The equivalence of ensembles for classical ... - Semantic Scholar

The equivalence of ensembles for classical systems of particles Hans-Otto Georgii

Mathematisches Institut der Universitat Munchen Theresienstr. 39, D{80333 Munchen

Abstract

For systems of particles in classical phase space with standard Hamiltonian, we consider (spatially averaged) microcanonical Gibbs distributions in nite boxes. We show that in nite{volume limits along suitable subsequences exist and are grand canonical Gibbs measures. On the way, we establish a variational formula for the thermodynamic entropy density, as well as a variational characterization of grand canonical Gibbs measures. Key words: Classical statistical mechanics; microcanonical ensemble; variational principle; entropy; pressure.

1 Introduction The equivalence of Gibbs ensembles is one of the central problems of statistical mechanics. As far as the thermodynamic functions are concerned, this question is already well understood. For example, it is well-known that, under suitable conditions on the interaction, the in nite{volume limits of the entropy per volume and of the Gibbs free energy per volume exist and are related to each other by a Legendre{Fenchel transform; see, for example, Refs. 3{5. A much deeper question is the equivalence of ensembles on the level of measures, which (in a possible but not optimal formulation) would mean that the microcanonical and the grandcanonical Gibbs distributions in nite boxes have the same in nite{volume limits. In this paper we prove a version of the last statement which is not aected by the possibility of phase transitions. We consider the standard setting of classical statistical mechanics. A particle in Euclidean space of dimension d  1 is characterized by a pair (x; p), where x 2 Rd is the position and p 2 Rd the momentum. A particle con guration (without multiple occupancies) is thus given by a set !  Rd  Rd of the form ! = f(x; px ) : x 2 !^ g; where !^ , the set of occupied positions, is a locally nite subset of Rd , and px 2 Rd is the momentum of the particle at position x. The set of all such con gurations ! is denoted by . We assume that the formal Hamiltonian has the standard form H (!)  H kin (!) + H pot (!) =

X

x2!^

jpxj2 + 21 1

X

x;y2!^ ;x6=y

'(x ? y) ;

(1.1)

where ' is a pair potential satisfying suitable stability conditions. (For convenience we assume throughout that the particle mass is equal to 1/2.) We are interested in the behavior of microcanonical Gibbs distributions in nite boxes in the in nite volume limit. For simplicity we only consider the half-open cubes n = [?n ? 1=2; n + 1=2[d of volume vn = (2n + 1)d ; n  0, which will often considered as tori by assuming periodic boundary conditions. We write MnjN;E;per for the microcanonical Gibbs distribution for N particles in n with the energy constraint Hn;per  E , where Hn;per is the Hamiltonian in n with periodic boundary condition; see (2.5). We also consider the microcanonical Gibbs distributions MnujN;E;per on the energy shells fE ? u < Hn;per  E g of thickness u > 0, and the microcanonical distributions Mn0jN;E;per on the energy surfaces fHn;per = E g; see (2.13), (2.15) and Lemma 6.3 for precise de nitions. We shall also deal with other than periodic boundary conditions. Here is an outline of our results. Under fairly general hypotheses on the pair interaction ' we will show that, in the limit as n ! 1 and N=vn ! ; E=vn ! " for an admissible pair (; ") of particle and energy densities, the sequence (MnjN;E;per)n0 is relatively (sequentially) compact in a fairly strong topology, and each accumulation point belongs to the class M;" of all translation invariant probability measures P on

which have expected particle density  and mean energy " and maximize the mean entropy under this constraint (Theorem 3.3). Clearly, M;" is contained in Gz; , the set of all P that minimize the mean free energy for a suitably chosen activity z > 0 and inverse temperature > 0. Our second main result is a variational principle stating that Gz; consists of all tempered, translation invariant (grand canonical) Gibbs measures for z; (Theorem 3.4). These two results together imply that every accumulation point of the microcanonical distributions MnjN;E;per is a Gibbs measure (in the grand canonical sense) for suitable parameters z; > 0. This proves the equivalence of ensembles on the level of measures in a formulation which still makes sense in the presence of phase transitions. If there is no phase transition, i.e., if Gz; contains a unique element P , then

MnjN;E;per ! P

in the above-mentioned topology. In fact, this convergence already holds when P is the unique tempered Gibbs measure with expected particle and energy densities (; "). A main step in the derivation of the preceding results will be to show that the thermodynamic entropy density can be characterized by a variational formula involving the mean entropy of translation invariant probability measures on (Theorem 3.2). As a matter of fact, it is the variational characterization of thermodynamic quantities in terms of states which will enable us to lift the equivalence of ensembles from the level of thermodynamic functions to the level of states. The same results will be obtained for microcanonical distributions on thick or thin energy shells, as well as for free or con gurational boundary conditions under an additional spatial averaging. As is easy to see from the proofs, our results can also be extended to the case when the kinetic energy is given by any positive power of the momenta. They remain also true for pure positional ensembles of particles without momentum, at least as long as no microcanonical distributions on energy shells are considered. Our proofs make essential use of some ideas from large deviation theory and rely, in 2

particular, on the developments in Refs. 1 and 2. These papers will be referred to as I and II. For example, Theorem II.1 stands for Theorem 1 of Ref. 2. In Section 2 we introduce our conditions on the pair interaction ' and describe the general setup. The main results are stated in Section 3. In Section 4 we investigate the mean entropy of translation invariant states, and in Section 5 we derive the variational characterization of the thermodynamic entropy density. The subject of Section 6 is the asymptotics of microcanonical distributions, whereas the nal Section 7 is devoted to the variational priciple for Gibbs measures. We conclude this introduction with a few bibliographical notes. A classical approach to the equivalence of ensembles ( rst proposed by Khinchin(6)) is to use a local central limit theorem for the particle number and energy. This was carried out in Refs. 7{10 both for continuous systems and for lattice systems. The drawback of this method is that it works only in situations where a good local limit theorem is available | which certainly requires, at least, the absence of phase transitions. For classical lattice systems, Martin{Lof(11) developed a \thermodynamic" approach to the equivalence of ensembles. It works also in the presence of phase transitions and makes essential use of translation invariance and variational principles, as we do here. A quite dierent approach is to de ne microcanonical resp. canonical Gibbs states directly in in nite volume in analogy to the familiar procedure in the grand canonical case, and to show that these are convex mixtures of the usual grand canonical Gibbs measures with different parameters. This idea was pursued in Refs. 12{17 both in the lattice and the continuous case. It is again not aected by the presence of phase transitions, but the underlying concept of equivalence | though natural in the context of in nite{volume time evolutions | is not the traditional concept which is the subject of the present paper.

2 The setting

2.1 The interaction

We start with our assumptions on the pair interaction ', an even measurable function from Rd to R [ f1g. (A1) ' is regular. That is, there exists a decreasing function : [0; 1[! [0; 1[ and a number r(') < 1 such that '(x)  ? (jxj) for all x 2 Rd ; (2.1) '(x)  (jxj) whenever jxj  r(') and Z1 (s)sd?1 ds < 1 : 0

(A2) Either ' = 1 on a neighborhood of the origin (hard core case). Or ' < 1 Lebesgue-almost everywhere, and ' is non-integrably divergent at the origin, in that there exists a decreasing function  : ]0; 1[! [0; 1[ such that '(x)  (jxj) whenever '(x)  0 (2.2) 3

and

Z1 0

(s)sd?1 ds = 1 :

As is well-known(3) , (A1) and (A2) together imply that ' is superstable. By de nition, this means that there exist constants a > 0; b < 1 such that for all n

Hn  a Tn ? b Nn :

(2.3)

In the above, Hn  Hn;free , the Hamiltonian in the box n with free boundary condition, is given by Hn (!) = H (!n ), where H is as in (1.1) and !n = ! \ (n  Rd ) is the restriction of ! 2 to n ; Nn (!) = card(^!n ) is the particle number in n ; and

Tn =

X

i2n \Zd

NC2 +i

(2.4)

is the sum of the squared particle numbers NC +i (!) = card(^! \ (C + i)) in the disjoint unit cells C + i in n , where C = 0 = [?1=2; 1=2[d stands for the centered unit cube. The superstability and regularity of ' will be sucient as long as we deal only with free or periodic boundary conditions. The latter lead to the periodic Hamiltonians

Hn;per(!) =

X

x2!^ n

jpxj2 + 21

= Hn (!) + 21

X

x2!^ n ;y2!^ (n) ;y6=x X

x2!^ n ;y2!^ (n) nn

'(y ? x)

'(y ? x):

(2.5)

Here !(n) = f(x + (2n + 1)i; p) : (x; p) 2 !n ; i 2 Zd g is the periodic continuation of !n. The non-integrable divergence of ' at the origin will become important as soon as con gurational boundary conditions are involved. For t > 0 we de ne

(t) = fTn =vn  t for all n  0g: S

(2.6)

The con gurations in  = t>0 (t) are called tempered. If ' even has a hard core we use the same symbols (t) and  to denote the set of all admissible hard-core con gurations. (This avoids a messy distinction of cases.) For each  2  and n  0 we let X '(y ? x) (2.7) Hn; (!) = Hn(!) + x2!^ n ;y2^nn

denote the Hamiltonian in n with tempered boundary condition  . The regularity of ' ensures that the last sums exists, cf. Lemma II.4.2.

