LIMITING R.

A.

GIBBS'

DISTRIBUTION

Minlos

INTRODUCTION In statistical physics the Gibbs distribution is given initially for s y s t e m s enclosed in a finite v e s s e l ~, and then the asymptotic p r o p e r t i e s of this distribution a r e investigated as I~}-- ~, where [~[ is the volume of the vessel. The question naturally a r i s e s as to the existence of a limiting Gibbs distribution, i.e., such a probability distribution on a manifold of states of an "infinite s y s t e m , " that it will be, in a definite sense, the limit of the Gibbs distribution as [~l-- ~ for finite v e s s e l s ~, and its p r o p e r t i e s will r e f l e c t the asymptotic p r o p e r t i e s of these d i s t r i b u t i o n s . In this, and subsequent papers, we show that such a limit actually exists in a large canonical ensemble of identical and pairwise interacting particles with definite values of the p a r a m e t e r s / 3 and z p r e scribing this distribution, and it p o s s e s s e s a number of r e m a r k a b l e p r o p e r t i e s (ergodicity, s t r o n g mixing). In this first paper, we complete only the f i r s t portion of this p r o g r a m , we establish the existing of the limiting m e a s u r e . We shall study its p r o p e r t i e s in the next paper. § 1.

Definition

of the

Limit

Space

and

Formulation

of the

Problem

1. S p a c e o f S t a t e s . Let us c o n s i d e r a s y s t e m with a variable number of indistinct p a r t i c l e s enclosed in a bounded domain (vessel) 12 C R v. By definition, any finite subset f~: c = {xi}, xi ~ ~ (i = 1, . . . . N) or points of the v e s s e l is the state of such a system. The number N = N(c) is called the number of p a r t i c l e s in the state c. We denote the set of all states of our s y s t e m by C ~. The set of states with fixed number N of particles will be denoted by C ~ , N ( C ~ , 0 = {@}). Evidently Co = U C~.x. .','>.

(i)

If ~N ( C RuN) denotes the uN-dimensional domain which is an N-tuple C a r t e s i a n product of domains ~, and ~N ( C fiN) is a subdomain in ~N consisting of sets {xt . . . . , XN} (xi E ~) with pairwise distinct e l e ments ( x i ~ x j , i ,e j), then

Cmv = ~IV/SN,

(2)

where SN is a group of commutations of N elements acting in ~N according to the formula c II {xI, . . . . XN} = {xii . . . . . XiN} (II = (i t . . . . . iN) ~ SN). The Lebesgue m e a s u r e : t

~ca.v (A) = 7., ~R,s' (A), can be introduced in C ~,N by using the r e p r e s e n t a t i o n (2), where the m e a s u r a b l e manifold ~ ~N is the prototype of the manifold A C t~,N with the factorization (2). Since a p N-dimensional Lebesgue m e a s u r e is defined in each of the C~,N, the space Cf~ is provided with a m e a s u r e X(.), in a natural m a n n e r by utilization of the decomposition (1), which we shall henceforth also designate as the Lebesgue m e a s u r e (X(q~) = X(Ca,~ = 1). The integral, in this m e a s u r e of m e a s u r a b l e functions in C~ is also defined in a natural way. R e m a r k . Let U be any m e a s u r a b l e submanifold in R v. As above, we can c o n s t r u c t a space CU of all finite submanifolds U provided with a Lebesgue m e a s u r e . Let us note that if U1 C Uz, then the mapping ~rU~, Ut: CU2---Cut is defined for which the submanifold ~U2,Ui(C) = c M U 1 E CUi c o r r e s p o n d s to each submanifold c E CU 2

Cu~ ~"u'-,Cu, \

/

(3)

~u,.u,'~ Cu / ~.,,.u, * While this w o r k was being p r e p a r e d for p r e s s , a p r e p r i n t of a w o r k by D. Ruelle appeared in which r e s u l t s analogous to ours w e r e obtained. Moscow State University. T r a n s l a t e d f r o m Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 1, No. 2, pp. 60-73, M a r c h - A p r i l , 1967. Original article submitted D e c e m b e r 20, 1966.

140

is c o m m u t a t i v e . In p a r t i c u l a r , the mapping CR v~rRv,U CU is defined for any u . 2. Hamilton Function and Gibbs Distribution. In the manifold C ~21et the Hamilton function H(c) be defined which is, a c c o r d i n g to the c u s t o m a r y r e p r e s e n t a t i o n

~,

u ( I x~ -- xi I ) whe~ ~¢ (c) >~ 2,

xiEc

0

when N ( c ) = O, 1.

P r o p e r t i e s of the function U(r), designate the b i n a r y i n t e r a c t i o n potential, will be d i s c u s s e d below. M o r e o v e r , the following p r o b a b i l i t y distribution in the manifold C ~2 is introduced in s t a t i s t i c a l p h y s i c s . Its density relative to Lebesgue m e a s u r e in C ~ equals p (c) = t Zx(~)e_~H(~), .=

(4)

w h e r e fl > 0 and z > 0 a r e p a r a m e t e r s d e s c r i b i n g the distribution, and E is a n o r m a l i z i n g f a c t o r , called the m a j o r s t a t i s t i c a l s u m . Cp.

