Content-Type: multipart/mixed; boundary="-------------9901150543748" This is a multi-part message in MIME format. ---------------9901150543748 Content-Type: text/plain; name="99-15.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-15.comments" PACS-Code: 75.10 Hk, 75.10 Jm, 05.70 Ce. ---------------9901150543748 Content-Type: text/plain; name="99-15.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-15.keywords" Classical and quantum unbounded spin systems, superstable interactions, Gibbs (DLR) measures, KMS states, entropy, pressure, Gibbs variational principle. ---------------9901150543748 Content-Type: application/x-tex; name="Spin.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Spin.tex" \documentclass[12pt,twoside]{article} %fancyheadings \usepackage{amstex, amssymb} \topmargin=0mm \oddsidemargin=2mm \evensidemargin=2mm %% \textwidth=160mm \textheight=240mm \voffset=-0.3 true cm \hoffset=-0.5 true cm \makeatletter \renewcommand{\thesection}{\arabic{section}.} \renewcommand{\section}{\@startsection{section}{1}{0mm}% the name {-\baselineskip}{0.5\baselineskip}{\large\bf}}% of section \renewcommand{\thesubsection}{\thesection\arabic{subsection}.} \pagestyle{myheadings} \markboth{ Yoo {\it et al}}{Statistical mechanics for unbounded spin systems} \begin{document} \newtheorem{Theorem}{Theorem}[section] \newtheorem{Definition}[Theorem]{Definition} \newtheorem{Assumption}[Theorem]{Assumption} \newtheorem{Proposition}[Theorem]{Proposition} \newtheorem{Lemma}[Theorem]{Lemma} \newtheorem{Corollary}[Theorem]{Corollary} \newtheorem{Remark}[Theorem]{Remark} \newtheorem{Example}[Theorem]{Example} \newtheorem{Condition}[Theorem]{Condition} \renewcommand{\theTheorem}{\thesection\arabic{Theorem}.} \parskip=0pt \baselineskip=18pt %\def\ba{\begin{array}} %\def\ea{\end{array}} \def\dsum{\displaystyle\sum} \def\dint{\displaystyle\int} \def\dsup{\displaystyle\sup} \def\dcup{\displaystyle\cup} \def\dcap{\displaystyle\cap} \def\lms{\longmapsto} \def\pf{% \par\vskip-15\p@ \trivlist \item[\hskip\labelsep\fontshape{n}\fontseries{bx}\selectfont\proofname]\ignorespaces} \def\endpf{\qed\endtrivlist} \@namedef{pf*}#1{\par\toks@\expandafter{\proofname}% \edef\restoreproofname{\def\noexpand\proofname{\the\toks@}}% \def\proofname{#1}\pf\restoreproofname\ignorespaces} \expandafter\let\csname endpf*\endcsname=\endpf \def\qedsymbol{\bbx} \def\qed{\RIfM@\else\unskip\nobreak\fi\quad\qedsymbol} \def\proofname{Proof.} \@addtoreset{equation}{section} \renewcommand{\theequation}{\thesection.\arabic{equation}} %symbols %-------------------------------------- \def\NN{{I\!\! 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F}} \def\IC{\ \hbox{\vrule width 0.6pt height 6pt depth 0pt \hskip -3.5 pt} C} \def\SIC{\ \hbox{\vrule width 0.4pt height 4pt depth 0pt \hskip -2.5 pt} C} \def\bbx{\rule{2.5mm}{2.5mm}} %%%%% \def\d{\delta} \def\nb{\nabla} \def\a{\alpha} \def\bt{\beta} \def\et{\eta} \def\d{\delta} \def\D{\Delta} \def\e{\varepsilon} \def\ph{\phi} \def\Ph{\Phi} \def\vph{\varphi} \def\g{\gamma} \def\G{\Gamma} \def\k{\kappa} \def\l{\lambda} \def\L{\Lambda} \def\m{\mu} \def\n{\nu} \def\r{\rho} \def\vr{\varrho} \def\om{\omega} \def\Om{\Omega} \def\p{\pi} \def\P{\Pi} \def\s{\sigma} \def\S{\Sigma} \def\th{\theta} \def\vth{\vartheta} \def\t{\tau} \def\ch{\chi} \def\x{\xi} \def\z{\zeta} \def\ps{\psi} \def\Ps{\Psi} \def\noi{\noindent} \def\fa{\forall} \def\ex{\exists} \def\nex{\nexists} \def\ub{\underbar} \def\ul{\underline} \def\ol{\overline} \def\pd{\prod} \def\se{\subseteq} \def\sbs{\subset} \def\ssbs{\sbs\sbs} \def\sps{\supset} \def\spe{\supseteq} \def\bs{\backslash} \def\wh{\widehat} \def\wt{\widetilde} \def\lt{\left} \def\rt{\right} \def\bls{\blacksquare} \def\wt{\widetilde} \def\wh{\widehat} \def\eqv{\equiv} \def\le{\leq} \def\ge{\geq} \def\ua{\uparrow} \def\ne{\neq} \def\da{\downarrow} \def\Tr{{\rm Tr}} \def\es{\emptyset} \def\usb{\undersetbrace} \def\us{\underset} \def\os{\overset} \def\ra{\rightarrow} \def\llra{\longleftrightarrow} %\def\BN{{\Bbb N}} \def\BZ{{\Bbb Z}} \def\BC{{\Bbb C}} %\def\BR{{\Bbb R}} \def\BE{\Bbb E} \def\Lra{\Longrightarrow} \def\lra{\longrightarrow} \def\lla{\longleftarrow} \def\ptl{\partial} \def\sm{\setminus} \def\Ol{\Om_{\log}} \def\Tr{\text{Tr}} \def\gm{\esG^\Ph(\Om)} \def\df{(\CE_\m, D(\CE_\m))} \def\ff{\CF C_b^\infty} \def\lf{\CF_{\loc}C_b^\infty} \def\hlf{\text{H-}\Om_{\text{loc}}} \def\({(\kern -2pt(} \def\){)\kern -2.2pt)} \def\<{<\kern -4.5pt<} \def\>{>\kern -4.7pt>} \def\erH{{\rm{\tt H}}} \def\mbA{{\mathbb A}} \def\mbB{{\mathbb B}} \def\mbC{{\mathbb C}} \def\mbD{{\mathbb D}} \def\mbE{{\mathbb E}} \def\mbF{{\mathbb F}} \def\mbG{{\mathbb G}} \def\mbH{{\mathbb H}} \def\mbI{{\mathbb I}} \def\mbJ{{\mathbb J}} \def\mbK{{\mathbb K}} \def\mbL{{\mathbb L}} \def\mbM{{\mathbb M}} \def\mbN{{\mathbb N}} \def\mbO{{\mathbb O}} \def\mbP{{\mathbb P}} \def\mbQ{{\mathbb Q}} \def\mbR{{\mathbb R}} \def\mbS{{\mathbb S}} \def\mbT{{\mathbb T}} \def\mbU{{\mathbb U}} \def\mbV{{\mathbb V}} \def\mbW{{\mathbb W}} \def\mbX{{\mathbb X}} \def\mbY{{\mathbb Y}} \def\mbZ{{\mathbb Z}} \def\mcA{{\mathcal A}} \def\mcB{{\mathcal B}} \def\mcC{{\mathcal C}} \def\mcD{{\mathcal D}} \def\mcE{{\mathcal E}} \def\mcF{{\mathcal F}} \def\mcG{{\mathcal G}} \def\mcH{{\mathcal H}} \def\mcI{{\mathcal I}} \def\mcJ{{\mathcal J}} \def\mcK{{\mathcal K}} \def\mcL{{\mathcal L}} \def\mcM{{\mathcal M}} \def\mcN{{\mathcal N}} \def\mcO{{\mathcal O}} \def\mcP{{\mathcal P}} \def\mcQ{{\mathcal Q}} \def\mcR{{\mathcal R}} \def\mcS{{\mathcal S}} \def\mcT{{\mathcal T}} \def\mcU{{\mathcal U}} \def\mcV{{\mathcal V}} \def\mcW{{\mathcal W}} \def\mcX{{\mathcal X}} \def\mcY{{\mathcal Y}} \def\mcZ{{\mathcal Z}} \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 $$ \Ps(r)\le Kr^{-\n-\e} \text{ for some }K \text{ and }\e>0 \text{ with }\dsum_{i\in \mbZ^\n}\Ps(|i|)0$, and $n\ge 1$, form a base of neighborhoods of $\m$. The following notion shall be useful throughout this paper. \begin{Definition}%2.2 A probability measure $\m$ on $(\Om,\mcF)$ is said to be regular if there exist $\ol A>0$ and $\ol \d$ so that for any $\L\ssbs \mbZ^\n$, the projection $\m_\L$ of $\m$ to $(\Om,\mcF_\L)$, which may be understood as a measure on $(\Om_\L,\mcF_\L^0)$, is absolutely continuous with respect to $ds_\L:=\pd_{i\in \L}ds_i$ with a Radon-Nikodym derivative $g(s_\L|\m)$ satisfying $$ 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)} $$ where $a_0:=\a_\m\,>\,0$. By the usual method we have $S_\L^{(0)}(\m)\ge S_{\L'}^{(0)}(\m)$ for $\L\sbs\L'$. Moreover, since $\m^{(0)}$ is a probability measure, we may define $S_\es^{(0)}(\m)\eqv 0$. By