2.2 Translation invariant states Next we consider the class of possible states of our particle system. The con guration space is equipped with the -algebra F which is generated by the counting variables N (B ) : ! ! card(! \ B ) for Borel sets B  Rd  Rd . It is well-known(18) that F is the Borel -algebra for the Polish topology on that is induced by the mappings N (h) : ! ! Px2!^ h(x; px ); h : Rd  Rd ! R continuous with compact support. 4

We consider the class P of all probability measures (\states") P on ( ; F ) satisfying Z

P (d!)

X

x2!^ n

(1 + jpx j2 ) < 1 for all n;

(2.8)

as well as the class P of all P 2 P which are invariant under the translation group  = (#x )x2Rd acting on via #x ! = f(y ? x; p) : (y; p) 2 !g. The mapping (x; !) ! #x ! is known(18) to be measurable. For each P 2 P there exists a number (P ) < 1, the intensity or mean particle number of P , such that P (N ) = (P )jj for all measurable   Rd ; here we write N = N (  Rd) for the counting variable associated to , jj is the Lebesgue measure R of , and P (f ) = fdP stands for the integral of a function f with respect to P . Also, for P 2 P there exists(18;1) a unique nite measure P  on Rd  , the Palm measure of P , such that Z

P (d!)

X

x2!^

Z

Z

F (x; px ; #x !) = dx P  (dp; d!)F (x; p; !)

(2.9)

for all measurable functions F  0 on Rd  Rd  . In particular, P  is supported on the set f(p; !) : (0; p) 2 !g, and the rst marginal P = P  (
 ) of P  satis es Z

P (d!)

X

Z

f (px) = vn f dP

x2!^ n Rd !

(2.10)

for all n  0 and measurable f : [0; 1[. We call P the momentum intensity measure. We introduce a topology L on P (which is much ner than the weak topology associated to the above-mentioned Polish topology on ) as follows. Let L denote the class of all measurable functions f : ! R which are local and tame in the sense that f (!) = f (!` ) and     X jf (!)j  c 1 + (1 + jpxj)  c 1 + N~`(!) (2.11) x2!^ `

for some `  0; c < 1 and all ! 2 . The topology L of local convergence is then de ned as the weak topology Ron P relative to L, i.e., as the smallest topology making the mappings P ! P (f )  fdP with f 2 L continuous. InR particular, the mean particle density (P ) and the mean momentum per particle p P (dp)=(P ) are continous functions of P 2 P .

2.3 The Gibbs ensembles

For a xed box n and particle number N we consider the set

njN = f! 2 : !^  n ; Nn (!) = N g

of all N {particle con gurations in n , and the associated \Lebesgue measure" (or \Liouville measure") Z Z 1 LnjN (A) = N ! N dx1 : : : dxN dN dp1 : : : dpN 1A (f(x1 ; p1 ); : : : ; (xN ; pN )g) ; (2.12) n R 5

A 2 F . The microcanonical distributions are obtained by conditioning LnjN on an energy shell of the form fE ? u < Hn;bc  E g for some E 2 R; 0 < u  1, and a boundary condition bc 2  [ fperg. That is, we set

MnujN;E;bc(d!) = 1fE ? u < Hn;bc  E g(!) LnjN (d!) / ZnujN;E;bc

(2.13)

whenever the parameters are chosen in such a way that the normalizing constant

ZnujN;E;bc = LnjN (E ? u < Hn;bc  E )

(2.14)

Mn0jN;E;bc = ulim Mu : !0 njN;E;bc

(2.15)

is positive. ((2.3) implies that it is always nite.) In the case u = 1 (which means that there is no lower bound on the energy) we shall omit the upper index u. ZnjN;E;bc is called the microcanonical partition function or the phase volume. We will also consider microcanonical distributions on thin energy shells fHn;bc = E g. Formally, these are de ned by It is well-known that these limits exists in the vague sense when Hn;bc is a C 2 function (which certainly holds if ' is C 2 with derivatives satisfying an analogue of (2.1)); see, for example, Khinchin(6) or Section 3.5 of Tjur(19) . However, we do not need any smoothness of '. Indeed, since we are only interested in the asymptotic behavior of Mn0jN;E;bc as n ! 1, we only need to de ne Mn0jN;E;bc for events in a microscopic subregion of n , and for such events the limit (2.15) exists without any additional assumption on ' (due to the smoothing eect of the kinetic energy). We defer the details until Section 6. Finally, for the nonperiodic boundary conditions bc 2  , we will have to deal with the averaged microcanonical distributions

M nujN;E;bc = vn?1

Z

n

dx MnujN;E;bc  #?x 1 :

(2.16)

This spatial averaging guarantees the asymptotic translation invariance we need. In the grand canonical ensemble, the parameters N and E are replaced by the positive parameters z , the activity, and , the inverse temperature. The grand canonical Gibbs distribution in n with boundary condition bc2  [ fperg and parameters z; is de ned by 1 Nn (!) exp[? H (!)] L (d!); (2.17) Gn;z; ;bc(d!) = ?n;z; ; n n;bc bc z P where Ln = N 0 LnjN is the Lebesgue measure on n = f! 2 : !^  n g, and n;z; ;bc = Ln (z Nn exp[? Hn;bc ])

is the grand canonical partition function (which is nite, cf. the estimates in the proof of Lemma 5.2). A measure P 2 P is called a tempered Gibbs measure for z; > 0 if P (  ) = 1 and, for all n  0 and measurable functions f  0 on , Z

Z

P (f ) = P (d ) Gn;z; ; (d!) f (!n [ ( n n)): 6

(2.18)

Clearly, the form of our Hamiltonian implies that, relative to all Gn;z; ; and thus all tempered Gibbs measures, the momenta of the particles are conditionally i.i.d. with a Maxwellian (i.e., normal) distribution when the set !^ of occupied positions is given. For functions f that depend only on !^ , (2.18) is equivalent to the equilibrium equations introduced by Ruelle(20) .

3 Results A basic ingredient of our results is the existence of the mean energy and mean entropy of any P 2 P . The mean energy is de ned by ?1 U (P ) = nlim !1 vn P (Hn ):

(3.1)

This limit exists in R [f1g. Indeed, (2.10) and (2.8) show that the kinetic contribution equals Z kin (3.2) U (P ) = jpj2 P (dp) = vn?1 P (Hnkin) < 1 independently of n. On the other hand, it was shown in Theorem II.1 (under the assumptions (2.1) and (2.3)) that the mean potential energy ?1 pot U pot (P ) = nlim !1 vn P (Hn )

(3.3)

exists in [a(P )2 ? b(P ); 1] and satis es Z

U pot (P ) = P  (dp; d!) 21

X 06=x2!^

'(x)

(3.4)

if P (NC2 ) < 1 and equals +1 otherwise. Moreover, (3.2), (3.4) and Theorem II.1 show that the functions U kin and U pot , and thus also U , on P are ane (even measure ane, in that they are ane with respect to convex mixtures formed by arbitrary probability measures on P ) and lower semicontinuous relative to L. The entropy of a state P 2 P in n is de ned by (

(log fn ) if Pn  Ln with density fn Sn(P ) = ?Pn?1 otherwise,

(3.5)

where Pn = P (f! 2 : !n 2
g) is the restriction of P to n . The following proposition on the mean entropy of invariant states will be proved in Section 4.

Proposition 3.1 For each P 2 P, the mean entropy ?1 S (P ) = nlim !1 vn Sn (P )

exists in R [ f1g and is a measure ane, upper semicontinuous function of P . The \energy-bounded" superlevel sets

fP 2 P : S (P )  ?c; U (P )  "g of S (with c; " 2 R) are compact and sequentially compact in L .

7

We emphasize that S fails to be upper semicontinuous with respect to the coarser topology Lb that is associated to the class Lb of all bounded local functions, see Example 4.3. Our rst main result concerns the existence and variational characterization of the thermodynamic entropy density. To state it we de ne for   0

"min() = inf fU (P ) : P 2 P ; (P ) = ; S (P ) > ?1g :

(3.6)

By (2.3), "min()  a2 ? b, and "min(
) is clearly convex. Also, "min(
) is nite and continuous on a maximal interval [0; max [, where max = 1 except when ' has a hard core. These facts are proved in Lemma II.7.1 (which extends without diculties to the present setting of particles with momentum). We introduce the convex set  = f(; ") : 0 <  < max; " > "min()g

(3.7)

and the abbreviation log? u = min(0; log u).