N>0

C~ ,.V

H e r e , HN(c) denotes the contraction of the function H(c) in the manifold C~2,N. Under those a s s u m p t i o n s r e l a t i v e to the potential U(r) which will be mentioned below, the i n t e g r a l s in each m e m b e r of the s e r i e s (5) c o n v e r g e for any fl > 0; the s e r i e s i t s e l f c o n v e r g e s for any z > 0. The distribution (4) is called the Gibbs distribution, and the s y s t e m of indistinct p a r t i c l e s t o g e t h e r with its distribution is the m a j o r canonical ensemble. 3. Limit Space. We shall be i n t e r e s t e d in the question of whether the Gibbs distribution h a s a l i m i t when the domain ~ is expanded without limit. We f i r s t c o n s t r u c t the limit s p a c e in which, as it t u r n s out, this limiting distribution is c o n c e n t r a t e d . Namely, let us c o n s i d e r a s p a c e Coo c o n s i s t i n g of all finite o r d e n u m e r a b l e submanifolds of the s p a c e R v. Only such d e n u m e r a b l e submanifolds a r e hence admitted which have only a finite n u m b e r of points in any bounded p a r t of the space, Let us note that CRv ~_ C= and a l s o CU~_ C= for any U. F u r t h e r m o r e , for any m e a s u r a b l e bounded s e t U C R v let us define the m a p p i n g ~ t s : Coo ~ C U which r e f e r s the p a r t in U to each submanifold c ~ Coo . Let us note that if U 1 ~_ U2, then the d i a g r a m

C~ \

~_u,-, Cu, / C~,

(6)

is c o m m u t a t i v e . F u r t h e r m o r e , we designate the manifold S C Coo cylindrical, if a m e a s u r a b l e bounded manifold U ~ R v and a m e a s u r a b l e manifold A C: CU a r e found such that S ( - SU,A) is a c o m p l e t e prototype of the manifold A for the mapping ~'U: SU,A = UrU)-I(A). We m a y analogously define c y l i n d r i c a l manifolds f r o m CRv by utilizing the mapping ~-RvU (U a m a n ifold): S~RA ~ = (lrltV,u)-t(A). Let us note that the set of c y l i n d r i c a l manifolds f r o m Coo g e n e r a t e s a finitely-additive a l g e b r a of manifolds (but not a cr-algebra!). Analogously, cylindrical manifo! Js f r o m CRY g e n e r a t e a finitely-additive a l g e b r a of manifolds. A s e t of c y l i n d r i c a l manifolds [SU,A} , w h e r e U is fixed and A ~ CU is a r b i t r a r y , g e n e r a t e s a or-algebra ~ u • Let us note that for a natural imbedding of the s p a c e CU in Coo (or in CRY) each m e a s u r a b l e manifold A c C U is contained in SU,A. We denote the m i n i m a l or-algebra of manifolds f r o m Coo g e n e r a t e d by cylindrical manifolds, by ~oo I evidently, ~ , C ~oo.

141

4. Properties of the Potential U(r). We s h a l l c o n s i d e r p o t e n t i a l s w h i c h d e c r e a s e s u f ficiently rapidly at infinity, and grow rapidly at zero. More accurately, we demand compliance with the

following conditions: I. T h e p o t e n t i a l U(r) p o s s e s s e s t h e p r o p e r t y of s t a b i l i t y :

~, U(Ix,--xjI)>--BN,

(7)

xi ~ x I

w h e r e B > 0 i s a n a b s o l u t e c o n s t a n t i n d e p e n d e n t o f the p o i n t s (x I . . . . .

xN) a n d of the n u m b e r N~

]1. T h e r e e x i s t s a f u n c t i o n r--*.o t e n d i n g m o n o t o n e l y to z e r o a s v(r) (-= v(r,/3)) s u c h t h a t

a)

f

] e-Z~"~l'"-- 1 Idx < ~ (r),

ixl>r

b) t h e r e e x i s t s a c o n s t a n t B 0 ( - B0(fl) s u c h t h a t d

I l e-~U¢) - - 11 v(d - - r) rv-I dr ~ Boy (d). o

I t c a n b e v e r i f i e d t h a t t h e s e c o n d i t i o n s w i l l b e s a t i s f i e d in the f o l l o w i n g two c a s e s : A. U(r) i s a f i n i t e function, w i t h a l o w e r bound a n d a s o l i d c o r e : U(r) = 0 f o r r > R and U(r) = co f o r r < ~ (in t h i s c a s e an e x p o n e n t i a l c a n b e t a k e n a s t h e function v(r). B. T h e f u n c t i o n U(r) g r o w s m o r e r a p i d l y a t z e r o than A r - ( ~ + e) a n d d e c r e a s e s m o r e r a p i d l y at i n f i n i t y t h a n B r - ( v + e ) . In t h i s c a s e the f u n c t i o n v(r) (with r > 1) c a n b e t a k e n a s the f u n c t i o n B~r - e . F o r t h e s e two c l a s s e s of p o t e n t i a l s c o n d i t i o n II i s v e r i f i e d b y d i r e c t c o m p u t a t i o n . C o n d i t i o n I i s e v i d e n t f o r c l a s s A . F o r c l a s s B i t h a s a l s o b e e n e s t a b l i s h e d ( s e e R. L. D o b r u s h i n [6] o r D. R u e l l e [5], say). 5. Construction of the Limiting Gibbs Measure i n C¢o. If it i s c o n v e n i e n t , the G i b b s d i s t r i b u t i o n in the e n s e m b l e ( C ~, H, fl, z) d e f i n e d in m e a s u r a b l e m a n i f o l d s A C C fh c a n be c o n s i d e r e d g i v e n on c y l i n d r i c a l m a n i f o l d s f r o m ~ fb if i t i s a s s u m e d t h a t P (Sn.d) = Pc n (A)

