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PACS-Code: 75.10 Hk, 75.10 Jm, 05.70 Ce.
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Classical and quantum unbounded spin systems, superstable interactions, Gibbs (DLR) measures, 
KMS states, entropy, pressure, Gibbs variational principle.
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\markboth{ Yoo {\it et al}}{Statistical mechanics for unbounded spin systems}

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\begin{titlepage}
\bigskip
\smallskip
\centerline{{\bf EQUILIBRIUM STATISTICAL MECHANICS FOR }}
\medskip
\centerline{{\bf CLASSICAL AND QUANTUM UNBOUNDED SPIN SYSTEMS}}
\vskip 1 true cm
\centerline{Hyun Jae Yoo\footnote{Department of
Mathematics, Kyungpook National University, Taegu 702-701, Korea. E-mail:
yoohj@gauss.kyungpook.ac.kr.}, Hung Hwan Lee\footnote{Department of
Mathematics, Kyungpook National University, Taegu 702-701, Korea. E-mail:
hhlee@kyungpook.ac.kr.},
Ho Il Kim\footnote{TGRC, Kyungpook National University, Taegu
702-701, Korea. E-mail: hikim@gauss.kyungpook.ac.kr.},}
\vskip 0.3 true cm
\centerline{Sang Gyu Jo\footnote{Department of Physics, Kyungpook National
University, Taegu 702-701, Korea. E-mail: sgjo@kyungpook.ac.kr.},
and Sang Don Choi\footnote{Department of
Physics, Kyungpook National University, Taegu 702-701, Korea. E-mail:
sdchoi@kyungpook.ac.kr.}}
\vskip 2 true cm
\begin{center}
\begin{minipage}{5in}
\centerline{\large\bf   Abstract}
\vspace{3ex}
{\rm We study the equilibrium statistical mechanics for classical and quantum
unbounded spin systems interacting via superstable and regular interactions. We
show that in classical systems any DLR measure satisfies the Gibbs variational
equality and the converse is also true if the interaction is of finite range.
For quantum systems, we show that the (weak) KMS state constructed via the
thermodynamic limit of finite volume Green's functions satisfies the Gibbs
variational equality.
}
\vskip 2 true cm
\noi {\bf Key words:} Classical and quantum unbounded spin systems, superstable
interaction, Gibbs (DLR) measures, KMS states, entropy, pressure,
Gibbs variational principle.
\vskip 1 true cm
\noi {\bf PACS-Code:} 75.10 Hk, 75.10 Jm, 05.70 Ce.
\end{minipage}
\end{center}
\end{titlepage}
\setcounter{page}{2}

\newpage

%\tableofcontents
\section{Introduction}
In this paper we study the equilibrium statistical mechanics for classical and
quantum unbounded spin systems. Mostly, we aim to show the equivalence of DLR
equilibrium conditions and Gibbs variational equality (for classical systems)
and to show that the (weak) KMS states satisfy the Gibbs variational equality
(for quantum systems).

In bounded spin systems (that is, the single spin states correspond to compact
sets and finite dimensional Hilbert spaces for classical and quantum systems,
respectively) it has been well-established that the equilibrium states for the
systems can be investigated via the tangent functionals to the pressure which is
a convex function of interactions, or via the Gibbs variational principle, or
via the DLR (resp. KMS) conditions, and these methods have turned out to be
equivalent to each other (see the references [Is, Si2, Ge, BR2] and the original
papers cited therein). So, it would be worthwhile  to
extend the theory to the unbounded systems. The study of unbounded (continuous)
spin systems, which we consider in this paper, draws also much attention from
its close connection with Euclidean quantum field theory [AH-K].

The interactions which we consider in this paper satisfy the superstability and
regularity in the sense of Ruelle [R2]. The classical model for these
interactions has been extensively investigated by Lebowitz and Presutti [LP]. In
[LP], the existence of the thermodynamic limit of the pressure (free energy) for
various boundary conditions has been proved. Moreover, the existence of equilibrium
measures satisfying DLR conditions has been also shown in [LP]. We will show
that the mean entropy for this system also exists and prove that any DLR measure
satisfies the Gibbs variational equality and that the converse is also true if
we further impose the finiteness of the interaction range. The quantum unbounded
spin systems has been studied e.g., in [P1] aiming at the survey of the
existence of the thermodynamic limit theory of pressure and the existence and
uniqueness of the equilibrium states satisfying the (weak) KMS condition. In
[PY], there has been proposed a characterization of Gibbs (equilibrium) states
for quantum unbounded spin systems. In this paper, like in the classical case,
we will show the existence of mean entropy and show that the (weak) KMS state
constructed in [P1] via the thermodynamic limit of finite volume Green's
functions satisfies the Gibbs variational principle.

We organize this paper as follows: In Section 2, we give necessary notations,
preliminaries, and main results. The classical model is explained in Subsection
2.1 and the quantum model in Subseciton 2.2. In Section 3, we give the proofs
for classical systems.  Section 4 is devoted to the proof for quantum systems
and in Appendix we prove some technical inequlities need in Section 4.

\section{Notations, Preliminaries, and Main Results}
\subsection{Classical Systems}

We consider the classical unbounded spin systems. Let $\mbZ^\n$ be the
$\n$-dimensional integer lattice. At each site $i\in \mbZ^\n$, there corresponds
a vector spin variable $s_i\in \mbR^d$. For
$x:=(x^1,\cdots, x^d)\in \mbR^d$ and $i:=(i_1,\cdots, i_\n)\in \mbZ^\n$ we
write
$$
|x|:=\big[\dsum_{l=1}^d(x^l)^2\big]^{1/2}, \qquad |i|:=\max_{1\le l\le
\n}|i_\n|.
$$
When a subset $\L\sbs\mbZ^\n$ is a finite set we will write $\L\ssbs\mbZ^\n$. We
will consider both interactions between the spins at different sites as well as
self interactions. Throughout this paper we will impose the following conditions
on the interaction:
\begin{Assumption}%2.1
The interaction $\Ph=(\Ph_\D)_{\D\ssbs\mbZ^\n}$ satisfies the following
conditions:

(a) $\Ph_\D$ is a Borel measurable function on $(\mbR^d)^\D$ for each $\D\ssbs
\mbZ^\n$.

(b) $\Ph_\D$ is invariant under translations of $\mbZ^\n$, i.e., for any $i\in
\mbZ^\n$, $\Ph_{\D+i}=\t_i\Ph_\D$, where $\t_i$ is the natural translation of
functions.

(c) (Superstability) There are $A>0$ and $c\in \mbR$ such that for every
$x_\L\in (\mbR^d)^\L$,
$$
V(x_\L):=\dsum_{\D\sbs\L}\Ph_\D(x_\D)\ge \dsum_{i\in \L}[Ax_i^2-c].
$$

(d) (Strong regularity) There exists a decreasing positive function $\Ps$ on the
natural numbers such that
$$
\Ps(r)\le Kr^{-\n-\e} \text{ for some }K \text{ and }\e>0  \text{ with
}\dsum_{i\in \mbZ^\n}\Ps(|i|)<A.
$$

Furthermore, if $\L_1$ and $\L_2$ are disjoint finite subsets of $\mbZ^\n$ and
if one writes
$$
V(x_{\L_1\cup\L_2})=V(x_{\L_1})+V(x_{\L_2})+W(x_{\L_1},x_{\L_2})
$$
then the bound
$$
|W(x_{\L_1},x_{\L_2})|\le \dsum_{i\in \L_1}\dsum_{i\in
\L_2}\Ps(|i-j|)\frac12(x_i^2+x_j^2)
$$
holds.
\end{Assumption}

 The Gibbs measures for the interaction $\Ph$ are defined on the configuration
 space $\Om:=(\mbR^d)^{\mbZ^\n}$ in the following way.
For each $\L\sbs\mbZ^\n$ let $\p_\L:\Om\to \Om_\L:=(\mbR^d)^\L$ be the
projection, i.e., for $s:=(s_i)_{i\in \mbZ^\n}\in \Om$,
$\p_\L(s):=s_\L:=(s_i)_{i\in \L}\in \Om_\L$. We let $\mcF_\L^0$ denote the Borel
$\s$-algebra on $\Om_\L$ and let $\mcF_\L:=\p_\L^{-1}(\mcF_\L^0)$. We simply
write $\mcF$ for $\mcF_{\mbZ^\n}$. Denote by $\mcP(\Om,\mcF)$ the set of all
probability measures on $(\Om,\mcF)$. We equip $\mcP(\Om,\mcF)$ with a topology,
so called the topology of local convergence [Ge]: For each $\m\in
\mcP(\Om,\mcF)$ the sets
$$
\{\n\in \mcP(\Om,\mcF)|\,\max_{1\le k\le n}|\n(A_k)-\m(A_k)|<\e\}
$$
with $A_1,\cdots,A_n\in \dcup_{\L\ssbs\mbZ^\n}\mcF_\L$, $\e>0$, and $n\ge 1$,
form a base of neighborhoods of $\m$.