Theorem 3.2 (a) Let (; ") 2  and t > 0. If n ! 1 and N; E; u, bc run through any sequences such that N=vn ! ; E=vn ! "; vn?1 log? u ! 0 and bc 2 (t) [ fperg,

the limit

s(; ") = lim vn?1 log ZnujN;E;bc

(3.8)

s(; ") = sup f S (P ) : P 2 P; (P ) = ; U (P )  " g :

(3.9)

exists and admits the variational characterization

(b) The function s(
;
) on [0; 1[R de ned by (3.9) is concave and upper semicontinuous. The set  is the interior of the eective domain fs(
;
) > ?1g, s(;
) is strictly increasing on ]"min (); 1[ for each , and the condition U (P )  " in (3.9) can be replaced by U (P ) = ". We note that the existence of the limit in (3.8) is already known for a long time, at least for bc = free (3;4) . So our main point is its variational characterization (3.9); it is the microcanonical counterpart of the more familiar variational formula for the pressure which appears in Proposition 7.1. For (; ") 2= , we still have the relation lim sup vn?1 log ZnujN;E;bc  s(; "); as is easy to see from the proof of Theorem 3.2 in Section 5. Theorem 3.2 is the essential step towards the following main result which will be proved in Section 6.

Theorem 3.3 Suppose that (; ") 2 ; t > 0; N=vn !  and E=vn ! " as n ! 1, and u 2 [0; 1] and bc 2 (t) [ fperg are allowed to vary with n. Then the sequences (MnujN;E;per)n0 and (M nujN;E;bc )n0 are (well-de ned for large n and) relatively sequentially compact in the topology L , and every accumulation point belongs to the set

M;" =f P 2 P : (P ) = ; U (P ) = "; S (P ) = s(; ") g : 8

Our nal task is to identify the \microcanonical equilibrium states" in M;" as grand canonical Gibbs measures. For (; ") 2  let = (; ") > 0 be the derivative of s(;
) at ". This derivative is known to exist, see Rechtmann and Penrose(21) or Remark 6.5 below. The dierentiability with respect to  seems to be unknown, but by concavity we can anyway nd a number z > 0 such that, for a suitable constant p(z; ) 2 R, the plane (0 ; "0 ) ! p(z; ) + "0 ? 0 log z is a tangent to s(
;
) at (; "). It then follows that for all P 2 P with U (P ) < 1

S (P )  s((P ); U (P ))  p(z; ) + U (P ) ? (P ) log z with equality when P 2 M;" . Introducing the mean free energy

Fz; (P ) = U (P ) ? (P ) log z ? S (P )

(3.10)

we nd that Fz; (P )  ?p(z; ) with equality for P 2 M;". Since M;" = 6 ;, we conclude that p(z; ) = ? min f Fz; (P ) : P 2 P g; (3.11) which is ( times) the pressure (see Proposition 7.1), and

M;"  Gz; f P 2 P : Fz; (P ) = ?p(z; ) g : (3.12) The following variational principle asserts that Gz; coincides with the set of all trans-

lation invariant tempered Gibbs measures. Together with Theorem 3.3 and (3.12) this completes the proof of the equivalence of ensembles.

Theorem 3.4 For all z; > 0, the set of all translation invariant tempered Gibbs measures for z; (as de ned around (2.18)) coincides with the set Gz; on which the mean free energy Fz; attains its minimum ?p(z; ). 7.

This result (which is well-known for lattice systems(22;23;24) ) will be proved in Section

Remark 3.5 The preceding results obviously imply that the set Gz; of translation invariant tempered Gibbs measures is non-empty | which was proved rst by Ruelle(20) and Dobrushin(21). In fact, this follows already from (the proof of) Proposition 3.1 together with Theorem 3.4: Fz; has compact sublevel sets fP 2 P : Fz; (P )  cg; c 2 R, and thus attains its minimum. Remark 3.6 Gz; also contains all accumulation points of the relatively compact sequence (Gn;z; ;per)n0 . This can be proved in complete analogy to Theorem 3.3; compare Proposition II.7.4. Thus, in the absence of phase transition when Gz; = fP g we have P = lim MnujN;E;per = lim Gn;z; ;per in L, and a similar statement holds for spatially averaged distributions with boundary conditions in (t) for some t. This expresses the equivalence of ensembles in terms of nite volume Gibbs distributions.

9

4 The mean entropy of invariant states In this section we prove Proposition 3.1. It is convenient to replace the in nite reference measures Ln in (3.5) by consistent probability measures, namely the (local restrictions of) Poisson point random elds. Let  : Rd ! R [ f1g be any function satisfying R ? (p) c( )  e dp < 1 and Q the Poisson point random eld with intensity measure  (dx; dp) = dx e? (p) dp. That is, Q is the unique measure in P relative to which the particle numbers N (B1 ); : : : ; N (Bk ) in disjoint phase space regions B1 ; : : : ; Bk  Rd  Rd are independent and Poisson distributed with parameters  (Bi ); 1  i  k. By de nition, its restriction Qn to n has the Radon{Nikodym density

qn (!) = exp[?

X

x2!^ n

 (px ) ? vnc( )]

(4.1)

relative to Ln . For any P 2 P we consider the relative entropy

I (Pn ; Qn ) =

(

Pn (log hn) if Pn  Qn with density hn

1

otherwise.

Clearly, Pn  Qn with a density hn if and only if Pn  Ln with density fn = hn qn . Hence we conclude from (3.5) and (2.10) that

Sn (P ) = ?I (Pn ; Qn) ? P (log qn ) = ?I (Pn ; Qn ) + vn P ( ) + vn c( ):

(4.2)

We will use the following fact.

Lemma 4.1

For all P 2 P , the mean relative entropy ?1  I (P ) = nlim !1 vn I (Pn ; Qn )

exists in [0; 1]. The function I on P is measure ane, and its sublevel sets fP 2 P : I (P )  cg; c  0, are compact and sequentially compact in the topology Lb that is induced by the class Lb of all bounded local functions. If  is such that Z

ecjpj? (p)dp < 1

for all c > 0 then the last assertion also holds in the topology L . Proof. The rst two sentences follow from the analogous result for lattice models d Z by identifying with ( 0 ) , cf. Theorems (15.12) and (15.20) of Ref. 24. The last assertion is proved in Proposition I.2.6. 2

We are particularly interested in the case when  (p) = jpj2 for some > 0. We then replace the index  by . In particular, Q is the ideal gas with particle density c( ) = (= )d=2 and Maxwellian momenta of variance 1=2 , and (4.2) takes the form

vn?1 Sn(P ) = ?vn?1 I (Pn ; Q n ) + U kin (P ) + c( ): 10

Since U kin is nite, we conclude from Lemma 4.1 that vn?1 Sn (P ) converges to

S (P ) = ?I (P ) + U kin (P ) + c( )

(4.3)

which is a measure ane function of P . It remains to establish the continuity properties of S . To this end, we note rst that a bound on the mean energy implies a bound on the mean kinetic energy. Indeed, since

U pot (P )  a(P )2 ? b(P )  ?b2 =4a by (2.3) and (3.3), we have the implication

U (P )  " =) U kin (P )  "~  " + b2 =4a:

(4.4)

The proof of Proposition 3.1 is therefore completed by the following lemma.

Lemma 4.2 The mean entropy S is upper semicontinous (relative to L), and the restricted superlevel sets fS  c; U kin  "~g, c; "~ 2 R, are compact and sequentially compact.

Proof. In view of (4.3), the set in question is contained in the set fI1  c~g with c~ = ?c + "~+ c(1), and the latter set is compact and sequentially compact by Lemma 4.1. We thus only need to show that fS  cg is closed, i.e., that S is upper semicontinuous. Applying (4.2) to the function  = j
j we see that

S (P ) = ?Ij
j(P ) + P (j
j) + c(j
j) for all P . But P ! P (j
j) is continuous by de nition of L , and Lemma 4.1 asserts that Ij
j is lower semicontinous relative to Lb  L. This gives the result. 2 We complete this section with an example showing that S fails to be upper semicontinuous relative to Lb .

Example 4.3 For each k  1 let Bk  Rd be a Borel set of volume ek2 =k which is disjoint from the unit cube C . Let  and k be such that e? = 1C and e?k = 2 1C + e?k 1Bk . qnk converges to qn Ln-almost everywhere, and thus in L1 (Ln ). Hence Qk converges in Lb (but not in L ) to Q . By (4.2), Z

S (Qk ) = ? k (p)e?k (p) dp + c(k ) = k + 1 + 1=k ! 1 as k ! 1, but on the other hand S (Q ) = 1. 2

5 The thermodynamic entropy density This section contains the proof of Theorem 3.2. To begin we consider the function s(; ") on [0; 1[R de ned by (3.9) (with the convention sup ; = ?1).