(8)

S i n c e U C ~ SU,A = S~,Tr-~,u(A) 6 ~ ~2, t h e p r o b a b i l i t i e s (8) a r e d e f i n e d in a l l m a n i f o l d s SU,A(U C ~2). L e t u s now c o n s i d e r a n y s e q u e n c e o f v e s s e l s ~21, S22, . . . . ~k, . . . . w h i c h t e n d to R v. T h i s l a t t e r m e a n s t h a t f o r a n y b o u n d e d m a n i f o l d U C R v t h e r e i s found a k(U) s u c h t h a t U C ~Jk f o r a l l k > k(U). A s e q u e n c e of p r o b a b i l i t i e s SU,A i s d e f i n e d f o r a n y c y l i n d r i c a l m a n i f o l d {Pk(SU,A)}, k > k(U). us now a s s u m e t h a t t h e r e e x i s t s a l i m i t

lim P~ (Su.A) = P(Su.A)

k--~v

Let

(9)

f o r e a c h c y l i n d r i c a l m a n i f o l d SU,A. I t i s e a s y to v e r i f y t h a t P(SU,A) i s i n d e p e n d e n t o f the c h o i c e of the p a i r (U,A) d e f i n i n g t h i s c y l i n d r i c a l m a n i f o l d , i . e . , if SU1,A 1 = SU2,A~, t h e n the l i m i t i s t h e s a m e :

P (Su,,A,) = P (Su~,A,) It i s a l s o e a s y to v e r i f y t h a t the p r o b a b i l i t i e s P(SU,A) f o r m a f i n i t e f i e l d o f p r o b a b i l i t i e s i n t h e a l g e b r a of c y l i n d r i c a l m a n i f o l d s . We a s k , c a n t h i s f i n i t e l y - a d d i t i v e m e a s u r e d e f i n e d on c y l i n d r i c a l m a n i f o l d s b e c o n t i n u e d to a c o m p l e t e l y a d d i t i v e m e a s u r e d e f i n e d in t h e q - a l g e b r a ~¢~ ? T h e a n s w e r i s g i v e n b y the f o l l o w ing theorem. T h e o r e m on C o n t i n u a t i o n . In t h e a l g e b r a o f c y l i n d r i c a l m a n i f o l d s l e t be d e f i n e d a f i n i t e l y - a d d i t i v e p r o b a b i l i s t i c m e a s u r e P(SU,A) s u c h t h a t i t s bound in a n y s u b a l g e b r a ~ u C ~ generates a.completely a d d i t i v e m e a s u r e . T h e n t h i s m e a s u r e m a y b e c o n t i n u e d to a c o m p l e t e l y a d d i t i v e m e a s u r e in t h e q - a l g e b r a ~ and uniquely besides.

142

To a known degree, this t h e o r e m is analogous to a well-known t h e o r e m of A. N. K o l m o g o r o v [1] on the continuation of consistent finite dimensional probability distributions to a d e n u m e r a b l y - a d d i t i v e m e a s u r e in functional s p a c e . The m o s t e s s e n t i a l point in the p r o o f is the v e r i f i c a t i o n that for a d e c r e a s i n g sequence of cylindrical manifolds

SU,,A, ~ Su,,A, ~ SU,,A, ~ . . . . having an e m p t y i n t e r s e c t i o n

(I0)

Su/.A~, = ~, the equality lira P (Su~.A~) = O. ~-,~o

(11)

is satisfied. As soon as the relation (11) is s a t i s f i e d , the continuation of the m e a s u r e is a c c o m p i i s h e d by a s t a n d a r d p r o c e d u r e , going back to Lebesgue and d e s c r i b e d in the books of A. N. K o l m o g o r o v [1], S. Saks [3], or Dunford and Schwartz [3], say. Let us prove the existence of the l i m i t (11). Let us a s s u m e the opposite, i.e., that lim P (Su~,.dk) = a ~ O, /e-,~o

and let us hence, deduce that N SUk, Ak ~ ~b. Without limiting the generality, we can consider the sequence of manifolds Uk to grow, i.e., U 1 ~_ U z ~-- U s . . . . and to be absorbing, i.e., any bounded manifold V is contained e n t i r e l y in Un s t a r t i n g with s o m e n(-V). F u r t h e r m o r e , for e v e r y two Ui and Uj (i > j) we define the manifold A~:) = ~xu~,ui(Ai ) ~ A / ~ A~/).