The following notion shall be useful throughout this paper.
\begin{Definition}%2.2
A probability measure $\m$ on $(\Om,\mcF)$ is said to be regular if there exist
$\ol A>0$ and $\ol \d$ so that for any $\L\ssbs \mbZ^\n$, the projection $\m_\L$
of $\m$ to $(\Om,\mcF_\L)$, which may be understood as a measure on
$(\Om_\L,\mcF_\L^0)$, is absolutely continuous with respect to $ds_\L:=\pd_{i\in
\L}ds_i$ with a Radon-Nikodym derivative $g(s_\L|\m)$ satisfying
$$
g(s_\L|\m)\le \exp[-\dsum_{i\in \L}(\ol As_i^2-\ol \d)].
$$
\end{Definition}
Before giving a definition of Gibbs measures we need to confine \lq\lq possible
confingurations". Let us define
$$
\begin{array}{rcl}
\mcS&:=&\dcup_{N\in \mbN}\mcS_N,\\[3mm]
\mcS_N&:=&\{s\in \Om|\,\fa\,l,\dsum_{|i|\le l}s_i^2\le
N^2(2l+1)^\n\}.\end{array}\eqno{(2.1)}
$$
It is easy to show that any regular measure is supported on $\mcS$. If $\m\in
\mcP(\Om,\mcF)$ satisfies $\m(\mcS)=1$, we say that $\m$ is {\it tempered}.

The partition function in $\L\ssbs\mbZ^\n$ for the interaction $\Ph$ with
boundary condition $\ol s\in \mcS$ is defined by
$$
Z^\Ph_\L(\ol s):=\dint ds_\L\exp[-V(s_\L)-W(s_\L,\ol s_{\L^c})],\eqno{(2.2)}
$$
where $ds_\L:=\pd_{i\in \L}ds_i$ and $ds_i$ means the Lebesgue measure on
$\mbR^d$ for each $i\in \mbZ^\n$.
The Gibbs specification $\g^\Ph=(\g_\L^\Ph)_{\L\ssbs\mbZ^\n}$ with respect to
$\mcS$ is defined by [Ge, Pr]:
$$
\g_\L^\Ph(A|\ol s)
:=\left\{\begin{array}{l}
\dfrac{1}{Z_\L^\Ph(\ol s)}\dint
ds_\L\exp[-V(s_\L)-W(s_\L,\ol s_{\L^c})]\,1_A(s_\L \ol s_{\L^c}),\quad
\text{ if }\ol s\in \mcS\\[3mm]
0,\quad \text{ if } \ol s\notin \mcS,
\end{array}\right.\eqno{(2.3)}
$$
where $A\in \mcF$ and $1_A$ is the indicator function on $A$ and $s_\L\ol
s_{\L^c}\in \Om$ coincides with $s_\L$ on $\L$ and with $\ol s_{\L^c}$ on
$\L^c$. It is easy to check that the Gibbs specification satisfies the
consistency condition [Ge, Pr]: For $\D\sbs\L$, $\ol s\in \mcS$,
$$
\begin{array}{rcl}
\g^\Ph_\L\g^\Ph_\D(A|\ol s)&:=&\dint_\mcS\g_\L^\Ph(ds'|\ol
s)\g_\D^\Ph(A|s')\\[3mm]
&=&\g_\L^\Ph(A|\ol s).
\end{array}\eqno{(2.4)}
$$
The Gibbs measures are defined as follows [Ge, Pr, Si2]:
\begin{Definition}%2.3
A probability measure $\m\in \mcP(\Om,\mcF)$ is said to be a Gibbs measure for
the interaction $\Ph$ if $\m$ is tempered and satisfies the following
equilibrium equations (DLR conditions):
$$
\m(A)=\dint\m(d\ol s)\g_\L^\Ph(A|\ol s), \quad \L\ssbs\mbZ^\n,\,\,\,A\in \mcF.
$$
We denote by $\mcG(\Ph)$ the family of all Gibbs measures and by
$\mcG^I(\Ph)$ the translationally invariant Gibbs measures.
\end{Definition}
For bounded functions $f$ on $\Om$,  we will sometimes use
the notation
$$
\g_\L^\Ph(f|\ol s):=\dint \g_\L^\Ph(ds'|\ol s)f(s').
$$
When $\Ph$ satisfies the conditions of Assumption 2.1, the existence of Gibbs
measures was proven in [LP]:
\begin{Theorem}%2.4
([LP, Theorem 4.5]) Suppose that the interaction $\Ph$ satisfies the hypotheses
of Assumption 2.1. Then, $\mcG(\Ph)\neq \es$.
\end{Theorem}
We notice that $\mcG^I(\Ph)$ is also nonempty. In fact, for any $\m\in
\mcG(\Ph)$ and $i\in \mbZ^\n$, it is easy to check that $\t_i\m\in \mcG(\Ph)$,
where $\t_i\m$ is the translation of $\m$. So, any (weak$^*$-)limit point of
$\{\frac{1}{|\L|}\sum_{i\in \L}\t_i\m\}$ as $\L$ increases to $\mbZ^\n$ belongs
to $\mcG^I(\Ph)$ (see [Si2, Corollary III.2.10]).
\vskip 0.3 true cm
The pressure, or the free energy, is defined in the following way.
We consider the free boundary
condition (b.c.), pure b.c., periodic b.c., and general b.c., respectively [LP].
The partition function in $\L$ with free b.c. is defined by
$$
Z_\L^\Ph:=\dint ds_\L\exp[-V(s_\L)]\eqno{(2.5)}
$$
and the (finite volume) pressure is defined by
$$
P_\L^\Ph:=\frac{1}{|\L|}\log Z_\L^\Ph,\eqno{(2.6)}
$$
where $|\L|$ means the volume (cardinality) of $\L\ssbs\mbZ^\n$. In order to
consider the pure b.c., for a technical reason, we need to introduce some
subsets of $\Om$ [LP]. For $a>0$ let
$$
X(a):=\{s\in \Om|\,s_i^2\le a\log|i|,\,\,|i|>1\}.\eqno{(2.7)}
$$
For $\ol s\in X(a)\dcap\mcS$, the partition function in the region $\L\ssbs
\mbZ^\n$ with pure b.c. $\ol s$ is given by the formula (2.2) and the pressure
is defined by
$$
P_\L^\Ph(\ol s):=\frac{1}{|\L|}\log Z_\L^\Ph(\ol s).\eqno{(2.8)}
$$
Let us also define
$$
\ol X(a):=\{s\in \Om|\,\ex\,\L(s)\ssbs\mbZ^\n,\,\,s_i^2\le a\log |i| \text{ for
}i\notin \L(s)\}.\eqno{(2.9)}
$$
It is not hard to show that if $\n\in \mcP(\Om)$ is a regular measure, then for
sufficiently large $a>0$ we have
$$
\n(\ol X(a))=1.\eqno{(2.10)}
$$
Now, let $\n\in \mcP(\Om)$ be a regular measure. The finite volume pressure for
the general b.c. $\n$ is defined by
$$
P_\L^\Ph(\n):=\dint\n(d\ol s)P_\L^\Ph(\ol s).\eqno{(2.11)}
$$
Finally, we introduce the periodic b.c.. Consider the \lq\lq torus"
$T_n:=(\mbZ/n\mbZ)^\n$ with the quotient map $q_n:\mbZ^\n\to T_n$. The
configuration space for $T_n$ is $\Om_{T_n}:=(\mbR^d)^{T_n}$, and $q_n$ induces
a map $r_n:\Om_{T_n}\to \Om$ (repeating the configuration across $\mbZ^\n$ with
period $n$ in each coordinate direction). The Hamiltonian for $T_n$ is defined
by [Is]
$$
_P\!H_{T_n}^\Ph:=\dsum_{[\D]}^{\quad *}\Ph_\D\circ r_n,\eqno{(2.12)}
$$
where $[\D]:=\{\D+nj|\,j\in \mbZ^\n\}$ is the equivalence class and
$\dsum^{\quad *}$
means that the sum is restricted to those $\D$ on which $q_n$ is one-to-one.
The pressure with periodic b.c. is defined by
$$
_P\!P_{T_n}^\Ph:=n^{-\n}\log\dint_{T_n}ds_{T_n}\exp[-_P\!H_{T_n}^\Ph(s_{T_n})].
\eqno{(2.13)}
$$
The thermodynamic limit of the pressure for any b.c. is proved in [LP]. Recall
that a sequence $\{\L_n\}$, $\L_n\ssbs \mbZ^\n$, is said to be
tending to $\mbZ^\n$ in the
sense of van Hove (we write $\lim_{\L_n\to \mbZ^\n \text{ (van Hove)}}$
hereafter) if: (a) $\L_{n+1}\sps\L_n$, (b) $\L_n\sps\D$ for any $\D\ssbs\mbZ^\n$
and some $n$, (c) given any parallelepiped $\G$ and the partition $\p(\G)$ of
$\mbZ^\n$ generated by translations of $\G$
$$
\lim N_\G^-(\L_n)=\infty,\quad \lim\frac{N_\G^-(\L_n)}{N_\G^+(\L_n)}=1,
$$
where $N_\G^-(\L_n)$ is the number of sets of $\p(\G)$ contained in $\L_n$,
$N_\G^+(\L_n)$ the number of sets with non-void intersection with $\L_n$.
\begin{Theorem}%2.5
([LP, Theorem 3.1]) Suppose that the hypotheses of Assumption 2.1 are satisfied
and $\L_n\to \mbZ^\n$ in the sense of van Hove. Then, $\{P_{\L_n}^\Ph\}$,
$\{P_{\L_n}^\Ph(\ol s)\}$, and  $\{P_{\L_n}^\Ph(\n)\}$ have the same
thermodynamic limit, say $P^\Ph$. Also, for the sequence of toruses $\{T_n\}$,
the sequence $\{_P\!P_{T_n}^\Ph\}$ converges to the same limit $P^\Ph$.
\end{Theorem}
We remark here that we have taken the definition for periodic b.c. from [Is],
which differs slightly from that of [LP]. Still, it is not hard to see that the
result of [LP] (for periodic b.c.) insists on to hold.
\vskip 0.3 true cm
We now define the mean entropy for Gibbs measures.
Let $\m\in \mcG(\Ph)$ and suppose that $f$ is a bounded
$\mcF_\L$-measurable function for some $\L\ssbs \mbZ^\n$. By Gibbs condition we
see that (we omit the superscript $\Ph$ in the notations if it involves no
confusion)
$$
\begin{array}{rcl}
\m(f)&=&\dint d\m(\ol s)\g_\L(f|\ol s)\\[3mm]
&=&\dint ds_\L f(s_\L)\dint d\m(\ol s)\frac{1}{Z_\L(\ol
s)}\exp[-V(s_\L)-W(s_\L,\ol s_{\L^c})]\\[3mm]
&=:&\dint ds_\L f(s_\L) \m^{(\L)}(s_\L).
\end{array}\eqno{(2.14)}
$$
Thus, the restriction $\m_\L$ of $\m$ to $\mcF_\L$, which we may understand as a
measure on $\Om_\L$, is absolutely continuous w.r.t. $ds_\L$ with a
Radon-Nikodym derivative $\m^{(\L)}(s_\L)$,
$$
\m^{(\L)}(s_\L)=\dint d\m(\ol s)\frac{1}{Z_\L(\ol
s)}\exp[-V(s_\L)-W(s_\L,\ol s_{\L^c})].\eqno{(2.15)}
$$
The entropy of $\m$ in $\L\ssbs \mbZ^\n$ is defined by [Is, Si2]
$$
S_\L(\m):=-\dint ds_\L\m^{(\L)}(s_\L)\log\m^{(\L)}(s_\L)=-\m(\log
\m^{(\L)}).\eqno{(2.16)}
$$
We will show that the mean entropy
$$
s(\m):=\lim_{\L\to \mbZ^\n\text{ (van Hove)}} |\L|^{-1}S_\L(\m)\eqno{(2.17)}
$$
exists. Furthermore, we will show that 
$\lim_{\L\to \mbZ^\n\text{ (van
Hove)}}$  $|\L|^{-1}\m(H_\L^\Ph)$, the limit of mean energy per unit
volume, also exists and we have the Gibbs variational
principle, which we state as a theorem:
\begin{Theorem}%2.6
Suppose that the interaction $\Ph$ satisfies the conditions of Assumption 2.1
and let $\m\in \mcG^I(\Ph)$. Then, the mean entropy $s(\m)$ and
$\lim_{\L\to \mbZ^\n\text{ (van Hove)}}|\L|^{-1}\m(H_\L^\Ph)$ exist and
satisfies the variational equality:
$$
s(\m)-\lim_{\L\to \mbZ^\n\text{ (van
Hove)}}|\L|^{-1}\m(H_\L^\Ph)=P^\Ph.\eqno{(2.18)}
$$
\end{Theorem}
The proof of Theorem 2.6 will be given in the next section.