putting $\L=\es$ in (3.6) we get $S_\es(\m)=0$. Using the property (b) we obtain (a). $\bbx$ \begin{Proposition}%3.2 Suppose $\m\in \mcG^I(\Ph)$. The mean entropy $$ 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_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] &=&_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)} $$ Since $\Ps$ is bounded and of finite range we see that the interaction $\Ph+\l \Ps$ is also superstable and regular for any $\l\in \mbR$. Recall the periodic boundary condition introduced in Subsection 2.1. Let $T_n:=(\mbZ/n\mbZ)^\n$ be a sufficiently large torus so that $T_n\sps \wt \L$. We let $<\cdot>_{n,\l}$ be the (local) Gibbs measure on $T_n$ with pure b.c. for the interaction $\Ph+\l\Ps$, i.e., $$ _{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)} $$ where the Hamiltonian $_P\!H_{T_n}^{\Ph+\l\Ps}$ is defined as in (2.12) and $Z_{T_n}^{(\l)}$ is the obvious normalizing factor. Given $\e>0$ we claim that there exist $\l_0>0$ and $\mcF_{T_n}$-measurable functions $F^{(n,\l)}$, $|\l|\le \l_0$, such that $$ _{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)} $$ It is not hard to show that $$ _{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-)\big]\\[3mm] &\le&\dsum_j\exp[]\quad \text{(by Jensen's inequality)}\\[3mm] &\le &\dsum_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:=\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)=_\a -_\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 \begin{description} \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). \item[[R2]] D. Ruelle, Probability estimates for continuous spin systems, {\it Commun. Math. Phys.} {\bf 50}, 189-194 (1976). \item[[Si1]] B. Simon, {\it Functional integration and quantum physics} (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} ---------------9901150543748 Content-Type: application/postscript; name="Spin.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Spin.ps" %!PS-Adobe-2.0 %%Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software %%Title: spin.dvi %%CreationDate: Fri Jan 15 20:13:23 1999 %%Pages: 21 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: C:\EMTEX\BIN\dvips32.EXE spin -pj:c:\temp\dv1.mfj %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1999.01.15:2012 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale 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2522 y(liminaries,)14 b(and)19 b(main)d(results.)25 b(The)18 b(classical)e(mo)q(del)h(is)g (explained)g(in)g(Subsection)g(2.1)h(and)h(the)-35 2596 y(quan)o(tum)c(mo)q(del)h(in)g(Subseciton)h(2.2.)24 b(In)17 b(Section)f(3,)h(w)o(e)g(giv)o(e)f(the)g(pro)q(ofs)j(for)e(classical)f (systems.)-35 2671 y(Section)h(4)i(is)f(dev)o(oted)g(to)g(the)h(pro)q (of)g(for)f(quan)o(tum)f(systems)h(and)g(in)g(App)q(endix)g(w)o(e)g (pro)o(v)o(e)f(some)-35 2746 y(tec)o(hnical)d(inequlities)g(need)i(in)g (Section)g(4.)p eop %%Page: 3 3 3 2 bop -35 14 a Fn(Statistical)15 b(mec)o(hanics)f(for)j(un)o(b)q (ounded)g(spin)f(systems)807 b Fs(3)-35 168 y Fq(2.)66 b(Notations,)22 b(Preliminaries,)i(and)f(Main)g(Results)-35 280 y(2.1.)65 b(Classical)23 b(Systems)-35 387 y Fs(W)l(e)17 b(consider)h(the)f(classical)h(un)o(b)q(ounded)g(spin)g(systems.)25 b(Let)18 b Fl(Z)1205 369 y Fk(\027)1242 387 y Fs(b)q(e)g(the)g Fj(\027)s Fs(-dimensional)e(in)o(teger)-35 462 y(lattice.)41 b(A)o(t)22 b(eac)o(h)h(site)g Fj(i)j Fi(2)g 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Fs(When)i(a)g(subset)g(\003)h Fi(\032)g Fl(Z)464 795 y Fk(\027)503 813 y Fs(is)f(a)h(\014nite)e(set)h(w)o(e)f (will)g(write)h(\003)g Fi(\032\032)h Fl(Z)1307 795 y Fk(\027)1326 813 y Fs(.)35 b(W)l(e)21 b(will)f(consider)g(b)q(oth)-35 888 y(in)o(teractions)d(b)q(et)o(w)o(een)g(the)h(spins)h(at)f (di\013eren)o(t)f(sites)h(as)h(w)o(ell)e(as)i(self)e(in)o(teractions.) 27 b(Throughout)-35 962 y(this)16 b(pap)q(er)h(w)o(e)e(will)g(imp)q (ose)h(the)g(follo)o(wing)g(conditions)g(on)h(the)f(in)o(teraction:)-35 1065 y Ft(Assumption)h(2.1.)24 b Fm(The)17 b(inter)n(action)i Fs(\010)14 b(=)f(\(\010)888 1072 y Fr(\001)920 1065 y Fs(\))939 1072 y Fr(\001)p Fg(\032\032)p Ff(Z)1049 1063 y Fe(\027)1084 1065 y Fm(satis\014es)18 b(the)g(fol)r(lowing)i(c)n (onditions:)38 1139 y(\(a\))d Fs(\010)155 1146 y Fr(\001)204 1139 y Fm(is)g(a)h(Bor)n(el)f(me)n(asur)n(able)h(function)g(on)g Fs(\()p Fl(R)996 1121 y Fk(d)1013 1139 y Fs(\))1032 1121 y Fr(\001)1081 1139 y Fm(for)f(e)n(ach)h Fs(\001)13 b Fi(\032\032)h 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Fr(\001)950 1472 y Fs(\))g Fi(\025)1035 1425 y Fh(X)1041 1530 y Fk(i)p Fg(2)p Fr(\003)1107 1472 y Fs([)p Fj(Ax)1186 1451 y Fr(2)1186 1484 y Fk(i)1216 1472 y Fi(\000)d Fj(c)p Fs(])p Fj(:)38 1612 y Fm(\(d\))21 b(\(Str)n(ong)h(r)n(e)n(gularity\))f(Ther)n(e)h(exists)h(a)e(de)n(cr)n (e)n(asing)h(p)n(ositive)g(function)h Fs(\011)f Fm(on)g(the)g(natur)n (al)-35 1687 y(numb)n(ers)17 b(such)h(that)275 1795 y Fs(\011\()p Fj(r)q Fs(\))c Fi(\024)f Fj(K)t(r)508 1775 y Fg(\000)p Fk(\027)r Fg(\000)p Fk(")619 1795 y Fm(for)k(some)g Fj(K)22 b Fm(and)c Fj(")13 b(>)h Fs(0)k Fm(with)1225 1748 y Fh(X)1223 1854 y Fk(i)p Fg(2)p Ff(Z)1284 1845 y Fe(\027)1308 1795 y Fs(\011\()p Fi(j)p Fj(i)p Fi(j)p Fs(\))13 b Fj(<)h(A:)38 1933 y Fm(F)l(urthermor)n(e,)i(if)h Fs(\003)415 1940 y Fr(1)452 1933 y Fm(and)h Fs(\003)581 1940 y Fr(2)618 1933 y Fm(ar)n(e)f(disjoint)h(\014nite)g(subsets)h(of)e Fl(Z)1260 1915 y Fk(\027)1296 1933 y Fm(and)h(if)f(one)h(writes)433 2042 y Fj(V)11 b Fs(\()p Fj(x)519 2049 y Fr(\003)543 2054 y 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w(s)1115 1815 y Fr(\003)1139 1806 y Fe(c)1158 1808 y Fs(\)])h(1)1223 1815 y Fk(A)1252 1808 y Fs(\()p Fj(s)1294 1815 y Fr(\003)p 1320 1781 V 1320 1808 a Fj(s)1343 1815 y Fr(\003)1367 1806 y Fe(c)1386 1808 y Fs(\))p Fj(;)73 b Fs(if)p 1536 1781 V 15 w Fj(s)14 b Fi(2)g(S)350 1904 y Fs(0)p Fj(;)73 b Fs(if)p 506 1876 V 16 w Fj(s)20 b(=)-30 b Fi(2)14 b(S)t Fj(;)1754 1843 y Fs(\(2)p Fj(:)p Fs(3\))-35 1990 y(where)20 b Fj(A)h Fi(2)g(F)k Fs(and)d(1)407 1997 y Fk(A)456 1990 y Fs(is)f(the)f(indicator)g(function)h(on)g Fj(A)f Fs(and)h Fj(s)1255 1997 y Fr(\003)p 1282 1963 V 1282 1990 a Fj(s)1305 1997 y Fr(\003)1329 1988 y Fe(c)1368 1990 y Fi(2)h Fs(\012)f(coincides) e(with)i Fj(s)1828 1997 y Fr(\003)-35 2065 y Fs(on)f(\003)g(and)h(with) p 304 2037 V 20 w Fj(s)327 2072 y Fr(\003)351 2063 y Fe(c)389 2065 y Fs(on)g(\003)495 2047 y Fk(c)512 2065 y Fs(.)33 b(It)20 b(is)g(easy)g(to)g(c)o(hec)o(k)e(that)j(the)f(Gibbs)g (sp)q(eci\014cation)g(satis\014es)h(the)-35 2140 y(consistency)15 b(condition)h([Ge,)g(Pr]:)k(F)l(or)d(\001)c Fi(\032)h Fs(\003,)p 898 2112 