(12)

(i') li) Since P(SUj,A~i)) ->P(Sui, Ai) -> o~and besides, Aj ~ A (i' > i), we obtain a sequence of imbedded ~ui manifolds S. ( i ) ~ S u ( i + ~ ) ~ S g ,~(/+2)~ . . . Ul, A] ~ I'A] 1,-/ -

-

-

-

with the nonempty i n t e r s e c t i o n 0 Sup.,::+'~ = S % gj,

(13)

w h e r e Bj = N A(: + s). Let us note that by virtue of (12) and (13) the manifolds Bj p o s s e s s the following p r o p e r t y B: = au:+~.u/(B:÷,)

(] = I, 2 .... ).

T h e r e f o r e , if we s e l e c t s o m e e l e m e n t c 1 ~ B 1 ~ CUt (c 1 C UI), then an e l e m e n t B 2 e x i s t s in B 2 which will i n t e r s e c t the manifold U 1 at cl; let us designate it the continuation c I. F u r t h e r m o r e , a continuation c 3 of the e l e m e n t c 2 is found in B3, etc. We t h e r e f o r e , c o n s t r u c t the i n c r e a s i n g sequence clCQCc3C

...

of finite manifolds f r o m R v. Evidently, t h e i r union is an e l e m e n t of the s p a c e C~. Let us note that ~'Uj(C) = c j ~ Bj C Aj for any mappingrTuj. We t h e r e f o r e , obtain that c ~ SUj, A] (J = 1 , 2 , . . . ) and c E N SUj, Aj, despite the a s s u m p t i o n (10). The t h e o r e m is proved. J If the l i m i t (9) of Gibbs probabi!ities P(SU,A) e x i s t s for any cylindrical manifold SU,A, where the p r o b a b i l i t i e s P(Su,A) p r e s c r i b e a d c a u m e r a b l y additive m e a s u r e in ~ v , then the limiting probability distribution they generate in ~ o ,rill be called the limiting Gibbs distribution. The a i m of this p a p e r is to e s t a b l i s h that under sufficiently g e n e r a l a s s u m p t i o n s on the potential U(r) and f o r definite values of the p a r a m e t e r s fl and z the limitir, g Gibbs distribution e x i s t s and has a n u m b e r of fine p r o p e r t i e s .

143

R e m a r k . In passing we shall see that the limit of probabilities of cylindrical m~nifolds is independent of the choice of the sequence of v e s s e l s ilk. § 2.

Correlation

Functions

and

Probabilities

of Cylindrical

Manifolds

We e x p r e s s the probability of manifolds Pfl(SU,A), evaluated in the ensemble (C~, H, fl, z), in t e r m s of s o - c a l l e d c o r r e l a t i o n functions. The function rk(ctlQ,

~, z) =--r,(x x. . . . . xkl a, B,z) = ~" I zk+'(C')e-~n(#"") dc='

(14)

Co

where c t = {xi, i = 1, . . . , k } , i s c a l l e d a k - t h o r d e r c o r r e l a t i o n function i n the ensemble (C~, H, f l , z). Now l e t us eonsider some U C ~. The mapping C~ )%Q,U..)CU

generates, as before, a probability distribution in CU according to the formula

Pcu (A) = Pc,, (~,u (A)). We now e x p r e s s the density of the probability distribution in CU with r e s p e c t to Lebesgue m e a s u r e in CU in t e r m s of the c o r r e l a t i o n function rk (c [ ~2, 7r, z}. F o r this purpose, let us note that if the manifold ~2 is separated into a sum of two nonintersecting sub-manifolds, then the space ~ = U [J U', is r e p r e s e n t e d as a Cartesian product of the spaces CU and CU', Cf~ = CU ® CU,, and the Lebesgue m e a s u r e in C~ is the product of the Lebesgue m e a s u r e s in CU and CU'. T h e r e f o r e , for any A C CU

Pcu(A)=~dc,

I zN(~*)+N"e--~H(c*'~')dc,.

A

Cu •

The meaning of this formula is that the density of the probability distribution generated b y the Gibbs distribution in ~U(Ct [~2,fl, z} equals

#u(ql f~, 1~,z) = z~v(~')s I zN(#:)e-~m~"~')dc~' CUO

or, written out in detail rw(x, .....

z , i . " l e_~m........~'Y'"'"Y")dy,. "" x~ l a , ~, z) = -zk ~ y,oo n~n=o

U'

dye.

(15)

U~

If the c h a r a c t e r i s t i c function XU'(y) of the manifold M' is introduced, then (15) may be r e w r i t t e n as.

#u ( x .

..

-~'~ . . . . . . . . . k, v,....,v,,)_= y , . . . d y , .

., x~.lQ, ~, z) = ~ ~' ~ f~

f$

t

Finally, n

n

H xu. (y~) = II [zu. (y~) 1

1 + II = i + ~ (x~.. (y~) -

1

1

1) +

y, [xo. ( y D -

l] [xu, (y~,) -

11 + . . . .

kt=~k=

whence

y, {&....ap}

144

. . . . is

q

t,:

=o

c,)H (xu.(y,)- ,)dc,. ifx

F i n a l l y , noting that X U , ( y ) - 1 = - X u ( y ) , we obtain

(--1) p ! r.v(~o+o(c, c~)dc~.