Lemma 5.1

(a) The supremum in (3.9) is attained whenever it is not ?1. 11

(b) s(
;
) is concave and upper semicontinuous, and its energy{bounded superlevel sets f(; ") : s(; ")  c1 ; "  c2 g; c1 ; c2 2 R, are compact. (c) The set  in (3.7) is the interior of fs(
;
) > ?1g. Proof. Assertion (a), the upper semicontinuity, and the compactness of the energy{bounded superlevel sets are immediate consequences of Proposition 3.1 and the continuity of (
). The concavity is clear because S , (
) and U are ane. Assertion (c) follows straight from the de nitions. 2

Our main task is the proof of the convergence (3.8) towards the limit s(; ") de ned by (3.9). We shall proceed in three stages. In the rst stage (which relies on Refs. I and II) we shall deal with \fattened" partition functions for which the particle number may range in a whole interval. In the second and third stage we shall remove the fattening by controlling the dependence of ZnjN;E;bc on the parameters N and E . The rst stage is based on techniques from large deviation theory. A central object in this theory is the translation invariant empirical eld

Rn;! = v?1 n

Z

 (n) dx 2 P n # x !

(5.1)

of a con guration ! in n . Here !(n) is as in (2.5). We shall take advantage from the formula Hn;per(!) = vn U (Rn;! ); (5.2) which follows from (3.2), (3.4) and the explicit formula (I.2.6) for the Palm measure  of Rn;! . The following lemma allows us to proceed from periodic to non-periodic Rn;! boundary conditions.

Lemma 5.2 (a) For any " < 1 and  > 0 there exists an integer k = k("; ) such that for all n

fHn  "vng \ n  fHn+k;per  (" + )vn g: (b) For given " < 1 and ; t > 0 there exists some n("; ; t) such that for all  2 (t) and n  n("; ; t) fHn;  "vng  fHn  (" + )vn g: Proof.

(a) If Hn  "vn then, by (2.3),

Nn2 =vn  Tn  vn "=a + b Nn =a and thus Tn  tvn for some t < 1 depending on " (and a; b). The result thus follows from Lemma II.4.3. (b) Here we use the non-integrability of ' at 0. By Lemma II.6.1 there exists an increasing function h : Z+ ! [0; 1[ with h(`)=`2 ! 1 as ` ! 1, and a number b0 > 0 such that Hn  Tnh ? b0 Nn (5.3) 12

for all n, where

Tnh =

X

i2n \Zd

h(NC +i ):

On the other hand, Lemma II.6.3 asserts that for given 0 > 0 and n larger than some n0 (0 ; t) Hn;  Hn ? 0 Tnh ? 0 vn (5.4) for all  2 (t). Suppose now that Hn;  "vn . (5.3) and (5.4) (with 0 = 1=2) together then show that Tnh  (2" + 1)vn + 2b0 Nn whenever n  n0 (1=2; t). Choosing q such that h(`)  `2 for `  q we see that

Nn2 =vn  Tn  Tnh + q2 vn: Combining this with the previous inequality we nd that Tnh  s vn for some s = s(") < 1 and all n  n0(1=2; t). Using (5.4) again with 0 = =(s + 1) we can conclude. 2 Here is the rst stage in the proof of (3.8). Proposition 5.3 Let " < 1 and D  [0; 1[ be a nondegenerate interval such that (; ") 2  for some  2 D. Let t > 0 and consider, for each n,

anjD;";bc  Ln(Nn 2 vn D; Hn;bc  vn ") =

X

N 2vn D

ZnjN;vn ";bc ;

where bc 2 (t) [ fperg may depend on n. Then ?1 log a

nlim !1 vn

njD;";bc = sup s(; "): 2D

Proof. 1) We rst derive the upper bound. We start with the case bc = per. Let > 0 be arbitrary. Eq. (4.1) with  = j
j2 , (5.2) and (4.4) yield 

anjD;";per = Q n 1=qn ; (Rn ) 2 D; U (Rn )  "  exp[vn "~ + vnc( )] Q n(Rn 2 A);



 U (P )  "g and D is the closure of D. Since A is where A = fP 2 P : (P ) 2 D; closed in L, we can use Theorem I.3.1 (with Q = Q ) to obtain lim sup vn?1 log anjD;";per  "~ + c( ) ? Pinf I (P ) 2A = "~ + sup [S (P ) ? U kin (P )] P 2A

 "~ + sup s(; "): 2D

The second step comes from (4.3), and in the last step we used the positivity of U kin and (3.9) together with the fact that (in view of our assumption on D and Lemma 5.1) the suprema of s(
; ") over D and D coincide. Letting ! 0 yields the upper bound in the case bc = per. 13

Turning to the case bc = free, we let D0 be any interval containing D in its interior,  > 0, and k be chosen according to Lemma 5.2 (a). Then

anjD;";free  an+kjD0;"+;per whenever n is so large that vn+k =vn is suciently close to 1. On the other hand, Lemma 5.1 (b) implies that

inf

sup s(; " + )  sup s(; ") :

D0 D;>0 2D0

2D

The upper bound for bc = free thus follows from that for bc = per. Finally, suppose bc =  2 (t) for some t > 0, and choose any  > 0. Lemma 5.2 (b) then shows that

anjD;";  anjD;"+;free

for suciently large n, and the desired upper bound follows from the previous case. 2) We now turn to the lower bound. Since U kin  0 we obtain as above

anjD;";per  exp[vn c( )] Q n (Rn 2 A ) for any > 0. Here A = f(
) 2 D ; U < "g and D is the interior of D. Note that A fails to be open because U is only lower semicontinuous. A direct application of

Theorem I.3.1 is therefore impossible. We rather need to extend the lower bound in Lemma II.7.2 to the present case of particles with momentum. To see that such an extension is valid we only need to control the additional kinetic term in Lemma II.5.1. (The  there is our U pot .) Using the notation in the proof of this lemma (in particular m = n + k for a xed k) we can write

U kin(P (n) ) = vm?1 P (n) (Hmkin) = vm?2

Z

m

P^ (n) (Hmkin  #x)dx :

Since Hmkin is a sum of single{particle terms and P^ (n) is m -periodic, the integrand above does not depend on x. Hence U kin(P (n) ) = vm?1 P^ (n) (Hmkin ) = vm?1 Pn (Hnkinj?q(n) )

 p?n 1vm?1 Pn(Hnkin ) = (vn=pnvm ) U kin (P ) and therefore lim supn!1 U kin (P (n) )  U kin (P ). This is all what is needed to extend

Lemmas II.5.1 and II.7.2 to the present case. The result is

lim inf vn?1 log anjD;";per  c( ) ? Pinf I (P ) n!1 2A = sup[S (P ) ? U kin (P )] P 2A  sup s(; ") ? "~ : 2D

In the last step we used (4.4) and the fact that the suprema of s(
; ") over D and D coincide. Letting ! 0 we obtain the lower bound in the case bc = per. The other boundary conditions can be treated by the same argument, with one addition: For bc = free we also have to use the last estimate in the proof of Proposition II.5.4, whereas in the case bc =  2 (t) we argue as in the lines leading to Eq. (II.6.6). 2 14

The second step in the proof of (3.8) consists in controlling the variation of the microcanonical partition functions ZnjN;E;bc with respect to N . We need to distinguish the two cases in assumption (A2). Lemma 5.4 Suppose that ' < 1 a.e.. Then for any given numbers ; t > 0 there exist constants c; u > 0 and n0 such that

ZnjN;E;bc  c ZnjN ?1;E?u;bc whenever N  vn; E 2 R, bc 2 (t) [ fperg and n  n0 . Proof. Let '+ = max('; 0) be the positive part of '. Since ' < 1 a.e., we can nd some q > 0 such that Z

min(1; '+ (x)=q)dx  1=4 :

Indeed, the integrand in the integral on the left side converges to 1f'+ =1g = 0 a.e. as

q ! 1, and the regularity assumption (A1) ensures that the dominated convergence theorem is applicable. Next, for each n; N; ! 2 njN ?1 and (x; p) 62 ! we can write X Hn;bc (! [ f(x; p)g) = Hn;bc (!) + jpj2 + '(y ? x) + hn;bc (x); y2!^

where, for example,

hn; (x) =

X

y2^nn

'(y ? x)

for  2 (t). Using (2.1) and the arguments in Lemma II.4.2 we see that there exists some r  r(') such that hn;bc (x)  1 whenever x 2 n?r , bc 2 (t) [ fperg and, in the case bc = per, ! 2 njN with N  vn . Let n0 be so large that vn?r  vn =2 when n  n0. Then we have for arbitrary n  n0 and ! 2 njN ?1

jfx 2 n?r :

X

y2!^

'(y ? x)  qgj 

Z



n?r

1?

X

y2!^



min(1; '+ (y ? x)=q) dx

 vn?r ? (N ? 1)=4  vn=4 :

Setting u = q + 2 we thus obtain the nal estimate

N ZnjN;E;bc 

Z

LnjN ?1 (d!)

Z

Z

n?r

dx dp

1fHn;bc(!)  E ? u; X '(y ? x)  q; jpj2  1g y2!^

 (c1 vn=4) ZnjN ?1;E?u;bc ;

where c1 is the volume of the unit ball in Rd . The lemma thus follows with c = c1 =4 .