As in the case of bounded spin systems, the converse of Theorem 2.6 also holds. However, for
a technical reason in the proof, we impose a further condition of finiteness of
interaction range.
\begin{Theorem}%2.7
Suppose that a finite range interaction $\Ph$ satisfies the conditions of
Assumption 2.1. Let $\m\in \mcP(\Om)$ be a translationally invariant regular measure (see Definition 2.2)
and suppose that the variational equality (2.18) holds. Then $\m\in
\mcG^I(\Ph)$.
\end{Theorem}
The proof will be given in the next section.

\subsection{Quantum Systems}
Let us now turn our attention to the quantum systems. It is generally accepted
that in quantum statistical mechanics the equilibrium states are those of KMS
states [BR1-2, Is, Si2]. For quantum unbounded spin systems, by using the
Green's function method [BR2], a state satisfying  (weak) KMS condition
has been constructed in [P1]. In [PY], there has been proposed a
characterization of Gibbs states by using the concept of Gibbs measures and the
conditional reduced density matrices. We will show that the (weak) KMS state
constructed in [P1] satisfies the Gibbs variational equality. Let us begin with
presenting necessary notations. We refer to [PY] for details.

For each bounded region $\L\ssbs\mbZ^\n$ the Hilbert space for the unbounded
spin systems is given by
$$
\begin{array}{rcl}
\mcH_\L&:=&\otimes_{i\in \L}L^2(\mbR^d,dx_i)\\[3mm]
&=&L^2((\mbR^d)^\L,dx_\L).
\end{array}\eqno{(2.19)}
$$
The Hamiltonian operator for the region $\L\ssbs\mbZ^\n$ is given by
$$
H_\L:=-\frac12 \dsum_{i\in \L}\D_i +V(x_\L),\eqno{(2.20)}
$$
where $\D_i$ is the Laplacian operator for the variable $x_i\in \mbR^d$ and
$V(x_\L)$ is the potential energy in the region $\L$. We assume that the
interaction $\Ph$ satisfies the hypotheses of Assumption 2.1.

For each bounded region $\L\ssbs \mbZ^\n$, the $C^*$-algebra for local
observables is defined by
$$
\mcA_\L:=\mcL(\mcH_\L),\eqno{(2.21)}
$$
where $\mcL(\mcH_\L)$ is the algebra of all bounded operators on $\mcH_\L$. If
$\L_1\dcap\L_2=\es$, then $\mcH_{\L_1\cup\L_2}=\mcH_{\L_1}\otimes \mcH_{\L_2}$
and $\mcA_{\L_1}$ is isomorphic to the $C^*$-algebra $\mcA_{\L_1}\otimes{\bf 1}
_{\L_2}$, where ${\bf 1}_{\L_2}$ denotes the identity operator on $\mcH_{\L_2}$.
In this way we identify $\mcA_\L$ as a sub-algebra of $\mcA_{\L'}$ when
$\L\sbs\L'$. Let
$$
\mcA:=\ol{\dcup_{\L\ssbs \mbZ^\n}\mcA_\L}\eqno{(2.22)}
$$
be the $C^*$-algebra of the quasilocal observables. Notice that $\mcA$ has an
identity.
\vskip 0.3 true cm
The partition function in a region $\L\ssbs \mbZ^\n$ is
given by
$$
Z_\L^\Ph:=\Tr_\L(e^{-H_\L}),\eqno{(2.23)}
$$
where $\Tr_\L$ means the trace on the Hilbert space $\mcH_\L$. Notice that by
the superstability condition of Assumption 2.1 (c), $e^{-H_\L}$ belongs to the
trace class and so $Z_\L^\Ph$ is well defined as a finite number. The finite
volume pressure is defined by
$$
P_\L^\Ph:=\frac{1}{|\L|}\log Z_\L^\Ph.\eqno{(2.24)}
$$
We will again suppress the superscript $\Ph$ from the notations.

