\documentstyle[11pt]{article}
\author{Hans-Otto Georgii \\
{\small Mathematisches Institut der Universit\"at M\"unchen}\\
{\small Theresienstr. 39, D--80333 M\"unchen}}
\title{The equivalence of ensembles\\for classical systems of particles}
\textwidth 14.5cm \textheight 23cm
\oddsidemargin 7mm \evensidemargin -1mm \topmargin -2mm
\parindent 0.5cm
\renewcommand{\baselinestretch}{1.05}
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\sloppy
%%%%%Kurzformen allgemeiner Befehle
\def\skip{\medskip\smallskip}
\def\ba{\begin{array}}
\def\ea{\end{array}}
\def\be{\begin{equation} \label}
\def\ee{\end{equation}}
\def\bea{\begin{eqnarray*}}
\def\eea{\end{eqnarray*}}
\def\beal{\begin{eqnarray} \label}
\def\eeal{\end{eqnarray}}
\def\rf#1{(\ref{#1})}
\def\mmbox#1{\enspace\mbox{#1}\enspace}
\def\proof{{\em Proof.} \quad}
\def\proposition#1{\noindent{\bf Proposition #1} \quad}
\def\lemma#1{\noindent{\bf Lemma #1} \quad}
\def\theorem#1{\noindent{\bf Theorem #1} \quad}
\def\corollary#1{\noindent{\bf Corollary #1} \quad}
\def\remark#1{\noindent{\it Remark #1} \quad}
%%%%%Allgemeine Symbole
\def\R{{\sf R}}
\def\Rd{{\sf R}^d}
\def\Z{{\sf Z}}
\def\Zd{{\sf Z}^d}
\def\ti{\to\infty}
%%%%%Konfigurationsraum und W.masse
\def\om{\omega}
\def\omh{\hat\om}
\def\omn{\om_n}
\def\omhn{\omh_n}
\def\xinom{x\in\omh}
\def\xinomn{x\in\omhn}
\def\Lan{\Lambda_n}
\def\La{\Lambda}
\def\uLan{\cap \Lan}
\def\vni{v_n^{-1}}
\def\Om{\Omega}
\def\ominOm{\om\in\Om}
\def\thx{\vartheta_x}
\def\Pinv{{\cal P}_{\Theta}}
\def\pinp{P \in\Pinv}
\def\Palm{P^{\circ}}
\def\taul{\tau_{\cal L}}
\def\LN{L_{n|N}}
%%%%%Potential und Hamiltonfunktion
\def\ph{\varphi}
\def\eh{\mbox{\small $\frac{1}{2}$}}
\def\kin{\mbox{\scriptsize kin}}
\def\pot{\mbox{\scriptsize pot}}
\def\Hkin{H^{\kin}}
\def\Hpot{H^{\pot}}
%%%%Randbedingungen
\def\free{\mbox{\scriptsize free}}
\def\per{\mbox{\scriptsize per}}
\def\bc{\mbox{\rm\scriptsize bc}}
\def\nfree{n,\free}
\def\nper{n,\per}
\def\nbc{n,\bc}
\def\nzet{n,\zeta}
\def\BC{\Om^*\cup\{\mbox{per}\}}
\def\BCt{\Om(t)\cup\{\mbox{per}\}}
%%%%%Mikrokanonische Groessen
\def\eps{\varepsilon}
\def\rhoeps{\rho,\eps}
\def\micro#1{M_{n|N,E,#1}}
\def\amicro#1{\bar{M}_{n|N,E,#1}}
\def\mZ#1{Z_{n|N,E,#1}}
\def\Mrhoeps{{\cal M}_{\rhoeps}}
%%%%%Grosskanonische Groessen
\def\zbet{z,\beta}
\def\nzbet{n,z,\beta}
\def\grand#1{G_{\nzbet,#1}}
\def\Gzbet{{\cal G}_{\zbet}}
\begin{document}
\date{}
\maketitle
\begin{abstract}
For systems of particles in classical phase space with standard Hamiltonian,
we consider (spatially averaged) microcanonical Gibbs distributions in finite
boxes. We show that infinite--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.
{\bf Key words:} Classical statistical mechanics; microcanonical ensemble;
variational principle; entropy; pressure.
\end{abstract}
\section{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 infinite--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 finite boxes have
the same infinite--volume limits. In this paper we prove a version of the last
statement which is not affected by the possibility of phase transitions.
We consider the standard setting of classical statistical mechanics. A
particle in Euclidean space of dimension $d\ge 1$ is characterized by a pair
$(x,p)$, where $x\in\Rd$ is the position and $p\in\Rd$ the momentum. A
particle configuration (without multiple occupancies) is thus given by a set
$\om\subset\Rd\times\Rd$ of the form
$$
\om = \{(x,p_x): \xinom \},
$$
where $\omh$, the set of occupied positions, is a locally finite subset of
$\Rd$, and $p_x\in\Rd$ is the momentum of the particle at position $x$. The set
of all such configurations $\om$ is denoted by $\Om$. We assume that the formal
Hamiltonian has the standard form
\beal{Hform}
H(\om) &\equiv& \Hkin(\om) + \Hpot(\om)\nonumber\\[2ex]
&=&\sum_{\xinom} |p_x|^2 + \eh\sum_{x,y\in\omh, x\ne y} \ph(x-y) \, ,
\eeal
where $\ph$ 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
finite boxes in the infinite volume limit. For simplicity we only consider
the half-open cubes $\Lan = [-n-1/2, n+1/2[^d$ of volume $v_n = (2n+1)^d, n\ge
0$, which will often considered as tori by assuming periodic boundary conditions.
We write $\micro{\per}$ for the microcanonical Gibbs distribution
for $N$ particles in $\Lan$ with the energy constraint $H_{\nper} \le E$, where
$H_{\nper}$ is the Hamiltonian in $\Lan$ with periodic boundary condition; see
\rf{Hper}. We also consider the microcanonical Gibbs distributions
$\micro{\per}^u$ on the energy shells $\{ E-u0$, and the microcanonical distributions $\micro{\per}^0$ on the energy
surfaces $\{H_{\nper} = E \}$; see \rf{micro}, \rf{micro0} and Lemma 6.3 for
precise definitions. 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 $\ph$ we will show that, in the limit as $n\ti$ and $N/v_n\to\rho,
E/v_n\to\eps$ for an admissible pair $(\rho, \eps)$ of particle and energy
densities, the sequence $(\micro{\per})_{n\ge 0}$ is relatively (sequentially)
compact in a fairly strong topology, and each accumulation point belongs to the
class $\Mrhoeps$ of all translation invariant probability measures
$P$ on $\Om$ which have expected particle density $\rho$ and mean energy
$\eps$ and maximize
the mean entropy under this constraint (Theorem 3.3). Clearly, $\Mrhoeps$
is contained in $\Gzbet$, the set of all $P$ that minimize the mean free energy
for a suitably chosen activity $z > 0$ and inverse temperature $\beta
> 0$. Our second main result is a variational principle stating that $\Gzbet$
consists of all tempered, translation invariant (grand canonical) Gibbs measures
for $\zbet$ (Theorem 3.4). These two results together imply that every
accumulation point of the microcanonical distributions $\micro{\per}$ is a
Gibbs measure (in the grand canonical sense) for suitable parameters $\zbet > 0$.
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 $\Gzbet$ contains a unique element
$P$, then
$$
\micro{\per}\to P
$$
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 $(\rhoeps)$.
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 $\Om$ (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
configurational 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 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 $\ph$
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 final 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 (first 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--L\"of$^{(11)}$ 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 different approach is to define microcanonical resp.\ canonical Gibbs
states directly in infinite 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 affected by the
presence of
phase transitions, but the underlying concept of equivalence --- though natural
in the context of infinite--volume time evolutions --- is not the traditional
concept which is the subject of the present paper.
\section{The setting}
\setcounter{equation}{0}
\subsection{The interaction}
We start with our assumptions on the pair interaction $\ph$, an even
measurable function from $\R^d$ to $\R\cup\{\infty\}$.\skip
\noindent (A1) \enspace $\ph$ is {\em regular}. That is, there exists a
decreasing function $\psi: [0,\infty[\to[0,\infty[$ and a number $r(\ph)<
\infty$ such that
\be{reg}\ba{rcrl}
\ph(x)&\ge& -\psi(|x|)&\mmbox{for all} x \in \R^d,\\[2ex]
\ph(x)&\le& \psi(|x|) &\mmbox{whenever} |x|\ge r(\ph)
\ea\ee
and
$$
\int_0^{\infty} \psi(s) s^{d-1} \,ds < \infty\,\,.
$$
\skip
\noindent (A2) \enspace Either $\ph =\infty$ on a neighborhood of the origin
({\em hard core case}). Or $\ph <\infty$ Lebesgue-almost everywhere, and $\ph$
is {\em non-integrably divergent at the origin}, in that there exists a decreasing
function $\chi:\,]0,\infty[\to[0,\infty[$ such that
\be{div}
\ph(x) \ge \chi(|x|) \mmbox{whenever} \ph(x)\ge 0
\ee
and
$$
\int_0^1 \chi(s)s^{d-1}\,ds = \infty\,\,.
$$
\skip
As is well-known$^{(3)}$, (A1) and (A2) together imply that $\ph$ is
{\em superstable}. By definition, this means that there exist constants
$a>0, b<\infty$ such that for all $n$
\be{ss}
H_n \ge a\,T_n - b\,N_n.
\ee
In the above, $H_n\equiv H_{\nfree}$ , the {\em Hamiltonian in the box $\Lan$
with free boundary condition}, is given by
$H_n(\om) = H(\om_n)$, where $H$ is as in \rf{Hform} and
$\om_n = \om\cap(\Lan\times\Rd)$ is the restriction of $\ominOm$ to $\Lan$;
$N_n(\om) = \mbox{card}(\omhn)$ is the particle number in $\Lan$; and
\be{Tn}
T_n = \sum_{i\in\Lan\cap\mbox{{\small\sf Z}}^d} \, N_{C+i}^2
\ee
is the sum of the squared particle numbers $N_{C+i}(\om)=\mbox{card}(\omh\cap
(C+i))$ in the disjoint unit cells $C+i$ in $\Lan$, where $C=\La_0=
[-1/2, 1/2[^d$ stands for the centered unit cube.
The superstability and regularity of $\ph$ will be sufficient as long as we
deal only with free or periodic boundary conditions. The latter lead to the
{\em periodic Hamiltonians}
\beal{Hper}
H_{\nper}(\om)&=&\sum_{\xinomn} |p_x|^2 + \eh\sum_{\xinomn, y\in\omh^{(n)}, y\ne x}
\ph(y-x)\nonumber\\[2ex]
&=&H_n(\om) + \eh\sum_{\xinomn, y\in\omh^{(n)}\setminus\Lan}\ph(y-x).
\eeal
Here $\om^{(n)}=\{(x+(2n+1)i, p) : (x,p)\in\omn, i\in\Zd\}$ is the periodic
continuation of $\omn$.
The non-integrable divergence of $\ph$ at the origin will become important
as soon as configurational boundary conditions are involved. For $t>0$ we define
\be{Omt}
\Om(t) = \{ T_n/v_n \le t \mmbox{for all} n\ge 0 \}.
\ee
The configurations in $\Om^* = \bigcup_{t>0}\Om(t)$ are called {\em tempered}.
If $\ph$ even has a hard core we use the same symbols $\Om(t)$ and $\Om^*$
to denote the set of all admissible hard-core configurations. (This avoids a
messy distinction of cases.) For each $\zeta \in \Om^*$ and $n\ge 0$ we let
\be{Hconf}
H_{\nzet}(\om) = H_n(\om) + \sum_{\xinomn, y\in\hat\zeta\setminus\Lan}\ph(y-x)
\ee
denote the {\em Hamiltonian in $\Lan$ with tempered boundary condition} $\zeta$.
The regularity of $\ph$ ensures that the last sums exists, cf. Lemma II.4.2.