2

If ' has a hard core we have an estimate in the opposite direction. Lemma 5.5 Suppose ' = 1 on a neighbourhood of 0. For given  > 0 and " < 1 there exist constants c; u > 0 and n0 such that

ZnjN;E;bc  c ZnjN +1;E?u;bc 15

for all n  n0 ; N  vn ; E  "vn , and bc 2  [ fperg. Proof. The hard{core property and the regularity (2.1) imply the existence of a number u > 0 such that X '(y ? x)  ?u y2!^

for all x 2 Rd and all admissible hard{core con gurations !^ . This gives the estimate (N + 1) ZnjN +1;E ?u;bc 

Z

= vn

LnjN (d!) Z

Z

Z

dx dp 1fHn;bc (!) + jpj2  E g dp ZnjN;E?jpj2;bc : n

In view of Lemma 5.7 below, the last expression is at most Z

vn dp e? jpj2 ZnjN;E;bc for some = (; ") > 0, provided N  vn ; E  "vn and n is large enough. The lemma is now obvious. 2 The next result marks the second stage in the proof of (3.8).

Proposition 5.6 Let (; ") 2  and t > 0. In the limit as n ! 1; N=vn ! ; E=vn ! " and bc = bc(n) 2 (t) [ fperg, vn?1 log ZnjN;E;bc converges to the function s(; ") de ned by (3.9). Proof. If D  [0; 1[ is a non-degenerate interval containing  in its interior and "0 > " then ZnjN;E;bc  anjD;"0;bc eventually. Proposition 5.3 and the upper semicontinuity of s(
;
) thus imply that lim sup vn?1 log ZnjN;E;bc  s(; ") : For the lower bound we consider rst the case when ' < 1 a.e.. We x any  >  and let u and 0 < c < 1 be as in Lemma 5.4. Let  > 0 be so small that  <  and " ? u > "min(). For suciently large n we have N=vn   and, by k{fold iteration of Lemma 5.4,

ZnjN;E;bc  ck ZnjN ?k;E?ku;bc : Averaging the right-hand side over all k between 0 and vn and choosing any "0 2 ]"min ; " ? [ and a non-degenerate closed interval D  ] ? ; [ we obtain ZnjN;E;bc  cvn (vn )?1 anjD;"0;bc when n is large enough. Proposition 5.3 thus gives lim inf vn?1 log ZnjN;E;bc   log c + s(0 ; "0 ) for any 0 2 D. Letting 0 ! ;  ! 0; "0 ! " in this order we can conclude. In the case when ' has a hard core we can proceed analogously, using Lemma 5.5 instead of 5.4. 2 16

The nal step in the proof of (3.8) is the consideration of energy shells. To this end we need to control the E {dependence of the ZnjN;E;bc's. Let L^ njN be the \Lebesgue measure" on the set ^ njN = f!^  n : card !^ = N g of position{con gurations of N particles in n (with the standard {algebra), that is, Z Z (5.5) f dL^ njN = N1 ! N dx1 : : : dxN f (fx1 ; : : : ; xN g) n for measurable f  0 on ^ njN . Note that Hn;potbc can be considered as a function of !^ . Lemma 5.7 For all n; N; E and bc we have Z Nd=2  ; ZnjN;E;bc = cN L^ njN (d!^ ) E ? Hn;potbc (^!) +

where cN is the volume of the unit ball in RNd . In particular, ZnjN;E;bc is dierentiable with respect to E with derivative   Zn0 jN;E;bc = cN (Nd=2) L^ njN (E ? Hn;potbc )+(Nd=2)?1 : (5.6)

which is sometimes called the structure function(6) . Also, for every  > 0 and " < 1 there exists a constant = (; "; t) > 0 such that ZnjN;E?u;bc  e? u ZnjN;E;bc (5.7) whenever N  vn ; E  "vn ; u  0, bc 2 (t) [ fperg for a given t > 0, and n is large enough. Proof. We only need to comment on (5.7). Let t > 0 be xed. By the proof of Lemma 5.2(b) there exists a number c > ?" (depending on " and t) such that, for all bc 2 (t) [fperg and suciently large n, Hn;potbc  ?c vn whenever Hn;potbc  " vn . Hence (E ? Hn;potbc )+  (" + c)vn and therefore

d 0 (5.8) dE log ZnjN;E;bc = ZnjN;E;bc / ZnjN;E;bc   d=2(" + c) : (5.7) now follows by integration. 2 It follows immediately from (5.7) and Proposition 5.6 that s(;
) is strictly increasing on ]"min(); 1[. This in turn shows that the condition \ "" in (3.9) can be replaced by \= "". (5.7) further implies that (1 ? e? u ) ZnjN;E;bc  ZnujN;E;bc  ZnjN;E;bc for a suitable > 0 whenever N=vn ! ; E=vn ! "; u > 0, bc 2 (t) [fperg for a given t, and n is large enough. Combining this inequality with Proposition 5.6 we arrive at (3.8). The proof of Theorem 3.2 is therefore complete. Remark 5.8 Under the conditions of (3.8) we also have that s(; ") = lim vn?1 log Zn0 jN;E;bc ; cf. (5.6). This follows from (5.8) and the inequality

2

Zn0 jN;E;bc 

Z1 0

du Zn0 jN;E+u;bc = Zn1jN;E+1;bc : 17

6 The asymptotics of microcanonical distributions This section contains the proof of Theorem 3.3. Let (; ") 2  and

N=vn ! ; E=vn ! "; and bc 2 (t) [ fperg (6.1) for some xed t > 0 as n ! 1. In a rst step, we shall prove Theorem 3.3 under the

additional hypothesis

vn?1 log? u ! 0 as n ! 1

(6.2) which appears in Theorem 3.2. In this case we follow the idea devised in Ref. 26. In a second step we then shall extend the result to the alternate case when u vanishes or tends to 0 exponentially with vn . Suppose (6.1) and (6.2) hold. For brevity we write Mn = MnujN;E;bc and M n = M nujN;E;bc. Theorem 3.2 implies that these measures are well-de ned for suciently large n. Let Mnper 2 P denote the measure relative to which the con gurations in the disjoint blocks n + (2n + 1)i; i 2 Zd , are independent with identical distribution Mn , and Z fn = vn?1 M Mnper  #?x 1 dx 2 P n

the associated invariant average. We shall derive the asymptotics of M n from that of fn . First, we get from (3.2) that M fn ) = vn?2 U kin(M

Z

n

Mnper (Hnkin  #x ) dx = vn?1 Mn(Hnkin ) :

(6.3)

The second equality follows from the periodicity of Mnper and the additive nature of Hnkin . Together with Lemma 5.2(b) and a version of (4.4) this implies that fn ) < 1: lim sup U kin (M

n!1

(6.4)

fn . The following key estimate follows from Next we consider the mean entropy of M Theorem 3.2. Proposition 6.1 Under conditions (6.1) and (6.2), lim infn!1 S (Mfn)  s(; ") : Proof. Let Q = Q1 and I = I1 be as in (4.3) for = 1. Lemma I.5.5 asserts that fn )  vn?1 I (Mn ; Qn ) : Together with (4.3), (6.3) and (4.2) this yields I (M fn )  ?vn?1 I (Mn ; Qn ) + vn?1 Mn (Hnkin ) + c(1) S (M = vn?1 Sn (Mn ) = vn?1 log ZnujN;E;bc ; and Theorem 3.2 gives the result. 2 Let us say that two sequences (P1;n ) and (P2;n ) in P are asymptotically equivalent, and write P1;n  P2;n as n ! 1, if for all f 2 L 



nlim !1 P1;n (f ) ? P2;n (f ) = 0 : RConsidering again the empirical Mn (d!)Rn+k;! 2 P .

elds Rn in (5.1) we de ne the measures Mn Rn+k = 18

Lemma 6.2

Assuming (6.1) and (6.2), we have for each k  0 fn  Mn Rn+k  M  n as n ! 1 : M

If bc = per then, in addition, M n  Mn as n ! 1. Proof. We only prove the rst asymptotic equivalence. A similar but simpler argument shows that Mn Rn  M n , and for bc = per one has Mn Rn (f ) = Mn (f ) when f 2 L and n is so large that f only depends on !n. Eqs. (4.3) and (6.4) together with Proposition 6.1 imply that fn ) < 1 ; lim sup I (M

n!1

(6.5)

where again I = I1 is the mean relative entropy with respect to the Poisson point fn  Mn Rn as n ! 1, which is random eld Q = Q1 . Lemma I.5.7 thus yields that M the case k = 0. For general k, the argument in the proof of this lemma shows that for each f 2 L and arbitrary a > 0 fn (f ) ? Mn Rn+k (f )j  o(1) + c M fn;k (N~` ; N~`  a) : (6.6) jvn?+1 k vnM