V 16 w Fj(s)f Fi(2)h(S)t Fs(,)514 2254 y Fj(\015)542 2236 y Fr(\010)539 2266 y(\003)570 2254 y Fj(\015)598 2236 y Fr(\010)595 2266 y(\001)627 2254 y Fs(\()p Fj(A)p Fi(j)p 697 2226 V Fj(s)o Fs(\))42 b(:=)873 2186 y Fh(Z)900 2299 y Fg(S)934 2254 y Fj(\015)962 2233 y Fr(\010)959 2266 y(\003)990 2254 y Fs(\()p Fj(ds)1057 2233 y Fg(0)1069 2254 y Fi(j)p 1083 2226 V Fj(s)p Fs(\))p Fj(\015)1153 2233 y Fr(\010)1150 2266 y(\001)1182 2254 y Fs(\()p Fj(A)p Fi(j)p Fj(s)1275 2233 y Fg(0)1285 2254 y Fs(\))786 2349 y(=)49 b Fj(\015)901 2331 y Fr(\010)898 2362 y(\003)928 2349 y Fs(\()p Fj(A)p Fi(j)p 998 2322 V Fj(s)p Fs(\))p Fj(:)1754 2289 y Fs(\(2)p Fj(:)p Fs(4\))-35 2439 y(The)16 b(Gibbs)g(measures)g(are)g(de\014ned)g(as)h(follo)o(ws)f ([Ge,)f(Pr,)h(Si2]:)-35 2538 y Ft(De\014nition)h(2.3.)24 b Fm(A)d(pr)n(ob)n(ability)g(me)n(asur)n(e)g Fj(\026)g Fi(2)h(P)t Fs(\(\012)p Fj(;)8 b Fi(F)d Fs(\))21 b Fm(is)g(said)g(to)g 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Fj(f)22 b Fs(on)17 b(\012,)f(w)o(e)g(will)f (sometimes)e(use)j(the)g(notation)608 289 y Fj(\015)636 269 y Fr(\010)633 301 y(\003)664 289 y Fs(\()p Fj(f)5 b Fi(j)p 726 262 23 2 v Fj(s)p Fs(\))14 b(:=)847 221 y Fh(Z)905 289 y Fj(\015)933 269 y Fr(\010)930 301 y(\003)961 289 y Fs(\()p Fj(ds)1028 269 y Fg(0)1040 289 y Fi(j)p 1054 262 V Fj(s)o Fs(\))p Fj(f)5 b Fs(\()p Fj(s)1166 269 y Fg(0)1178 289 y Fs(\))p Fj(:)-35 410 y Fs(When)16 b(\010)h(satis\014es)g(the)g(conditions)f(of)h(Assumption)f(2.1,)h(the) f(existence)f(of)i(Gibbs)g(measures)f(w)o(as)-35 485 y(pro)o(v)o(en)f(in)h([LP]:)-35 592 y Ft(Theorem)g(2.4.)24 b Fm(\([LP,)15 b(The)n(or)n(em)g(4.5]\))g(Supp)n(ose)h(that)g(the)g (inter)n(action)h Fs(\010)f Fm(satis\014es)g(the)g(hyp)n(othe-)-35 667 y(ses)h(of)h(Assumption)f(2.1.)23 b(Then,)17 b Fi(G)s Fs(\(\010\))d Fi(6)p Fs(=)g Fi(;)p Fm(.)-35 775 y Fs(W)l(e)j(notice)g (that)g Fi(G)334 757 y Fk(I)354 775 y Fs(\(\010\))h(is)f(also)h (nonempt)o(y)l(.)23 b(In)17 b(fact,)g(for)h(an)o(y)f Fj(\026)f Fi(2)g(G)s Fs(\(\010\))i(and)g Fj(i)d Fi(2)h Fl(Z)1613 757 y Fk(\027)1632 775 y Fs(,)h(it)g(is)g(easy)-35 849 y(to)h(c)o(hec)o(k)e(that)i Fj(\034)287 856 y Fk(i)301 849 y Fj(\026)f Fi(2)f(G)s Fs(\(\010\),)i(where)f Fj(\034)697 856 y Fk(i)711 849 y Fj(\026)h Fs(is)g(the)f(translation)i(of)f Fj(\026)p Fs(.)26 b(So,)18 b(an)o(y)g(\(w)o(eak)1571 831 y Fg(\003)1590 849 y Fs(-\)limit)d(p)q(oin)o(t)-35 924 y(of)h Fi(f)63 904 y Fr(1)p 50 913 45 2 v 50 941 a Fg(j)p Fr(\003)p Fg(j)107 887 y Fh(P)160 939 y Fk(i)p Fg(2)p Fr(\003)230 924 y Fj(\034)251 931 y Fk(i)266 924 y Fj(\026)p Fi(g)g Fs(as)h(\003)f(increases)g(to)g Fl(Z)746 906 y Fk(\027)781 924 y Fs(b)q(elongs)h(to)f Fi(G)1048 906 y Fk(I)1068 924 y Fs(\(\010\))g(\(see)g([Si2,)f(Corollary)i(I)q(I)q (I.2.10]\).)38 1034 y(The)f(pressure,)f(or)i(the)f(free)f(energy)l(,)g (is)h(de\014ned)g(in)f(the)h(follo)o(wing)g(w)o(a)o(y)l(.)21 b(W)l(e)16 b(consider)f(the)h(free)-35 1109 y(b)q(oundary)e(condition)f (\(b.c.\),)f(pure)h(b.c.,)f(p)q(erio)q(dic)h(b.c.,)f(and)h(general)g (b.c.,)f(resp)q(ectiv)o(ely)f([LP].)h(The)-35 1184 y(partition)k (function)g(in)g(\003)g(with)g(free)f(b.c.)21 b(is)16 b(de\014ned)g(b)o(y)633 1305 y Fj(Z)670 1284 y Fr(\010)666 1317 y(\003)712 1305 y Fs(:=)777 1237 y Fh(Z)835 1305 y Fj(ds)883 1312 y Fr(\003)918 1305 y Fs(exp[)p Fi(\000)p Fj(V)10 b Fs(\()p Fj(s)1126 1312 y Fr(\003)1153 1305 y Fs(\)])568 b(\(2)p Fj(:)p Fs(5\))-35 1429 y(and)16 b(the)g(\(\014nite)g(v)o(olume\))e(pressure)i(is)g(de\014ned)g(b)o(y) 722 1551 y Fj(P)760 1530 y Fr(\010)753 1563 y(\003)802 1551 y Fs(:=)891 1517 y(1)p 872 1539 62 2 v 872 1585 a Fi(j)p Fs(\003)p Fi(j)947 1551 y Fs(log)9 b Fj(Z)1055 1530 y Fr(\010)1051 1563 y(\003)1083 1551 y Fj(;)657 b Fs(\(2)p Fj(:)p Fs(6\))-35 1678 y(where)14 b Fi(j)p Fs(\003)p Fi(j)g Fs(means)g(the)h(v)o(olume)d(\(cardinalit)o(y\))i(of)h (\003)f Fi(\032\032)f Fl(Z)1087 1660 y Fk(\027)1106 1678 y Fs(.)21 b(In)14 b(order)h(to)g(consider)g(the)f(pure)h(b.c.,)-35 1753 y(for)h(a)h(tec)o(hnical)d(reason,)j(w)o(e)f(need)f(to)i(in)o(tro) q(duce)f(some)f(subsets)i(of)f(\012)h([LP].)f(F)l(or)g Fj(a)d(>)h Fs(0)j(let)498 1867 y Fj(X)t Fs(\()p Fj(a)p Fs(\))d(:=)f Fi(f)p Fj(s)h Fi(2)g Fs(\012)p Fi(j)8 b Fj(s)874 1847 y Fr(2)874 1880 y Fk(i)908 1867 y Fi(\024)13 b Fj(a)8 b Fs(log)h Fi(j)p Fj(i)p Fi(j)p Fj(;)24 b Fi(j)p Fj(i)p Fi(j)13 b Fj(>)h Fs(1)p Fi(g)p Fj(:)433 b Fs(\(2)p Fj(:)p Fs(7\))-35 1982 y(F)l(or)p 52 1954 23 2 v 16 w Fj(s)14 b Fi(2)g Fj(X)t Fs(\()p Fj(a)p Fs(\))c Fi(\\)h(S)t Fs(,)16 b(the)g(partition)g(function)f(in)h(the)g(region)g(\003)e Fi(\032\032)f Fl(Z)1301 1964 y Fk(\027)1336 1982 y Fs(with)j(pure)g (b.c.)p 1655 1954 V 20 w Fj(s)g Fs(is)g(giv)o(en)-35 2056 y(b)o(y)f(the)h(form)o(ula)f(\(2.2\))i(and)g(the)f(pressure)g(is)g (de\014ned)g(b)o(y)661 2178 y Fj(P)699 2158 y Fr(\010)692 2191 y(\003)727 2178 y Fs(\()p 746 2151 V Fj(s)p Fs(\))e(:=)891 2145 y(1)p 872 2167 62 2 v 872 2212 a Fi(j)p Fs(\003)p Fi(j)947 2178 y Fs(log)9 b Fj(Z)1055 2158 y Fr(\010)1051 2191 y(\003)1083 2178 y Fs(\()p 1102 2151 23 2 v Fj(s)p Fs(\))p Fj(:)596 b Fs(\(2)p Fj(:)p Fs(8\))-35 2303 y(Let)16 b(us)h(also)f(de\014ne)p 288 2377 45 2 v 288 2417 a Fj(X)5 b Fs(\()p Fj(a)p Fs(\))13 b(:=)g Fi(f)p Fj(s)h Fi(2)g Fs(\012)p Fi(j)8 b(9)g Fs(\003\()p Fj(s)p Fs(\))14 b Fi(\032\032)f Fl(Z)914 2397 y Fk(\027)932 2417 y Fj(;)25 b(s)994 2397 y Fr(2)994 2430 y Fk(i)1027 2417 y Fi(\024)14 b Fj(a)8 b Fs(log)h Fi(j)p Fj(i)p Fi(j)15 b Fs(for)i Fj(i)i(=)-30 b Fi(2)14 b Fs(\003\()p Fj(s)p Fs(\))p Fi(g)p Fj(:)223 b Fs(\(2)p Fj(:)p Fs(9\))-35 2532 y(It)18 b(is)g(not)i(hard)f (to)g(sho)o(w)g(that)h(if)e Fj(\027)j Fi(2)d(P)t Fs(\(\012\))h(is)g(a)g (regular)g(measure,)e(then)i(for)g(su\016cien)o(tly)d(large)-35 2606 y Fj(a)d(>)h Fs(0)i(w)o(e)g(ha)o(v)o(e)771 2681 y Fj(\027)s Fs(\()p 817 2641 V Fj(X)5 b Fs(\()p Fj(a)p Fs(\)\))13 b(=)h(1)p Fj(:)682 b Fs(\(2)p Fj(:)p Fs(10\))-35 2778 y(No)o(w,)16 b(let)g Fj(\027)i Fi(2)d(P)t Fs(\(\012\))i(b)q(e)g(a) g(regular)g(measure.)22 b(The)17 b(\014nite)f(v)o(olume)e(pressure)j (for)g(the)g(general)f(b.c.)