\subsection{Translation invariant states}
Next we consider the class of possible states of our particle system. The
configuration space $\Om$ is equipped with the $\sigma$-algebra ${\cal F}$
which is generated by the counting variables $N(B):\om\to\mbox{card}(\om\cap B)$
for Borel sets $B\subset\Rd\times\Rd$. It is well-known$^{(18)}$ that
${\cal F}$ is the Borel $\sigma$-algebra for the Polish topology on $\Om$ that
is induced by the mappings $N(h):\om\to\sum_{\xinom}h(x,p_x), h:\Rd\times\Rd
\to\R$ continuous with compact support.
We consider the class ${\cal P}$ of all probability measures (``states'') $P$
on $(\Om,{\cal F})$ satisfying
\be{P}
\int P(d\om)\sum_{\xinomn}(1+|p_x|^2) < \infty \mmbox{for all} n,
\ee
as well as the class $\Pinv$ of all $P\in{\cal P}$ which are invariant under
the translation group $\Theta = (\thx)_{x\in\mbox{{\small\sf R}}^d}$ acting on $\Om$ via
$\thx\om=\{(y-x,p):(y,p)\in\om\}$. The mapping $(x,\om)\to\thx\om$ is known$^{(18)}$
to be measurable.
For each $\pinp$ there exists a number
$\rho(P)<\infty$, the {\em intensity} or {\em mean particle number} of $P$,
such that $P(N_{\La})=\rho(P)|\La|$ for all measurable $\La\subset
\Rd$; here we write $N_{\La}=N(\La\times\Rd)$ for the counting variable
associated to $\La$, $|\La|$ is the Lebesgue measure
of $\La$, and $P(f)=\int f dP$ stands for the integral of a function $f$
with respect to $P$.
Also, for $\pinp$ there exists$^{(18,1)}$ a unique finite measure $\Palm$ on
$\Rd\times\Om$, the {\em Palm measure} of $P$, such that
\be{Palm}
\int P(d\om)\sum_{\xinom}F(x,p_x,\thx\om) = \int dx \int \Palm(dp,d\om)
F(x,p,\om)
\ee
for all measurable functions $F\ge0$ on $\Rd\times\Rd\times\Om$. In particular,
$\Palm$ is supported on the set
$\{(p,\om): (0,p)\in\om\}$, and the first marginal $\mu_P = \Palm(\cdot\times
\Om)$ of $\Palm$ satisfies
\be{muP}
\int P(d\om) \sum_{\xinomn} f(p_x) = v_n \int f\, d\mu_P
\ee
for all $n\ge 0$ and measurable $f:\Rd\to[0,\infty[$. We call $\mu_P$ the
{\em momentum intensity measure}.
We introduce a topology $\taul$ on ${\cal P}$ (which is much finer than
the weak topology associated to the above-mentioned Polish topology on $\Om$)
as follows. Let ${\cal L}$ denote the class of all measurable functions
$f:\Om\to\R$ which are {\em local} and {\em tame} in the sense that
$f(\om)=f(\om_{\ell})$ and
\be{Lf}
|f(\om)| \le c \Bigl(1 + \sum_{\xinom_{\ell}} (1 + |p_x|)\Bigr)\equiv c\left(
1+\tilde{N}_{\ell}(\om)\right)
\ee
for some $\ell\ge 0, c < \infty$ and all $\ominOm$. The {\em topology $\taul$
of local convergence} is then defined as the weak$^*$ topology on ${\cal P}$
relative to ${\cal L}$, i.e., as the smallest topology making the mappings
$P\to P(f)\equiv\int f dP$ with $f\in{\cal L}$ continuous. In particular,
the mean particle density $\rho(P)$ and the mean momentum per particle $\int p\,
\mu_P(dp)/\rho(P)$ are continous functions of $\pinp$.
\subsection{The Gibbs ensembles}
For a fixed box $\Lan$ and particle number $N$ we consider the set
$$
\Om_{n|N} = \left\{\ominOm: \omh\subset\Lan, N_n(\om) = N \right\}
$$
of all $N$--particle configurations in $\Lan$, and the associated ``Lebesgue
measure'' (or ``Liouville measure'')
\be{LnN}
\LN(A) = \frac{1}{N!}\int_{\Lan^N} dx_1\ldots dx_N \int_{\mbox{{\small\sf R}}^{dN}}
dp_1\ldots
dp_N \, 1_A\left(\{(x_1, p_1),\ldots,(x_N, p_N)\}\right),
\ee
$A\in{\cal F}$. The {\em microcanonical distributions} are obtained by
conditioning $\LN$ on an energy shell of the form $\{ E-u0$
if $P(\Om^*)=1$ and, for all $n\ge0$ and measurable functions $f\ge0$ on $\Om$,
\be{GM}
P(f) = \int P(d\zeta)\int \grand{\zeta} (d\om)\, f(\omn\cup(\zeta\setminus
\zeta_n)).
\ee
Clearly, the form of our Hamiltonian implies that, relative to all $\grand{\zeta}$
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 $\omh$ of occupied positions is given. For functions $f$ that depend
only on $\omh$, \rf{GM} is equivalent to the equilibrium equations introduced by
Ruelle$^{(20)}$.
\section{Results}
\setcounter{equation}{0}
A basic ingredient of our results is the existence of the mean energy and mean
entropy of any $\pinp$. The {\em mean energy} is defined by
\be{U}
U(P) = \lim_{n\ti} \vni P(H_n).
\ee
This limit exists in $\R\cup\{\infty\}$. Indeed, \rf{muP} and \rf{P} show that
the kinetic contribution equals
\be{Ukin}
U^{\kin}(P) = \int |p|^2\mu_P(dp) = \vni P(H_n^{\kin}) < \infty
\ee
independently of $n$. On the other hand, it was shown in Theorem II.1 (under
the assumptions \rf{reg} and \rf{ss}) that the mean potential energy
\be{Upot1}
U^{\pot}(P) = \lim_{n\ti} \vni P(\Hpot_n)
\ee
exists in $[a\rho(P)^2-b\rho(P), \infty]$ and satisfies
\be{Upot2}
U^{\pot}(P) = \int\Palm(dp,d\om) \,\eh\sum_{0\ne\xinom}\ph(x)
\ee
if $P(N_C^2)<\infty$ and equals $+\infty$ otherwise. Moreover, \rf{Ukin},
\rf{Upot2} and Theorem II.1 show that the functions $U^{\kin}$ and $U^{\pot}$,
and thus also $U$, on $\Pinv$ are affine (even measure affine, in that they
are affine with respect to convex mixtures formed by arbitrary
probability measures on $\Pinv$)
and lower semicontinuous relative to $\taul$.
The {\em entropy} of a state $P\in{\cal P}$ in $\Lan$ is defined by
\be{Sn}
S_n(P) = \left\{\ba{cl} - P_n(\log f_n) & \mbox{if}\,\, P_n\ll L_n \mmbox{with
density} f_n\\ -\infty&\mbox{otherwise,}\ea\right.
\ee
where $P_n = P(\{\ominOm:\omn\in\cdot\})$ is the restriction of $P$ to $\Lan$.
The following proposition on the mean entropy of invariant states will be
proved in Section 4.
\skip
\proposition{3.1}{\em For each $\pinp$, the mean entropy}
$$
S(P) = \lim_{n\ti} \vni S_n(P)
$$
{\em exists in $\R\cup\{\infty\}$ and is a measure affine, upper semicontinuous
function of $P$. The ``energy-bounded" superlevel sets}
$$
\left\{\pinp : S(P)\ge -c, U(P)\le \eps\right\}
$$
{\em of $S$ (with $c,\eps\in\R$) are compact and sequentially compact in $\taul$.}\skip
We emphasize that $S$ fails to be upper semicontinuous with respect to the
coarser topology $\tau_{{\cal L}^b}$ that is associated to the class
${\cal L}^b$ of all {\em bounded}\, local functions, see
Example 4.3.
Our first main result concerns the existence and variational characterization
of the thermodynamic entropy density. To state it we define for $\rho\ge 0$
\be{eps0}
\eps_{\min}(\rho) = \inf \left\{ U(P): \pinp, \rho(P)=\rho, S(P)>-\infty\right\}.
\ee
By \rf{ss}, $\eps_{\min}(\rho)\ge a \rho^2 -b \rho$, and $\eps_{\min}(\cdot)$ is
clearly
convex. Also, $\eps_{\min}(\cdot)$ is finite and continuous on a maximal interval
$[0,\rho_{\max}[$, where $\rho_{\max}=\infty$ except when $\ph$ has a hard core.
These facts are proved in Lemma II.7.1 (which extends without difficulties to
the present setting of particles with momentum). We introduce the convex set
\be{Sigma}
\Sigma = \left\{(\rhoeps): 0 < \rho < \rho_{\max}, \eps > \eps_{\min}(\rho)\right\}
\ee
and the abbreviation $\log_- u = \min(0,\log u)$.
\skip
\theorem{3.2} (a)\quad {\em Let $(\rhoeps)\in\Sigma$ and $t>0$. If $n\ti$ and
$N, E, u$,} bc {\em run through any sequences such that $N/v_n\to\rho,
E/v_n\to\eps
, \vni\log_-u\to0$ and} bc $\in\BCt$, {\em the limit}
\be{slim}
s(\rhoeps) = \lim \vni \log \mZ{\bc}^u
\ee
{\em exists and admits the variational characterization}
\be{svar}
s(\rhoeps) = \sup\bigm\{ S(P): \pinp, \rho(P)=\rho, U(P)\le\eps\bigm\}.
\ee
(b) \quad {\em The function $s(\cdot,\cdot)$ on $[0,\infty[\times\R$ defined
by \rf{svar} is concave and upper semicontinuous. The set $\Sigma$ is the
interior of the effective domain $\{s(\cdot,\cdot)>-\infty\}$, $s(\rho,\cdot)$
is strictly increasing on $]\eps_{\min}(\rho), \infty[$ for each $\rho$, and the
condition $U(P)\le\eps$ in \rf{svar} can be replaced by $U(P)=\eps$.}
\skip
We note that the existence of the limit in \rf{slim} is already known for a
long time, at least for bc = free $^{(3,4)}$. So
our main point is its variational characterization \rf{svar}; it is the
microcanonical counterpart of the more familiar variational formula for
the pressure which appears in Proposition 7.1. For $(\rhoeps)
\notin\Sigma$, we still have the relation
$$
\limsup \vni\log \mZ{\bc}^u \le s(\rhoeps),
$$
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.
\skip
\theorem{3.3} {\em Suppose that $(\rhoeps)\in\Sigma, t>0, N/v_n\to\rho$ and
$E/v_n\to\eps$ as $n\ti$, and $u\in[0,\infty]$ and} bc $\in\BCt$ {\em are allowed
to vary with $n$. Then the sequences $(\micro{\per}^u)_{n\ge0}$ and
$(\amicro{\bc}^u)_{n\ge0}$ are (well-defined for large $n$ and) relatively
sequentially compact in the topology $\taul$, and every accumulation point
belongs to the set}
$$
\Mrhoeps = \bigm\{ \pinp: \rho(P)=\rho, U(P)=\eps, S(P)=s(\rhoeps)\bigm\}.
$$
\skip
Our final task is to identify the ``microcanonical equilibrium states''
in $\Mrhoeps$ as grand canonical
Gibbs measures. For $(\rhoeps)\in\Sigma$ let $\beta=\beta(\rhoeps)>0$ be
the derivative of $s(\rho,\cdot)$ at $\eps$. This derivative is known to
exist, see Rechtmann and Penrose$^{(21)}$ or Remark 6.5 below. The
differentiability with
respect to $\rho$ seems to be unknown, but by concavity we can anyway
find a number $z>0$ such that, for a suitable constant $p(\zbet)\in\R$, the plane
$$
(\rho',\eps')\to p(\zbet) + \beta\eps' - \rho' \log z
$$
is a tangent to $s(\cdot,\cdot)$ at $(\rhoeps)$. It then follows that for all
$\pinp$ with $U(P)<\infty$
$$
S(P) \le s(\rho(P),U(P)) \le p(\zbet) + \beta U(P) - \rho(P) \log z
$$
with equality when $P\in\Mrhoeps$. Introducing the {\em mean free energy}
\be{F}
F_{\zbet}(P) = \beta\,U(P) - \rho(P) \log z - S(P)
\ee
we find that $F_{\zbet}(P)\ge -p(\zbet)$ with equality for $P\in\Mrhoeps$.