Here ` and c are such that f depends only on !` and (2.11) holds (where N~` was 1 fn;k 2 P is the average of the measures M per  #? de ned), and M x with x 2 n+k , n;k per where Mn;k is the measure relative to which the con gurations in the disjoint blocks fn;0 = n+k + (2n + 2k + 1)i; i 2 Zd , are i.i.d. with distribution Mn . (In particular, M per f Mn , and for k  1 there are Mn;k {almost surely no particles in the corridors at the boundaries of the blocks.) To complete the proof we therefore need to show that a can be chosen in such a way that the last term in (6.6) becomes arbitrarily small, uniformly in n. By Lemma I.5.2, fn;k )n0 in the place this would follow if we knew that (6.5) holds for the sequence (M f of (Mn )n0 . But this is indeed the case because

dQn / dQn+k = 1fNn+k ?Nk =0g exp[(vn+k ? vn )c(1)] and therefore fn;k )  I (Mn ; Qn+k ) = I (Mn ; Qn ) + (vn+k ? vn )c(1) : v n I (M fn;k )n0 thus follows as in the case k = 0. 2 Assertion (6.5) for the sequence (M

We are now prepared for the rst step in the proof of Theorem 3.3. Proof of Theorem 3.3 under condition (6.2). By (6.4) and Lemmas 6.1 and 4.2, the fn )n0 is sequentially compact, and every accumulation point P satis es sequence (M fn ) = N=vn !  and (
) is continuous, we have (P ) = . S (P )  s(; "). Since (M To show that U (P )  " we use Lemma 6.2. For each k, P is also an accumulation point of (Mn Rn+k )n0 , and

U (Mn Rn+k ) = Mn(U (Rn+k )) = vn?+1 k Mn(Hn+k;per) because of (5.2). By Lemma 5.2, for each  > 0 we can nd some k such that Mn (Hn+k;per)  (" + )vn for suciently large n. Since U is lower semicontinuous, 19

it follows that U (P )  ". In fact, U (P ) = " because otherwise S (P ) < s(; ") by the strict monotonicity of s(;
). Hence P 2 M;" . fn ) carry over to the asympBy Lemma 6.2, the above properties of the sequence (M totically equivalent sequence (M n ) and, in the case bc = per, to (Mn ). This establishes Theorem 3.3 under assumption (6.2). 2 We now turn to the case when (6.2) fails, which means that u tends to zero exponentially with vn . Since this includes the case u = 0, we rst need a proper de nition of the microcanonical distributions Mn0jN;E;bc on the energy surfaces fHn;bc = E g. As we have said in Section 2.3, it is sucient to have a de nition of Mn0jN;E;bc(f ) when f 2 L is xed and n is suciently large. Such a de nition can be obtained in analogy to (5.6). We still assume (6.1).

Lemma 6.3 For each m  1 there exists a number n(m) such that, for all n  n(m) and 0 < u  1, MnujN;E;bc is well-de ned and the following holds: For any f 2 L which

only depends on the particle con guration in the union  = (f ) of at most m unit cells, the limit Mn0jN;E;bc(f ) = ulim Mu (f ) !0 njN;E;bc exists and equals Zn0 jN;E;bc(f )=Zn0 jN;E;bc , where X

Zn0 jN;E;bc(f ) = Z

kN ?4=d

Z

L\n jk (d!) f (!) cN ?k (N ? k)(d=2)

(N ?k)d=2?1  : (6.7)  L^ nnjN ?k (d^) E ? H kin(!) ? Hn;potbc(^! [ ^) +

In the above, the Lebesgue measures L\n jk and L^ n njN ?k are de ned in obvious analogy to (2.12) and (5.5), and cN ?k is as in (5.6). Proof. Note rst that, by Remark 5.8, ZnujN;E;bc  u Zn0 jN;E ?u;bc > 0 when 0 < u  1 and n is suciently large. This means that for large n the measures MnujN;E;bc are well-de ned. It follows straight from the de nitions that ZE

0

Z (f ) ds = LnjN (f ; E ? u < Hn;bc  E; N  N ? 4=d) E ?u njN;s;bc

for all u > 0, and the condition k  N ? 4=d in (6.7) ensures that Zn0 jN;s;bc(f ) is a continuous function of s. The lemma will therefore be proved once we have shown that

fHn;bc  E g \ njN  fN  N ? 4=dg for n large enough. So let ! 2 njN and suppose that N (!) > N ? 4=d  N .Then Tn(!)  N 2 =m and therefore Hn(!)  a N 2 =m? b N . By an estimate similar to Lemma II.4.2, we nd a constant c > 0 such that

Hn;bc (!) ? Hn (!) 

(

?cNt if bc 2 (t) ?cN 2 =vn if bc = per.

20

Combining these estimates with (6.1) we see that Hn;bc (!) > E when n is large enough, and the lemma follows. 2 Our strategy for the extension of Theorem 3.3 to the case when (6.2) fails is a comparison of MnujN;E;bc with MnsjN;E;bc for some s  u satisfying vn?1 log? s ! 0. This comparison is based on the monotonicity properties of Zn0 jN;E;bc(f ) and the next lemma.

Lemma 6.4 Suppose that u ! 0 as n ! 1 and (6.1) holds. Then Zn0 jN;E+u;bc / Zn0 jN;E;bc ! 1 as n ! 1: Proof. The ratio under consideration is certainly larger than 1. To obtain an upper estimate we consider the function

n(s) = log Zn0 jN;s;bc for values of n and s for which Zn0 jN;s;bc > 0. Computing the derivatives as in Lemma 5.7 and using the Cauchy{Schwarz inequality we obtain the estimate (1=n0 )0  2=(Nd ? 2)  aN ; cf. Ref. 21. Hence

1=n0 (s) ? 1=n0 (t)  aN (s ? t) when s  t, and the mean value theorem gives

vn

.

(6.8)



n(E + vn ) ? n(E ) ? 1=n0 (E )  vn aN

for each  > 0. Letting n ! 1 and  ! 0 we thus obtain from Remark 5.8 lim sup n0 (E )  @+ s(; ")=@" :

(6.9)

The expression on the right side is the right derivative with respect to " (which exists by concavity). Averaging over t in (6.8) we obtain 1=n0 (E + u) ? u / (n (E + u) ? n (E ))  u aN :

(6.10)

Together with (6.9) (for E + u in place of E ) this gives lim sup(n (E + u) ? n (E ))  lim sup u n0 (E + u) = 0 which is the desired estimate. 2 Remark 6.5 (6.10) also gives a counterpart to (6.9) which, together with the concavity of s(;
), implies that s(;
) is dierentiable and

d log Z 0 @s(; ")=@" = lim dE njN;E;bc :

The same argument also shows that the last identity holds without the prime on the righthand side. 2 21

The following proposition asserts that microcanonical distributions on thin and on slightly thicker energy shells are asymptotically equivalent.

Proposition 6.6 Suppose (6.1) holds and u; s 2 [0; 1] vary with n in such a way that 0  u  s ! 0 and vn?1 log s ! 0. Then M nujN;E;bc  M nsjN;E;bc and MnujN;E;per  MnsjN;E;per as n ! 1 : Proof. We drop all indices which are not necessary. We need to show that, for given f 2 L, M nujbc (f ) ? M nsjbc (f ) ! 0 and Mnujper (f ) ? Mnsjper(f ) ! 0 as n ! 1. We can assume without loss that f  0. Let ` and c be such that f (!) = f (!` ) and (2.11) holds, n(v`+1 ) be chosen according to Lemma 6.3, and n  n(v`+1 ). We de ne

an (u; f )  anjE (u; f ) =

Z1 0

Zn0 jE?ru(f ) dr

and an (u) = an (u; 1) for the constant function 1. Then Mnu (f ) = an (u; f )=an (u). (Note that an (u)  Zn0 jE ?u > 0 for large n, by Remark 5.8.) It follows from (6.7) that Zn0 jE (f ) is increasing in E and its increments are increasing in f . Hence 0 



.

an (u; g) ? an (s; g) an(u)  .  an(0; h) ? an(s; h) an(s)

whenever g  h. Let a > 0 be a constant to be chosen later. We apply the preceding inequality to the two functions g1 = f 1fN~` ag and g2 = f 1fN~` >ag and the corresponding h1 and h2 that are obtained by estimating f according to (2.11). After a spatial averaging we arrive at the estimate 0  M nu (f ) ? rn M ns (f )  c(1 + a)(qn ? 1) + c q~n M nsjE+s(N~`; N~` > a) : Here 1  rn  an (s)=an (u)  Zn0 jE ?s=Zn0 jE ! 1 as n ! 1 by Lemma 6.4, and similarly 1  qn  an (0)=an (s)  Zn0 jE =Zn0 jE ?s ! 1 and q~n  Zn0 jE +s=Zn0 jE ?s ! 1 as n ! 1. By Lemma 6.2, fs lim sup M nsjE +s(N~` ; N~` > a)  lim sup M njE +s(N~` ; N~` > a) ;

and (6.5) and Lemma I.5.2 imply that the last expression tends to zero as a ! 1. Combining these estimates we arrive at the desired conclusion. In the case bc = per we simply omit the spatial averaging in the preceding argument. 2 Rest of the proof of Theorem 3.3. It is sucient to consider suitable sub{ subsequences of arbitrary subsequences. We can therefore assume that min(u; 1) converges to a limit q. If q > 0 we are in the case (6.2) which was already treated. In the alternative case, we can choose a sequence (s) as in the hypothesis of Proposition 6.6, and the validity of the theorem for the sequences (M nsjbc ) and (Mnsjper ) carries over to the sequences (M nujbc ) and (Mnujper ). 2

22

7 The variational principle for Gibbs measures This nal section contains the proof of Theorem 3.4. To obtain a convenient expression for the excess mean free energy we rst need the grand canonical analogue of Theorem 3.2, namely the existence and variational characterization of the pressure.