-35 2853 y Fj(\027)j Fs(is)d(de\014ned)g(b) o(y)648 2935 y Fj(P)686 2914 y Fr(\010)679 2947 y(\003)714 2935 y Fs(\()p Fj(\027)s Fs(\))e(:=)859 2867 y Fh(Z)917 2935 y Fj(\027)s Fs(\()p Fj(d)p 988 2907 23 2 v(s)p Fs(\))p Fj(P)1068 2914 y Fr(\010)1061 2947 y(\003)1096 2935 y Fs(\()p 1115 2907 V Fj(s)p Fs(\))p Fj(:)559 b Fs(\(2)p Fj(:)p Fs(11\))p eop %%Page: 6 6 6 5 bop -35 14 a Fs(6)1673 b Fn(Y)l(o)q(o)17 b Fm(et)h(al)-35 168 y Fs(Finally)l(,)i(w)o(e)g(in)o(tro)q(duce)h(the)f(p)q(erio)q(dic)h (b.c..)34 b(Consider)21 b(the)f(\\torus")j Fj(T)1353 175 y Fk(n)1397 168 y Fs(:=)f(\()p Fl(Z)-11 b Fj(=n)p Fl(Z)d Fs(\))1631 150 y Fk(\027)1671 168 y Fs(with)21 b(the)-35 243 y(quotien)o(t)e(map)h Fj(q)296 250 y Fk(n)340 243 y Fs(:)h Fl(Z)411 225 y Fk(\027)450 243 y Fi(!)g Fj(T)550 250 y Fk(n)573 243 y Fs(.)34 b(The)21 b(con\014guration)g (space)g(for)g Fj(T)1268 250 y Fk(n)1311 243 y Fs(is)f(\012)1399 250 y Fk(T)1420 254 y Fe(n)1465 243 y Fs(:=)g(\()p Fl(R)1595 225 y Fk(d)1612 243 y Fs(\))1631 225 y Fk(T)1652 229 y Fe(n)1675 243 y Fs(,)h(and)g Fj(q)1831 250 y Fk(n)-35 317 y Fs(induces)14 b(a)i(map)e Fj(r)306 324 y Fk(n)343 317 y Fs(:)f(\012)405 324 y Fk(T)426 328 y Fe(n)463 317 y Fi(!)h Fs(\012)h(\(rep)q(eating)h(the)e(con\014guration)j(across)f Fl(Z)1367 299 y Fk(\027)1400 317 y Fs(with)f(p)q(erio)q(d)h Fj(n)f Fs(in)g(eac)o(h)-35 392 y(co)q(ordinate)h(direction\).)21 b(The)16 b(Hamiltonian)e(for)j Fj(T)930 399 y Fk(n)969 392 y Fs(is)f(de\014ned)g(b)o(y)g([Is])691 551 y Fk(P)712 544 y Fj(H)756 523 y Fr(\010)752 556 y Fk(T)773 560 y Fe(n)810 544 y Fs(:=)927 481 y Fg(\003)875 496 y Fh(X)887 604 y Fr([\001])955 544 y Fs(\010)990 551 y Fr(\001)1033 544 y Fi(\016)11 b Fj(r)1091 551 y Fk(n)1115 544 y Fj(;)601 b Fs(\(2)p Fj(:)p Fs(12\))-35 756 y(where)18 b([\001])g(:=)g Fi(f)p Fs(\001)13 b(+)g Fj(nj)s Fi(j)8 b Fj(j)22 b Fi(2)d Fl(Z)599 738 y Fk(\027)618 756 y Fi(g)g Fs(is)g(the)g(equiv)m(alence)e (class)j(and)1328 694 y Fg(\003)1276 709 y Fh(X)1368 756 y Fs(means)e(that)h(the)g(sum)g(is)-35 831 y(restricted)12 b(to)i(those)h(\001)e(on)h(whic)o(h)f Fj(q)640 838 y Fk(n)677 831 y Fs(is)h(one-to-one.)21 b(The)14 b(pressure)g(with)g(p)q (erio)q(dic)f(b.c.)20 b(is)14 b(de\014ned)-35 906 y(b)o(y)471 997 y Fk(P)492 990 y Fj(P)530 969 y Fr(\010)523 1002 y Fk(T)544 1006 y Fe(n)581 990 y Fs(:=)f Fj(n)675 969 y Fg(\000)p Fk(\027)733 990 y Fs(log)804 922 y Fh(Z)832 1035 y Fk(T)853 1039 y Fe(n)884 990 y Fj(ds)932 997 y Fk(T)953 1001 y Fe(n)984 990 y Fs(exp[)p Fi(\000)1112 997 y Fk(P)1132 990 y Fj(H)1176 969 y Fr(\010)1172 1002 y Fk(T)1193 1006 y Fe(n)1216 990 y Fs(\()p Fj(s)1258 997 y Fk(T)1279 1001 y Fe(n)1302 990 y Fs(\)])p Fj(:)381 b Fs(\(2)p Fj(:)p Fs(13\))-35 1110 y(The)20 b(thermo)q(dynamic)d(limit) h(of)i(the)g(pressure)g(for)h(an)o(y)f(b.c.)32 b(is)20 b(pro)o(v)o(ed)g(in)g([LP].)f(Recall)g(that)i(a)-35 1185 y(sequence)14 b Fi(f)p Fs(\003)225 1192 y Fk(n)248 1185 y Fi(g)p Fs(,)h(\003)336 1192 y Fk(n)374 1185 y Fi(\032\032)e Fl(Z)501 1167 y Fk(\027)520 1185 y Fs(,)i(is)g(said)h(to)g(b)q(e)f (tending)h(to)g Fl(Z)1092 1167 y Fk(\027)1126 1185 y Fs(in)f(the)g(sense)h(of)g(v)m(an)g(Ho)o(v)o(e)e(\(w)o(e)h(write)-35 1260 y(lim)32 1267 y Fr(\003)56 1271 y Fe(n)78 1267 y Fg(!)p Ff(Z)138 1258 y Fe(\027)166 1267 y Fr(\(v)n(an)c(Ho)o(v)o(e\)) 356 1260 y Fs(hereafter\))17 b(if:)23 b(\(a\))18 b(\003)761 1267 y Fk(n)p Fr(+1)846 1260 y Fi(\033)e Fs(\003)935 1267 y Fk(n)958 1260 y Fs(,)i(\(b\))g(\003)1107 1267 y Fk(n)1146 1260 y Fi(\033)e Fs(\001)h(for)h(an)o(y)g(\001)e Fi(\032\032)g Fl(Z)1616 1241 y Fk(\027)1652 1260 y Fs(and)i(some)-35 1334 y Fj(n)p Fs(,)f(\(c\))f(giv)o(en)h(an)o(y)g(parallelepip)q(ed)f (\000)h(and)h(the)f(partition)g Fj(\031)r Fs(\(\000\))g(of)h Fl(Z)1277 1316 y Fk(\027)1313 1334 y Fs(generated)f(b)o(y)f (translations)-35 1409 y(of)g(\000)515 1486 y(lim)6 b Fj(N)635 1465 y Fg(\000)630 1499 y Fr(\000)664 1486 y Fs(\(\003)717 1493 y Fk(n)741 1486 y Fs(\))14 b(=)f Fi(1)p Fj(;)57 b Fs(lim)1027 1452 y Fj(N)1071 1432 y Fg(\000)1066 1466 y Fr(\000)1101 1452 y Fs(\(\003)1154 1459 y Fk(n)1177 1452 y Fs(\))p 1027 1475 170 2 v 1027 1521 a Fj(N)1071 1501 y Fr(+)1066 1535 y(\000)1101 1521 y Fs(\(\003)1154 1528 y Fk(n)1177 1521 y Fs(\))1215 1486 y(=)13 b(1)p Fj(;)-35 1609 y Fs(where)i Fj(N)149 1589 y Fg(\000)144 1623 y Fr(\000)178 1609 y Fs(\(\003)231 1616 y Fk(n)255 1609 y Fs(\))g(is)g(the)g(n)o(um)o(b)q(er)f(of)i(sets)f(of)h Fj(\031)r Fs(\(\000\))f(con)o(tained)g(in)g(\003)1224 1616 y Fk(n)1247 1609 y Fs(,)g Fj(N)1320 1589 y Fr(+)1315 1623 y(\000)1350 1609 y Fs(\(\003)1403 1616 y Fk(n)1426 1609 y Fs(\))h(the)f(n)o(um)o(b)q(er)f(of)h(sets)-35 1684 y(with)h(non-v)o(oid)g(in)o(tersection)f(with)h(\003)683 1691 y Fk(n)706 1684 y Fs(.)-35 1803 y Ft(Theorem)g(2.5.)24 b Fm(\([LP,)18 b(The)n(or)n(em)f(3.1]\))i(Supp)n(ose)f(that)h(the)h (hyp)n(otheses)e(of)g(Assumption)i(2.1)e(ar)n(e)-35 1878 y(satis\014e)n(d)g(and)h Fs(\003)281 1885 y Fk(n)321 1878 y Fi(!)d Fl(Z)423 1860 y Fk(\027)460 1878 y Fm(in)j(the)h(sense)g (of)e(van)i(Hove.)27 b(Then,)19 b Fi(f)p Fj(P)1240 1860 y Fr(\010)1233 1891 y(\003)1257 1895 y Fe(n)1281 1878 y Fi(g)p Fm(,)g Fi(f)p Fj(P)1403 1860 y Fr(\010)1396 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 y Fm(have)f(the)g(same)g(thermo)n(dynamic)g(limit,)g(say)f Fj(P)881 1935 y Fr(\010)909 1953 y Fm(.)23 b(A)o(lso,)c(for)e(the)h(se) 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 2040 y Fk(T)261 2044 y Fe(n)284 2027 y Fi(g)e Fm(c)n(onver)n(ges)h(to)g (the)g(same)f(limit)h Fj(P)961 2009 y Fr(\010)989 2027 y Fm(.)-35 2147 y Fs(W)l(e)c(remark)e(here)i(that)h(w)o(e)f(ha)o(v)o(e) g(tak)o(en)f(the)i(de\014nition)e(for)i(p)q(erio)q(dic)f(b.c.)20 b(from)13 b([Is],)h(whic)o(h)f(di\013ers)-35 2221 y(sligh)o(tly)i(from) h(that)i(of)g([LP].)e(Still,)g(it)g(is)h(not)h(hard)g(to)f(see)g(that)g (the)g(result)g(of)g([LP])g(\(for)h(p)q(erio)q(dic)-35 2296 y(b.c.\))i(insists)c(on)h(to)g(hold.)38 2406 y(W)l(e)e(no)o(w)g (de\014ne)g(the)g(mean)f(en)o(trop)o(y)g(for)i(Gibbs)f(measures.)20 b(Let)c Fj(\026)e Fi(2)g(G)s Fs(\(\010\))h(and)h(supp)q(ose)g(that)-35 2481 y Fj(f)23 b Fs(is)18 b(a)h(b)q(ounded)g Fi(F)343 2488 y Fr(\003)370 2481 y Fs(-measurable)e(function)h(for)h(some)e (\003)g Fi(\032\032)g Fl(Z)1219 2463 y Fk(\027)1238 2481 y Fs(.)27 b(By)18 b(Gibbs)h(condition)f(w)o(e)g(see)-35 2556 y(that)e(\(w)o(e)g(omit)f(the)h(sup)q(erscript)g(\010)h(in)e(the)h (notations)i(if)d(it)h(in)o(v)o(olv)o(es)e(no)j(confusion\))168 2692 y Fj(\026)p Fs(\()p Fj(f)5 b Fs(\))49 b(=)399 2624 y Fh(Z)458 2692 y Fj(d\026)p Fs(\()p 531 2665 V Fj(s)p Fs(\))p Fj(\015)598 2699 y Fr(\003)625 2692 y Fs(\()p Fj(f)5 b Fi(j)p 687 2665 V Fj(s)p Fs(\))313 2814 y(=)399 2746 y Fh(Z)458 2814 y Fj(ds)506 2821 y Fr(\003)532 2814 y Fj(f)g Fs(\()p Fj(s)603 2821 y Fr(\003)630 2814 y Fs(\))657 2746 y Fh(Z)716 2814 y Fj(d\026)p Fs(\()p 789 2786 V Fj(s)p Fs(\))884 2780 y(1)p 836 2802 121 2 v 836 2848 a Fj(Z)869 2855 y Fr(\003)896 2848 y Fs(\()p 915 2820 23 2 v Fj(s)p Fs(\))970 2814 y(exp[)p Fi(\000)p Fj(V)10 b Fs(\()p Fj(s)1178 2821 y Fr(\003)1205 2814 y Fs(\))h Fi(\000)g Fj(W)c Fs(\()p Fj(s)1380 2821 y Fr(\003)1406 2814 y Fj(;)p 1428 2786 V 8 w(s)1451 2821 y Fr(\003)1475 2812 y Fe(c)1493 2814 y Fs(\)])306 2935 