Since $\Mrhoeps\ne\emptyset$, we conclude that
\be{p}
p(\zbet) = - \min\bigm\{ F_{\zbet}(P) : \pinp\bigm\},
\ee
which is ($\beta$ times) the {\em pressure} (see Proposition 7.1), and
\be{incl}
\Mrhoeps \subset \Gzbet \equiv \bigm\{\pinp : F_{\zbet}(P) = -p(\zbet)\bigm\}.
\ee
The following variational principle asserts that $\Gzbet$ coincides with
the set of all translation invariant tempered Gibbs measures. Together with
Theorem 3.3
and \rf{incl} this completes the proof of the equivalence of ensembles.
\skip
\theorem{3.4} {\em For all $\zbet > 0$, the set of all translation
invariant tempered Gibbs measures for $\zbet$ (as defined around \rf{GM})
coincides with
the set $\Gzbet$ on which the mean free energy $F_{\zbet}$ attains its
minimum $-p(\zbet)$.} \skip
This result (which is well-known for lattice systems$^{(22,23,24)}$)
will be proved in Section 7.
\skip
\remark{3.5} The preceding results obviously imply that the set $\Gzbet$ of
translation invariant tempered Gibbs measures is non-empty --- which was
proved first by Ruelle$^{(20)}$ and Dobrushin$^{(21)}$. In fact, this
follows already from (the proof of) Proposition 3.1 together with
Theorem 3.4: $F_{\zbet}$ has compact sublevel sets $\{\pinp:
F_{\zbet}(P)\le c\}, c\in\R$, and thus attains its minimum.
\skip
\remark{3.6} $\Gzbet$ also contains all accumulation points of the relatively
compact sequence $(\grand{\per})_{n\ge 0}$. This can be proved in complete
analogy to Theorem 3.3; compare Proposition II.7.4. Thus, in the absence of
phase transition when $\Gzbet = \{P\}$ we have
$$
P = \lim \micro{\per}^u = \lim \grand{\per}
$$
in $\taul$, and a similar statement holds for spatially averaged distributions
with boundary conditions in $\Om(t)$ for some $t$. This expresses the
equivalence of ensembles in terms of finite volume Gibbs distributions.
\section{The mean entropy of invariant states}
\setcounter{equation}{0}
In this section we prove Proposition 3.1. It is convenient to replace the
infinite reference measures $L_n$ in \rf{Sn} by consistent probability
measures, namely the (local restrictions of) Poisson point random fields.
Let $\tau:\Rd\to\R\cup\{\infty\}$ be any function satisfying $c(\tau)
\equiv\int e^{-\tau(p)} dp <\infty$ and
$Q^{\tau}$ the {\em Poisson point random field} with intensity measure
$\mu^{\tau}(dx,dp) = dx\,
e^{-\tau(p)}dp$. That is, $Q^{\tau}$ is the unique measure in $\Pinv$
relative to which the particle numbers $N(B_1),\ldots,N(B_k)$ in disjoint
phase space regions $B_1,\ldots,B_k\subset\Rd\times\Rd$ are independent and
Poisson distributed with parameters $\mu^{\tau}(B_i), 1\le i\le k$.
By definition, its restriction $Q_n^{\tau}$ to $\Lan$ has the Radon--Nikodym
density
\be{qn}
q_n^{\tau}(\om) = \exp[-\sum_{\xinomn}\tau(p_x) - v_n c(\tau)]
\ee
relative to $L_n$.
For any $\pinp$ we consider the relative entropy
$$
I(P_n;Q_n^{\tau}) = \left\{\ba{cl} P_n(\log h_n) &\mbox{if} \,P_n\ll Q_n^{\tau}
\mmbox{with density}h_n\\
\infty&\mbox{otherwise.}\ea\right.
$$
Clearly, $P_n\ll Q_n^{\tau}$ with a density $h_n$ if and only if $P_n\ll L_n$
with density $f_n = h_n q_n^{\tau}$. Hence we conclude from \rf{Sn} and
\rf{muP} that
\beal{SnI}
S_n(P) &=& -I(P_n;Q_n^{\tau}) - P(\log q_n^{\tau})\nonumber\\
&=& -I(P_n;Q_n^{\tau}) + v_n \mu_P(\tau)+v_n c(\tau).
\eeal
We will use the following fact. \skip
\lemma{4.1} {\em For all $\pinp$, the mean relative entropy}
$$
I_{\tau}(P) = \lim_{n\ti}\vni I(P_n;Q_n^{\tau})
$$
{\em exists in $[0,\infty]$. The function $I_{\tau}$ on $\Pinv$ is measure
affine, and its sublevel sets $\{\pinp:I_{\tau}(P)\le c\}, c\ge 0$, are
compact and sequentially compact in the topology $\tau_{{\cal L}^b}$ that
is induced by the class ${\cal L}^b$ of all bounded local functions. If
$\tau$ is such that}
$$
\int e^{c|p|-\tau(p)}dp < \infty
$$
{\em for all $c>0$ then the last assertion also holds in the topology $\taul$.}
\skip
\proof The first two sentences follow
from the analogous result for lattice models by identifying $\Om$ with
$(\Om_0)^{\mbox{{\small\sf Z}}^d}$, cf. Theorems (15.12) and (15.20) of
Ref.\ 24. The
last assertion is proved in Proposition I.2.6. $\Box$
\skip
We are particularly interested in the case when $\tau(p)=\beta|p|^2$
for some $\beta>0$. We then replace the index $\tau$ by $\beta$. In particular,
$Q^{\beta}$ is the ideal gas with particle density
$c(\beta) = (\pi/\beta)^{d/2}$ and Maxwellian momenta of variance $1/2\beta$,
and \rf{SnI} takes the form
$$
\vni\,S_n(P) = -\vni I(P_n;Q_n^{\beta}) + \beta\,U^{\kin}(P) + c(\beta).
$$
Since $U^{\kin}$ is finite, we conclude from Lemma 4.1 that
$\vni S_n(P)$ converges to
\be{SI}
S(P) = -I_{\beta}(P) +\beta\,U^{\kin}(P) + c(\beta)
\ee
which is a measure affine function of $P$.
It remains to establish the continuity properties of $S$. To this end,
we note first that a bound on the mean energy implies a bound on the mean kinetic
energy. Indeed, since
$$
U^{\pot}(P)\ge a\rho(P)^2-b\rho(P)\ge -b^2/4a
$$
by \rf{ss} and \rf{Upot1}, we have the implication
\be{epstilde}
U(P)\le\eps \Longrightarrow U^{\kin}(P)\le\tilde\eps\equiv \eps + b^2/4a.
\ee
The proof of Proposition 3.1 is therefore completed by the following lemma.
\skip
\lemma{4.2}{\em The mean entropy $S$ is upper semicontinous (relative to
$\taul$), and the restricted superlevel sets $\{ S\ge c, U^{\kin}\le\tilde
\eps\}$, $c, \tilde\eps\in\R$, are compact and sequentially compact.}\skip
\proof In view of \rf{SI}, the set in question is contained in the set
$\{I_{1}\le \tilde c\}$ with $\tilde c = -c+\tilde\eps+c(1)$, and the
latter set is compact and sequentially compact by Lemma 4.1. We thus
only need to show that
$\{S\ge c\}$ is closed, i.e., that $S$ is upper semicontinuous.
Applying \rf{SnI} to the function $\tau=|\cdot|$ we see that
$$
S(P) = -I_{|\cdot|}(P) + \mu_P(|\cdot|) + c(|\cdot|)
$$
for all $P$. But $P\to\mu_{P}(|\cdot|)$ is continuous by definition of $\taul$,
and Lemma 4.1 asserts that $I_{|\cdot|}$ is lower semicontinous relative to
$\tau_{{\cal L}^b}\subset\taul$. This gives the result. $\Box$\skip
We complete this section with an example showing that
$S$ fails to be upper semicontinuous relative to $\tau_{{\cal L}^b}$.\skip
\noindent{\em Example 4.3}\quad For each $k\ge 1$ let $B_k\subset\Rd$ be a Borel
set of volume $e^{k^2}/k$ which is disjoint from the unit cube $C$. Let $\tau$
and $\tau_k$ be such that $e^{-\tau}=1_C$ and $e^{-\tau_k}=1_C+e^{-k^2}1_{B_k}$.
$q_n^{\tau_k}$ converges to $q_n^{\tau}$
$L_n$-almost everywhere, and thus in $L^1(L_n)$. Hence $Q^{\tau_k}$ converges
in $\tau_{{\cal L}^b}$ (but not in $\taul$) to $Q^{\tau}$. By \rf{SnI},
\bea
S(Q^{\tau_k})&=&-\int\tau_k(p) e^{-\tau_k(p)}dp +c(\tau_k) \\
&=&k+1+1/k \ti
\eea
as $k\ti$, but on the other hand $S(Q^{\tau})=1$. $\Box$
\section{The thermodynamic entropy density}
\setcounter{equation}{0}
This section contains the proof of Theorem 3.2. To begin we consider the
function $s(\rhoeps)$ on $[0,\infty[\times\R$ defined by \rf{svar} (with
the convention $\sup\emptyset=-\infty$).\skip
\lemma{5.1} (a)\quad {\em The supremum in \rf{svar} is attained
whenever it is not $-\infty$.}
(b)\quad {\em $s(\cdot,\cdot)$ is concave and upper semicontinuous,
and its energy--bounded superlevel sets $\{(\rhoeps):s(\rhoeps)\ge c_1,
\eps \le c_2\}, c_1,c_2\in\R$, are compact.}
(c)\quad{\em The set $\Sigma$ in \rf{Sigma} is the interior of
$\{s(\cdot,\cdot)>-\infty\}$.}
\skip
\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 $\rho(\cdot)$. The concavity
is clear because $S$, $\rho(\cdot)$ and $U$ are affine.
Assertion (c) follows straight from the definitions. $\Box$\skip
Our main task is the proof of the convergence \rf{slim} towards the limit
$s(\rhoeps)$ defined by \rf{svar}. We shall proceed in three stages. In the
first 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 $\mZ{\bc}$ on the parameters $N$ and $E$.
The first stage is based on techniques from large deviation theory. A central
object in this theory is the {\em translation invariant empirical field}
\be{Rn}
R_{n,\om} = \vni\int_{\Lan}\delta_{\thx\om^{(n)}} dx \in\Pinv
\ee
of a configuration $\om$ in $\Lan$. Here $\om^{(n)}$ is as in \rf{Hper}.
We shall take advantage from the formula
\be{HperU}
H_{n,\per}(\om) = v_n\,U(R_{n,\om}),
\ee
which follows from \rf{Ukin}, \rf{Upot2} and the explicit formula (I.2.6)
for the Palm measure $R_{n,\om}^{\circ}$ of $R_{n,\om}$. The following
lemma allows us to proceed from periodic to
non-periodic boundary conditions.\skip
\lemma{5.2} (a) \quad{\em For any $\eps<\infty$ and $\delta>0$ there
exists an integer $k=k(\eps,\delta)$ such that for all $n$}
$$
\{H_n\le\eps v_n\}\cap\Om_n\subset\{H_{n+k,\per}\le(\eps+\delta)v_n\}.
$$
(b)\quad{\em For given $\eps<\infty$ and $\delta, t>0$ there exists some
$n(\eps,\delta,t)$ such that
for all $\zeta\in\Om(t)$ and $n\ge n(\eps,\delta,t)$}
$$
\{H_{n,\zeta}\le\eps v_n\}\subset\{H_n\le(\eps+\delta)v_n\}.
$$
\skip
\proof (a) If $H_n\le\eps v_n$ then, by \rf{ss},
$$
N_n^2/v_n \le T_n \le v_n\,\eps/a + b\,N_n/a
$$
and thus $T_n\le tv_n$ for some $t<\infty$ depending on $\eps$ (and $a, b$).