We notice that by the Feynman-Kac formula [Si1], the operator $e^{-H_\L}$ has
its integral kernel function
$$
e^{-H_\L}(x_\L,y_\L)=\dint
P_{x_\L,y_\L}(ds_\L)\exp\big[-\dint_0^1V(s_\L(\t))\,d\t\big],\eqno{(2.25)}
$$
where $x_\L$ and $y_\L$ are points in $(\mbR^d)^\L$, $s_\L\in
(C([0,1];\mbR^d))^\L$,  and $P_{x_\L,y_\L}(ds_\L)$ is the conditional Wiener
measure on the path space
$$
\Om_{x_\L,y_\L}:=\{s_\L\in
(C([0,1];\mbR^d))^\L|\,s_\L(0)=x_\L,\,\,s_\L(1)=y_\L\}.
$$
(See e.g. [Si1] for details.) We will simply write $V(s_\L)$ for
$\int_0^1V(s_\L(\t))\,d\t$.  Thus, the partition function $Z_\L$ can be
rewritten as an integral over a path space:
$$
Z_\L=\dint_{(\mbR^d)^\L}dx_\L\dint_{\Om_{x_\L,x_\L}}P_{x_\L,x_\L}(ds_\L)
e^{-V(s_\L)}.\eqno{(2.26)}
$$
Since, by a translation $y_\L\mapsto y_\L-x_\L$, the measure
$P_{x_\L,x_\L}(ds_\L)$ on $\Om_{x_\L,x_\L}$ becomes the measure
$P_{0,0}(ds_\L)=: P(ds_\L)$ on $\Om_{0,0}$, by letting $\l(ds_\L):=dx_\L
P(ds_\L)$ on $(\mbR^d)^\L\times\Om_{0,0}$, the expression in (2.26) can be
further simplified:
$$
Z_\L=\dint\l(ds_\L)e^{-V(s_\L)}.\eqno{(2.27)}
$$
By using the above Wiener integral formalism and Ruelle-type probability
estimates, Park has shown the existence of the thermodynamic limit of the
pressure [P1]. We remark here that by introducing the concept of \lq\lq Gibbs
specifications" (and hence Gibbs measures) for quantum spin systems as in [PY],
we can also define the finite volume pressures with different kind of boundary
conditions like in the classical case (see (2.6), (2.8), and (2.11)). By
extending the method of [LP] to the quantum case we can also prove that the
thermodynamic limits of the pressure with different b.c. are the same (cf.
Theorem 2.5). We need, however, only the result for the free b.c., which we
state as a Theorem.
\begin{Theorem}%2.8
([P1, Theorem 2.2]) Suppose that the hypotheses of Assumption 2.1 hold.
Then, the limit
$$
\lim_{\L\to \mbZ^\n\text{ (van Hove)}}P_\L^\Ph=P^\Ph
$$
exits.
\end{Theorem}
\vskip 0.3 true cm
We now consider the equilibrium states for quantum systems.
Define the finite volume Green's functions [BR2] by
$$
G_\L(A,B;t):=\om_\L(A\a_t^\L(B)), \quad A,\,\,B\in \mcA_\L,\eqno{(2.28)}
$$
where $\om_\L$ is the local Gibbs state on $\mcA_\L$:
$$
\om_\L(A):=\frac{1}{Z_\L}\Tr_\L(e^{-H_\L}A), \quad A\in \mcA_\L,\eqno{(2.29)}
$$
and  $\a_t^\L$ is the time evolution automorphism on $\mcA_\L$:
$$
\a_t^\L(B):=e^{itH_\L}Be^{-itH_\L},\quad B\in \mcA_\L.\eqno{(2.30)}
$$
By Hahn-Banach theorem the state $\om_\L$ can be extended to a state on $\mcA$
and we use the same notation for the extension. Notice that $G_\L$ is bounded as
$$
|G_\L(A,B;t)|\le \|A\|\,\|B\|.\eqno{(2.31)}
$$
Thus by Tychonoff's theorem, there exists a subnet
$\{\L_n\}\sbs\mbZ^\n$ such that
$$
G_\L(A,B;t):=\lim_{n\to \infty}G_{\L_n}(A,B;t)\eqno{(2.32)}
$$
exists for all $A,\,B\in \mcA$, $t\in \mbR$ [BR2]. Clearly, the values
$$
\om(A):=G(A,B;t),\quad A\in \mcA,\eqno{(2.33)}
$$
determine a state $\om$ over the quasilocal algebra $\mcA$. That is, $\om$ is a
positive-definite linear functional with norm one on $\mcA$. Let us impose a
further condition on the interaction:
\begin{Assumption}%2.9
(Polynomial boundedness of interaction) There exist a constant $D>0$ and a
natural number $n$ such that the one-body interaction $\Ph_{\{i\}}(x_i)\eqv
P(x_i)$ satisfies the following bound:
$$
P(x_i)\le D(|x_i|^{2n}+1),\quad i\in \mbZ^\n.
$$
\end{Assumption}
Under the conditions of Assumption 2.1 and Assumption 2.9, Park has shown that
the state $\om$ in (2.33) satisfies a (weak) KMS
condition. We refer to [P1] for the details. We take the state $\om$ as a probe
for our purpose.
\vskip 0.5 true cm
The mean entropy for the states of quantum systems is defined as follows.
Let $\r$ be any state on $\mcA$ and $\r_\L$ the restriction
of $\r$ to $\mcA_\L$, $\L\ssbs\mbZ^\n$.
Suppose that $\r_\L$ is a normal state on $\mcA_\L$, i.e.,
there exists a density matrix $\r^{(\L)}\in \mcA_\L$ such that
$$
\r_\L(A)=\Tr_\L(\r^{(\L)}A),\quad A\in \mcA_\L.\eqno{(2.34)}
$$
The entropy of $\r$ in $\L$ is defined by [BR2, Is, Si2]
$$
\begin{array}{rcl}
S_\L(\r)&:=&-\Tr_\L(\r^{(\L)}\log\r^{(\L)})\\[3mm]
&=&-\r_\L(\log\r^{(\L)}).
\end{array}\eqno{(2.35)}
$$
From the definition, we see that (we omit $\Ph$ from the notations)
$$
S_\L(\om_\L)-\om_\L(H_\L)=|\L|P_\L.\eqno{(2.36)}
$$
In general we have the following properties:
\begin{Proposition}%2.10
Suppose that $\r$ is a state on $\mcA$ such that the restriction $\r_\L$ of $\r$
to any  $\mcA_\L$,
$\L\ssbs\mbZ^\n$,  is a normal state. Then, the following properties hold:

(a) For any $\L\ssbs \mbZ^\n$,
$$
S_\L(\r)-\r(H_\L)\le |\L|P_\L.\eqno{(2.37)}
$$

(b) The mean entropy
$$
s(\r):=\lim_{a\to \infty}a^{-\n}S_{C_a}(\r)\eqno{(2.38)}
$$
exists, where $C_a$ is a cube of sides $a$.
\end{Proposition}
We will give a proof of Proposition 2.10 in Section 4. As a corollary to
Proposition 2.10 we have the following variational inequality:
\begin{Corollary}%2.11
Suppose the hypotheses of Proposition 2.10 hold. Then, the inequality
$$
s(\r)-\liminf_{a\to \infty}a^{-\n}\r(H_{C_a})\le P^\Ph\eqno{(2.39)}
$$
holds.
\end{Corollary}
One of the main purpose of this paper is to show that for the state $\om$ given
in (2.33) the equality holds in (2.39):
\begin{Theorem}%2.12
Suppose the hypotheses of Assumption 2.1 and 2.9 hold. Let $\om$ be the
equilibrium state given in (2.33). Then, the mean energy per unit volume $\lim_{a\to
\infty}a^{-\n}\om(H_{C_a})$ exists and the equality
$$
s(\om)-\lim_{a\to \infty}a^{-\n}\om(H_{C_a})=P^\Ph
$$
holds.
\end{Theorem}
The proof of the above theorem will be given in Section 4.

\section{Proofs of Main Results: Classical Systems}
\subsection{DLR measures satisfy the variational equality}
In this subsection we will prove Theorem 2.6. Let $\m\in \mcG(\Ph)$ be a Gibbs
measure. Recall the definition of the entropy $S_\L(\r)$ in (2.16). We first
give some properties of $S_\L(\m)$.
\begin{Lemma}%3.1
Let $\m\in \mcG(\Ph)$ and $S_\L(\m)$, $\L\ssbs\mbZ^\n$, be defined by (2.16).
Then, the following properties hold:

(a) $S_\L(\m)\le 0$ (negativity).

(b) $\L\sbs\L'\Lra S_{\L'}(\m)\le S_\L(\m)$ (decrease).