y(=:)399 2867 y Fh(Z)458 2935 y Fj(ds)506 2942 y Fr(\003)532 2935 y Fj(f)e Fs(\()p Fj(s)603 2942 y Fr(\003)630 2935 y Fs(\))p Fj(\026)678 2914 y Fr(\(\003\))733 2935 y Fs(\()p Fj(s)775 2942 y Fr(\003)801 2935 y Fs(\))p Fj(:)1730 2814 y Fs(\(2)p Fj(:)p Fs(14\))p eop %%Page: 7 7 7 6 bop -35 14 a Fn(Statistical)15 b(mec)o(hanics)f(for)j(un)o(b)q (ounded)g(spin)f(systems)807 b Fs(7)-35 168 y(Th)o(us,)20 b(the)f(restriction)f Fj(\026)458 175 y Fr(\003)504 168 y Fs(of)i Fj(\026)g Fs(to)g Fi(F)711 175 y Fr(\003)737 168 y Fs(,)g(whic)o(h)e(w)o(e)h(ma)o(y)f(understand)i(as)g(a)g(measure) e(on)i(\012)1761 175 y Fr(\003)1788 168 y Fs(,)g(is)-35 243 y(absolutely)c(con)o(tin)o(uous)g(w.r.t.)k Fj(ds)624 250 y Fr(\003)667 243 y Fs(with)c(a)h(Radon-Nik)o(o)q(dym)e(deriv)m (ativ)o(e)g Fj(\026)1437 225 y Fr(\(\003\))1491 243 y Fs(\()p Fj(s)1533 250 y Fr(\003)1559 243 y Fs(\),)350 377 y Fj(\026)379 356 y Fr(\(\003\))434 377 y Fs(\()p Fj(s)476 384 y Fr(\003)502 377 y Fs(\))f(=)587 309 y Fh(Z)645 377 y Fj(d\026)p Fs(\()p 718 350 23 2 v Fj(s)p Fs(\))814 343 y(1)p 765 366 121 2 v 765 411 a Fj(Z)798 418 y Fr(\003)825 411 y Fs(\()p 844 384 23 2 v Fj(s)p Fs(\))899 377 y(exp[)p Fi(\000)p Fj(V)d Fs(\()p Fj(s)1108 384 y Fr(\003)1134 377 y Fs(\))g Fi(\000)g Fj(W)c Fs(\()p Fj(s)1309 384 y Fr(\003)1335 377 y Fj(;)p 1357 350 V 8 w(s)1380 384 y Fr(\003)1404 375 y Fe(c)1423 377 y Fs(\)])p Fj(:)260 b Fs(\(2)p Fj(:)p Fs(15\))-35 515 y(The)16 b(en)o(trop)o(y)f (of)i Fj(\026)f Fs(in)g(\003)e Fi(\032\032)f Fl(Z)576 497 y Fk(\027)610 515 y Fs(is)j(de\014ned)g(b)o(y)g([Is,)f(Si2])337 649 y Fj(S)367 656 y Fr(\003)394 649 y Fs(\()p Fj(\026)p Fs(\))f(:=)f Fi(\000)587 581 y Fh(Z)646 649 y Fj(ds)694 656 y Fr(\003)720 649 y Fj(\026)749 629 y Fr(\(\003\))804 649 y Fs(\()p Fj(s)846 656 y Fr(\003)872 649 y Fs(\))8 b(log)i Fj(\026)1000 629 y Fr(\(\003\))1054 649 y Fs(\()p Fj(s)1096 656 y Fr(\003)1123 649 y Fs(\))j(=)h Fi(\000)p Fj(\026)p Fs(\(log)c Fj(\026)1395 629 y Fr(\(\003\))1449 649 y Fs(\))p Fj(:)248 b Fs(\(2)p Fj(:)p Fs(16\))-35 781 y(W)l(e)16 b(will)f(sho)o(w)i(that)f(the)g(mean)f(en)o(trop)o(y)563 905 y Fj(s)p Fs(\()p Fj(\026)p Fs(\))f(:=)121 b(lim)733 939 y Fr(\003)p Fg(!)p Ff(Z)817 929 y Fe(\027)845 939 y Fr(\(v)n(an)11 b(Ho)o(v)o(e\))1023 905 y Fi(j)p Fs(\003)p Fi(j)1085 885 y Fg(\000)p Fr(1)1132 905 y Fj(S)1162 912 y Fr(\003)1189 905 y Fs(\()p Fj(\026)p Fs(\))474 b(\(2)p Fj(:)p Fs(17\))-35 1047 y(exists.)28 b(F)l(urthermore,)17 b(w)o(e)h(will)g(sho)o(w)h(that)g(lim)893 1054 y Fr(\003)p Fg(!)p Ff(Z)978 1045 y Fe(\027)1006 1054 y Fr(\(v)n(an)10 b(Ho)o(v)o(e\))1196 1047 y Fi(j)p Fs(\003)p Fi(j)1258 1029 y Fg(\000)p Fr(1)1305 1047 y Fj(\026)p Fs(\()p Fj(H)1397 1029 y Fr(\010)1393 1059 y(\003)1425 1047 y Fs(\),)19 b(the)f(limit)e(of)k(mean)-35 1121 y(energy)14 b(p)q(er)g(unit)g(v)o (olume,)e(also)j(exists)f(and)h(w)o(e)f(ha)o(v)o(e)g(the)g(Gibbs)h(v)m (ariational)f(principle,)f(whic)o(h)h(w)o(e)-35 1196 y(state)i(as)h(a)g(theorem:)-35 1312 y Ft(Theorem)f(2.6.)24 b Fm(Supp)n(ose)16 b(that)g(the)h(inter)n(action)g Fs(\010)g Fm(satis\014es)f(the)h(c)n(onditions)g(of)f(Assumption)h(2.1)-35 1387 y(and)h(let)i Fj(\026)c Fi(2)g(G)257 1369 y Fk(I)276 1387 y Fs(\(\010\))p Fm(.)26 b(Then,)19 b(the)g(me)n(an)g(entr)n(opy)f Fj(s)p Fs(\()p Fj(\026)p Fs(\))g Fm(and)h Fs(lim)1194 1395 y Fr(\003)p Fg(!)p Ff(Z)1279 1385 y Fe(\027)1308 1395 y Fc(\(van)13 b(Hove\))1492 1387 y Fi(j)p Fs(\003)p Fi(j)1554 1369 y Fg(\000)p Fr(1)1601 1387 y Fj(\026)p Fs(\()p Fj(H)1693 1369 y Fr(\010)1689 1400 y(\003)1721 1387 y Fs(\))18 b Fm(exist)-35 1462 y(and)f(satis\014es)h(the)g (variational)g(e)n(quality:)489 1586 y Fj(s)p Fs(\()p Fj(\026)p Fs(\))12 b Fi(\000)121 b Fs(lim)641 1620 y Fr(\003)p Fg(!)p Ff(Z)725 1610 y Fe(\027)754 1620 y Fc(\(van)13 b(Hove\))937 1586 y Fi(j)p Fs(\003)p Fi(j)999 1566 y Fg(\000)p Fr(1)1045 1586 y Fj(\026)p Fs(\()p Fj(H)1137 1566 y Fr(\010)1133 1599 y(\003)1166 1586 y Fs(\))g(=)h Fj(P)1288 1566 y Fr(\010)1316 1586 y Fj(:)400 b Fs(\(2)p Fj(:)p Fs(18\))-35 1722 y(The)16 b(pro)q(of)h(of)g(Theorem)e(2.6)h (will)f(b)q(e)i(giv)o(en)e(in)h(the)g(next)g(section.)38 1796 y(As)23 b(in)h(the)f(case)h(of)g(b)q(ounded)h(spin)e(systems,)h (the)g(con)o(v)o(erse)e(of)j(Theorem)d(2.6)i(also)g(holds.)-35 1871 y(Ho)o(w)o(ev)o(er,)15 b(for)j(a)g(tec)o(hnical)e(reason)j(in)e (the)g(pro)q(of,)i(w)o(e)e(imp)q(ose)g(a)h(further)f(condition)h(of)g (\014niteness)-35 1946 y(of)e(in)o(teraction)f(range.)-35 2062 y Ft(Theorem)h(2.7.)24 b Fm(Supp)n(ose)c(that)g(a)g(\014nite)h(r)n (ange)f(inter)n(action)h Fs(\010)f Fm(satis\014es)g(the)h(c)n (onditions)f(of)g(As-)-35 2137 y(sumption)15 b(2.1.)21 b(L)n(et)15 b Fj(\026)f Fi(2)g(P)t Fs(\(\012\))h Fm(b)n(e)h(a)f(tr)n (anslational)r(ly)h(invariant)g(r)n(e)n(gular)e(me)n(asur)n(e)h(\(se)n (e)g(De\014nition)-35 2211 y(2.2\))i(and)g(supp)n(ose)g(that)h(the)g (variational)g(e)n(quality)g(\(2.18\))f(holds.)22 b(Then)c Fj(\026)d Fi(2)f(G)1486 2193 y Fk(I)1505 2211 y Fs(\(\010\))p Fm(.)-35 2328 y Fs(The)i(pro)q(of)h(will)e(b)q(e)i(giv)o(en)e(in)h(the) g(next)g(section.)-35 2472 y Fq(2.2.)65 b(Quan)n(tum)23 b(Systems)-35 2579 y Fs(Let)18 b(us)g(no)o(w)g(turn)g(our)g(atten)o (tion)g(to)g(the)g(quan)o(tum)e(systems.)25 b(It)18 b(is)f(generally)g (accepted)g(that)i(in)-35 2654 y(quan)o(tum)14 b(statistical)g(mec)o (hanics)f(the)j(equilibrium)11 b(states)16 b(are)g(those)f(of)h(KMS)f (states)h([BR1-2,)f(Is,)-35 2729 y(Si2].)20 b(F)l(or)13 b(quan)o(tum)g(un)o(b)q(ounded)i(spin)e(systems,)g(b)o(y)g(using)h(the) g(Green's)f(function)h(metho)q(d)f([BR2],)-35 2803 y(a)i(state)h (satisfying)f(\(w)o(eak\))g(KMS)g(condition)g(has)h(b)q(een)f (constructed)h(in)f([P1].)20 b(In)15 b([PY],)f(there)h(has)-35 2878 y(b)q(een)g(prop)q(osed)h(a)g(c)o(haracterization)e(of)h(Gibbs)h (states)g(b)o(y)e(using)i(the)f(concept)f(of)i(Gibbs)f(measures)-35 2953 y(and)j(the)f(conditional)g(reduced)g(densit)o(y)g(matrices.)23 b(W)l(e)17 b(will)f(sho)o(w)i(that)g(the)g(\(w)o(eak\))f(KMS)g(state)p eop %%Page: 8 8 8 7 bop -35 14 a Fs(8)1673 b Fn(Y)l(o)q(o)17 b Fm(et)h(al)-35 168 y Fs(constructed)e(in)h([P1])g(satis\014es)g(the)g(Gibbs)g(v)m (ariational)g(equalit)o(y)l(.)22 b(Let)17 b(us)g(b)q(egin)g(with)g (presen)o(ting)-35 243 y(necessary)f(notations.)22 b(W)l(e)16 b(refer)f(to)i([PY])e(for)i(details.)38 317 y(F)l(or)f(eac)o(h)g(b)q (ounded)i(region)e(\003)e Fi(\032\032)g Fl(Z)757 299 y Fk(\027)792 317 y Fs(the)i(Hilb)q(ert)f(space)i(for)g(the)f(un)o(b)q (ounded)h(spin)g(systems)-35 392 y(is)f(giv)o(en)f(b)o(y)639 455 y Fi(H)681 462 y Fr(\003)750 455 y Fs(:=)41 b Fi(\012)882 462 y Fk(i)p Fg(2)p Fr(\003)943 455 y Fj(L)976 437 y Fr(2)996 