Proposition 7.1

Let z; > 0 and p(z; ) be de ned by (3.11). Then ?1 log n;z; ;bc = p(z; )

nlim !1 vn

for each bc 2  [ fperg, and the convergence is uniform in bc 2 (t) [ fperg for each t > 0. Proof. In the pure positional setting of particles without momentum, this result was established in Ref. II. The present case can be reduced to the positional one as follows. For P 2 P we let P^ = P (! : !^ 2
) denote the distribution of the con guration of particle positions in ^ , the set of all locally nite subsets !^ of Rd . In particular, if Qz; is the2 Poisson point random eld on Rd  Rd with intensity measure (dx; dp) = z dx e? jpj dp (which amounts to setting  (p) = ? log z + jpj2 in (4.1)) then Q^ z; is the Poisson point random eld on Rd with particle density c(z; ) = z c( ), and we can write pot n;z; ;bc = exp[vn c(z; )] Q^ z; n (exp[? Hn;bc ]) because Hn;potbc(!) is a function of !^ . Theorems II.2 and II.3 therefore imply that h

i

pot ?1 ^ z; U pot (P^ ) + I^z; (P^ ) ; nlim !1 vn log Qn (exp[? Hn;bc ]) = ? Pmin 2P 

(7.1)

where I^z; (P^ ) = limn!1 vn?1 I (P^n ; Q^ z; n ) is de ned in analogy to Lemma 4.1. On the other hand, (3.10) and (4.2) show that h

i

p(z; ) = c(z; ) ? Pmin U pot (P ) + Iz; (P ) ; 2P 

(7.2)

where Iz; is the mean relative entropy with respect to Qz; . We thus only need to show that the minima on the right-hand sides of (7.1) and (7.2) coincide. This will follow once we have shown that, for each P 2 P , I^z; (P^ )  Iz; (P ) with equality when P is Maxwellian, in that Z P = P^ (d!^ ) ? (^!;
) : Here ? (^!;
) is the conditional distribution of Qz; given !^ , i.e., the distribution of f(x; Yx ) : x 2 !^ g, where the Yx are i.i.d. centered normal with variance 1=2 . To verify this inequality we can assume that Iz; (P ) < 1. Then, for all n, Pn  Qz; n !) = ? (^!; fn ). Thus, by Jensen's with a density fn , whence P^n  Q^ z; n with density f^n (^ inequality, ^ z; (f^ log f^n ) I (P^n; Q^ z; n ) = Q Zn n !; fn log fn) = I (Pn; Qz;  Q^ z; n (d!^ ) ? (^ n ) 23

with equality when fn(!) only depends on !^ . This gives the desired relationship between I^z; and Iz; . 2 The preceding proposition shows that the excess mean free energy can be characterized as a mean relative entropy relative to the grand canonical Gibbs distributions with free boundary condition.

Corollary 7.2

For all z; > 0 and P 2 P ,

?1 Fz; (P )  Fz; (P ) + p(z; ) = nlim !1 vn I (Pn ; Gn;z; ;free) :

Proof. we have

Since Gn;z; ;free is equivalent to Ln with the density appearing in (2.17),

I (Pn ; Gn;z; ;free) = ?Sn(P ) ? P (Nn ) log z + P (Hn) + log n;z; ;free for all n, so that the corollary follows from Proposition 3.1, Eqs. (3.1) and (3.10), and Proposition 7.1. 2 We now turn to the proof of Theorem 3.4. We can assume without loss that

z = = 1 (because this only amounts to a rescaling of the position and momentum spaces together with a rescaling of '), and we shall drop all indices referring to these parameters. In particular, we write Gn resp. Gn; for the grand canonical Gibbs distribution in n with free boundary condition resp. bc =  . We shall also need the Gibbs distributions in more general sets  in place of n , which will be denoted by G resp. G; . The sets  and
considered below will always be nite unions of the unit cells C + i; i 2 Zd . Our rst aim is the proof of that part of the variational principle which is essential for the equivalence of ensembles, namely that all minimizers of the free energy are Gibbsian. For this we shall need an estimate of the expected variational distance Z

P (d ) kG0; ? G0;k k

(relative to certain measures P ) between the Gibbs distributions in 0 = C with boundary conditions  and k =  \ (k  Rd ). Clearly, if ' has nite range then this distance vanishes as soon as k exceeds the range of '. The general case will be treated in Lemma 7.4; the next lemma serves as preparation. For each i 2 Zd we write i = (d(C; C + i)), where is as in assumption (A1) and d(C; C + i) is the distance of the cells C and C + i. We also set = Pi2Zd i and ~i = i =2 so that P d ~i = 1=2. Note that ~0 = 0. i2Z

Lemma 7.3

For every exponent  1 there exists a constant a < 1 such that

G0; (N0 )  a + for all  2  .

X

i2Zd

24

~i NC +i ( )

Proof. This is a version of Lemma 2 of Dobrushin(25). For completeness we P indicate the proof. Let b = i2Zd ~i NC +i ( ). It is easy to see that b < 1 for  2  (cf. Lemma II.4.2), and we only need to show that

G0; (N0 ; N0 > b )  a :

(7.3)

In view of (5.3) and (2.1), 0 H0pot ;  h(N0 ) ? b N0 ? 2 N0 b :

On the other hand, 0;  1. The left-hand side of (7.3) is therefore at most h i X ` exp ?h(`) + b0 ` + 2 `2 c(1)` `>b

/ `! ;

and the corresponding sum over all `  0 is nite because h(`)=`2 ! 1 as ` ! 1. 2

Lemma 7.4

Let P 2 P be such that tP  P (NC2 ) < 1. Then for any  > 0 we can nd a number k0 such that for all k  k0 Z

P (d ) kG0; ? G0;k k < 

and, for each , the same inequality holds when P is replaced by Z

PG  P (d) G; ( :  [ ( n   Rd ) 2
) : Proof. A standard estimate (see Ref. 24, p. 33, proof of (b)) and a symmetry argument yield for each  2  

.

kG0; ? G0;k k  2 L0 j exp[?H0; ] ? exp[?H0;k ]

max(0; ; 0:k ):

Decomposing the integral above into parts according to the sign of H0; ? H0;k and using (2.1) we obtain the upper bound 2 [G0; (N0 ) + G0;k (N0 )]

X

i2Zd n

i NC +i ( ) k

when k  r('). Applying Lemma 7.3 for = 1, integrating over  and using the Cauchy-Schwarz inequality we nd Z

P (d ) kG0; ? G0;k k  (4a1 t1P=2 + 2tP )k ;

P

where k  i2Zd nk i tends to zero as k ! 1. This proves the rst assertion. The same kind of estimate also gives the second assertion, provided we can nd a constant t < 1 such that PG (NC2 +i )  t for all  and all i 2 Zd . To show this we use again an idea of Dobrushin(25) . Since PG(NC2 +i ) = tP when i 2 Zd n  we only need to consider the case when i 2 Zd \ . We de ne

t = max PG(NC2 +i ) : d i2Z \

25

In view of the consistency of the Gibbs distributions and by Lemma 7.3 we have for each i 2 Zd \ 

PG(NC2 +i )

Z

PG (d ) GC +i; (NC2 +i ) X ~i?j PG (NC2 +j )  a2 +

=

j 2Zd

and therefore t  a2 + 21 max(t ; tP ). On the other hand, t < 1 because Z

t  P (d ) G; (N2 )  a0 +

X

i2Zd n

(d(C + i; )) P (NC2 )

for some a0 = a0 () < 1 by an obvious variant of Lemma 7.3. It follows that t  t  max(tP ; 2a2 ), and the proof is complete. 2 After these preparations we can enter into the proof that the minimizers of the mean free energy are tempered Gibbs measures. We follow the well-known idea of Preston(23;24) . Let P 2 P be such that F (P ) = 0. Then U pot (P ) < 1 and thus tP = P (NC2 ) < 1. Hence Lemma 7.4 is applicable. Moreover, the ergodic theorem (for the multidimensional discrete translation group (#i )i2Zd ) shows that P (  ) = 1, i.e., P is tempered. On the other hand, we conclude from Corollary 7.3 that Pn  Gn when n is large enough. By translation invariance this means that, for all suciently large cubes , P is absolutely continuous with respect to G with a density g . Here P is the restriction of P to the events in , i.e., the image of P under the mapping ! ! ! = ! \ (  Rd ), and G is the Gibbs distribution in  with free boundary condition. In particular, for any
  we consider the restriction G;
= (G)
of G to the events in
, and we have that P
 G;
with density Z

g;
( ) = Gn
;
(d!) g (! [ 
) : The crucial consequence of the assumption F (P ) = 0 is the following.