(c) $S_{\L\cup\L'}(\m)+S_{\L\cap\L'}(\m)\le S_\L(\m)+S_{\L'}(\m)$ (strong
subadditivity).
\end{Lemma}
{\bf Proof}. The proofs of (b) and (c) can be done by the same method given,
e.g., in the proof of Lemma II.2.1 of [Is]. The property (a) will not be used in
this paper, but we have given it as a property of the entropy. Since the a
priori measure, the Lebesgue measure, is not a finite measure we can not simply
take $S_\es(\m)\eqv 0$ as done in the proof of Lemma II.2.1 of [Is] for bounded
spin systems. However, we
can make a detour. Let us introduce a probability measure
$$
d\m^{(0)}(s_i):=N^{-1}\exp(-\a s_i^2)ds_i,\eqno{(3.1)}
$$
where $\a>0$ is a sufficiently small number and $N:=(\p/\a)^{d/2}$ is the
normalization constant. For each $\L\ssbs\mbZ^\n$ let us define
$$
\wt V(s_\L):=V(s_\L)-\dsum_{i\in \L}\a s_i^2.\eqno{(3.2)}
$$
We then have, by Gibbs condition, for $f\in L^1(\Om;d\m)$
$$
\begin{array}{rcl}
\dint fd\m&=&\dint d\m(\ol s)\frac{1}{Z_\L(\ol s)}\dint
ds_\L\exp[-V(s_\L)-W(s_\L,\ol s_{\L^c})]f(s_\L\ol s_{\L^c})\\[3mm]
&=&\dint d\m(\ol s)\frac{1}{Z_\L^{(0)}(\ol s)}\dint
d\m^{(0)}(s_\L)\exp[-\wt V(s_\L)-W(s_\L,\ol s_{\L^c})]f(s_\L\ol s_{\L^c}),
\end{array}\eqno{(3.3)}
$$
where
$$
Z_\L^{(0)}(\ol s):=\dint d\m^{(0)}(s_\L)\exp[-\wt V(s_\L)-W(s_\L,\ol s_{\L^c})]
=N^{-|\L|}Z_\L(\ol s).
$$
Let us define (see (2.15))
$$
\begin{array}{rcl}
\m_\L^{(0)}(s_\L)&:=&\dint d\m(\ol s)\frac{1}{Z_\L^{(0)}(\ol s)}
\exp[-\wt V(s_\L)-W(s_\L,\ol s_{\L^c})]\\[3mm]
&=&N^{|\L|}\exp[\a\sum_{i\in \L}s_i^2]\,\m^{(\L)}(s_\L).\end{array}\eqno{(3.4)}
$$
We notice that $\m_\L^{(0)}(s_\L)$ is a Radon-Nikodym derivative of $\m_\L$, the
restriction of $\m$ to $\mcF_\L$, with respect to $\m^{(0)}(\eqv
(\m^{(0)})^\L)$. Define
$$
S_\L^{(0)}(\m):=-<\m_\L^{(0)}\log\m_\L^{(0)}>^{(0)},\eqno{(3.5)}
$$
where $<\cdot>^{(0)}$ is, obviously, the expectation w.r.t. $\m^{(0)}$. Then,
using (3.4) we have that
$$
S_\L(\m)=|\L|[\log N+a_0]+S_\L^{(0)}(\m),\eqno{(3.6)}
$$
where $a_0:=\a<s_i^2>_\m\,>\,0$. By the usual method we have $S_\L^{(0)}(\m)\ge
S_{\L'}^{(0)}(\m)$ for $\L\sbs\L'$. Moreover, since $\m^{(0)}$ is a probability
measure, we may define $S_\es^{(0)}(\m)\eqv 0$. By putting $\L=\es$ in (3.6) we
get $S_\es(\m)=0$. Using the property (b) we obtain (a). $\bbx$
\begin{Proposition}%3.2
Suppose $\m\in \mcG^I(\Ph)$. The mean entropy
$$
s(\m):=\lim_{\L\to \mbZ^\n\text{ (van Hove)}}|\L|^{-1}S_\L(\m)
$$
exists.
\end{Proposition}
{\bf Proof}. It follows from Lemma 3.1 and the same method given in [Is, Theorem
II.2.2]. $\bbx$
\vskip 0.5 true cm
\noi{\bf Proof of Theorem 2.6}. Let us now suppose $\m\in \mcG^I(\Ph)$. By
Jensen's inequality we see that
$$
\begin{array}{rcl}
S_\L(\m)-\m(H_\L)&=&\m(-\log\m^{(\L)}-H_\L)\\[3mm]
&\le &\log\m(\frac{e^{-H_\L}}{\m^{(\L)}})\\[3mm]
&=&\log<e^{-H_\L}>_0=|\L|P_\L,
\end{array}\eqno{(3.7)}
$$
where $<\cdot>_0$ is the expectation with respect to the Lebesgue measure on
$\Om_\L$. Dividing both sides of (3.7) by $|\L|$, taking limit supremum, and
using Theorem 2.5 and Proposition 3.2, we get
$$
s(\m)-\liminf_{\L\to \mbZ^\n\text{ (van Hove)}}\frac{1}{|\L|}\m(H_\L)\le
P^\Ph.\eqno{(3.8)}
$$
Next, for each $\L\ssbs \mbZ^\n$ and $\ol s\in \mcS\cap\ol X(a)$ let us define
a state $\m_\L^{(\ol s)}$ on $C(\Om_\L)$ as follows: for any $f\in C(\Om_\L)$
$$
\begin{array}{rcl}
\m_\L^{(\ol s)}(f)&:=&\dfrac{1}{Z_\L(\ol s)}\dint ds_\L\exp[-V(s_\L)-W(s_\L,\ol
s_{\L^c})]f(s_\L)\\[3mm]
&\eqv&\dint ds_\L\,\m^{(\L|\ol s)}(s_\L)f(s_\L).\end{array}\eqno{(3.9)}
$$
Notice that the function $0\le t\mapsto g(t):=-t\log t$ is concave. Thus,
$$
\begin{array}{rcl}
S_\L(\m)&=&-<\m^{(\L)}\log\m^{(\L)}>_0\\[3mm]
&=&<g(\int d\m(\ol s)\m^{(\L|\ol s)})>_0\\[3mm]
&\ge&-\dint d\m(\ol s)<\m^{(\L|\ol s)}\log\m^{(\L|\ol s)}>_0\\[3mm]
&=&\dint d\m(\ol s)\big[\log Z_\L(\ol s)+\m_\L^{(\ol s)}(V(\cdot)+W(\cdot,\ol
s_{\L^c}))\big]\\[3mm]
&=&|\L|P_\L(\m)+\dint d\m(s)[H_\L(s_\L)+W(s_\L,s_{\L^c})],\end{array}\eqno{(3.10)}
$$
where we have used the notation (2.11) and the Gibbs condition to get the last
equality. We assert that
$$
\lim_{\L\to \mbZ^\n\text{ (van Hove)}}\frac{1}{|\L|}\dint
d\m(s)W(s_\L,s_{\L^c})=0.\eqno{(3.11)}
$$
Under the condition of (3.11), after dividing both sides of (3.10) by $|\L|$ and
taking limsup, and using Theorem 2.5 and Proposition 3.2 again, we get
$$
s(\m)\ge P^\Ph+\limsup_{\L\to \mbZ^\n\text{ (van
Hove)}}\frac{1}{|\L|}\m(H_\L).\eqno{(3.12)}
$$
This, together with the inequality (3.8), completes the proof. So, it remains
only to prove the assertion (3.11). In fact, by using the regularity of the
interaction, monotone convergence theorem, and regularity of the Gibbs measure
(see Definition 2.2) we see that
$$
\begin{array}{rcl}
|\L|^{-1}\dint d\m(s)|W(s_\L,s_{\L^c})|&\le &|\L|^{-1}\dint d\m(s)\dsum_{i\in
\L}\dsum_{j\in \L^c}\Ps(|i-j|)\frac12(s_i^2+s_j^2)\\[3mm]
&\le &c|\L|^{-1}\dsum_{i\in\L}\dsum_{j\in \L^c}\Ps(|i-j|),\end{array}\eqno{(3.13)}
$$
for some constant $c>0$. Let us define for $r>0$
$$
\ptl_r^-(\L):=\{i\in \L|\,\text{dist}\,(i,\L^c)\le r\}.\eqno{(3.14)}
$$
Notice that given $\e>0$, there exists  $r>0$ such that
$$
\dsum_{j\in \mbZ^\n:\,|j|\ge r}\Ps(|j|)\le \e.\eqno{(3.15)}
$$
We fix such a  number $r>0$. If $\L$ is sufficiently large, it is obvious that
$$
|\L|^{-1}|\ptl _r^-(\L)|\le \e.\eqno{(3.16)}
$$
Splitting the sum $\dsum_{i\in \L}=\dsum_{i\in \L\sm\ptl_r^-(\L)}+\dsum_{i\in
\ptl_r^-(\L)}$ in (3.13) we see that the right hand side of (3.13) is bounded by
$(c+J)\e$, where $J:=\dsum_{j\in \mbZ^n}\Ps(|j|)$. Since $\e>0$ is arbitrary,
this proves the assertion (3.11) and the proof of Thoerem 2.6 is now completed.
$\bbx$
\begin{Remark}%3.3
When the interaction $\Ph$ is one-body and two-body interaction, i.e.,
$\Ph_\D=0$ if $|D|\ge 3$, then one easily checks that the statement of Theorem
2.6 can be replaced by the usual form [Is, Si2]:
$$
s(\m)-\m(A_\Ph)=P^\Ph,\eqno{(3.16)}
$$
where $A_\Ph:=\dsum_{X\ni 0}\frac1{|X|}\Ph_X$.
\end{Remark}
\subsection{Variational equality implies the DLR condition}
In this subsection we will prove Theorem 2.7. We assume that the interaction
$\Ph$ is of finite range, i.e., there exists $R>0$ such that $\Ph_X=0$ whenever
$\text{diam}(X)\ge R$, where $\text{diam}(X):=\max\{|i-j|\,|\,i,j\in X\}$ is the
diameter of $X$. Suppose that a  translationally invariant measure $\m\in
\mcP(\Om)$ is regular (see  Definition 2.2) and satisfies the variational
equality (2.18). We want to show that $\m\in \mcG^I(\Ph)$. The key idea of the
proof we have borrowed from that of Theorem III.2.1 of [Is].
\vskip 0.3 true cm
\noi{\bf Proof of Theorem 2.7}. We have to show that for any bounded
$\mcF$-measurable $f:\Om\to \mbR$,
$$
\m(f-\g_\L f)=0,\quad  \forall\,\,\L\ssbs \mbZ^\n.\eqno{(3.17)}
$$
It is enough to show (3.17) only for local functions. So, suppose that $f$ is
$\mcF_{\L'}$-measurable for some $\L'\ssbs \mbZ^\n$. Let us put $F:=f-\g_\L f$.
Since the interaction is of finite range $F$ is also a local function, i.e., $F$
is $\mcF_{\wt \L}$-measurable for some finite $\wt \L\sps\L\cup\L'$. Let us
define an auxiliary interaction $\Ps$ as follows:
$$
\left\{\begin{array}{l}
\Ps_{\wt \L+i}=\t_iF \quad\text{ for }i\in \mbZ^\n\\[3mm]
\Ps_Y=0 \quad \text{ if }Y \text{ is not a translate of }\wt
\L.\end{array}\right.\eqno{(3.18)}
$$
Since $\Ps$ is bounded and of finite range we see that the interaction $\Ph+\l
\Ps$ is also superstable and regular for any $\l\in \mbR$. Recall the periodic
boundary condition introduced in Subsection 2.1. Let $T_n:=(\mbZ/n\mbZ)^\n$ be a
sufficiently large torus so that $T_n\sps \wt \L$. We let $<\cdot>_{n,\l}$ be
the (local) Gibbs measure on $T_n$ with pure b.c. for the interaction
$\Ph+\l\Ps$, i.e.,
$$
<A>_{n,\l}:=\frac1{Z_{T_n}^{(\l)}}\dint
ds_{T_n}\exp[-_P\!H_{T_n}^{\Ph+\l\Ps}(s_{T_n})]A(s_{T_n}),\eqno{(3.19)}
$$
where the Hamiltonian $_P\!H_{T_n}^{\Ph+\l\Ps}$ is defined as in (2.12) and
$Z_{T_n}^{(\l)}$ is the obvious normalizing factor. Given $\e>0$ we claim that
there exist $\l_0>0$ and $\mcF_{T_n}$-measurable functions $F^{(n,\l)}$, $|\l|\le
\l_0$, such that
$$
<F^{(n,\l)}>_{n,\l}=0\text{ and }\|F^{(n,\l)}-F\|_{\infty}\le \e.\eqno{(3.20)}
$$
In fact, for each $t\in \Om_{T_n}$ (we regard $T_n$ as a subset of $\mbZ^\n$ if
necessary) let us define
$$
\g_\L^{(n,\l)}f(t):=\frac1{Z_\L^{(n,\l)}(t)}\dint ds'_\L\exp\big[-
_P\!H_{T_n}^{\Ph+\l\Ps}(s'_{\L}t_{T_n\sm\L})-\l\dsum_{(i+\wt \L)\cap\L\ne
\es}(\t_iF)(s'_\L t_{T_n\sm\L})\big]f(s'_\L t_{T_n\sm\L}),\eqno{(3.21)}
$$
where $Z_\L^{(n,\l)}(t)$ is a normalization factor. We define
$$
F^{(n,\l)}(t):=f(t)-\g_\L^{(n,\l)}f(t),\quad t\in \Om_{T_n}.\eqno{(3.22)}
$$
It is not hard to show that
$$
<F^{(n,\l)}>_{n,\l}=0.\eqno{(3.23)}
$$
In order to show that $\|F^{(n,\l)}-F\|_{\infty}\le \e$,  without loss,
we may assume $f\ge 0$ since $\g_\L$ and
$\g_\L^{(n,\l)}$ are positivity preserving, i.e., $\g_\L f\ge 0$ and
$\g_\L^{(n,\l)}\ge 0$ whenever $f\ge 0$.
We notice that since the interaction $\Ph$ is of finite range, if $\g=0$ then,
$$
\g_\L^{(n,0)}f(t)=\g_\L f(t),\quad t\in \Om_{T_n}.\eqno{(3.24)}
$$
On the other hand, since $F$ is bounded, it is easy to see that
$$
e^{-2|\l|\,|\L|\,|\wt \L|\,\|F\|_\infty}\g_\L^{(n,0)}f(t)
\le \g_\L^{(n,\l)}f(t)\le
e^{2|\l|\,|\L|\,|\wt \L|\,\|F\|_\infty}\g_\L^{(n,0)}f(t)\eqno{(3.25)}
$$
From (3.24)-(3.25) we see that there exists $\l_0>0$ such that
$\|F^{(n,\l)}-F\|_\infty\le \e$ if $|\l|\le \l_0$. Thus, our assertion (3.20) is
proved.