The result thus follows from Lemma II.4.3.
(b) Here we use the non-integrability of $\ph$ at $0$. By Lemma II.6.1 there
exists an increasing function $h:\Z_+\to[0,\infty[$ with $h(\ell)/\ell^2\to
\infty$ as $\ell\ti$, and a number $b'>0$ such that
\be{sss}
H_n\ge T_n^h - b'\,N_n
\ee
for all $n$, where
$$
T_n^h = \sum_{i\in\Lan\cap\mbox{{\small\sf Z}}^d}h(N_{C+i}).
$$
On the other hand, Lemma II.6.3 asserts that for given $\delta'>0$ and $n$
larger than some $n_0(\delta',t)$
\be{Hlow}
H_{n,\zeta}\ge H_n-\delta'T_n^h-\delta'v_n
\ee
for all $\zeta\in\Om(t)$. Suppose now that $H_{n,\zeta}\le\eps v_n$. \rf{sss}
and \rf{Hlow} (with $\delta'=1/2$) together then show that
$$
T_n^h\le(2\eps+1)v_n+2b'N_n
$$
whenever $n\ge n_0(1/2,t)$. Choosing $q$ such that $h(\ell)\ge\ell^2$ for $\ell
\ge q$ we see that
$$
N_n^2/v_n\le T_n\le T_n^h+q^2v_n.
$$
Combining this with the previous inequality we find that $T_n^h\le s\,v_n$ for
some $s=s(\eps)<\infty$ and all $n\ge n_0(1/2,t)$. Using \rf{Hlow} again
with $\delta'=\delta/(s+1)$ we can conclude. $\Box$ \skip
Here is the first stage in the proof of \rf{slim}.\skip
\proposition{5.3}{\em Let $\eps<\infty$ and $D\subset[0,\infty[$ be a
nondegenerate interval such that $(\rhoeps)\in\Sigma$ for some $\rho\in D$.
Let $t>0$ and consider, for each $n$,}
$$
a_{n|D,\eps,\bc}\equiv L_n(N_n\in v_nD, H_{\nbc}\le v_n\eps)
=\sum_{N\in v_nD}Z_{n|N,v_n\eps,\bc}\,\,,
$$
{\em where} bc$\,\in\BCt$ {\em may depend on $n$. Then}
$$
\lim_{n\ti}\vni\log a_{n|D,\eps,\bc} =\sup_{\rho\in D} s(\rhoeps).
$$
\skip
\proof 1)\enspace We first derive the upper bound. We start with the
case bc = per. Let $\beta>0$ be arbitrary. Eq.\ \rf{qn} with $\tau=
\beta|\cdot|^2$, \rf{HperU} and \rf{epstilde} yield
\bea
a_{n|D,\eps,\per} &=& Q_n^{\beta}\Bigl(1/q_n^{\beta}\,;\, \rho(R_n)\in D, U(R_n)
\le\eps\Bigr)\\
&\le&\exp[v_n\beta\tilde\eps+v_nc(\beta)]\,\,Q_n^{\beta}(R_n\in A),
\eea
where $A = \bigl\{\pinp:\rho(P)\in\bar D, U(P)\le\eps\bigr\}$ and
$\bar D$ is the closure of $D$. Since $A$ is closed in $\taul$, we
can use Theorem I.3.1 (with $Q=Q^{\beta}$) to obtain
\bea
\limsup\vni\log a_{n|D,\eps,\per}&\le&\beta\tilde\eps+c(\beta)-
\inf_{P\in A} I_{\beta}(P)\\
&=&\beta\tilde\eps + \sup_{P\in A}[S(P)-\beta\,U^{\kin}(P)]\\
&\le&\beta\tilde\eps + \sup_{\rho\in D} s(\rhoeps).
\eea
The second step comes from \rf{SI}, and in the last step we used
the positivity of $U^{\kin}$ and \rf{svar} together with the fact
that (in view of our assumption on $D$ and Lemma 5.1) the suprema
of $s(\cdot,\eps)$ over $D$ and $\bar D$
coincide. Letting $\beta\to0$ yields the upper bound in the case
bc = per.
Turning to the case bc = free, we let $D'$ be any interval
containing $D$ in its interior, $\delta>0$, and $k$ be chosen
according to Lemma 5.2 (a). Then
$$
a_{n|D,\eps,\free}\le a_{n+k|D',\eps+\delta,\per}
$$
whenever $n$ is so large that $v_{n+k}/v_n$ is sufficiently close
to 1. On the other hand, Lemma 5.1 (b) implies that
$$
\inf_{D'\supset D, \delta>0} \sup_{\rho\in D'} s(\rho,\eps+\delta) \le
\sup_{\rho\in D} s(\rhoeps)\,\,.
$$
The upper bound for bc = free thus follows from that for bc = per.
Finally, suppose bc = $\zeta \in\Om(t)$ for some $t>0$, and choose
any $\delta>0$. Lemma 5.2 (b) then shows that
$$
a_{n|D,\eps,\zeta}\le a_{n|D,\eps+\delta,\free}
$$
for sufficiently large $n$, and the desired upper bound follows from the previous case.
2)\enspace We now turn to the lower bound. Since $U^{\kin}\ge 0$ we obtain
as above
$$
a_{n|D,\eps,\per}\ge\exp[v_nc(\beta)]\,Q_n^{\beta}(R_n\in A^{\circ})
$$
for any $\beta>0$. Here $A^{\circ}=\{\rho(\cdot)\in D^{\circ}, U<\eps\}$ and
$D^{\circ}$ is the interior of $D$.
Note that $A^{\circ}$ 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 $\Phi$ there is our $U^{\pot}$.) Using
the notation
in the proof of this lemma (in particular $m=n+k$ for a fixed $k$) we can write
$$
U^{\kin}(P^{(n)}) = v_m^{-1}P^{(n)}(H_m^{\kin}) = v_m^{-2}\int_{\La_m}
\hat P^{(n)}(H_m^{\kin}\circ\thx)dx\,\,.
$$
Since $H_m^{\kin}$ is a sum of single--particle terms and $\hat P^{(n)}$ is
$\La_m$-periodic, the integrand above does not depend on $x$. Hence
$$\ba{rcccl}
U^{\kin}(P^{(n)}) &=& v_m^{-1}\hat P^{(n)}(H_m^{\kin})&=&v_m^{-1}P_n(H_n^{\kin}|
\Gamma_{q(n)})\\[2ex]
&\le&p_n^{-1}v_m^{-1}\,P_n(H_n^{\kin})&=&(v_n/p_nv_m)\,U^{\kin}(P)
\ea$$
and therefore $\limsup_{n\ti}U^{\kin}(P^{(n)})\le U^{\kin}(P)$. This is all
what is needed to extend Lemmas II.5.1 and II.7.2 to the present case. The result is
\bea
\liminf_{n\ti}\vni\log a_{n|D,\eps,\per}&\ge&c(\beta)-\inf_{P\in A^{\circ}}
I_{\beta}(P)\\
&=&\sup_{P\in A^{\circ}}\bigl[S(P)-\beta\,U^{\kin}(P)\bigr]\\
&\ge&\sup_{\rho\in D} s(\rhoeps) - \beta\,\tilde\eps\,\,.
\eea
In the last step we used \rf{epstilde} and the fact that the suprema of
$s(\cdot,\eps)$ over $D^{\circ}$ and $D$ coincide. Letting $\beta\to0$ 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 $= \zeta\in\Om(t)$ we argue as in the lines leading to Eq.\
(II.6.6). $\Box$
\skip
The second step in the proof of \rf{slim} consists in controlling the
variation of the
microcanonical partition functions $\mZ{\bc}$ with respect to $N$.
We need to distinguish the two cases in assumption (A2). \skip
\lemma{5.4} {\em Suppose that $\ph<\infty$ a.e.. Then for any given numbers
$\nu, t >0$ there exist constants $c, u>0$ and $n_0$ such that}
$$
\mZ{\bc}\ge c\, Z_{n|N-1,E-u,\bc}
$$
{\em whenever $N\le\nu v_n, E\in\R$}, bc $\in\BCt$ {\em and} $n\ge n_0$.\skip
\proof Let $\ph_+ =\max(\ph,0)$ be the positive part of $\ph$. Since $\ph<\infty$
a.e., we can find some $q>0$ such that
$$
\int \min(1,\ph_+(x)/q)dx\le 1/4\nu\,\,.
$$
Indeed, the integrand in the integral on the left side converges to
$1_{\{\ph_+=\infty\}}=0$ a.e. as $q\ti$, and the regularity assumption
(A1) ensures that the dominated convergence theorem is applicable. Next,
for each $n, N, \om\in\Om_{n|N-1}$ and
$(x,p)\not\in\om$ we can write
$$
H_{n,\bc}(\om\cup\{(x,p)\}) = H_{n,\bc}(\om)+|p|^2+\sum_{y\in\omh}\ph(y-x)
+h_{n,\bc}(x),
$$
where, for example,
$$
h_{n,\zeta}(x)=\sum_{y\in\hat\zeta\setminus\Lan}\ph(y-x)
$$
for $\zeta\in\Om(t)$. Using \rf{reg} and the arguments in Lemma II.4.2 we see
that there exists some $r\ge r(\ph)$ such that $h_{n,\bc}(x)\le1$ whenever
$x\in\La_{n-r}$, bc $\in\BCt$ and, in the case bc = per, $\om\in\Om_{n|N}$
with $N\le\nu v_n$. Let $n_0$ be so large that $v_{n-r}\ge v_n/2$ when $n\ge
n_0$. Then we have for arbitrary $n\ge n_0$ and $\om\in\Om_{n|N-1}$
\bea
\bigl|\bigl\{x\in\La_{n-r}: \sum_{y\in\omh}\ph(y-x)\le q\bigr\}\bigr|
&\ge&\int_{\La_{n-r}} \Bigl(1-\sum_{y\in\omh}\min(1,\ph_+(y-x)/q)\Bigr) dx\\
&\ge&v_{n-r}-(N-1)/4\nu \enspace\ge\enspace v_n/4\,\,.
\eea
Setting $u=q+2$ we thus obtain the final estimate
\bea
N\,\mZ{\bc}&\ge& \int L_{n|N-1}(d\om) \int_{\La_{n-r}} dx \int dp\\
&&\quad \raisebox{1ex}{1}\bigl\{H_{n,\bc}(\om)\le E-u, \sum_{y\in\omh}
\ph(y-x)\le q, |p|^2\le1\bigr\}\\
&\ge&(c_1v_n/4)\,Z_{n|N-1,E-u,\bc}\,\,,
\eea
where $c_1$ is the volume of the unit ball in $\Rd$. The lemma thus follows with
$c=c_1/4\nu$. $\Box$ \skip
If $\ph$ has a hard core we have an estimate in the opposite direction.\skip
\lemma{5.5} {\em Suppose $\ph=\infty$ on a neighbourhood of $0$. For given
$\nu>0$ and $\eps<\infty$ there exist constants $c, u >0$ and $n_0$ such that}
$$
\mZ{\bc}\ge c\,Z_{n|N+1,E-u,\bc}
$$
{\em for all $n\ge n_0, N\ge\nu v_n, E\le\eps v_n$, and} bc $\in\BC$.
\skip
\proof The hard--core property and the regularity \rf{reg} imply the existence
of a number $u>0$ such that
$$
\sum_{y\in\omh}\ph(y-x)\ge-u
$$
for all $x\in\Rd$ and all admissible hard--core configurations $\omh$. This
gives the estimate
\bea
(N+1)\,Z_{n|N+1,E-u,\bc}&\le&\int L_{n|N}(d\om)\int_{\Lan}dx\int dp\,\,1 \mbox{\raisebox{-1ex}{$\bigl\{H_{n,\bc}(\om)+|p|^2\le E\bigr\}$}}\\
&=&v_n\int dp \,\,Z_{n|N,E-|p|^2,\bc}\,\,.
