hence the vector with comp onents 1 1 , . . . , n n is orthogonal to the hyp erplane and consequently it is collinear to the vector = (1 , . . . , n ). Hence 1 = . . . = n . Thus in p oints of the Poincar´ set the derivatives e


f f , ..., are equal to each other. It H1 Hn
0

means obviously the dep endence of functions

0

and

.

Remark. Since functions and are assumed to be only once continuously differentiable, then the Fourier series (5.2), (5.6)-(5.8) may diverge. In this case relations (5.9) can be deduced in another way. For this purpose we multiply the derivative with respect to time = +...+ 1 n on exp i(k , ), apply the operation of averaging over T 1 (2 )
n T
n

n

n

( · ) 1 . . . dn ,

and integrate by parts. In result we obtain the left part of relation (5.9) up to the factor -i. The right part of (5.9) is obtained similarly.

Let us intro duce the additional condition B) The Poincar´ set P is everywhere dense in any parallelepip ed K i . e The prop osition 3 imply the following corollary
REGULAR AND CHAOTIC DYNAMICS, V. 7,
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1, 2002







{

,

}=







0








Well known that { with Hamiltonian obtain

, 1 } is a derivative of 0 . The Poisson bracket {

1

by virtue of the initial system of differential equations 0, 1 } has similar sense. Besides by formula (4.2) we f {Hs , Hs
1

}.

(16)

(17)

(18)

7


V. V. KOZLOV

Hence we immediately obtain the following representation on any parallelepip ed K i :
0


= Fi (

0

),

n-1

0


n-1

Deduction of Gibbs distribution. Now let tend to zero. At the limit we obtain the n unjoined subsystems moving indep endently: the change of the initial data p (l) , q (l) for l = s do es not affect the dynamics of subsystem with numb er s. In order to b e consistent we shall also assume at = 0 that the subsystem with numb er s b eing in some fixed state (p (s) , q (s) ) M is a random event. The following natural condition plays the main role in the pro cess of deduction of Gibbs distribution C) These random events are indep endent. If s (p(s) , q (s) ), 1 s n is a density of probability distribution of the subsystem with numb er s, and (P , Q, ) = 0 (P , Q) + O () is a density of probability distribution in the initial system with Hamiltonian (1.2), then using condition C and the rules of multiplication of probabilities of indep endent events we obtain as 0 the following equality: 0 = 1 . . . n . (22) Equality (6.1) is also called the Gibbs hyp othesis on the preservation of thermo dynamic equilibrium of subsystems at vanishing interaction ([1], see also [13]). The sense of this term will b e explained b elow. Our main result is the following theorem Theorem. Suppose conditions A, B and C are fulfil led. Then s = c s e for al l 1 s n.
Hs k - Hs k

In particular, according to (6.1), 0 = c 0 e
-

From (6.1) we see that all separate subsystems have the same distribution. We compute the factors c using the normalizing condition s dn p dn q = 1.
M

Proof of theorem. First note that s is a function of Hamiltonian Hs only, and it is continuously differentiable in all op en intervals (a1 , a2 ),
8

¡



Since functions 0 and on H1 , . . . , Hn-1 .


0

are dep endent, then the right part of this equality actually do es not dep end

,

,

(a2 , a3 ), . . . ,

REGULAR AND CHAOTIC DYNAMICS, V. 7,



= fi (H1 , . . . , H

,

0

- H1 - . . . - H

cs = const > 0

c0 = c1 . . . c n .
s



where Fi is some continuously differentiable function. Indeed, intro ducing the new variables H 1 , . . . , H write down formula (4.2) in the another form:

,

0

=

Hs instead of H1 , . . . , Hn we can ).

n-1

(ar , ).
1, 2002





Corollary. If conditions A and B are fulfil led, then the functions dependent.