.~u(qlf~, [3, z) p=o

T h e r e f o r e , f o r any A

CUd,'

D CU (16) k

s

Ak

CU,s

w h e r e Ak = A C CU,k. (Let us r e c a l l that CU,k is a s e t of s t a t e s with fixed n u m b e r k of p a r t i c l e s . ) p a r t i c u l a r , if A = CU,n, then pcc.(Cu~[~)=.

In

d c ( - - 1 ) ' i r"+s(cx'c:)dc~ = ~" ~ ( - - ) I .l-~ C ~n C ., rt(Q)dcz.

~ ,d

¢J

s CU,a

CU,s

,=n

Cu,i

Hence, in o r d e r to i n v e s t i g a t e the e x i s t e n c e o f l i m i t s f o r the p r o b a b i l i t i e s PCU (AI~) a s ~ -,co, it is s u f ficient to study the l i m i t i n g b e h a v i o r of the c o r r e l a t i o n functions rk(Xl, . . . . Xk [~, % z) a s ~ ~ ~. In the next s e c t i o n we shall e s t a b l i s h the e x i s t e n c e of l i m i t i n g c o r r e l a t i o n functions rk(x 1. . . . . oo, fl, z) as well as the r a t e of c o n v e r g e n c e of the c o r r e l a t i o n functions to the l i m i t i n g function. § 3.

Investigation

of Correlation

Xk [

Functions

As is s e e n f r o m the p r e c e d i n g s e c t i o n , in o r d e r to e s t a b l i s h the e x i s t e n c e of l i m i t i n g p r o b a b i l i t i e s f o r c y l i n d r i c a l manifolds, it is sufficient to e s t a b l i s h the e x i s t e n c e of l i m i t i n g c o r r e l a t i o n functions rk(c I ~, fl, z) ----rk(x 1. . . . . Xk [ ~, % z) and to e s t i m a t e the r a t e of c o n v e r g e n c e of the c o r r e l a t i o n functions f o r a finite d o m a i n g to t h e s e l i m i t i n g functions as ~ --*~. M o r e o v e r , we l a t e r need an i m p o r t a n t p r o p e r t y of c o r r e l a t i o n functions, the m u l t i p l i c a t i v i t y p r o p e r t y . We obtain all t h e s e r e s u l t s by u t i l i z i n g a s y s t e m of equations c o n n e c t i n g the c o r r e l a t i o n functions, the c o r r e l a t i o n equations. Ruelle [4] f i r s t applied this method, h o w e v e r , we need s o m e r e f i n e m e n t s of his r e s u l t s . As is e a s i l y p e r c e i v e d f r o m (14), the c o r r e l a t i o n functions rk(x 1. . . . . s a t i s f y the following s y s t e m of e q u a t i o n s : r,,.(cl-Q, ~, z) :.~ ze

o

Xk I~, fl, z) = rk(c l l2, fl, z)

(e ,c(,,, s;,I)__ 1)r/,,+,_a(c,Q)dc,

r,¢_l(cla, ~, z) -: ~1

1,

(17)

n = l C~,lz i ~ l

w h e r e ~ = (x 2. . . . .

xk). Hence, f o r k -- 1 we should put r 0 = 1.

T h i s s y s t e m of equations m a y be modified s o m e w h a t by u s i n g the s y m m e t r y o f rk(x 1. . . . . in its a r g u m e n t s . Let us utilize the s t a b i l i t y p r o p e r t y

Xk t~,fl, z)

k

U ( I x ~ - - xJI) > - Bk, i~e]=l

w h e r e B is an absolute c o n s t a n t . found s u c h that

It, h e n c e , follows that in any s e t (xl, . . . .

xk) at l e a s t one xi = x m i n is

~. U(Ix"i'~--x/l)>--B. Selecting such an x min for each set, we can define a measurable function R vk in x min = xmin(xI. . .xld. The s y s t e m of equations (17) may evidently be rewritten as follows by putting xI = xmin: --,; ~ U(I ~mi~ - x i I)

~,~(cIQ, ~, z) : ~e J~

~

Ir~-l(~JQ,~,z) + ~ n=l

w h e r e ff = (x 2. . . . .

n

~ r~+,,_,(~, q ) I [ (e-~''x'-~''~- OdcO, C~,~ i=l

xlO.

145

Now we e x a m i n e a r a t h e r m o r e g e n e r a l s y s t e m .

Let the function ~(x) be positive, and let !~ ~ (x)dx

< 0% Utilizing this function we define the p r o b a b i l i t y d i s t r i b u t i o n in the s p a c e CRv of finite s u b m a n i f o l d s of the s p a c e R v b y m e a n s of the f o r m u l a n

p (x~ . . . . .

--I~ ~, Ullxi--xjll

i=1

x. I t, ~, t0 =

w h e r e p(x 1. . . . , Xn [~, fl, g) = p(c [~, f l , / 4 is the d e n s i t y of this d i s t r i b u t i o n with r e s p e c t to the L e b e s g u e m e a s u r e in CRY, and

N~o

CRy, .V

xEc

It is s e e n f r o m the e s t i m a t e (18) that this s e r i e s will c o n v e r g e f o r any function ~(x) LI(RV). Let us note t h a t f o r ~(x) = X£(x) we a r r i v e at the c u s t o m a r y Gibbs d i s t r i b u t i o n in the e n s e m b l e (C£, H, fl, z). As b e f o r e , we m a y i n t r o d u c e c o r r e l a t i o n functions rk(cl[~, fl, #) = rk(x 1. . . . . Xk [~, fl, ~) by m e a n s of the formula k

zk 1-[ g (xi) f e--~H(qx:)z'V(c'41-[ g (x) .dc: i=l

r~ (x. . . . . .