\eea
In view of Lemma 5.7 below, the last expression is at most
$$
v_n\int dp\,\,e^{-\gamma|p|^2}\,\mZ{\bc}
$$
for some $\gamma=\gamma(\nu,\eps)>0$, provided $N\ge\nu v_n, E\le\eps v_n$
and $n$ is large enough.
The lemma is now obvious. $\Box$\skip
The next result marks the second stage in the proof of \rf{slim}.\skip
\proposition{5.6} {\em Let $(\rhoeps)\in\Sigma$ and $t>0$. In the limit as
$n\ti, N/v_n\to\rho, E/v_n\to\eps$ and} bc = bc$(n) \in\BCt$, $\vni\log\mZ{\bc}$
{\em converges to the function $s(\rhoeps)$ defined by \rf{svar}.}
\skip
\proof If $D\subset[0,\infty[$ is a non-degenerate interval containing
$\rho$ in its interior and $\eps'>\eps$ then $\mZ{\bc}\le a_{n|D,\eps',\bc}$
eventually. Proposition 5.3 and the upper semicontinuity of $s(\cdot,\cdot)$
thus imply that
$$
\limsup\vni\log\mZ{\bc}\le s(\rhoeps)\,\,.
$$
For the lower bound we consider first the case when $\ph<\infty$ a.e.. We
fix any $\nu>\rho$ and let $u$ and $00$ be so small that $\delta<\rho$ and $\eps-u\delta>\eps_{\min}(\rho)$.
For sufficiently large $n$ we have $N/v_n\le\nu$ and, by $k$--fold iteration of Lemma 5.4,
$$
\mZ{\bc}\ge c^k\,Z_{n|N-k,E-ku,\bc}\,\,.
$$
Averaging the right-hand side over all $k$ between 0 and $\delta v_n$ and
choosing any $\eps'\in]\eps_{\min},\eps-\delta[$ and a non-degenerate closed
interval $D\subset\, ]\rho-\delta,\rho[$ we obtain
$$
\mZ{\bc}\ge c^{\delta v_n}(\delta v_n)^{-1}\,a_{n|D,\eps',\bc}
$$
when $n$ is large enough. Proposition 5.3 thus gives
$$
\liminf\vni\log\mZ{\bc}\ge\delta\log c + s(\rho',\eps')
$$
for any $\rho'\in D$. Letting $\rho'\to\rho, \delta\to0, \eps'\to\eps$
in this order we can conclude. In the case when $\ph$ has a hard core
we can proceed analogously, using Lemma 5.5 instead of 5.4. $\Box$\skip
The final step in the proof of \rf{slim} is the consideration of energy shells.
To this end we need to control the $E$--dependence of the $\mZ{\bc}$'s. Let
$\hat L_{n|N}$ be the ``Lebesgue measure'' on the set $\hat\Om_{n|N}=\{\omh\subset
\Lan: \mbox{card}\,\omh=N\}$ of position--configurations of $N$ particles in
$\Lan$ (with the standard $\sigma$--algebra), that is,
\be{LnNhat}
\int f\,d\hat L_{n|N}= \frac{1}{N!}\int_{\Lan^N}dx_1\ldots dx_N f(\{x_1,\ldots,x_N\})
\ee
for measurable $f\ge0$ on $\hat \Om_{n|N}$. Note that $\Hpot_{n,\bc}$ can be
considered as a function of $\omh$.\skip
\lemma{5.7} {\em For all $n, N, E$ and} bc {\em we have}
$$
\mZ{\bc} = c_N \int \hat L_{n|N}(d\omh) \Bigl(E-\Hpot_{n,\bc}(\omh)\Bigr)_+^{Nd/2}\,\,,
$$
{\em where $c_N$ is the volume of the unit ball in $\R^{Nd}$. In particular,
$\mZ{\bc}$ is differentiable with respect to $E$ with derivative}
\be{Zstrich}
\mZ{\bc}' = c_N(Nd/2)\,\,\hat L_{n|N}\Bigl((E-\Hpot_{n,\bc})_+^{(Nd/2)-1}\Bigr)\,\,.
\ee
{\em which is sometimes called the structure function$^{(6)}$. Also, for every
$\nu>0$ and $\eps<\infty$ there exists a constant $\gamma= \gamma(\nu,\eps,t)>0$ such that}
\be{gamma}
Z_{n|N,E-u,\bc}\le e^{-\gamma u}\,\mZ{\bc}
\ee
{\em whenever $N\ge\nu v_n, E\le\eps v_n, u\ge0$}, bc $\in\BCt$ {\em for a
given $t>0$, and $n$ is large enough.}\skip
\proof We only need to comment on \rf{gamma}. Let $t>0$ be fixed. By the proof
of Lemma 5.2(b) there exists a number $c>-\eps$ (depending on $\eps$ and $t$)
such that, for all bc $\in\BCt$ and sufficiently large $n$,
$\Hpot_{n,\bc}\ge-c\,v_n$ whenever $\Hpot_{n,\bc}\le\eps\,v_n$. Hence
$(E-\Hpot_{n,\bc})_+ \le(\eps+c)v_n$
and therefore
\be{Abl}
\frac{d}{dE}\log\mZ{\bc} = \mZ{\bc}'\bigm/ \mZ{\bc}\ge\gamma\equiv\nu d/2(\eps+c)\,\,.
\ee
\rf{gamma} now follows by integration. $\Box$\skip
It follows immediately from \rf{gamma} and Proposition 5.6 that $s(\rho,\cdot)$
is strictly increasing on $]\eps_{\min}(\rho),\infty[$. This in turn shows that
the condition ``$\le\eps$'' in \rf{svar} can be replaced by ``$=\eps$''.
\rf{gamma} further implies that
$$
(1-e^{-\gamma u})\,\mZ{\bc}\le \mZ{\bc}^u \le \mZ{\bc}
$$
for a suitable $\gamma>0$ whenever $N/v_n\to\rho, E/v_n \to\eps, u>0$, bc $\in
\BCt$ for a given $t$, and $n$ is large enough. Combining this inequality with
Proposition 5.6 we arrive at \rf{slim}. The proof of Theorem 3.2 is therefore
complete.\skip
\remark{5.8} Under the conditions of \rf{slim} we also have that
$$
s(\rhoeps)=\lim\vni\log\mZ{\bc}'\,\,;
$$
cf. \rf{Zstrich}. This follows from \rf{Abl} and the inequality
$$
\mZ{\bc}'\le\int_0^1 du\,Z_{n|N,E+u,\bc}' = Z_{n|N,E+1,\bc}^1\,\,.
$$
$\Box$
\section{The asymptotics of microcanonical distributions}
\setcounter{equation}{0}
This section contains the proof of Theorem 3.3. Let $(\rhoeps)\in\Sigma$ and
\be{lim}
N/v_n\to\rho, E/v_n\to\eps, \,\,\mbox{and \enspace bc}\,\in\BCt
\ee
for some fixed $t>0$ as $n\ti$. In a first step, we shall prove Theorem 3.3
under the additional hypothesis
\be{u}
\vni\log_{-}u\to 0\mmbox{as}n\ti
\ee
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 $v_n$.
Suppose \rf{lim} and \rf{u} hold. For brevity we write $M_n=\micro{\bc}^u$
and $\bar{M}_n=\amicro{\bc}^u$. Theorem 3.2 implies that these measures are
well-defined for sufficiently large $n$. Let $M_n^{\per}\in{\cal P}$ denote
the measure relative to which the configurations in the disjoint blocks
$\Lan+(2n+1)i, i\in\Zd$, are independent with identical distribution $M_n$, and
$$
\widetilde{M}_n = \vni\int_{\Lan} M_n^{\per}\circ\thx^{-1}\,dx \in \Pinv
$$
the associated invariant average. We shall derive the asymptotics of
$\bar{M}_n$ from that of $\widetilde{M}_n$. First, we get from \rf{Ukin} that
\be{UkinM}
U^{\kin}(\widetilde{M}_n) = v_n^{-2}\int_{\Lan}M_n^{\per}(\Hkin_n\circ\thx)\,dx
= \vni M_n(\Hkin_n)\,.
\ee
The second equality follows from the periodicity of $M_n^{\per}$ and the
additive nature of $\Hkin_n$. Together with Lemma 5.2(b) and a version of
\rf{epstilde} this implies that
\be{lUkinM}
\limsup_{n\ti} U^{\kin}(\widetilde{M}_n) <\infty.
\ee
Next we consider the mean entropy of $\widetilde{M}_n$. The following key estimate
follows from Theorem 3.2.\skip
\proposition{6.1} {\em Under conditions \rf{lim} and \rf{u},}
$\liminf_{n\ti} S(\widetilde{M}_n) \ge s(\rhoeps)\,\,.$
\skip
\proof Let $Q=Q^1$ and $I=I_1$ be as in \rf{SI} for $\beta=1$. Lemma I.5.5
asserts that
$I(\widetilde{M}_n)\le\vni I(M_n;Q_n)\,.$
Together with \rf{SI}, \rf{UkinM} and \rf{SnI} this yields
\bea
S(\widetilde{M}_n)&\ge&-\vni I(M_n;Q_n)+\vni M_n(\Hkin_n)+c(1)\\\vspace{1ex}
&=&\vni S_n(M_n)\enspace=\enspace\vni\log\mZ{\bc}^u\,\,,
\eea
and Theorem 3.2 gives the result. $\Box$\skip
Let us say that two sequences $(P_{1,n})$ and $(P_{2,n})$ in ${\cal P}$ are
{\em asymptotically equivalent}, and write $P_{1,n}\sim P_{2,n}$ as $n\ti$, if
for all $f\in{\cal L}$
$$
\lim_{n\ti}\Bigl(P_{1,n}(f) - P_{2,n}(f)\Bigr) = 0\,\,.
$$
Considering again the empirical fields $R_n$ in \rf{Rn} we define the measures
$M_n R_{n+k}=\int M_n(d\om)R_{n+k,\om}\in\Pinv$.\skip
\lemma{6.2} {\em Assuming \rf{lim} and \rf{u}, we have for each $k\ge0$}
$$
\widetilde{M}_n\sim M_nR_{n+k}\sim \bar{M}_n\mmbox{as}n\ti\,.
$$
{\em If} bc = per {\em then, in addition, $\bar{M}_n\sim M_n$ as $n\ti$.}\skip
\proof We only prove the first asymptotic equivalence. A similar but simpler
argument shows that $M_nR_n\sim\bar{M}_n$, and for bc = per one has
$M_nR_n(f) = M_n(f)$ when $f\in{\cal L}$ and $n$ is so large that
$f$ only depends on $\omn$.
Eqs.\ \rf{SI} and \rf{lUkinM} together with Proposition 6.1 imply that
\be{lsI}
\limsup_{n\ti} I(\widetilde{M}_n) < \infty\,,
\ee
where again $I=I_1$ is the mean relative entropy with respect to the
Poisson point random field $Q=Q^1$.
Lemma I.5.7 thus yields that $\widetilde{M}_n\sim M_nR_n$ as $n\ti$,
which is the case $k=0$. For general $k$, the argument in the proof
of this lemma shows that for each $f\in{\cal L}$ and arbitrary $a>0$
\be{Ungl}
\bigl|v_{n+k}^{-1}v_n\widetilde{M}_n(f)-M_nR_{n+k}(f)\bigr|\le o(1)+
c\,\widetilde{M}_{n,k}(\tilde{N}_{\ell}; \tilde{N}_{\ell}\ge a)\,\,.
\ee
Here $\ell$ and $c$ are such that $f$ depends only on $\om_{\ell}$
and \rf{Lf} holds (where $\tilde{N}_{\ell}$ was defined), and
$\widetilde{M}_{n,k}\in\Pinv$ is the average of the measures
$M_{n,k}^{\per}\circ\thx^{-1}$ with $x\in\La_{n+k}$, where
$M_{n,k}^{\per}$ is the measure relative to which the configurations
in the disjoint blocks $\La_{n+k}+(2n+2k+1)i, i\in\Zd$, are i.i.d.
with distribution $M_n$. (In particular, $\widetilde{M}_{n,0}=
\widetilde{M}_n$, and for $k\ge1$ there are $M_{n,k}^{\per}$--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 \rf{Ungl} becomes arbitrarily small,
uniformly in $n$.