Now, suppose that the equality (2.18) holds.
Recall the notion of the pressure for general b.c. in (2.13). We see that
$$
\begin{array}{rcl}
\dfrac{d}{d\l}\,_P\!P_{T_n}^{\Ph+\l\Ps}&=&<\exp[-_P\!H_{T_n}^{\Ph+\l\Ps}]>_0^{-1}
<-F\exp[-_P\!H_{T_n}^{\Ph+\l\Ps}]>_0\\[3mm]
&=&<-F>_{n,\l}.\end{array}\eqno{(3.26)}
$$
Thus, by using (3.20) we get
$$
|\frac{d}{d\l}\,_P\!P_{T_n}^{\Ph+\l\Ps}|\le \e\quad \text{ for }|\l|\le
\l_0.\eqno{(3.27)}
$$
This implies that
$$
_P\!P_{T_n}^{\Ph+\l\Ps}\le\, _P\!P_{T_n}^{\Ph}+\e|\l|\quad\text{ for }|\l|\le
\l_0.\eqno{(3.28)}
$$
By letting $n$ go to infinity and using Theorem 2.5 we then have
$$
P^{\Ph+\l\Ps}\le P^{\Ph}+\e|\l|\quad\text{ for }|\l|\le
\l_0.\eqno{(3.29)}
$$
On the other hand, by the same method used in the proof of (3.8), and using
(2.18) we see that for any $\l\in \mbR$
$$
\begin{array}{rcl}
P^{\Ph+\l\Ps}&\ge&s(\m)-\liminf_{\L\to \mbZ^\n\text{ (van
Hove)}}\frac1{|\L|}\m(H_\L^{\Ph+\l\Ps})\\[3mm]
&=&s(\m)-\lim_{\L\to \mbZ^\n\text{ (van
Hove)}}\frac1{|\L|}\m(H_\L^{\Ph})-\l\m(F)\\[3mm]
&=&P^\Ph-\l\m(F).\end{array}\eqno{(3.30)}
$$
Combining (3.29) and (3.30) we have
$$
|\m(F)|\le \e.\eqno{(3.31)}
$$
Since $\e>0$ is arbitrary this implies $\m(F)=0$, which is the equation (3.17)
that was to be shown. $\bbx$

\section{Proofs of Main Results: Quantum Systems}
In this section we prove Theorem 2.12, that is, the state $\om$ on $\mcA$
defined in (2.33), which satisfies the (weak) KMS condition, satisfies the Gibbs
variational equality. Let us begin with some lemmas that will be needed.
\begin{Lemma}%4.1
(Jensen's inequality) Let $\r$ be any state on $\mcA$. Then for any self-adjoint
$B\in \mcA$, $e^{\r(B)}\le \r(e^B)$.
\end{Lemma}
The proof of the above lemma can be found, e.g., in [Is, Lemma I.3.1].

Recall the definition of entropy for states in (2.35).
\begin{Lemma}%4.2
Suppose that $\r$ is a state on $\mcA$ such that the restriction $\r_\L$ of $\r$
to any $\mcA_\L$, $\L\ssbs\mbZ^\n$, is a normal state with a density matrix
$\r^{(\L)}\in \mcA_\L$.
Then, the following properties hold:

(a) $S_\L(\r)\ge 0$ (positivity).