CRy

x6c..

x~l ~. ~. , ) =

(19)

-- (~, 3, z)

w h e r e c I = {xi} k. T h e s e c o r r e l a t i o n functions a l s o s a t i s f y a s y s t e m of equations (x 1 = x min)

r~(cl~,~,~)=z~(xOe

f rk+"-'(c'ql~'~'z) l-[ (e-~U(m-YJ~)--l)dc'

rk-~(cl~,~,z) + y '

;>~'

n=t

w h e r e ~" = (x 2. . . . .

' (20)

YJEct

CR,,# ,n

Xk).

We show now that a unique solution e x i s t s f o r this equation f o r s u f f i c i e n t l y s m a l l z, and we e s t i m a t e it. To this end, we i n t r o d u c e a B a n a c h s p a c e ~ d , c o n s i s t i n g of infinite lines of functions q~ = {~l(x~), % ( x v x2) . . . . .

~.(x~ . . . . .

x.) . . . . } = {~(c)}, c E C a l ,

s u c h that tl

I ~.1 < c a " H ~(x~), 1

and we a s s u m e , a s usual, I1~ ~. The o p e r a t o r A a c t i n g a c c o r d i n g to the f o r m u l a (x 1 = x m i n , e i = (x i . . . . . x l d , ~' = (x2, . . . .

xn))

(AO).(x l, . . . , x.) = z[(xOe

i>.z

cry

(20')

ueq

(for n = 1, r 0 = 1) is, as we now show, a bounded o p e r a t o r in ~d f o r sufficiently s m a l l z.

In fact, let It~ U-< 1, i.e., n

I ~.(x, . . . . .

x.)l < a " [ [ ~ ( x i )

( n - - 1 , 2 . . . . );

1

then

( AqJ). (x,

"

x.)l< IzJa"-'e "B IIl (x,)l 1

146

{

+

[s RV

I(e-"""-'"- J ( )ley

]'}
"

BII 1

where C ([3, ~) = I I (e-aUto-u) - - 1)1 I ~ (Y) I dy. Rv

T h e r e f o r e , IIAl[d--- e/3B Izl d - l e C ( f l , ~ ) d If lzl is s u f f i c i e n t l y s m a l l , so that z l e 3 B i < m a x x e - C ( f l , 0 x , than a d m a y be s e l e c t e d such that IIA lid < 1. Now, let us note that (20) m a y be r e w r i t t e n a s P = AP q- A (~), w h e r e P is a line of c o r r e l a t i o n functions, and A(~) = (z~ (x), 0, 0 . . . . }. We hence obtain that a unique s o l u t i o n of Eq. (20) e x i s t s , and h e n c e I[P[Id < I z l / d ( 1 - I I A I[ d) = B0, i.e., ttk(xl .....

xkl~,

~,z)[(d'~ BotH ~(xi)l .

(21)

Since the line of c o r r e l a t i o n functions P = (rk(x 1. . . . . Xk I~, fl, z)} belongs to ~d , as is s e e n f r o m (19), the unique solution of (20) a g r e e s with this line, and we obtain the e s t i m a t e (21) f o r the c o r r e l a t i o n functions. Let us note that although the initial f o r m u l a (19) is m e a n i n g f u l only f o r ~?(x) E LI(RO and ~(x)> 0, the s p a c e R e , Eq. (20), and all o u r e s t i m a t e s r e m a i n valid f o r any bounded ~(x) (I ~(x) I< s f o r s u f f i c i e n t l y s m a l l s, w h e r e all the e s t i m a t e s a r e u n i f o r m in ~(x) f o r fixed s). We, hence, obtain that a solution of (20) e x i s t s f o r ~(x) =1. It is n a t u r a l to call the functions thus o b t a i n e d rk(x 1. . . . . Xk 10% fl, z) the l i m i t i n g c o r r e l a t i o n functions. Let us note that C ([3, ~) . ( max [ ~ I I ] e-~(Ixl) - - 1 I dx ----max I ~ ] C ([3). RV

T h e r e f o r e , f o r all ~(x) such that m a x JUx)l < s, II AIla
E ~d

s a t i s f y the e s t i m a t e

[ (Pa(xl . . . . .

xn)[ < Dv Imin d (xi, S)] d n ~-[ [ ~ (xl) 1, ~

(22)

w h e r e S C R v is s o m e manifold, d(x, S) the d i s t a n c e b e t w e e n the point x and S, v(r), the e s t i m a t i n g f u n c tion i n t r o d u c e d in s e c t i o n 3.

(AO),, (xl . . . . . x,,)~ uDv [mind (x,., S)] d" 1-[ I [ (xi) l, e0e 0 < u <1. l Proof.

Let k > 1. T h e n --1~ ~ U(Ix,--x i I)

l(AOh(c) t - e

:>t

I~(x0l Izll~R

II(e-~UCl~'-Yl~--l)~('[,cOdcl.