By Lemma I.5.2, this would follow if we knew that \rf{lsI} holds for
the sequence $(\widetilde{M}_{n,k})_{n\ge0}$ in the place of
$(\widetilde{M}_n)_{n\ge0}$. But this is indeed the case because
$$
dQ_n\bigm/dQ_{n+k} = 1_{\{N_{n+k}-N_k=0\}}\exp[(v_{n+k}-v_n)c(1)]
$$
and therefore
$$
v_n I(\widetilde{M}_{n,k})\le I(M_n;Q_{n+k}) = I(M_n;Q_n)+(v_{n+k}-v_n)c(1)\,\,.
$$
Assertion \rf{lsI} for the sequence $(\widetilde{M}_{n,k})_{n\ge0}$ thus follows as
in the case $k=0$. $\Box$\skip
We are now prepared for the first step in the proof of Theorem 3.3.
\skip
{\em Proof of Theorem 3.3 under condition \rf{u}.}\quad By \rf{lUkinM}
and Lemmas 6.1 and 4.2, the sequence $(\widetilde{M}_n)_{n\ge0}$ is
sequentially compact, and every accumulation point $P$ satisfies $S(P)
\ge s(\rhoeps)$. Since $\rho(\widetilde{M}_n)=N/v_n\to\rho$ and
$\rho(\cdot)$ is continuous, we have $\rho(P)=\rho$.
To show that $U(P)\le\eps$ we use Lemma 6.2. For each $k$, $P$ is
also an accumulation point of $(M_nR_{n+k})_{n\ge0}$, and
$$
U(M_nR_{n+k})=M_n\bigl( U(R_{n+k})\bigr) = v_{n+k}^{-1} M_n(H_{n+k,\per})
$$
because of \rf{HperU}. By Lemma 5.2, for each $\delta>0$ we can find
some $k$ such that $M_n(H_{n+k,\per})\le(\eps+\delta)v_n$ for sufficiently large $n$.
Since $U$ is lower semicontinuous, it follows that $U(P)\le\eps$. In fact,
$U(P)=\eps$ because otherwise $S(P)~~0$
when $0~~__0$,
and the condition $k\le N-4/d$ in \rf{Zstrichf} ensures that $Z_{n|N,s,\bc}'(f)$
is a continuous function of $s$. The lemma will therefore be proved once we have shown that
$$
\{H_{n,\bc}\le E\}\cap \Om_{n|N}\subset\{N_{\La}\le N-4/d\}
$$
for $n$ large enough. So let $\ominOm_{n|N}$ and suppose that $N_{\La}(\om)
>N-4/d\equiv\underline{N}$.Then $T_n(\om)\ge\underline{N}^2/m$ and therefore
$H_n(\om)\ge a\,\underline{N}^2/m-b\,N$. By an estimate similar to
Lemma II.4.2, we find a constant $c>0$ such that
\bea
H_{n,\bc}(\om)-H_n(\om)&\ge&\left\{\ba{lr}-cNt&\mbox{if bc}\,\in\Om(t)\\
-cN^2/v_n&\mbox{if bc = per.}\ea\right.
\eea
Combining these estimates with \rf{lim} we see that $H_{n,\bc}(\om)>E$
when $n$ is large enough, and the lemma follows. $\Box$\skip
Our strategy for the extension of Theorem 3.3 to the case when \rf{u} fails is
a comparison of $\micro{\bc}^u$ with $\micro{\bc}^s$ for some $s\ge u$
satisfying $\vni\log_{-} s\to0$. This comparison is based on the monotonicity
properties of $\mZ{\bc}'(f)$ and the next lemma.\skip
\lemma{6.4}{\em Suppose that $u\to0$ as $n\ti$ and \rf{lim} holds. Then}
$$
Z_{n|N,E+u,\bc}'\bigm/Z_{n|N,E,\bc}'\to 1 \mmbox{as} n\ti.
$$
\proof The ratio under consideration is certainly larger than 1.
To obtain an upper estimate we consider the function
$$
\sigma_n(s)=\log Z_{n|N,s,\bc}'
$$
for values of $n$ and $s$ for which $Z_{n|N,s,\bc}'>0$. Computing
the derivatives as in Lemma 5.7 and using the Cauchy--Schwarz inequality
we obtain the estimate
$$
(1/\sigma_n')'\le 2/(Nd-2)\equiv a_N\,\,;
$$
cf. Ref.\ 21. Hence
\be{6.4.1}
1/\sigma_n'(s) - 1/\sigma_n'(t) \le a_N(s-t)
\ee
when $s\ge t$, and the mean value theorem gives
$$
\delta v_n\Bigm/\Bigl(\sigma_n(E+\delta v_n)-\sigma_n(E)\Bigr) -
1/\sigma_n'(E)\le\delta v_na_N
$$
for each $\delta>0$. Letting $n\ti$ and $\delta\to0$ we thus obtain from Remark 5.8
\be{6.4.2}
\limsup \sigma_n'(E)\le \partial_+s(\rhoeps)/\partial\eps\,\,.
\ee
The expression on the right side is the right derivative with respect to $\eps$
(which exists by concavity). Averaging over $t$ in \rf{6.4.1} we obtain
\be{6.4.3}
1/\sigma_n'(E+u)-u\bigm/\bigl(\sigma_n(E+u)-\sigma_n(E)\bigr)\le u\,a_N\,.
\ee
Together with \rf{6.4.2} (for $E+u$ in place of $E$) this gives
$$
\limsup\bigl(\sigma_n(E+u)-\sigma_n(E)\bigr)\le\limsup u\,\sigma_n'(E+u)=0
$$
which is the desired estimate. $\Box$\skip
\remark{6.5} \rf{6.4.3} also gives a counterpart to \rf{6.4.2} which,
together with the concavity of $s(\rho,\cdot)$, implies that $s(\rho,\cdot)$
is differentiable and
$$
\partial s(\rhoeps)/\partial\eps = \lim\frac{d}{dE}\log\mZ{\bc}'\,\,.
$$
The same argument also shows that the last identity holds without
the prime on the righthand side. $\Box$\skip
The following proposition asserts that microcanonical distributions
on thin and on slightly thicker energy shells are asymptotically equivalent.\skip
\proposition{6.6}{\em Suppose \rf{lim} holds and $u, s\in [0,\infty]$
vary with $n$ in such a way that $0\le u\le s\to0$ and $\vni\log s\to0$. Then}
$$
\amicro{\bc}^u\sim\amicro{\bc}^s \mmbox{and} \micro{\per}^u\sim\micro{\per}^s
\mmbox{as}n\ti\,\,.
$$
\proof We drop all indices which are not necessary. We need to show
that, for given $f\in{\cal L}$, $\bar M^u_{n|\bc}(f)-\bar
M^s_{n|\bc}(f)\to0$ and
$M^u_{n|\per}(f)-M^s_{n|\per}(f)\to0$ as $n\ti$. We can
assume without loss that $f\ge0$. Let $\ell$ and $c$ be such that
$f(\om)=f(\om_{\ell})$ and \rf{Lf} holds, $n(v_{\ell+1})$ be chosen
according to Lemma 6.3, and $n\ge n(v_{\ell+1})$. We define
$$
a_n(u,f)\equiv a_{n|E}(u,f)=\int_0^1Z_{n|E-ru}'(f)\,dr
$$
and $a_n(u)=a_n(u,1)$ for the constant function 1. Then
$M_n^u(f)=a_n(u,f)/a_n(u)$. (Note that $a_n(u)\ge Z_{n|E-u}'>0$ for
large $n$, by Remark 5.8.) It follows from \rf{Zstrichf} that
$Z_{n|E}'(f)$ is increasing in $E$ and its increments are increasing in $f$. Hence
\bea
0&\le&\Bigl(a_n(u,g)-a_n(s,g)\Bigr)\Bigm/a_n(u)\\
&\le&\Bigl(a_n(0,h)-a_n(s,h)\Bigr)\Bigm/a_n(s)
\eea
whenever $g\le h$. Let $a>0$ be a constant to be chosen later. We apply the
preceding inequality to the two functions $g_1=f\,1_{\{\tilde N_{\ell}\le a\}}$
and $g_2 = f\,1_{\{\tilde N_{\ell}> a\}}$ and the corresponding $h_1$ and $h_2$
that are obtained by estimating $f$ according to \rf{Lf}. After a spatial
averaging we arrive at the estimate
\bea
0&\le&\bar{M}_n^u(f)-r_n\,\bar{M}_n^s(f) \\
&\le&c(1+a)(q_n-1)+c\,\tilde q_n\,\bar{M}^s_{n|E+s}(\tilde N_{\ell};
\tilde N_{\ell}> a)\,\,.
\eea
Here $1\ge r_n\equiv a_n(s)/a_n(u)\ge Z_{n|E-s}'/Z_{n|E}'\to1$ as
$n\ti$ by Lemma 6.4, and similarly
$$
1\le q_n\equiv a_n(0)/a_n(s)\le Z_{n|E}'/Z_{n|E-s}' \to1
$$
and $\tilde q_n\equiv Z_{n|E+s}'/Z_{n|E-s}'\to1$ as $n\ti$. By Lemma 6.2,
$$
\limsup\bar{M}^s_{n|E+s}(\tilde N_{\ell};\tilde N_{\ell}>a)
\le \limsup\widetilde{M}^s_{n|E+s}(\tilde N_{\ell};\tilde N_{\ell}>a)\,\,,
$$
and \rf{lsI} and Lemma I.5.2 imply that the last expression tends to zero as
$a\ti$. Combining these estimates we arrive at the desired conclusion.
In the case bc = per we simply omit the spatial averaging in the
preceding argument. $\Box$\skip
{\em Rest of the proof of Theorem 3.3.} \quad It is sufficient 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 \rf{u} 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 $(\bar{M}_{n|\bc}^s)$ and $({M}_{n|\per}^s)$
carries over to the sequences $(\bar{M}_{n|\bc}^u)$ and $({M}_{n|\per}^u)$. $\Box$
\section{The variational principle for Gibbs measures}
\setcounter{equation}{0}
This final section contains the proof of Theorem 3.4. To obtain a
convenient expression for the excess mean free energy we first need
the grand canonical analogue of Theorem 3.2, namely the existence and
variational characterization of the pressure.\skip
\proposition{7.1} {\em Let $\zbet>0$ and $p(\zbet)$ be defined by \rf{p}. Then}
$$
\lim_{n\ti}\vni\log\Xi_{\nzbet,\bc} = p(\zbet)
$$
{\em for each} bc $\in\BC$, {\em and the convergence is uniform in} bc
$\in\BCt$ {\em for each $t>0$.} \skip
\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 $\pinp$ we let $\hat P = P(\om:
\omh\in\cdot)$ denote the distribution of the configuration of particle
positions in $\hat\Om$, the set of all locally finite subsets $\omh$ of
$\Rd$. In particular, if $Q^{\zbet}$ is the Poisson point random field
on $\Rd\times\Rd$ with intensity measure $\mu(dx,dp)=
z\,dx\,e^{-\beta|p|^2}dp$ (which amounts to setting $\tau(p)=
-\log z+\beta|p|^2$ in \rf{qn}) then $\hat Q^{\zbet}$ is the Poisson
point random field on $\Rd$ with particle density $c(\zbet)=z\,c(\beta)$,
and we can write
$$
\Xi_{\nzbet,\bc}= \exp[v_nc(\zbet)]\,\hat Q_n^{\zbet}(\exp[-\beta\,\Hpot_{n,\bc}])
$$
because $\Hpot_{n,\bc}(\om)$ is a function of $\omh$. Theorems II.2 and II.3
therefore imply that
\be{7a}
\lim_{n\ti}\vni\log\hat Q_n^{\zbet}(\exp[-\beta\Hpot_{n,\bc}])
= - \min_{\pinp}\Bigl[\beta\,U^{\pot}(\hat P)+\hat I_{\zbet}(\hat P)\Bigr]\,\,,
\ee
where $\hat I_{\zbet}(\hat P)=\lim_{n\ti}\vni I(\hat P_n;\hat Q_n^{\zbet})$
is defined in analogy to Lemma 4.1. On the other hand, \rf{F} and \rf{SnI} show that
\be{7b}
p(\zbet)=c(\zbet)-\min_{\pinp}\Bigl[\beta\,U^{\pot}(P)+I_{\zbet}(P)\Bigr]\,\,,
\ee
where $I_{\zbet}$ is the mean relative entropy with respect to $Q^{\zbet}$.