455 y Fs(\()p Fl(R)1054 437 y Fk(d)1071 455 y Fj(;)8 b(dx)1146 462 y Fk(i)1160 455 y Fs(\))756 550 y(=)49 b Fj(L)876 532 y Fr(2)895 550 y Fs(\(\()p Fl(R)972 532 y Fk(d)990 550 y Fs(\))1009 532 y Fr(\003)1035 550 y Fj(;)8 b(dx)1110 557 y Fr(\003)1137 550 y Fs(\))p Fj(:)1730 503 y Fs(\(2)p Fj(:)p Fs(19\))-35 642 y(The)16 b(Hamiltonian)e(op)q (erator)k(for)e(the)g(region)h(\003)c Fi(\032\032)h Fl(Z)1022 624 y Fk(\027)1057 642 y Fs(is)i(giv)o(en)f(b)o(y)626 778 y Fj(H)666 785 y Fr(\003)706 778 y Fs(:=)f Fi(\000)816 744 y Fs(1)p 816 767 25 2 v 816 812 a(2)853 731 y Fh(X)859 836 y Fk(i)p Fg(2)p Fr(\003)933 778 y Fs(\001)974 785 y Fk(i)999 778 y Fs(+)d Fj(V)g Fs(\()p Fj(x)1134 785 y Fr(\003)1161 778 y Fs(\))p Fj(;)536 b Fs(\(2)p Fj(:)p Fs(20\))-35 945 y(where)18 b(\001)149 952 y Fk(i)181 945 y Fs(is)h(the)g(Laplacian)g(op)q(erator)h(for)f(the)f(v)m(ariable)h Fj(x)1121 952 y Fk(i)1153 945 y Fi(2)f Fl(R)1243 927 y Fk(d)1279 945 y Fs(and)h Fj(V)12 b Fs(\()p Fj(x)1463 952 y Fr(\003)1489 945 y Fs(\))19 b(is)f(the)h(p)q(oten)o(tial)-35 1020 y(energy)i(in)g(the)h(region)f(\003.)38 b(W)l(e)21 b(assume)g(that)h(the)g(in)o(teraction)f(\010)h(satis\014es)g(the)f(h)o (yp)q(otheses)h(of)-35 1095 y(Assumption)15 b(2.1.)38 1169 y(F)l(or)h(eac)o(h)g(b)q(ounded)h(region)f(\003)e Fi(\032\032)f Fl(Z)756 1151 y Fk(\027)774 1169 y Fs(,)j(the)g Fj(C)927 1151 y Fg(\003)946 1169 y Fs(-algebra)h(for)g(lo)q(cal)f (observ)m(ables)g(is)g(de\014ned)g(b)o(y)759 1301 y Fi(A)799 1308 y Fr(\003)840 1301 y Fs(:=)d Fi(L)p Fs(\()p Fi(H)1000 1308 y Fr(\003)1027 1301 y Fs(\))p Fj(;)670 b Fs(\(2)p Fj(:)p Fs(21\))-35 1432 y(where)23 b Fi(L)p Fs(\()p Fi(H)208 1439 y Fr(\003)235 1432 y Fs(\))h(is)f(the)h(algebra)g(of)g(all)g(b)q (ounded)h(op)q(erators)g(on)f Fi(H)1290 1439 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y Fj(e)349 2912 y Fg(\000)p Fk(H)405 2918 y Fd(\003)430 2933 y Fs(\()p Fj(x)477 2940 y Fr(\003)503 2933 y Fj(;)8 b(y)549 2940 y Fr(\003)575 2933 y Fs(\))14 b(=)659 2865 y Fh(Z)718 2933 y Fj(P)749 2940 y Fk(x)769 2946 y Fd(\003)792 2940 y Fk(;y)819 2946 y Fd(\003)844 2933 y Fs(\()p Fj(ds)911 2940 y Fr(\003)938 2933 y Fs(\))8 b(exp)1048 2892 y Fh(\002)1080 2933 y Fi(\000)1129 2865 y Fh(Z)1179 2878 y Fr(1)1157 2978 y(0)1207 2933 y Fj(V)j Fs(\()p Fj(s)1288 2940 y Fr(\003)1315 2933 y Fs(\()p Fj(\034)6 b Fs(\)\))i Fj(d\034)1459 2892 y Fh(\003)1480 2933 y Fj(;)236 b Fs(\(2)p Fj(:)p Fs(25\))p eop %%Page: 9 9 9 8 bop -35 14 a Fn(Statistical)15 b(mec)o(hanics)f(for)j(un)o(b)q (ounded)g(spin)f(systems)807 b Fs(9)-35 168 y(where)16 b Fj(x)134 175 y Fr(\003)177 168 y Fs(and)i Fj(y)297 175 y Fr(\003)340 168 y Fs(are)f(p)q(oin)o(ts)g(in)g(\()p Fl(R)685 150 y Fk(d)702 168 y Fs(\))721 150 y Fr(\003)748 168 y Fs(,)f Fj(s)801 175 y Fr(\003)843 168 y Fi(2)f Fs(\()p Fj(C)t Fs(\([0)p Fj(;)8 b Fs(1];)g 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Fj(\025)p Fs(\()p Fj(ds)945 1111 y Fr(\003)973 1104 y Fs(\))p Fj(e)1015 1084 y Fg(\000)p Fk(V)7 b Fr(\()p Fk(s)1100 1090 y Fd(\003)1123 1084 y Fr(\))1139 1104 y Fj(:)577 b Fs(\(2)p Fj(:)p Fs(27\))-35 1231 y(By)11 b(using)i(the)f(ab)q(o)o(v)o(e)g(Wiener)g(in)o(tegral)f (formalism)f(and)j(Ruelle-t)o(yp)q(e)d(probabilit)o(y)h(estimates,)g(P) o(ark)-35 1305 y(has)j(sho)o(wn)h(the)e(existence)f(of)j(the)e(thermo)q (dynamic)e(limit)g(of)k(the)e(pressure)h([P1].)20 b(W)l(e)14 b(remark)e(here)-35 1380 y(that)21 b(b)o(y)f(in)o(tro)q(ducing)g(the)g (concept)h(of)f(\\Gibbs)i(sp)q(eci\014cations")f(\(and)g(hence)f(Gibbs) g(measures\))-35 1455 y(for)k(quan)o(tum)f(spin)h(systems)f(as)i(in)f ([PY],)f(w)o(e)g(can)i(also)g(de\014ne)f(the)g(\014nite)f(v)o(olume)f (pressures)-35 1529 y(with)16 b(di\013eren)o(t)h(kind)f(of)h(b)q (oundary)h(conditions)g(lik)o(e)d(in)h(the)h(classical)f(case)h(\(see)g (\(2.6\),)g(\(2.8\),)g(and)-35 1604 y(\(2.11\)\).)k(By)13 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Fr(\003)843 2075 y Fs(\()p Fj(\032)887 2055 y Fr(\(\003\))941 2075 y Fj(A)p Fs(\))p Fj(;)56 b(A)14 b Fi(2)g(A)1205 2082 y Fr(\003)1231 2075 y Fj(:)485 b Fs(\(2)p Fj(:)p Fs(34\))-35 2199 y(The)16 b(en)o(trop)o(y)f(of)i Fj(\032)f Fs(in)g(\003)g(is)g(de\014ned)g(b)o(y) g([BR2,)f(Is,)h(Si2])587 2308 y Fj(S)617 2315 y Fr(\003)643 2308 y Fs(\()p Fj(\032)p Fs(\))42 b(:=)f Fi(\000)p Fs(T)l(r)930 2315 y Fr(\003)956 2308 y Fs(\()p Fj(\032)1000 2290 y Fr(\(\003\))1063 2308 y Fs(log)9 b Fj(\032)1159 2290 y Fr(\(\003\))1213 2308 y Fs(\))755 2405 y(=)48 b Fi(\000)p Fj(\032)905 2412 y Fr(\003)931 2405 y Fs(\(log)10 b Fj(\032)1047 2387 y Fr(\(\003\))1101 2405 y Fs(\))p Fj(:)1730 2357 y Fs(\(2)p Fj(:)p Fs(35\))-35 2514 y(F)l(rom)15 b(the)h(de\014nition,)f (w)o(e)h(see)g(that)g(\(w)o(e)g(omit)f(\010)h(from)f(the)h(notations\)) 623 2638 y Fj(S)653 2645 y Fr(\003)680 2638 y Fs(\()p Fj(!)729 2645 y Fr(\003)755 2638 y Fs(\))c Fi(\000)e Fj(!)865 2645 y Fr(\003)892 2638 y Fs(\()p Fj(H)951 2645 y Fr(\003)978 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Fs(\))17 b Fm(\(str)n(ong)h(sub)n(additivity\).)-35 2803 y Ft(Pro)r(of)p Fs(.)32 b(The)20 b(pro)q(ofs)h(of)f(\(b\))g(and)h (\(c\))e(can)h(b)q(e)g(done)h(b)o(y)e(the)h(same)f(metho)q(d)g(giv)o (en,)g(e.g.,)h(in)g(the)-35 2878 y(pro)q(of)14 b(of)g(Lemma)e(I)q (I.2.1)h(of)h([Is].)19 b(The)14 b(prop)q(ert)o(y)f(\(a\))h(will)f(not)h (b)q(e)f(used)h(in)f(this)h(pap)q(er,)g(but)g(w)o(e)f(ha)o(v)o(e)-35 2953 y(giv)o(en)j(it)g(as)i(a)f(prop)q(ert)o(y)g(of)g(the)g(en)o(trop)o (y)l(.)22 b(Since)16 b(the)h(a)g(priori)f(measure,)g(the)g(Leb)q(esgue) i(measure,)p eop %%Page: 12 12 12 11 bop -35 14 a Fs(12)1649 b Fn(Y)l(o)q(o)17 b Fm(et)h(al)-35 168 y Fs(is)e(not)h(a)g(\014nite)f(measure)f(w)o(e)h(can)h(not)g (simply)e(tak)o(e)h Fj(S)999 176 y Fg(;)1018 168 y Fs(\()p Fj(\026)p Fs(\))f Fi(\021)f Fs(0)j(as)g(done)g(in)g(the)f(pro)q(of)i (of)e(Lemma)-35 243 y(I)q(I.2.1)e(of)i([Is])e(for)h(b)q(ounded)h(spin)f (systems.)20 b(Ho)o(w)o(ev)o(er,)13 b(w)o(e)i(can)g(mak)o(e)e(a)j (detour.)21 b(Let)15 b(us)h(in)o(tro)q(duce)-35 317 y(a)g(probabilit)o (y)g(measure)581 395 y Fj(d\026)635 375 y Fr(\(0\))683 395 y Fs(\()p Fj(s)725 402 y Fk(i)739 395 y Fs(\))e(:=)f Fj(N)881 375 y Fg(\000)p Fr(1)937 395 y Fs(exp)o(\()p Fi(\000)p Fj(\013s)1123 375 y Fr(2)1123 408 y Fk(i)1143 395 y Fs(\))p Fj(ds)1210 402 y Fk(i)1224 395 y Fj(;)516 b Fs(\(3)p Fj(:)p Fs(1\))-35 501 y(where)23 b Fj(\013)k(>)f Fs(0)f(is)e(a)h(su\016cien)o(tly)e(small)g(n)o(um)o(b)q(er)g(and)i Fj(N)32 b Fs(:=)26 b(\()p Fj(\031)r(=\013)p Fs(\))1334 483 y Fk(d=)p Fr(2)1414 501 y Fs(is)d(the)h(normalization)-35 576 y(constan)o(t.)d(F)l(or)c(eac)o(h)e(\003)f Fi(\032\032)f Fl(Z)553 558 y Fk(\027)587 576 y Fs(let)j(us)h(de\014ne)635 