V b~Ct

T h i s e x p r e s s i o n d o e s not e x c e e d

III e - ~ U ( I x ' - y i l ~ -

d'*-'e~nl~ I~(xe)lDIzl v [mina(xi, S)I -t- ~,, d k i=l

XiE¢

k~l

CRv,k

i=l

l lIIlK(Y31v[min(d(x. S), e(.qz, S))lacl , i=1

Xi6C" Yi f: ct

147

which, in turn, i s l e s s than IM~-llz'e~S~I]~(x')l{v[m~

!>

nd(x''S)]q-~'d~v[mind(x''S)l k;=z

lfZ

4(!/i, 3

i

~ , ' e-~clx~-yj 0 -

1 ' ' ~(yj)' dcl

d(x~,,S) =

i=t,...,k

As h a s b e e n m e n t i o n e d in s e c t i o n 3,

I e - ~ ( ~ - ~ -- 1 [ I ~ (Y)] ~ (d - - I x -- y ] ) dy + fz--
S

t e-PU(tx'-ut)-- 11 I ~ (Y) ] dy • B,v (d).

Ix--yl>d

w h e r e B t is an absolute constant.

We thus obtain n

] (A~)~ (Xl. . . . .

xn) I < De t~B [ z l dn-' 1-[ [ ~ (xi) l v [mind (xl, S)I eac(~'~)(1 +dB1 max [ ~ (x)I). i

i=l

x

U n d e r the condition 1~ I < s we m a y a l w a y s s e l e c t s o s m a l l a z that f o r the d = d(s) s e l e c t e d above e ~s j z I edC(;~)sd-z [ 1 -~ Bzsd] = u < 1.

(23)

We, hence, indeed obtain the a s s e r t i o n of l e m m a 1. Corollary.

L e t the v e c t o r ~ p o s s e s s t h e p r o p e r t y

(22). T h e n the solution of the equation

~F= (D q- AT

(24)

under the condition (23) will, as before, sat~fy the estimate

x.)[ < B~d"[I I~ (xl)Iv [m:n d (x,, S)l,,

] v~n(x, . . . . . where B2 = B(1-u) -t.

T h i s r e s u l t is o b t a i n e d by expanding the solution of (24) in the s e r i e s ~ = ~ -t- A ~ + A2~ q -- ...

and a p p l y i n g l e m m a 1 to e a c h m e m b e r of the s e r i e s . Now, let us show that r k ( x 1. . . . . Xk [~,/3, z) c o n v e r g e to rk(xt, . . . . Xk [=o,/3, z), a s f~ ~=o, and l e t us e s t a b l i s h the r a t e of this c o n v e r g e n c e . L e t us i n t r o d u c e the o p e r a t o r X ~ in ~a: (X~ ~). (x~. . . . .

x.) = I I x~ (x~)% (x~. . . . .

x.),

w h e r e Xf~(x) is the c h a r a c t e r i s t i c function of the v e s s e l ~2. E q u a t i o n (14) now b e c o m e s P~ = X= A(1) -~- X~ AP~, w h e r e we have put ~(x) -- 1, and Pf2 is a c h a i n of c o r r e l a t i o n functions in the e n s e m b l e (C~, H,/3, z). Let us introduce the d i f f e r e n c e AG = P ~ 2 - X ~ P.o. T h i s d i f f e r e n c e e v i d e n t l y s a t i s f i e s the equation A~ : X , A ( E - - X~)Poo -I- X~AA~.

L e t us show that the v e c t o r ~ = X f f f k ( E - X ~ P ~ o s a t i s f i e s the e s t i m a t e (22). We have --~ ~ U(lxt--xi t)

oo

c,n=,=~

148

(25)

w h e r e ~ = (x 2. . . . .

xk), ~2' is the c o m p l e m e n t to the v e s s e l ~. Hence, oo


k=l

i

( q ~ d e n o t e s the b o u n d a r y of the v e s s e l i~), s i n c e as h a s b e e n m e n t i o n e d above, by v i r t u e of the s e l e c t i o n of v(r)

l

(e-;~u(ix-ul)

1) dy ~. v (d).

I x - - y l :-. d

Applying the c o r o l l a r y of l e m m a 1 to (25), we obtain the e s t i m a t e

xn) [~ Aod~ "."[rain d (xi, 0ff)l,

I (A~)n (x I . . . . .

i

w h e r e A 0 = B 0 [z left B e d C ( f l ) ( 1 - u ) -1. Let us now define the l i m i t i n g p r o b a b i l i t y P C u ( A l ~o) f o r A ~__ CU by r e p l a c i n g the c o r r e l a t i o n functions in (16) by the l i m i t s :

Pcu(A [ o c ) = s~] ~

S r~+s(Q, Q[ oc)dc.r

I, (--1)'dQ ."

(26)

CU, s

It follows f r o m the e s t i m a t e (22) that the s e r i e s (26) c o n v e r g e s . Furthermore,

we have

- - rk+~ (c,, Q [-Q)Ideldc2< ~ ~ I~u I'*+~ Aod~+Sv [rain xeu d (x, 0fl)] = v [rain ~eu d (x, Og~)]Aoe*l~r, k

w h e r e ]U

(27)

s

is the v - d i m e n s i o n a l m e a s u r e of the m a n i f o l d U.