We thus only need to show that the minima on the right-hand sides of \rf{7a}
and \rf{7b} coincide. This will follow once we have shown that, for each
$\pinp$, $\hat I_{\zbet}(\hat P)\le I_{\zbet}(P)$ with equality when $P$
is Maxwellian, in that
$$
P=\int \hat P(d\omh)\,\Gamma_{\beta}(\omh,\cdot)\,\,.
$$
Here $\Gamma_{\beta}(\omh,\cdot)$ is the conditional distribution of $Q^{\zbet}$
given $\omh$, i.e., the distribution of $\{(x,Y_x):\xinom\}$, where the
$Y_x$ are i.i.d. centered normal with variance $1/2\beta$.
To verify this inequality we can assume that $I_{\zbet}(P)<\infty$.
Then, for all $n$, $P_n\ll Q_n^{\zbet}$ with a density $f_n$, whence
$\hat P_n\ll \hat Q_n^{\zbet}$ with density $\hat f_n(\omh)=
\Gamma_{\beta}(\omh,f_n)$. Thus, by Jensen's inequality,
\bea
I(\hat P_n; \hat Q_n^{\zbet})&=&\hat Q_n^{\zbet}(\hat f_n\log\,\hat f_n)\\
&\le&\int \hat Q_n^{\zbet}(d\omh)\,\Gamma_{\beta}(\omh, f_n\log\,f_n)
\enspace=\enspace I(P_n;Q^{\zbet}_n)
\eea
with equality when $f_n(\om)$ only depends on $\omh$. This gives the
desired relationship between $\hat I_{\zbet}$ and $I_{\zbet}$. $\Box$\skip
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.\skip
\corollary{7.2} {\em For all $\zbet>0$ and $\pinp$,}
$$
\delta F_{\zbet}(P)\equiv F_{\zbet}(P)+p(\zbet)=\lim_{n\ti}\vni
I(P_n;\grand{\free})\,\,.
$$
\proof Since $\grand{\free}$ is equivalent to $L_n$ with the
density appearing in \rf{grand}, we have
$$
I(P_n;\grand{\free}) = -S_n(P)-P(N_n)\log z +\beta\,P(H_n)+
\log\Xi_{\nzbet,\free}
$$
for all $n$, so that the corollary follows from Proposition 3.1,
Eqs.\ \rf{U} and \rf{F}, and Proposition 7.1. $\Box$\skip
We now turn to the proof of Theorem 3.4. We can assume without loss
that $z=\beta=1$ (because this only amounts to a rescaling of the
position and momentum spaces together with a rescaling of $\ph$),
and we shall drop all indices referring to these parameters. In
particular, we write $G_n$ resp.\ $G_{n,\zeta}$ for the grand
canonical Gibbs distribution in $\Lan$ with free
boundary condition resp.\ bc = $\zeta$. We shall also need
the Gibbs distributions in more general sets $\La$ in place of
$\Lan$, which will be denoted by $G_{\La}$ resp.\ $G_{\La,\zeta}$.
The sets $\La$ and $\Delta$ considered below will always be finite
unions of the unit cells $C+i, i\in\Zd$.
Our first 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
$$
\int P(d\zeta)\,\|G_{0,\zeta}-G_{0,\zeta_k}\|
$$
(relative to certain measures $P$) between the Gibbs distributions
in $\La_0 =C$ with boundary conditions $\zeta$ and $\zeta_k=\zeta
\cap(\La_k\times\Rd)$. Clearly, if $\ph$ has finite range then this
distance vanishes as soon as $k$ exceeds the range of $\ph$. The
general case will be treated in Lemma 7.4; the next lemma serves as preparation.
For each $i\in\Zd$ we write $\psi_i=\psi(d(C,C+i))$, where $\psi$ is as
in assumption (A1) and $d(C,C+i)$ is the distance of the cells $C$ and
$C+i$. We also set $\Psi=\sum_{i\in\mbox{{\small\sf Z}}^d}\psi_i$ and
$\tilde\psi_i=\psi_i/2\Psi$ so that $\sum_{i\in\mbox{{\small\sf Z}}^d}
\tilde\psi_i=1/2$. Note that $\tilde\psi_0=0$.\skip
\lemma{7.3} {\em For every exponent $\gamma\ge1$ there exists a constant
$a_{\gamma}<\infty$ such that}
$$
G_{0,\zeta}(N_0^{\gamma}) \le a_{\gamma}+ \sum_{i\in\mbox{{\small\sf Z}}^d}
\tilde\psi_i\,N_{C+i}(\zeta)^{\gamma}
$$
{\em for all $\zeta\in\Om^*$.}\skip
\proof This is a version of Lemma 2 of Dobrushin$^{(25)}$. For
completeness we indicate the proof. Let $b_{\zeta}= \sum_{i\in
\mbox{{\small\sf Z}}^d} \tilde\psi_i\, N_{C+i}(\zeta)$. It is easy
to see that $b_{\zeta}<\infty$ for $\zeta\in\Om^*$ (cf. Lemma II.4.2),
and we only need to show that
\be{agamma}
G_{0,\zeta}(N_0^{\gamma}; N_0 > b_{\zeta}) \le a_{\gamma}\,\,.
\ee
In view of \rf{sss} and \rf{reg},
$$
\Hpot_{0,\zeta}\ge h(N_0)-b'\,N_0-2\Psi\,N_0\,b_{\zeta}\,\,.
$$
On the other hand, $\Xi_{0,\zeta}\ge1$. The left-hand side of \rf{agamma}
is therefore at most
$$
\sum_{\ell>b_{\zeta}}\ell^{\gamma}\exp\left[-h(\ell)+b'\ell+2\Psi\ell^2\right]
c(1)^{\ell}\bigm/\ell!\,,
$$
and the corresponding sum over all $\ell\ge0$ is finite because
$h(\ell)/\ell^2\ti$ as $\ell\ti$. $\Box$\skip
\lemma{7.4} {\em Let $\pinp$ be such that $t_P\equiv P(N_C^2)<\infty$.
Then for any $\delta>0$ we can find a number $k_0$ such that for all $k\ge k_0$}
$$
\int P(d\zeta)\,\|G_{0,\zeta}-G_{0,\zeta_k}\| <\delta
$$
{\em and, for each $\La$, the same inequality holds when $P$ is
replaced by}
$$
PG_{\La}\equiv\int P(d\eta)\,G_{\La,\eta}(\zeta: \zeta \cup
(\eta \setminus\La\times\Rd)\in\cdot)\,\,.
$$
\proof A standard estimate (see Ref.\ 24, p.\ 33, proof of (b)) and a
symmetry argument yield for each $\zeta\in\Om^*$
$$
\|G_{0,\zeta}-G_{0,\zeta_k}\|
\le 2\,L_0\Bigl(|\exp[-H_{0,\zeta}]-\exp[-H_{0,\zeta_k}]\Bigr) \Bigm/
\max(\Xi_{0,\zeta},\Xi_{0.\zeta_k}).
$$
Decomposing the integral above into parts according to the sign of
$H_{0,\zeta} - H_{0,\zeta_k}$ and using \rf{reg} we obtain the upper bound
$$
2\,\bigl[G_{0,\zeta}(N_0)+G_{0,\zeta_k}(N_0)\bigr] \sum_{i\in
\mbox{{\small\sf Z}}^d \setminus \La_k} \psi_iN_{C+i}(\zeta)
$$
when $k\ge r(\ph)$. Applying Lemma 7.3 for $\gamma=1$, integrating
over $\zeta$ and using the Cauchy-Schwarz inequality we find
$$
\int P(d\zeta)\,\|G_{0,\zeta}-G_{0,\zeta_k}\| \le (4a_1t_P^{1/2}+2t_P)\delta_k\,,
$$
where $\delta_k\equiv \sum_{i\in\Zd\setminus \La_k} \psi_i$ tends to
zero as $k\ti$. This proves the first assertion.
The same kind of estimate also gives the second assertion, provided
we can find a constant $t<\infty$ such that $PG_{\La}(N_{C+i}^2)\le t$
for all $\La$ and all $i\in\Zd$. To show this we use again an idea of
Dobrushin$^{(25)}$.
Since $PG_{\La}(N_{C+i}^2)=t_P$ when $i\in\Zd\setminus\La$ we only need
to consider the case when $i\in\Zd\cap\La$. We define
$$
t_{\La} = \max_{i\in\mbox{{\small\sf Z}}^d\cap\La}PG_{\La}(N_{C+i}^2)\,\,.
$$
In view of the consistency of the Gibbs distributions and by Lemma 7.3 we
have for each $i\in\Zd\cap\La$
\bea
PG_{\La}(N_{C+i}^2) &=&\int PG_{\La}(d\zeta)\,G_{C+i,\zeta}(N_{C+i}^2)\\
&\le&a_2+\sum_{j\in\mbox{{\small\sf Z}}^d} \tilde\psi_{i-j}\,PG_{\La}(N_{C+j}^2)
\eea
and therefore $t_{\La}\le a_2+\eh\max(t_{\La},t_P)$. On the other hand,
$t_{\La} <\infty$ because
$$
t_{\La}\le\int P(d\zeta)\,G_{\La,\zeta}(N_{\La}^2) \le a'+ \sum_{i\in
\mbox{{\small\sf Z}}^d \setminus\La} \psi(d(C+i,\La))\,P(N_C^2)
$$
for some $a'=a'(\La)<\infty$ by an obvious variant of Lemma 7.3. It follows that
$t_{\La}\le t\equiv\max(t_P,2a_2)$, and the proof is complete. $\Box$\skip
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 $\pinp$ be such that $\delta F(P)=0$. Then
$U^{\pot}(P)<\infty$ and thus $t_P=P(N_C^2)<\infty$. Hence Lemma 7.4 is applicable.
Moreover, the ergodic theorem (for the multidimensional discrete translation group
$(\vartheta_i)_{i\in\Zd}$) shows that $P(\Om^*)=1$, i.e., $P$ is tempered. On the
other hand, we conclude from Corollary 7.3 that $P_n\ll G_n$ when $n$ is large
enough. By translation invariance this means that, for all sufficiently large
cubes $\La$, $P_{\La}$ is absolutely continuous with respect to $G_{\La}$ with
a density $g_{\La}$. Here $P_{\La}$ is the restriction of $P$ to the events in
$\La$, i.e., the image of $P$ under the mapping $\om\to\om_{\La} = \om\cap
(\La\times\Rd)$, and $G_{\La}$ is the Gibbs distribution in $\La$ with free
boundary condition. In particular, for any $\Delta\subset\La$ we consider the
restriction $G_{\La,\Delta} =(G_{\La})_{\Delta}$ of $G_{\La}$ to the events
in $\Delta$, and we have that
$P_{\Delta}\ll G_{\La,\Delta}$ with density
$$
g_{\La,\Delta}(\zeta) = \int G_{\La\setminus\Delta,\zeta_{\Delta}}(d\om)\,
g_{\La}(\om\cup\zeta_{\Delta})\,\,.
$$
The crucial consequence of the assumption $\delta F(P)=0$ is the following.