691 y Fh(e)629 704 y Fj(V)11 b Fs(\()p Fj(s)710 711 y Fr(\003)737 704 y Fs(\))i(:=)h Fj(V)d Fs(\()p Fj(s)916 711 y Fr(\003)942 704 y Fs(\))g Fi(\000)1022 656 y Fh(X)1028 762 y Fk(i)p Fg(2)p Fr(\003)1103 704 y Fj(\013s)1157 683 y Fr(2)1157 716 y Fk(i)1177 704 y Fj(:)563 b Fs(\(3)p Fj(:)p Fs(2\))-35 866 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1351 y Fe(c)1236 1353 y Fs(\)])13 b(=)h Fj(N)1378 1332 y Fg(\000j)p Fr(\003)p Fg(j)1452 1353 y Fj(Z)1485 1360 y Fr(\003)1512 1353 y Fs(\()p 1531 1325 V Fj(s)p Fs(\))p Fj(:)-35 1468 y Fs(Let)i(us)h(de\014ne)e(\(see)h (\(2.15\)\))314 1606 y Fj(\026)343 1580 y Fr(\(0\))343 1619 y(\003)390 1606 y Fs(\()p Fj(s)432 1613 y Fr(\003)459 1606 y Fs(\))41 b(:=)612 1538 y Fh(Z)670 1606 y Fj(d\026)p Fs(\()p 743 1578 V Fj(s)q Fs(\))851 1572 y(1)p 791 1594 146 2 v 791 1651 a Fj(Z)828 1625 y Fr(\(0\))824 1664 y(\003)875 1651 y Fs(\()p 894 1623 23 2 v Fj(s)p Fs(\))949 1606 y(exp[)p Fi(\000)1083 1593 y Fh(e)1077 1606 y Fj(V)10 b Fs(\()p Fj(s)1157 1613 y Fr(\003)1184 1606 y Fs(\))h Fi(\000)g Fj(W)c Fs(\()p Fj(s)1359 1613 y Fr(\003)1385 1606 y Fj(;)p 1407 1578 V 8 w(s)1430 1613 y Fr(\003)1454 1604 y Fe(c)1472 1606 y Fs(\)])526 1707 y(=)48 b Fj(N)656 1689 y Fg(j)p Fr(\003)p Fg(j)711 1707 y Fs(exp[)p Fj(\013)839 1670 y Fh(P)891 1722 y Fk(i)p Fg(2)p Fr(\003)962 1707 y Fj(s)985 1689 y Fr(2)985 1719 y Fk(i)1004 1707 y Fs(])8 b Fj(\026)1055 1689 y Fr(\(\003\))1110 1707 y Fs(\()p Fj(s)1152 1714 y Fr(\003)1178 1707 y Fs(\))p Fj(:)1754 1644 y Fs(\(3)p Fj(:)p Fs(4\))-35 1833 y(W)l(e)16 b(notice)g(that)g Fj(\026)327 1807 y Fr(\(0\))327 1846 y(\003)375 1833 y Fs(\()p Fj(s)417 1840 y Fr(\003)443 1833 y Fs(\))h(is)f(a)h (Radon-Nik)o(o)q(dym)e(deriv)m(ativ)o(e)g(of)i Fj(\026)1243 1840 y Fr(\003)1269 1833 y Fs(,)f(the)h(restriction)e(of)i Fj(\026)g Fs(to)f Fi(F)1814 1840 y Fr(\003)1841 1833 y Fs(,)-35 1907 y(with)g(resp)q(ect)g(to)g Fj(\026)331 1889 y Fr(\(0\))379 1907 y Fs(\()p Fi(\021)d Fs(\()p Fj(\026)498 1889 y Fr(\(0\))546 1907 y Fs(\))565 1889 y Fr(\003)591 1907 y Fs(\).)21 b(De\014ne)571 2035 y Fj(S)604 2010 y Fr(\(0\))601 2049 y(\003)652 2035 y Fs(\()p Fj(\026)p Fs(\))14 b(:=)f Fi(\000)h Fj(<)f(\026)931 2010 y Fr(\(0\))931 2049 y(\003)987 2035 y Fs(log)c Fj(\026)1087 2010 y Fr(\(0\))1087 2049 y(\003)1149 2035 y Fj(>)1187 2015 y Fr(\(0\))1234 2035 y Fj(;)506 b Fs(\(3)p Fj(:)p Fs(5\))-35 2163 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Fs(,)j(suc)o(h)g(that)473 1436 y Fj(<)e(F)564 1415 y Fr(\()p Fk(n;\025)p Fr(\))658 1436 y Fj(>)696 1443 y Fk(n;\025)750 1436 y Fs(=)g(0)i(and)h Fi(k)p Fj(F)1001 1415 y Fr(\()p Fk(n;\025)p Fr(\))1093 1436 y Fi(\000)11 b Fj(F)c Fi(k)1207 1443 y Fg(1)1257 1436 y Fi(\024)14 b Fj(":)383 b Fs(\(3)p Fj(:)p Fs(20\))-35 1534 y(In)16 b(fact,)f(for)i(eac)o(h)f Fj(t)d Fi(2)h Fs(\012)433 1541 y Fk(T)454 1545 y Fe(n)493 1534 y Fs(\(w)o(e)i(regard)h Fj(T)765 1541 y Fk(n)804 1534 y Fs(as)g(a)g(subset)f(of)h Fl(Z)1146 1516 y Fk(\027)1181 1534 y Fs(if)f(necessary\))g(let)f(us)i (de\014ne)-35 1660 y Fj(\015)-7 1634 y Fr(\()p Fk(n;\025)p Fr(\))-10 1673 y(\003)74 1660 y Fj(f)5 b Fs(\()p Fj(t)p Fs(\))14 b(:=)318 1626 y(1)p 243 1648 174 2 v 243 1705 a Fj(Z)280 1679 y Fr(\()p Fk(n;\025)p Fr(\))276 1718 y(\003)361 1705 y Fs(\()p Fj(t)p Fs(\))430 1592 y Fh(Z)488 1660 y Fj(ds)536 1639 y Fg(0)536 1672 y Fr(\003)571 1660 y Fs(exp)654 1619 y Fh(\002)675 1660 y Fi(\000)714 1667 y Fk(P)735 1660 y Fj(H)779 1639 y 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1953 y Fr(\()p Fk(n;\025)p Fr(\))611 1974 y Fs(\()p Fj(t)p Fs(\))d(:=)h Fj(f)5 b Fs(\()p Fj(t)p Fs(\))11 b Fi(\000)g Fj(\015)920 1948 y Fr(\()p Fk(n;\025)p Fr(\))917 1987 y(\003)1001 1974 y Fj(f)5 b Fs(\()p Fj(t)p Fs(\))p Fj(;)56 b(t)14 b Fi(2)g Fs(\012)1270 1981 y Fk(T)1291 1985 y Fe(n)1314 1974 y Fj(:)402 b Fs(\(3)p Fj(:)p Fs(22\))-35 2090 y(It)15 b(is)h(not)h(hard)g(to)g(sho)o(w)f(that)726 2165 y Fj(<)e(F)817 2144 y Fr(\()p Fk(n;\025)p Fr(\))911 2165 y Fj(>)949 2172 y Fk(n;\025)1003 2165 y Fs(=)g(0)p Fj(:)637 b Fs(\(3)p Fj(:)p Fs(23\))-35 2263 y(In)13 b(order)h(to)g(sho) o(w)g(that)g Fi(k)p Fj(F)490 2245 y Fr(\()p Fk(n;\025)p Fr(\))576 2263 y Fi(\000)6 b Fj(F)h Fi(k)685 2270 y Fg(1)735 2263 y Fi(\024)14 b Fj(")p Fs(,)f(without)h(loss,)g(w)o(e)g(ma)o(y)e (assume)h Fj(f)19 b Fi(\025)13 b Fs(0)h(since)f Fj(\015)1735 2270 y Fr(\003)1776 2263 y Fs(and)-35 2338 y Fj(\015)-7 2312 y Fr(\()p Fk(n;\025)p Fr(\))-10 2351 y(\003)92 2338 y Fs(are)18 b(p)q(ositivit)o(y)f(preserving,)h(i.e.,)e Fj(\015)764 2345 y Fr(\003)791 2338 y Fj(f)23 b Fi(\025)16 b Fs(0)j(and)g Fj(\015)1061 2312 y Fr(\()p Fk(n;\025)p Fr(\))1058 2351 y(\003)1159 2338 y Fi(\025)e Fs(0)i(whenev)o(er)d Fj(f)23 b Fi(\025)17 b Fs(0.)27 b(W)l(e)18 b(notice)-35 2413 y(that)e(since)g(the)g(in)o(teraction)f(\010)h(is)h(of)f(\014nite) g(range,)g(if)g Fj(\015)h Fs(=)c(0)k(then,)592 2529 y Fj(\015)620 2503 y Fr(\()p Fk(n;)p Fr(0\))617 2542 y(\003)698 2529 y Fj(f)5 b Fs(\()p Fj(t)p Fs(\))14 b(=)g Fj(\015)874 2536 y Fr(\003)900 2529 y Fj(f)5 b Fs(\()p Fj(t)p Fs(\))p Fj(;)57 b(t)13 b Fi(2)i Fs(\012)1170 2536 y Fk(T)1191 2540 y Fe(n)1214 2529 y Fj(:)502 b Fs(\(3)p Fj(:)p Fs(24\))-35 2645 y(On)16 b(the)g(other)g(hand,)h(since)e Fj(F)23 b Fs(is)16 b(b)q(ounded,)h(it)e(is)h(easy)h(to)f(see)g(that)258 2762 y Fj(e)281 2741 y Fg(\000)p Fr(2)p Fg(j)p Fk(\025)p Fg(j)t(j)p Fr(\003)p Fg(j)6 b(j)433 2733 y Fa(e)431 2741 y Fr(\003)q Fg(j)f(k)p Fk(F)g Fg(k)534 2745 y Fb(1)568 2762 y Fj(\015)596 2736 y Fr(\()p Fk(n;)p Fr(0\))593 2775 y(\003)675 2762 y Fj(f)g Fs(\()p Fj(t)p Fs(\))13 b Fi(\024)h Fj(\015)854 2736 y Fr(\()p Fk(n;\025)p Fr(\))851 2775 y(\003)935 2762 y Fj(f)5 b Fs(\()p Fj(t)p Fs(\))14 b Fi(\024)g Fj(e)1110 2741 y Fr(2)p Fg(j)p Fk(\025)p Fg(j)t(j)p Fr(\003)p Fg(j)6 b(j)1235 2733 y Fa(e)1233 2741 y Fr(\003)p Fg(j)g(k)p Fk(F)f Fg(k)1336 2745 y Fb(1)1370 2762 y Fj(\015)1398 2736 y Fr(\()p Fk(n;)p Fr(0\))1395 2775 y(\003)1476 2762 y Fj(f)g Fs(\()p Fj(t)p Fs(\))169 b(\(3)p Fj(:)p Fs(25\))-35 2878 y(F)l(rom)12 b(\(3.24\)-\(3.25\))j(w)o (e)e(see)g(that)g(there)g(exists)g Fj(\025)898 2885 y Fr(0)932 2878 y Fj(>)h Fs(0)f(suc)o(h)h(that)f Fi(k)p Fj(F)1295 2860 y Fr(\()p Fk(n;\025)p Fr(\))1381 2878 y Fi(\000)5 b Fj(F)i Fi(k)1489 2885 y Fg(1)1539 2878 y Fi(\024)14 b Fj(")f Fs(if)g Fi(j)p Fj(\025)p Fi(j)h(\024)g Fj(\025)1821 2885 y Fr(0)1841 2878 y Fs(.)