0

and

0

are everywhere

(21)

(23)

(24)


ON JUSTIFICATION OF GIBBS DISTRIBUTION

More exactly the numb er of such functions is equal to the numb er of the connected comp onents of level surface (hs ), when the energy hs changes in each of intervals (6.3). Some of these functions may coincide. Indeed, for almost all hs a1 the Hamiltonian system with Hamiltonian H s has everywhere dense set of nondegenerate p erio dic tra jectories on energy surfaces (hs ). Otherwise condition A is not fulfilled b ecause of the identity of separate subsystems. Then, according to Poincar´ [11], the p oints e of nondegenerate p erio dic tra jectories are stationary for the restriction of any first integral on (hs ). The continuity imply that the first integrals of Hamiltonian system with Hamiltonian function H s are constant on the connected comp onents of (hs ). At last we should note that s is the first integral and use the (simplified) arguments of section 4. Now let us consider again equation (6.1) true on any parallelepip ed K i : 0 (H1 + . . . + Hn ) = 1 (H1 ) . . . n (Hn ). (25)
n

This functional equation is easily solved. We differentiate (6.4) sequentially with resp ect to H 1 , . . . , H and divide the result on the pro duct 1 . . . n . In result we obtain the following chain of equations 1 n 1 = . . . = n = - . Here is some constant indep endent on the numb er s. Hence s = c s e
- H
s

,

cs = const.

(26)

The dimension of constant is equal to the inverse energy dimension. Usually one supp ose that = (k )-1 , where is the absolute temp erature, and k is the Boltzmann constant. We should note that formula (6.5) may dep end on the choice of multiindex i = (i 1 , . . . , in ). More precisely, any connected comp onent of the set i has its own set of the factors and c s in (6.5). However, we can easily show that the constant has an universal character. Indeed, supp ose the constants in formula (6.5) are equal to the values 1 , . . . , n on some connected comp onent of domain i with some index i. Then, according to (6.1), in this domain 0 = c 0 e


s H

s

.


(27)

If some s are not equal to each other, then functions (6.6) and 0 = (Hs ) are indep endent. But this statement contradicts to the corollary of prop osition 3. This argument has the evident physical meaning: at thermo dynamic equilibrium all comp onents of system have identical temp erature. The last remaining p ossibility is that constants c s in formula (6.5) are different on the different intervals of values of energy (6.3). But in reality this p ossibility is not realized b ecause of the continuity prop erty of functions s : M R. The theorem is completely proved. In conclusion of the work we shall make one imp ortant remark. The Gibbs theory presented in [1] do es not imply that the densities of probability distributions 1 , . . . , n should b e continuous functions on M . We can consider more general situation and assume, for example, that functions s are continuously differentiable only on those domains of phase space M , in which energy is contained b etween its neighb oring critical values ar < H s < a
r +1

(r = 1, . . . , p;

a

p+1

= ).

(28)

Naturalness of such assumption is already evident if we consider the example of mathematical p endulum: the separatrices on phase cylinder separate the domains with essentially different typ e of motion (fluctuations and rotations).
REGULAR AND CHAOTIC DYNAMICS, V. 7,
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9


V. V. KOZLOV

Applying the develop ed ab ove metho d we again obtain formula (6.2), but the constants c s have different values in different domains (6.7). Moreover, these constants may b e different for different connected comp onents of domains (6.7). It is quite p ossible that such generalized discontinuous Gibbs distribution could b e useful for the study of concrete thermo dynamic systems. Let us assume, for example, that the phase space M has only two domains M + and M- of form (6.7). In domains M± we have the following densities of distributions c± e We calculate the constants c
+ - H k

.

and c- using the normalizing condition e
+

c

-

+ M

H k

dn p d n q + c

- M
-

e

-

H k

dn p dn q = 1.

If the difference = c+ - c- is given, then factors c c
+ M

±

are uniquely defined by this equality: e
-

e

-

H k

dn p d n q = 1 +
M

-

H k

dn p d n q ,

c

- M

e

H - k

dn p d n q = 1 -
M
+

e

-

H k

dn p d n q .

Since c± > 0, then the right parts of these equalities are p ositive. It happ ens with certainty if the jump satisfies the following inequality || <
M

e

-

H k

-1

d pd q

n

n

.

Could we obtain the probability distribution with piecewise smo oth function of distribution using the Fowler ­ Darwin metho d? The pap er is prepared with the financial supp ort of RFBR(99-01-0196) and INTAS.

References
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