Lemma 7.5 Let P 2 P be such that F (P ) = 0, and let k  1 and  > 0 be given. Then there exist two sets
;   Rd such that k 
  and 



G jg;
? g;
nC j <  : Proof. In view of the argument in Ref. 24, p. 324, Step 2, it is sucient to show that for any  > 0 we can nd
and  such that k 
  and

I
(P ; G ) ? I
nC (P ; G ) <  :

(7.4)

Here we write I
(
;
) for the relative entropy of measures that are restricted to the events in
. Using Corollary 7.2 we see that there exists some n  k such that vn?1 I (Pn ; Gn)  =2d vk . As in Ref. 24, p. 324, Step 1, we choose an m with md vk  26

vn  (2m)d vk and disjoint translates k (j ) =  + i(j )  n; i(j ) 2 Zd ; 1  j  md . Writing
(j ) = k (1) [ : : : [ k (j ) we nd md h

X m?d

j =1

i

I
(j) (P ; Gn ) ? I
(j)n(C +i(j)) (P ; Gn)

 m?d I
(md )(P ; Gn ) <  : So for at least one j the corresponding term in the sum above is less than , and by translation invariance we obtain (7.4) for  = n ? i(j ) and
=
(j ) ? i(j ). 2 We are now ready for the rst direction of the variational principle. Proposition 7.6 Each P 2 P with F (P ) = 0 is a tempered Gibbs measure. Proof. We saw already that P is tempered and tP < 1. We need to establish Eq. (2.18) for any measurable f  0. We can clearly assume that f 2 L and 0  f  1. For given  > 0 we let k be so large that f only depends on the particles in k and the conclusions of Lemma 7.4 hold, and we determine
and  according to Lemma 7.5. Consider the term on the left-hand side of Eq. (2.18) for n = 0. We replace its inner integral Z G0 f ( )  G0; (d!) f (!0 [  n 0) by f~k ( )  G0 f (k ). Since Rf only depends on the particles in k , this replacement leads to an error of at most P (d ) kG0; ? G0;k k < . Since f~k only depends on the particles in k n C 
n C and P  G on the events which occur in this set, P (f~k ) = G (g;
nC f~k ) : Changing f~k in the last term back into G0 f gives an error of at most Z

G(d ) g;
nC ( )kG0; ? G0;k k =

Z

PGn(
nC ) (d )kG0; ? G0;k k <  :

(For the equality above we rst use the fact that Gn(
nC );!
nC is the conditional distribution of G given the con guration !
nC in
n C , and then that P has density g;
nC relative to G on events occurring in
n C .) As a next step we observe that

G (g;
nC G0 f ) = G(g;
nC f ) because g;
nC does not depend on the particles in 0 = C and G0; is the suitable conditional distribution of G. According to the choice of
and  we may nally replace g;
nC by g;
with an error less than . In this way we arrive at the term G(g;
f ) = P (f ). We have thus shown that

jP (G0f ) ? P (f )j < 3 ; and this gives (2.18) for n = 0. To obtain (2.18) for general n we can either apply Theorem (1.33) of Ref. 24, or simply repeat the preceding argument on a new spatial scale. 2 27

We nally come to the converse part of the variational principle.

Proposition 7.7

F vanishes on the set of tempered Gibbs measures.

Proof. We might deduce this result from the large deviation principle for tempered Gibbs measures in Ref. II by exploiting the fact that a good rate function in a large deviation principle vanishes at the underlying measure whenever this ergodic. But we prefer a more direct argument which requires only the somewhat weaker Proposition 7.1. Let P be a tempered Gibbs measure. We need that tP = P (NC2 ) < 1. This follows either from subtle direct analysis, namely the superstability estimates of Ruelle(20) . Or from general theory by noting that (by the ergodic decomposition of translation invariant tempered Gibbs measures(24) and the fact that F is measure ane) P can be assumed to be ergodic, in which case tP < 1 follows from the temperedness by means of the ergodic theorem. We need to show that the limit in Corollary 7.2 vanishes. We x any n  0. By (2.17) and (2.18), Pn  Gn with density

gn(!) = =

Z

Z

P (d ) dGn; =dGn(!) P (d ) exp[Hn(!) ? Hn; (!)] n / n; :

Therefore we can write, using Jensen's inequality, (2.18), and the lower regularity of ',

I (Pn ; Gn) = Gn(Z gn log gn)   Gn P (d ) dGn; =dGn log dGn; =dGn =

Z

 tP

P (d!) [Hn (!) ? Hn;! (!) + log n ? log n;! ] X

i2Zd \n ;j 2Zd nn

i?j + log n ?

Z

P (d ) log n; :

After division by vn , the rst of the last three terms converges to 0, and Proposition 7.1 ensures that the second term converges to p(1; 1). On the other hand we have n;  1, so that the same proposition together with Fatou's lemma and the temperedness of P implies that Z ?1 P (d ) log n;  p(1; 1) : lim inf v n!1 n It follows that F (P ) = 0, and the proof is complete. 2

References 1. H.O. Georgii and H. Zessin, Large deviations and the maximum entropy principle for marked point random elds, Probab. Theory Relat. Fields 96:177{204 (1993). 2. H.O. Georgii, Large deviations and the equivalence of ensembles for Gibbsian particle systems with superstable interaction, Probab. Theory Relat. Fields 99:171{195 (1994). 3. D. Ruelle, Statistical Mechanics. Rigorous results (Benjamin, New York, 1969).

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4. O.E. Lanford, Entropy and equilibrium states in classical statistical mechanics, in Statistical Mechanics and Mathematical Problems, A. Lenard, ed., Lect. Notes Phys. 20 (Springer, Berlin, 1973). 5. A. Martin-Lof, Statistical Mechanics and the Foundations of Thermodynamics, Lect. Notes Phys. 101 (Springer, Berlin, 1979). 6. A.E. Khinchin, Mathematical foundations of statistical mechanics (Dover, 1949). 7. A.M. Hal na, The limiting equivalence of the canonical and grand canonical ensembles (low density case), Math. USSR Sbornik 9:1{52 (1969). 8. R.A. Minlos and A. Haitov, Limiting equivalence of thermodynamic ensembles in case of one-dimensional systems, Trans. Moscow Math. Soc. 32:143{180 (1975). 9. R.L. Dobrushin and B. Tirozzi, The central limit theorem and the problem of equivalence of ensembles, Commun. Math. Phys. 54:173{192 (1977). 10. M. Campanino, G. Del Grosso, and B. Tirozzi, Local limit theorem for Gibbs random elds of particles and unbounded spins, J. Math. Phys. 20:1752-1758 (1979). 11. A. Martin{Lof, The equivalence of ensembles and the Gibbs phase rule for classical lattice systems, J. Statist. Phys. 20:557-569 (1979). 12. R.L. Thompson, Equilibrium states on thin energy shells, Memoirs Amer. Math. Soc. 150 (1974). 13. H.O. Georgii, Canonical Gibbs measures, Lect. Notes Math. 760 (Springer, Berlin, 1979) 14. M. Aizenman, S. Goldstein, and J.L. Lebowitz, Conditional equilibrium and the equivalence of microcanonical and and grandcanonical ensembles in the thermodynamic limit, Commun. Math. Phys. 62:279{302 (1978). 15. C. Preston, Canonical and microcanonical Gibbs states, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 46:125{158 (1979). 16. M. Pirlot, Generalized canonical states, Ann. Sci. Univ. Clermont-Ferrand II. Probab. Appl. 4:69{91 (1985). 17. P. Vanheuverzwijn, Discrete lattice systems and the equivalence of microcanonical, canonical and grand canonical Gibbs states, Commun. Math. Phys. 101:153{172 (1985). 18. K. Matthes, J. Kerstan, and J. Mecke, In nitely Divisible Point Processes (Wiley, Chichester, 1978). 19. T. Tjur, Probability based on Radon measures (Wiley, Chichester, 1980). 20. D. Ruelle, Superstable interactions in classical statistical mechanics, Commun. Math. Phys. 18:127{159 (1970). 21. R. Rechtmann and O. Penrose, Continuity of the temperature and derivation of the Gibbs canonical distribution in classical statistical mechanics, J. Statist. Phys. 19:359{ 366 (1978). 22. O.E. Lanford and D. Ruelle, Observables at in nity and states with short range correlations in statistical mechanics, Commun. Math. Phys. 9:327{338 (1968). 23. C. Preston, Random elds, Lect. Notes Math. 534 (Springer, Berlin, 1976) 24. H.O. Georgii, Gibbs Measures and Phase Transitions (de Gruyter, Berlin, 1988). 25. R.L. Dobrushin, Gibbsian random elds for particles without hard core, Theor. Math. Phys. 4:705{719 (1970). 26. H.O. Georgii, Large deviations and maximum entropy principle for interacting random elds on Zd, Ann. Probab. 21:1845{1875 (1993).

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