(b) $\L\cap \L'=\es\Lra S_{\L\cup\L'}(\r)\le S_\L(\r)+S_{\L'}(\r)$
(subadditivity).
\end{Lemma}
{\bf Proof}. (a) is obvious since $\Tr_\L(\r^{(\L)})=1$. For (b), we notice that
the bound
$$
\Tr(A\log A)-\Tr(A\log B)\ge \Tr(A-B)\eqno{(4.1)}
$$
holds for any trace class non-singular positive operators $A$ and $B$. By
letting $A:=\r^{(\L\cup\L')}$ and $B:=\r^{(\L)}\otimes\r^{(\L')}$ in (4.1) we
get
$$
\begin{array}{rcl}
S_{\L\cup\L'}(\r)&=&-\Tr_{(\L\cup\L')}(A\log A)\\[3mm]
&\le &-\Tr_{(\L\cup\L')}(A\log B)\\[3mm]
&=&-\Tr_{\L}(\r^{(\L)}\log \r^{(\L)})-\Tr_{\L'}(\r^{(\L')}\log \r^{(\L')})
\\[3mm]
&=&S_\L(\r)+S_{\L'}(\r).\quad \bbx
\end{array}
$$
\vskip 0.5 true cm
\noi{\bf Proof of Proposition 2.10}. (b) is a simple consequence of the
subadditivity of entropy in Lemma 4.2 (b). In order to prove (a) let $\{e_j\}$
be the normalized eigen-vectors of $\r^{(\L)}$ with $\r^{(\L)}e_j=\l_je_j$. We
notice that $\dsum_j\l_j=1$. Then,
$$
\begin{array}{rcl}
e^{S_\L(\r)-\r(H_\L)}&=&e^{\Tr_\L(\r^{(\L)}(-\log\r^{(\L)}-H_\L))}\\[3mm]
&=&\exp\big[\dsum_j\l_j(-\log\l_j-<e_j,H_\L e_j>)\big]\\[3mm]
&\le&\dsum_j\exp[<e_j,-H_\L e_j>]\quad \text{(by Jensen's inequality)}\\[3mm]
&\le &\dsum_j<e_j,e^{-H_\L}e_j>\quad\text{(by Lemma 4.1)}\\[3mm]
&=&\Tr_\L(e^{-H_\L})=e^{|\L|P_\L}.\quad \bbx
\end{array}
$$
We are now going to prove Theorem 2.12. For that purpose we need to define some
notations.
Recall the local Gibbs states $\om_\L$ defined in (2.29). We simply write
$\om_n$ for $\om_{\L_n}$, where $\{\L_n\}$ is a net in (2.32). We notice
that the state $\om$ in (2.33) is a weak limit of $\{\om_n\}$. Let us write
$$
K^{(n)}:=\frac1{Z_{\L_n}}\exp(-H_{\L_n})\eqno{(4.2)}
$$
and $\om^{(\L)}$ the density matrix of the restriction of $\om$ to $\mcA_\L$.
When $\mcH_1$  and $\mcH_2$ are two Hilbert spaces and if $A$ is a trace class
operator on $\mcH_1\otimes\mcH_2$ we denote by
$\Tr_{(\mcH_1\otimes\mcH_2|\mcH_1)} (A)$ the partial trace of $A$ on $\mcH_2$,
i.e., $\Tr_{(\mcH_1\otimes\mcH_2|\mcH_1)} (A)$ is a trace class operator on
$\mcH_1$ such that
$$
\Tr_{\mcH_1}(\Tr_{(\mcH_1\otimes\mcH_2|\mcH_1)} (A))=\Tr_{\mcH_1\otimes\mcH_2}
 (A).
$$
When $\L\sbs\L'$, we simply write $\Tr_{(\L'|\L)}(A)$ for $\Tr_{(\mcH_{\L'}|
\mcH_\L)}(A)$. One notes that
$$
\om^{(\L)}=\text{w-}\lim_{n\to \infty}\Tr_{(\L'|\L)}K^{(n)},\eqno{(4.3)}
$$
where w-limit means that
$$
\Tr_\L(\om^{(\L)}A)=\lim_{n\to \infty}\Tr_\L((\Tr_{(\L_n|\L)}K^{(n)})A), \quad
A\in \mcA_\L.\eqno{(4.4)}
$$
We further simplify the notation $\Tr_{(\L_n|\L)}K^{(n)}$ by $K_\L^{(n)}$ and
$\Tr_{(\L_n|\L_n\sm\L)}K^{(n)}$ by $K_{\L_n\sm\L}^{(n)}$.
\vskip 0.3 true cm
\noi {\bf Proof of Theorem 2.12.} By Corollary 2.11, it is enough to show
$$
s(\om)\ge \limsup_{a\to \infty}a^{-\n}\om(H_{C_a})+P^\Ph.\eqno{(4.5)}
$$
By the inequality (4.1),  the subadditivity of entropy in Lemma 4.2 (b), and
(2.36)-(2.37),  we see that
$$
\begin{array}{rcl}
S_\L(\om)&=&-\Tr_\L(\om^{(\L)}\log \om^{(\L)})\\[3mm]
&=&-\lim_{n\to \infty}\Tr_\L(K_\L^{(n)}\log \om^{(\L)})\\[3mm]
&\ge&-\limsup_{n\to \infty}\Tr_\L(K_\L^{(n)}\log K_\L^{(n)})\\[3mm]
&=&\limsup_{n\to \infty}S_\L(\om_n)\\[3mm]
&\ge&\limsup_{n\to \infty}[S_{\L_n}(\om_n)-S_{\L_n\sm\L}(\om_n)]\\[3mm]
&\ge &\limsup_{n\to
\infty}[\om_n(H_{\L_n})+|\L_n|P_{\L_n}-\om_n(H_{\L_n\sm\L})-|\L_n\sm\L|
P_{\L_n\sm\L}].\end{array}\eqno{(4.6)}
$$
Recall the
definition of the measure $d\l$ on the path spaces appeared in (2.27) and define
for $0\le \a\le 1$,
$$
Z_{\L_n}(\a):=\dint d\l(s_{\L_n})\exp[-V(s_\L)-V(s_{\L_n\sm\L})-\a
W(s_\L,s_{\L_n\sm\L})].\eqno{(4.7)}
$$
Notice that
$$
\begin{array}{l}
\log Z_{\L_n}(1)=\log Z_{\L_n}=|\L_n|P_{\L_n}\\[3mm]
\log Z_{\L_n}(0)=\log Z_\L +\log Z_{\L_n\sm
\L}=|\L|P_\L+|\L_n\sm\L|P_{\L_n\sm\L}.
\end{array}\eqno{(4.8)}
$$
Using (4.8) in the last expression of (4.6) we get
$$
S_\L(\om)\ge \limsup_{n\to
\infty}[\om_n(H_{\L})+|\L|P_{\L}+\om_n(W(x_\L,x_{\L_n\sm\L}))+\log
Z_{\L_n}(1)-\log Z_{\L_n}(0)].\eqno{(4.9)}
$$
In Appendix, we will show that there exists $c>0$ such that the bounds
$$
\begin{array}{rcl}
\limsup_{n\to \infty}|\om_n(W(x_\L,x_{\L_n\sm\L}))|&\le& c\dsum_{i\in
\L}\dsum_{j\in \L^c}\Ps(|i-j|),\\[3mm]
\limsup_{n\to \infty}|\log Z_{\L_n}(1)-\log Z_{\L_n}(0)|&\le& c\dsum_{i\in
\L}\dsum_{j\in \L^c}\Ps(|i-j|)
\end{array}\eqno{(4.10)}
$$
hold. Using (4.10) in (4.9) we get
$$
S_\L(\om)\ge \om(H_\L)+|\L|P_\L -2c\dsum_{i\in \L}\dsum_{j\in
\L^c}\Ps(|i-j|).\eqno{(4.11)}
$$
We take $\L$ to be the cubes $C_a$ of sides $a$ and divide both sides of (4.11)
by $a^{-\n}$, the volume of $C_a$, and use (2.38), Theorem 2.8, and the method
used to show (3.11) to get the inequality (4.5). The proof is now completed.
$\bbx$

\vskip 1 true cm
\noi{\bf {\large Appendix}.}

\noi We show the inequlities in (4.10).
We first prove the second inequality of (4.10). By the
mean value theorem there exists $\a_1\in (0,1)$ such that
$$
\begin{array}{l}
\log Z_{\L_n}(1)-\log Z_{\L_n}(0)\\[3mm]
=\dfrac{d}{d\a}\log Z_{\L_n}(\a)\Big|_{\a=\a_1}\\[3mm]
=\dfrac{1}{Z_{\L_n}(\a_1)}\dint d\l(s_{\L_n})\exp[-V(s_\L)-V(s_{\L_n\sm\L})-\a_1
W(s_\L,s_{\L_n\sm\L})](-W(s_\L,s_{\L_n\sm\L}))\\[3mm]
=:<-W(s_\L,s_{\L_n\sm\L})>_{\a_1},
\end{array}\eqno{(A.1)}
$$
where the expectation $<\cdot>_\a$, $0\le \a\le 1$, is defined by
$$
<A>_\a:=\frac{1}{Z_{\L_n}(\a)}\dint d\l(s_{\L_n})\exp[-V(s_\L)-V(s_{\L_n\sm\L})-\a
W(s_\L,s_{\L_n\sm\L})]A(s_{\L_n}).\eqno{(A.2)}
$$
We then have
$$
\frac{d^2}{d\a^2}\log Z_{\L_n}(\a)=<W(s_\L,s_{\L_n\sm\L})^2>_\a
-<W(s_\L,s_{\L_n\sm\L})>_\a^2\,\ge\, 0.
$$
That is, the function $\a\mapsto \frac{d}{d\a}\log Z_{\L_n}(\a)$ is a monotone
function and hence we have
$$
|\log Z_{\L_n}(1)-\log Z_{\L_n}(0)|\le \max\Big\{
\big|\frac{d}{d\a}\log Z_{\L_n}(\a)\big|_{\a=0}\big|,\,\,\big|
\frac{d}{d\a}\log Z_{\L_n}(\a)\big|_{\a=1}\big|\Big\}.\eqno{(A.3)}
$$
We calculate $\frac{d}{d\a}\log Z_{\L_n}(\a)\big|_{\a=1}$. The other one can
be done similary. First, notice that by a superstable estimate established in
[PY, Proposition 2.5], there exist $A^*>0$ and $\d>0$ such that for any
$\D\sbs\L_n$,
$$
\frac{1}{Z_{\L_n}}\dint d\l(s_{\L_n\sm\D})e^{-V(s_{\L_n})}\le \exp\big
[\dsum_{i\in \D}(-A^*s_i^2+\d)\big].\eqno{(A.4)}
$$
Then, by the regularity of the interaction given in Assumption 2.1 (d) we see
that
$$
\begin{array}{l}
\big|\dfrac{d}{d\a}\log Z_{\L_n}(\a)\big|_{\a=1}\big|\\[3mm]
=\big|\dfrac1{Z_{\L_n}}\dint
d\l(s_{\L_n})e^{-V(s_{\L_n})}W(s_\L,s_{\L_n\sm\L})\big|\\[3mm]
\le \dsum_{i\in \L}\dsum_{j\in \L_n\sm\L}\frac12\Ps(|i-j|)\dfrac1{Z_{\L_n}}
\dint d\l(s_{\L_n})e^{-V(s_{\L_n})}(s_i^2+s_j^2)\\[3mm]
\le \dsum_{i\in \L}\dsum_{j\in \L_n\sm\L}\frac12\Ps(|i-j|)\dint
d\l(s_{i})d\l(s_j)e^{-A^*(s_i^2+s_j^2)+2\d}(s_i^2+s_j^2)\\[3mm]
\le c \dsum_{i\in \L}\dsum_{j\in \L_n\sm\L}\Ps(|i-j|).
\end{array}
$$

