Poisson Analysis And Statistical Mechanics

The report presents new Poisson integral representations of grand canonical Gibbs ensemble correlation functions for equilibrium and nonequilibrium continuous systems of classical statistical mechanics. These representations give an opportunity to apply technique, which is well developed for lattice systems. Some new results are announced.


Introduction
In this short report I am going to give some new examples of application of Poisson analysis in physics.There are many well-known examples of application of Gaussian functional integrals for treating models of statistical mechanics.These are the the works by D. C. Brydges and P. Federbush 5], T. Z. Imbrie 2], A. I. Pilyavsky and A. L. Rebenko 3] in which Sine-Gordon transformation was used for Gaussian representation of correlation functions and the work of I. R. Yukhnovskii, M. F. Holovko and their collaborators (see 21]) in which the method of collective variables was applied.
But these methods work only in the case of two-body interaction potential V (x ?y), which satis es the following assumptions: 1.Positive de nition: (f; V f) = Z f(x)V (x ?y)f(y)dxdy 0; 2.Regularity condition: V (0) < 1.
As to assumption 1., a wide class of interactions (including Yukawa and Coulomb interaction with hard-core), does satisfy 1., but the second assumption 2. is not accepted from physical point of view as the repulsion forces of two particles should increase to in nity when they collide, i.e.V (0) = +1.
In the paper 15] a new representation for distribution functions of classical statistical mechanics was proposed.It was based on Poisson measure integral representation.This new form of integral representation for distribution functions gives an opportunity to simplify considerably the construction of cluster expansion and proof of its convergence.In the later work of 16] this method was extended to wide class of interaction potentials, which satisfy only stability condition: 3. U N (x 1 ; :::; x N ) ?NB, and some integrability property.
The main goal of this talk is a short description of this method and announcing some new results, which can be obtained using Poisson measure representation.
A short contents of this report is following.In Section 2 we brie y state some notions and formulas of the Poisson analysis, needed for later exposition.In Sections 3 and 4 we obtain representations of correlation functions of classical systems for equilibrium and nonequilibrium cases respectively.And nally in the conclusion we enumerate some new results and discuss some perspectives.

Some remarks on the Poisson analysis
Detailed exposition of di erent aspects of the Poisson analysis may be found in 6{11 ,18].In this section we are going to remind only some of the most useful de nitions and formulas which will be used later.
Remark Of course, formula ( 1) is correct even in case when = < 3 but we are going to apply this formalism to the problems of equilibrium statistical mechanics and therefore will start from the nite set embedded into < 3 .Notation 1 Let L 2 = L 2 (S 0 ; dP z ) be the space of functions square inte- grable with respect to the measure dP z .
For proof see Theorem 3 in 7].
Notation 2 Let x = ( ?x).Lemma 2 For any F q] 2 L 2 the following formula is true: Z S 0 dP z (q)F q] = e ?zj j ( Proof.Let us expand the exponent on the right-hand side of (1) in series, taking into account that j j < 1: Z S 0 dP z (q)e i<';q> = e ?zj j By Lemma 1 the latter formula may be extended to any F(q) 2 L 2 . 2 Corollary The set 8 < : n X j=1 (x ?x j ) 9 = ; with Poisson distribution law for x 1 ; : : : ; x n 2 (x i 6 = x j if i 6 = j) is carrie set for the measure dP z ( ).See 16] for more details.

The Wick theorems
The formulas (10) express the Wick polynomials in terms of usual polynomials.If we inverse them we obtain 17] the Poisson analog of well-known in the Quantum eld theory and Gaussian analysis Wick theorem 1] which gives the rule of expressing the usual polynomials in terms of the Wick polynomials.To formulate it we need rst the following de nition: De nition 4 Let the `pairing' of the n Poisson elds be the following: < ' 1 ; j q >< ' 2 ; j q > : : : < ' k ; j q >=< ' 1 ' 2 : : : ' k ; q > : (12) Theorem 1 The usual product of the Poisson elds is equal to the sum of all corresponding normal products with all possible `pairings' including the normal product without any `pairing'.The proof of Theorem 1 uses generalized Wiener-Itô-Segal isomorphism 8] under which the operator of multiplication by the q(x) goes over into the operator q(x) = (a + (x) + 1) (a ?(x) + 1) on the Fock space F = F (L 2 (< 3 )), where a + (x) and a ?(x) are usual creation and annihilation operators: a ?(x); a + (x 0 )] = (x ?x 0 ): The normal ordering of the product of such operators in the Fock space means their usual product in which all creation operators are placed on the left and annihilation operators are placed on the right.Therefore, after applying Wiener-Itô-Segal isomorphism the proof of Theorem 1 merely turns into a combinatoric exercise (see 17] and 12] for details).
In the same way it may be proved the following generalized Wick theorem which in connection with formula (11) appears to be very useful for calculation of some averages by the measure dP z .
Poisson analysis and statistical mechanics 123 Theorem 2 The product of normally ordered products of the Poisson elds is equal to the sum of all corresponding normal products with all possible `pairings' which connect the Poisson elds from initially di erent normal products including the normal product without any `pairing'.