It is s e e n f r o m (27) that lira Pcu(A I .O-k)~ Pcu(A [ ~), w h e r e i n the c o n v e r g e n c e is u n i f o r m r e l a t i v e to m e a s u r a b l e m a n i f o l d s A f r o m CU. It h e n c e follows e a s i l y that the n u m b e r s P C u ( A I ~ ) define a d e n u m e r a b l y - a d d i t i v e field of p r o b a b i l i t i e s in CU. T h e l i m i t i n g d i s t r i b u t i o n in CU (coincident with the l i m i t i n g d i s t r i b u t i o n in ~ u ) is given by the density ~(q[oo)= ~(--1) ~ i s

r"=~(c"c"-I~)dQ

(N(c) = k ) .

(27')

CU, s

By v i r t u e of all the above, the p r o b a b i l i t i e s PCu(AI¢~ g e n e r a t e a d e n u m e r a b l y - a d d i t i v e m e a s u r e in the ( r - a l g e b r a ~ ¢ , i.e., the l i m i t i n g Gibbs d i s t r i b u t i o n e x i s t s . Now, let SU,A be s o m e c y l i n d r i c a l m a n i f o l d . L e t us c o n s i d e r the conditional Gibbs p r o b a b i l i t y d i s t r i b u t i o n defined on the nmnifold SU,A. F u r t h e r m o r e , let us fix s o m e e l e m e n t c o ~ A C CU and let us c o n s i d e r the l i m i t i n g conditional d i s t r i b u t i o n in the manifold SU,c0 of such e l e m e n t s c E C ~ which i n t e r s e c t the nmnifold U in the e l e m e n t c 0. T h e f o r m u l a

P (K'Su.A) = I P (K/'Su,c~) nu(C0]~) [P (Su.A)l-l dco• A

is e v i d e n t l y valid f o r an a r b i t r a r y m a n i f o l d K. Although the e x i s t e n c e of the l i m i t i n g conditional d i s t r i b u tion P(./SU,c0) r e s u l t s (for a l m o s t all c 0) f r o m the g e n e r a l t h e o r e m s of m e a s u r e t h e o r y , it will be i m p o r t a n t f o r us l a t e r to e s t a b l i s h its e x i s t e n c e d i r e c t l y , and to c o n n e c t it to the conditional c o r r e l a t i o n functions. Let us note that we can define the conditional Gibbs d i s t r i b u t i o n on a s e t of s u b n m n i f o l d s

149

c C C [2, which i n t e r s e c t the manifold U C ~2 at the element c o E U, by means of the formula

e--f3H(¢)zN(C) p,, (C/Co) =

where Z (0, [1, Z/Co) =

I

e-om~''*)zN(~°+~v(~*)d q ,

CQ--U

ande={%c0},c0e

U,c~ ~ a - U .

T h e c o r r e l a t i o n functions rk(x~ . . . . , x k / c 0) for the conditional Gibbs distribution evidently equal r ( e / e ~ = r(e)/r(c0). If the manifold ~ - U is examined, it is then evident that r (c/e o [ O) = r ( q I ~, 0 ~ U), c = {cl, co},

where --~. Y. U(Ix--Yil )

[(x):

e

y~ec°

T h e r e f o r e , the c o r r e l a t i o n functions r k ( c / c 0) satisfy the s y s t e m of equations (where x 1 = xmin(ct)) k --~ ~ U(Ixt--xit)

r~(C/Cot~, ~, z) --

z~(xOe

i=2

~

l-[ (e-~U(Ir'-y/I)- l)r ~, Q/co)d Q,

¢¢[..~ YIEcl

and the limiting c o r r e l a t i o n functions satisfy an analogous s y s t e m of equations. F o r any manifold V lying outside the manifold U, we may define the limiting probabilities PCV(A/c 0) (A C CV), which generate a m e a s u r e in the a - a l g e b r a @~ /SU,c0, which a g r e e s with the conditional Gibbs distribution. In the next paper we shall show that this conditional distribution PCv(A/c0) is close to the absolute distribution Pcv(A) in cylindrical manifolds SV,A, if the manifold V is sufficiently r e m o t e f r o m the manifold U. I take the opportunity to e x p r e s s my gratitude to-R. L. Dobrushin and Ya. G. Sinai for useful d i s cussions of the questions c o n s i d e r e d herein. LITERATURE lp

2. 3. 4. 5. 6.

150

CITED

A. N. Kolmogorov, Fundamental concepts of probability theory. Moscow, ONTI (1936). S. Saks, T h e o r y of the integral. Moscow, IL (1949). N. Dunford and J. Sschwartz, Linear o p e r a t o r s , pt. I, Moscow, IL (1962). D. Ruelle, C o r r e l a t i o n functions, Ann. Phys., 2.~5, 109-120 (1963). D. Ruelle, Classical statistical mechanics, [Russian translation], Helv. Phys. Acta, 36.__,1 8 3 - 2 0 0 1 9 6 3 ) ; "Matematika, w 1_00:4(1966). R. L. Dobrushin, Existence of the configuration integral. T h e o r y of Probability and its Application, 9_ 626-638 (1964).