In order to prove the first inequality of (4.10) we notice, by using the
regularity of the interaction and (A.4), that
$$
\begin{array}{l}
|\om_n(W(x_\L,x_{\L_n\sm\L}))|\\[3mm]
=\big|\dfrac1{Z_{\L_n}}\dint
d\l(s_{\L_n})e^{-V(s_{\L_n})}W(x_\L,x_{\L_n\sm\L})\big|\\[3mm]
\le \dsum_{i\in \L}\dsum_{j\in \L_n\sm\L}\frac12\Ps(|i-j|)\dint
d\l(s_{i})d\l(s_j)e^{-A^*(s_i^2+s_j^2)+2\d}(s_i(0)^2+s_j(0)^2).
\end{array}
\eqno{(A.5)}
$$
Notice that
$$
\dint d\l(s_i)e^{-A^*s_i^2}s_i(0)^2=\dint dx_i \,x_i^2\dint
P_{x_i,x_i}(ds_i)e^{-A^*s_i^2}.\eqno{(A.6)}
$$
By using the same argument used in the proof of the bounds in (A.13)-(A.15)
of [PY], we get that the r.h.s. of (A.6) is bounded by
$$
\dint dx_i\,x_i^2 e^{-A'x_i^2+\d'}\le c'<\infty,\eqno{(A.7)}
$$
for some constants $A'$, $\d'$, and $c'$. Inserting (A.7) into (A.5) we
finish the proof of (4.10).

\vskip 1 true cm
\noi{\bf ACKNOWLEDGMENTS}.  We thank Prof. Y. M. Park for valuable discussions.
H. J. Yoo was supported by grant of Post-Doc. Program from Kyungpook
National University (1998).
\vskip 1 true cm
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\item[[AH-K]] S. Albeverio and R. H\o egh-Krohn, Homogeneous random fields and
statistical mechanics, {\it J. Funct. Anal.} {\bf 19}, 242-272 (1975).
\item[[BR1]] O. Bratteli and D. W. Robinson, {\it Operator algebras and quantum
statistical mechanics, 1} (Springer-Verlag, New York/Heidelberg/Berlin, 1979).
\item[[BR2]] O. Bratteli and D. W. Robinson, {\it Operator algebras and quantum
statistical mechanics, 2} (Springer-Verlag, New York/Heidelberg/Berlin, 1981).
\item[[Ge]] H. O. Georgii, {\it Gibbs measures and phase transitions} (de
Gruyter, Berlin, 1988).
\item[[Is]] R. B. Israel, {\it Convexity in the theory of lattice gases}
(Princeton University Press, Princeton, 1979).
\item[[LP]] J. L. Lebowitz and E. Presutti, Statistical mechanics of systems of
unbounded spins, {\it Commun. Math. Phys.} {\bf 50}, 195-218 (1976). Erratum,
{\it Commun. Math. Phys.} {\bf 78}, 151 (1980).
\item[[P1]] Y. M. Park, Quantum statistical mechanics of unbounded continuous
spin systems, {\it J. Korean Math. Soc.} {\bf 22} (1), 43-74 (1985).
\item[[PY]] Y. M. Park and H. J. Yoo, A characterization of Gibbs states of
lattice boson systems, {\it J. Stat. Phys.} {\bf 75}(1/2), 215-239 (1994).
\item[[Pr]] C. Preston, {\it Random fields}. Lecture Notes in Mathmatics vol.
534 (Springer-Verlag, Berlin, 1976).
\item[[R1]] D. Ruelle, {\it Statistical meahanics. Rigorous results} (Benjamim,
New York, 1969).
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Commun. Math. Phys.} {\bf 50}, 189-194 (1976).
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(Academic Press, New York, 1979).
\item[[Si2]] B. Simon, {\it The statistical mechanics of lattice gases, vol 1}
(Princeton University Press, Princeton, 1993).
\end{description}
\end{document}
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1891 y(\003)1420 1895 y Fe(n)1444 1878 y Fs(\()p 1463
1851 23 2 v Fj(s)p Fs(\))p Fi(g)p Fm(,)g(and)g Fi(f)p
Fj(P)1723 1860 y Fr(\010)1716 1891 y(\003)1740 1895 y
Fe(n)1764 1878 y Fs(\()p Fj(\027)s Fs(\))p Fi(g)-35 1953
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n(quenc)n(e)h(of)f(toruses)g Fi(f)p Fj(T)1710 1960 y
Fk(n)1733 1953 y Fi(g)p Fm(,)g(the)-35 2027 y(se)n(quenc)n(e)h
Fi(f)188 2034 y Fk(P)209 2027 y Fj(P)247 2009 y Fr(\010)240
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%%Page: 9 9
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b Fj(\026)1055 1689 y Fr(\(\003\))1110 1707 y Fs(\()p
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1840 y Fr(\003)1269 1833 y Fs(,)f(the)h(restriction)e(of)i
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y Fs(,)-35 1907 y(with)g(resp)q(ect)g(to)g Fj(\026)331
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Fj(S)604 2010 y Fr(\(0\))601 2049 y(\003)652 2035 y Fs(\()p
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y Fr(\(0\))931 2049 y(\003)987 2035 y Fs(log)c Fj(\026)1087
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13 12 bop -35 14 a Fn(Statistical)15 b(mec)o(hanics)f(for)j(un)o(b)q
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Ft(22)c Fs(\(1\),)f(43-74)i(\(1985\).)-35 1123 y Ft([PY)24
b Fs(])d(Y.)g(M.)f(P)o(ark)i(and)g(H.)f(J.)g(Y)l(o)q(o,)i(A)e(c)o
(haracterization)f(of)i(Gibbs)g(states)g(of)g(lattice)e(b)q(oson)87
1197 y(systems,)14 b Fm(J.)j(Stat.)23 b(Phys.)e Ft(75)p
Fs(\(1/2\),)c(215-239)i(\(1994\).)-35 1314 y Ft([Pr)24
b Fs(])13 b(C.)g(Preston,)h Fm(R)n(andom)g(\014elds)p
Fs(.)22 b(Lecture)13 b(Notes)g(in)h(Mathmatics)e(v)o(ol.)19
b(534)c(\(Springer-V)l(erlag,)87 1388 y(Berlin,)f(1976\).)-35
1505 y Ft([R1)23 b Fs(])16 b(D.)g(Ruelle,)e Fm(Statistic)n(al)19
b(me)n(ahanics.)j(R)o(igor)n(ous)16 b(r)n(esults)h Fs(\(Benjamim)o(,)c
(New)j(Y)l(ork,)f(1969\).)-35 1621 y Ft([R2)23 b Fs(])g(D.)f(Ruelle,)g
(Probabilit)o(y)g(estimates)f(for)i(con)o(tin)o(uous)f(spin)h(systems,)
g Fm(Commun.)40 b(Math.)87 1696 y(Phys.)21 b Ft(50)p
Fs(,)16 b(189-194)i(\(1976\).)-35 1812 y Ft([Si1)23 b
Fs(])12 b(B.)g(Simon,)f Fm(F)l(unctional)16 b(inte)n(gr)n(ation)e(and)g
(quantum)h(physics)d Fs(\(Academic)d(Press,)k(New)f(Y)l(ork,)87
1887 y(1979\).)-35 2003 y Ft([Si2)23 b Fs(])d(B.)f(Simon,)g
Fm(The)i(statistic)n(al)g(me)n(chanics)h(of)f(lattic)n(e)h(gases,)g
(vol)f(1)f Fs(\(Princeton)g(Univ)o(ersit)o(y)87 2077
y(Press,)c(Princeton,)f(1993\).)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF
---------------9901150543748--