\skip
\lemma{7.5}{\em Let $\pinp$ be such that $\delta F(P)=0$, and let $k\ge1$ and
$\delta>0$ be given. Then there exist two sets $\Delta, \La\subset\Rd$ such that
$\La_k\subset\Delta\subset\La$ and}
$$
G_{\La}\Bigl(|g_{\La,\Delta}-g_{\La,\Delta\setminus C}|\Bigr) <\delta\,\,.
$$
\skip
\proof In view of the argument in Ref.\ 24, p.\ 324, Step 2, it is sufficient
to show that for any $\delta>0$ we can find $\Delta$ and $\La$ such that
$\La_k\subset\Delta\subset\La$ and
\be{Preston}
I_{\Delta}(P;G_{\La})-I_{\Delta\setminus C}(P;G_{\La}) <\delta\,\,.
\ee
Here we write $I_{\Delta}(\cdot\,;\cdot)$ for the relative entropy of measures
that are restricted to the events in $\Delta$. Using Corollary 7.2 we see that
there exists some $n\ge k$ such that $\vni I(P_n;G_n)\le\delta/2^dv_k$. As in
Ref.\ 24, p.\ 324, Step 1, we choose an $m$ with $m^dv_k\le v_n\le(2m)^dv_k$
and disjoint translates $\La_k(j)=\La+i(j)\subset \Lan, i(j)\in\Zd, 1\le j\le
m^d$. Writing $\Delta(j)=\La_k(1)\cup\ldots\cup\La_k(j)$ we find
\bea
&&m^{-d}\sum_{j=1}^{m^d}\Bigl[ I_{\Delta(j)}(P;G_n)- I_{\Delta(j)
\setminus(C+i(j))}(P;G_n)\Bigr]\\
&&\qquad\le\enspace m^{-d}\,I_{\Delta(m^d)}(P;G_n)\enspace<\enspace\delta\,\,.
\eea
So for at least one $j$ the corresponding term in the sum above is less
than $\delta$, and by translation invariance we obtain \rf{Preston} for
$\La=\Lan-i(j)$ and $\Delta=\Delta(j)-i(j)$. $\Box$\skip
We are now ready for the first direction of the variational principle.\skip
\proposition{7.6}{\em Each $\pinp$ with $\delta F(P)=0$ is a tempered Gibbs measure.}\skip
\proof We saw already that $P$ is tempered and $t_P<\infty$. We need to
establish Eq.\ \rf{GM} for any measurable $f\ge0$. We can clearly assume
that $f\in{\cal L}$ and $0\le f\le1$. For given $\delta>0$ we let $k$ be
so large that $f$ only depends on the particles in $\La_k$ and the conclusions
of Lemma 7.4 hold, and we determine $\Delta$ and $\La$ according to Lemma 7.5.
Consider the term on the left-hand side of Eq.\ \rf{GM} for $n=0$. We replace
its inner integral
$$
G_0f(\zeta)\equiv\int G_{0,\zeta}(d\om)\,f(\om_0\cup\zeta\setminus\zeta_0)
$$
by $\tilde f_k(\zeta)\equiv G_0f(\zeta_k)$. Since $f$ only depends on the
particles in $\La_k$, this replacement leads to an error of at most
$\int P(d\zeta)\,\|G_{0,\zeta}-G_{0,\zeta_k}\|<\delta$. Since $\tilde f_k$
only depends on the particles in $\La_k\setminus C\subset\Delta\setminus C$
and $P\ll G_{\La}$ on the events which occur in this set,
$$
P(\tilde f_k)=G_{\La}(g_{\La,\Delta\setminus C}\tilde f_k)\,\,.
$$
Changing $\tilde f_k$ in the last term back into $G_0f$ gives an error of at most
\bea
&&\int G_{\La}(d\zeta)\,g_{\La,\Delta\setminus C}(\zeta) \|G_{0,\zeta}-G_{0,\zeta_k}\|\\
&&\qquad=\enspace\int PG_{\La\setminus(\Delta\setminus C)}(d\zeta)\|G_{0,\zeta}-G_{0,\zeta_k}\|\enspace<\enspace\delta\,\,.
\eea
(For the equality above we first use the fact that $G_{\La\setminus
(\Delta\setminus C),\om_{\Delta\setminus C}}$ is the conditional distribution
of $G_{\La}$ given the configuration $\om_{\Delta\setminus C}$ in $\Delta\setminus C$,
and then that $P$ has density $g_{\La,\Delta\setminus C}$ relative to $G_{\La}$ on
events occurring in $\Delta\setminus C$.) As a next step we observe that
$$
G_{\La}(g_{\La,\Delta\setminus C} G_0f) = G_{\La}(g_{\La,\Delta\setminus C}f)
$$
because $g_{\La,\Delta\setminus C}$ does not depend on the particles in $\La_0=C$
and $G_{0,\zeta}$ is the suitable conditional distribution of $G_{\La}$. According
to the choice of $\Delta$ and $\La$ we may finally replace $g_{\La,\Delta\setminus C}$
by $g_{\La,\Delta}$ with an error less than $\delta$. In this way we arrive at the term $ G_{\La}(g_{\La,\Delta}f)=P(f)$. We have thus shown that
$$
| P(G_0f)-P(f)|<3\delta\,\,,
$$
and this gives \rf{GM} for $n=0$. To obtain \rf{GM} for general $n$ we can either
apply Theorem (1.33) of Ref.\ 24, or simply repeat the preceding argument on a
new spatial scale. $\Box$\skip
We finally come to the converse part of the variational principle.\skip
\proposition{7.7} {\em $\delta F$ vanishes on the set of tempered Gibbs measures.}\skip
\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 $t_P=P(N_C^2)<\infty$. 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 $\delta F$ is measure affine) $P$ can be assumed to be ergodic,
in which case $t_P<\infty$ follows from the temperedness by means of the ergodic theorem.
We need to show that the limit in Corollary 7.2 vanishes. We fix any $n\ge0$.
By \rf{grand} and \rf{GM}, $P_n\ll G_n$ with density
\bea
g_n(\om)&=&\int P(d\zeta)\,dG_{n,\zeta}/dG_n(\om)\\
&=&\int P(d\zeta)\,\exp[H_n(\om)-H_{n,\zeta}(\om)]\,\,\Xi_n\bigm/\Xi_{n,\zeta}\,\,.
\eea
Therefore we can write, using Jensen's inequality, \rf{GM}, and the lower regularity of $\ph$,
\bea
I(P_n;G_n) &=&G_n(g_n\log g_n)\\
&\le&G_n\left(\int P(d\zeta)\,dG_{n,\zeta}/dG_n \log dG_{n,\zeta}/dG_n\right)\\
&=&\int P(d\om)\left[H_n(\om)-H_{n,\om}(\om)+ \log\Xi_n - \log\Xi_{n,\om} \right]\\
&\le&t_P\sum_{i\in\mbox{{\small\sf Z}}^d\cap\Lan, j\in\mbox{{\small\sf Z}}^d
\setminus\Lan} \psi_{i-j} + \log\Xi_n -\int P(d\zeta)\,\log\Xi_{n,\zeta}\,\,.
\eea
After division by $v_n$, the first 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 $\Xi_{n,\zeta}\ge1$, so that the same proposition
together with Fatou's lemma and the temperedness of $P$ implies that
$$
\liminf_{n\ti}\vni\int P(d\zeta)\log\Xi_{n,\zeta}\ge p(1,1)\,\,.
$$
It follows that $\delta F(P)=0$, and the proof is complete. $\Box$
\skip
\vspace{2ex}
\noindent{\Large\bf References}
\skip
\renewcommand{\baselinestretch}{0.97}
{\small\begin{enumerate}
\item H.O. Georgii and H. Zessin, Large deviations and the maximum
entropy principle for marked point random fields, {\em Probab. Theory Relat.
Fields} {\bf96}:177--204 (1993).
\item H.O. Georgii, Large deviations and the equivalence of ensembles
for Gibbsian particle systems with superstable interaction, {\em Probab.
Theory Relat. Fields} {\bf99}:171--195 (1994).
\item D. Ruelle, {\em Statistical Mechanics. Rigorous results}
(Benjamin, New York, 1969).
\item O.E. Lanford, Entropy and equilibrium states in classical
statistical mechanics, in {\em Statistical Mechanics and Mathematical
Problems}, A. Lenard, ed., Lect. Notes Phys. 20 (Springer, Berlin, 1973).
\item A. Martin-L\"of, {\em Statistical Mechanics and the Foundations of
Thermodynamics}, Lect. Notes Phys. 101 (Springer, Berlin, 1979).
\item A.E. Khinchin, {\em Mathematical foundations of statistical
mechanics} (Dover, 1949).
\item A.M. Halfina, The limiting equivalence of the canonical and grand
canonical ensembles (low density case), {\em Math. USSR Sbornik} {\bf9}:1--52 (1969).
\item R.A. Minlos and A. Haitov, Limiting equivalence of thermodynamic
ensembles in case of one-dimensional systems, {\em Trans. Moscow Math. Soc.}
{\bf32}:143--180 (1975).
\item R.L. Dobrushin and B. Tirozzi, The central limit theorem and the
problem of equivalence of ensembles, {\em Commun. Math. Phys.} {\bf54}:173--192 (1977).
\item M. Campanino, G. Del Grosso, and B. Tirozzi, Local limit theorem for
Gibbs random fields of particles and unbounded spins, {\em J. Math. Phys.}
{\bf20}:1752-1758 (1979).
\item A. Martin--L\"of, The equivalence of ensembles and the Gibbs phase rule
for classical lattice systems, {\em J. Statist. Phys.} {\bf20}:557-569 (1979).
\item R.L. Thompson, {\em Equilibrium states on thin energy shells}, Memoirs
Amer. Math. Soc. 150 (1974).
\item H.O. Georgii, {\em Canonical Gibbs measures}, Lect. Notes Math. 760
(Springer, Berlin, 1979)
\item M. Aizenman, S. Goldstein, and J.L. Lebowitz, Conditional equilibrium
and the equivalence of microcanonical and and grandcanonical
ensembles in the thermodynamic limit, {\em Commun. Math. Phys.} {\bf62}:279--302 (1978).
\item C. Preston, Canonical and microcanonical Gibbs states, {\em Z.
Wahrscheinlichkeitstheorie Verw. Gebiete} {\bf46}:125--158 (1979).
\item M. Pirlot, Generalized canonical states, {\em Ann. Sci. Univ.
Clermont-Ferrand II. Probab. Appl.} {\bf4}:69--91 (1985).
\item P. Vanheuverzwijn, Discrete lattice systems and the equivalence of
microcanonical, canonical and grand canonical Gibbs states, {\em Commun.
Math. Phys.} {\bf101}:153--172 (1985).
\item K. Matthes, J. Kerstan, and J. Mecke, {\em Infinitely Divisible Point
Processes} (Wiley, Chichester, 1978).
\item T. Tjur, {\em Probability based on Radon measures} (Wiley, Chichester,
1980).
\item D. Ruelle, Superstable interactions in classical statistical
mechanics, {\em Commun. Math. Phys.} {\bf18}:127--159 (1970).
\item R. Rechtmann and O. Penrose, Continuity of the temperature and derivation
of the Gibbs canonical distribution in classical statistical mechanics,
{\em J. Statist. Phys.} {\bf19}:359--366 (1978).
\item O.E. Lanford and D. Ruelle, Observables at infinity and states with
short range correlations in statistical mechanics, {\em Commun. Math. Phys.}
{\bf9}:327--338 (1968).
\item C. Preston, {\em Random fields}, Lect. Notes Math. 534
(Springer, Berlin, 1976)
\item H.O. Georgii, {\em Gibbs Measures and Phase Transitions}
(de Gruyter, Berlin, 1988).
\item R.L. Dobrushin, Gibbsian random fields for particles without hard
core, {\em Theor. Math. Phys.} {\bf4}:705--719 (1970).
\item H.O. Georgii, Large deviations and maximum entropy principle for
interacting random fields on ${\rm Z}^d$, {\em Ann. Probab.} {\bf21}:1845--1875 (1993).
\end{enumerate}}
\end{document}
__