-35 2953 y(Th)o(us,)i(our)g (assertion)h(\(3.20\))g(is)f(pro)o(v)o(ed.)p eop %%Page: 16 16 16 15 bop -35 14 a Fs(16)1649 b Fn(Y)l(o)q(o)17 b Fm(et)h(al)38 168 y Fs(No)o(w,)k(supp)q(ose)g(that)f(the)g(equalit)o(y)f(\(2.18\))i (holds.)36 b(Recall)20 b(the)h(notion)h(of)f(the)g(pressure)g(for)-35 243 y(general)16 b(b.c.)k(in)c(\(2.13\).)22 b(W)l(e)16 b(see)g(that)263 350 y Fj(d)p 248 372 54 2 v 248 417 a(d\025)315 390 y Fk(P)337 383 y Fj(P)375 363 y Fr(\010+)p Fk(\025)p Fr(\011)368 397 y Fk(T)389 401 y Fe(n)519 383 y Fs(=)42 b Fj(<)13 b Fs(exp[)p Fi(\000)778 390 y Fk(P)798 383 y Fj(H)842 363 y Fr(\010+)p Fk(\025)p Fr(\011)838 397 y Fk(T)859 401 y Fe(n)946 383 y Fs(])g Fj(>)1011 363 y Fg(\000)p Fr(1)1011 395 y(0)1058 383 y Fj(<)h Fi(\000)p Fj(F)g Fs(exp[)p Fi(\000)1323 390 y Fk(P)1343 383 y Fj(H)1387 363 y Fr(\010+)p Fk(\025)p Fr(\011)1383 397 y Fk(T)1404 401 y Fe(n)1490 383 y Fs(])g Fj(>)1556 390 y Fr(0)519 479 y Fs(=)42 b Fj(<)13 b Fi(\000)p Fj(F)20 b(>)779 486 y Fk(n;\025)847 479 y Fj(:)1730 418 y Fs(\(3)p Fj(:)p Fs(26\))-35 594 y(Th)o(us,)c(b)o(y)f(using)i(\(3.20\))g(w)o(e)f(get)572 735 y Fi(j)605 701 y Fj(d)p 591 723 V 591 769 a(d\025)658 742 y Fk(P)679 735 y Fj(P)717 714 y Fr(\010+)p Fk(\025)p Fr(\011)710 748 y Fk(T)731 752 y Fe(n)820 735 y Fi(j)e(\024)f Fj(")65 b Fs(for)17 b Fi(j)p Fj(\025)p Fi(j)d(\024)f Fj(\025)1213 742 y Fr(0)1233 735 y Fj(:)483 b Fs(\(3)p Fj(:)p Fs(27\))-35 865 y(This)16 b(implies)e(that)511 952 y Fk(P)532 945 y Fj(P)570 925 y Fr(\010+)p Fk(\025)p Fr(\011)563 959 y Fk(T)584 963 y Fe(n)687 945 y Fi(\024)748 952 y Fk(P)769 945 y Fj(P)807 925 y Fr(\010)800 958 y Fk(T)821 962 y Fe(n)856 945 y Fs(+)d Fj(")p Fi(j)p Fj(\025)p Fi(j)65 b Fs(for)16 b Fi(j)p Fj(\025)p Fi(j)e(\024)g Fj(\025)1274 952 y Fr(0)1294 945 y Fj(:)422 b Fs(\(3)p Fj(:)p Fs(28\))-35 1054 y(By)15 b(letting)h Fj(n)g Fs(go)h(to)g (in\014nit)o(y)e(and)h(using)h(Theorem)e(2.5)i(w)o(e)e(then)h(ha)o(v)o (e)541 1184 y Fj(P)579 1164 y Fr(\010+)p Fk(\025)p Fr(\011)696 1184 y Fi(\024)e Fj(P)787 1164 y Fr(\010)826 1184 y Fs(+)d Fj(")p Fi(j)p Fj(\025)p Fi(j)65 b Fs(for)16 b Fi(j)p Fj(\025)p Fi(j)e(\024)g Fj(\025)1244 1191 y Fr(0)1264 1184 y Fj(:)452 b 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y(with)g(some)f(lemm)o(as)f(that)j(will)e(b)q(e)h(needed.)-35 2679 y Ft(Lemm)o(a)g(4.1.)24 b Fm(\(Jensen)-5 b('s)22 b(ine)n(quality\))f(L)n(et)g Fj(\032)g Fm(b)n(e)h(any)e(state)i(on)f Fi(A)p Fm(.)33 b(Then)22 b(for)e(any)h(self-adjoint)-35 2754 y Fj(B)16 b Fi(2)e(A)p Fm(,)j Fj(e)160 2736 y Fk(\032)p Fr(\()p Fk(B)r Fr(\))249 2754 y Fi(\024)d Fj(\032)p Fs(\()p Fj(e)369 2736 y Fk(B)399 2754 y Fs(\))p Fm(.)-35 2878 y Fs(The)i(pro)q(of)h(of)g(the)f(ab)q(o)o(v)o(e)g(lemm)o(a)e(can)j(b)q (e)f(found,)g(e.g.,)f(in)h([Is,)f(Lemma)f(I.3.1].)38 2953 y(Recall)h(the)h(de\014nition)g(of)g(en)o(trop)o(y)g(for)g(states) h(in)f(\(2.35\).)p eop %%Page: 17 17 17 16 bop -35 14 a Fn(Statistical)15 b(mec)o(hanics)f(for)j(un)o(b)q (ounded)g(spin)f(systems)783 b Fs(17)-35 168 y Ft(Lemm)o(a)16 b(4.2.)24 b Fm(Supp)n(ose)e(that)h Fj(\032)g Fm(is)f(a)h(state)g(on)g Fi(A)g Fm(such)g(that)f(the)i(r)n(estriction)e Fj(\032)1555 175 y Fr(\003)1604 168 y Fm(of)h Fj(\032)g Fm(to)f(any)-35 243 y Fi(A)5 250 y Fr(\003)31 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Fd(1)740 2961 y Fg(\012H)797 2966 y Fd(2)814 2961 y Fg(jH)854 2966 y Fd(1)872 2961 y Fr(\))887 2953 y Fs(\()p Fj(A)p Fs(\)\))14 b(=)f(T)l(r)1097 2960 y Fg(H)1127 2965 y Fd(1)1144 2960 y Fg(\012H)1201 2965 y Fd(2)1221 2953 y Fs(\()p Fj(A)p Fs(\))p Fj(:)p eop %%Page: 18 18 18 17 bop -35 14 a Fs(18)1649 b Fn(Y)l(o)q(o)17 b Fm(et)h(al)-35 168 y Fs(When)e(\003)e Fi(\032)f Fs(\003)241 150 y Fg(0)253 168 y Fs(,)i(w)o(e)h(simply)e(write)i(T)l(r)685 176 y Fr(\(\003)723 166 y Fb(0)734 176 y Fg(j)p Fr(\003\))784 168 y Fs(\()p Fj(A)p Fs(\))g(for)h(T)l(r)1000 176 y Fr(\()p Fg(H)1044 185 y Fd(\003)1065 178 y Fb(0)1078 176 y Fg(jH)1118 182 y Fd(\003)1140 176 y Fr(\))1156 168 y Fs(\()p Fj(A)p Fs(\).)k(One)16 b(notes)g(that)624 289 y Fj(!)656 268 y Fr(\(\003\))724 289 y Fs(=)e(w-)21 b(lim)836 319 y Fk(n)p Fg(!1)936 289 y Fs(T)l(r)986 297 y Fr(\(\003)1024 287 y Fb(0)1035 297 y Fg(j)p Fr(\003\))1085 289 y Fj(K)1130 268 y Fr(\()p Fk(n)p Fr(\))1181 289 y Fj(;)559 b Fs(\(4)p Fj(:)p Fs(3\))-35 410 y(where)15 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(established)h(in)h([PY,)f(Prop)q(osition)i(2.5],)f(there)f(exist)h Fj(A)1521 2657 y Fg(\003)1554 2675 y Fj(>)e Fs(0)i(and)h Fj(\016)e(>)f Fs(0)-35 2750 y(suc)o(h)i(that)g(for)h(an)o(y)f(\001)d Fi(\032)h Fs(\003)488 2757 y Fk(n)511 2750 y Fs(,)403 2879 y(1)p 375 2901 82 2 v 375 2946 a Fj(Z)408 2953 y Fr(\003)432 2957 y Fe(n)469 2844 y Fh(Z)527 2912 y Fj(d\025)p Fs(\()p Fj(s)622 2920 y Fr(\003)646 2924 y Fe(n)669 2920 y Fg(n)p Fr(\001)718 2912 y Fs(\))p Fj(e)760 2892 y Fg(\000)p Fk(V)7 b Fr(\()p Fk(s)845 2898 y Fd(\003)866 2902 y Fe(n)889 2892 y Fr(\))919 2912 y Fi(\024)14 b Fs(exp)1055 2872 y Fh(\002)1084 2865 y(X)1087 2971 y Fk(i)p Fg(2)p Fr(\001)1156 2912 y Fs(\()p Fi(\000)p Fj(A)1251 2892 y Fg(\003)1270 2912 y Fj(s)1293 2892 y Fr(2)1293 2925 y Fk(i)1323 2912 y Fs(+)d Fj(\016)r Fs(\))1415 2872 y Fh(\003)1435 2912 y Fj(:)293 b Fs(\()p Fj(A:)p Fs(4\))p eop %%Page: 20 20 20 19 bop -35 14 a Fs(20)1649 b Fn(Y)l(o)q(o)17 b Fm(et)h(al)-35 168 y Fs(Then,)d(b)o(y)h(the)g(regularit)o(y)f(of)i(the)f(in)o (teraction)f(giv)o(en)h(in)f(Assumption)h(2.1)g(\(d\))g(w)o(e)g(see)g (that)260 266 y Fh(\014)260 296 y(\014)297 275 y Fj(d)p 281 297 57 2 v 281 343 a(d\013)351 308 y Fs(log)10 b Fj(Z)456 315 y Fr(\003)480 319 y Fe(n)504 308 y Fs(\()p Fj(\013)p Fs(\))573 266 y Fh(\014)573 296 y(\014)590 328 y Fk(\013)p Fr(=1)660 266 y Fh(\014)660 296 y(\014)260 430 y Fs(=)312 387 y Fh(\014)312 417 y(\014)362 396 y Fs(1)p 333 418 82 2 v 333 464 a Fj(Z)366 471 y Fr(\003)390 475 y Fe(n)428 362 y Fh(Z)486 430 y Fj(d\025)p Fs(\()p Fj(s)581 437 y Fr(\003)605 441 y Fe(n)629 430 y Fs(\))p Fj(e)671 409 y Fg(\000)p Fk(V)e Fr(\()p Fk(s)757 415 y Fd(\003)778 419 y Fe(n)801 409 y Fr(\))817 430 y Fj(W)f Fs(\()p Fj(s)912 437 y Fr(\003)938 430 y Fj(;)h(s)983 437 y Fr(\003)1007 441 y Fe(n)1028 437 y Fg(n)p Fr(\003)1073 430 y Fs(\))1092 387 y Fh(\014)1092 417 y(\014)260 551 y Fi(\024)312 504 y Fh(X)318 609 y Fk(i)p Fg(2)p Fr(\003)420 504 y Fh(X)393 612 y Fk(j)r 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Fr(+)p Fk(s)1138 1389 y Fd(2)1138 1411 y Fe(j)1156 1400 y Fr(\)+2)p Fk(\016)1234 1421 y Fs(\()p Fj(s)1276 1428 y Fk(i)1290 1421 y Fs(\(0\))1352 1401 y Fr(2)1383 1421 y Fs(+)g Fj(s)1455 1428 y Fk(j)1474 1421 y Fs(\(0\))1536 1401 y Fr(2)1556 1421 y Fs(\))p Fj(:)1742 1328 y Fs(\()p Fj(A:)p Fs(5\))-35 1589 y(Notice)k(that)358 1602 y Fh(Z)416 1670 y Fj(d\025)p Fs(\()p Fj(s)511 1677 y Fk(i)526 1670 y Fs(\))p Fj(e)568 1649 y Fg(\000)p Fk(A)621 1638 y Fb(\003)640 1649 y Fk(s)656 1638 y Fd(2)656 1660 y Fe(i)675 1670 y Fj(s)698 1677 y Fk(i)712 1670 y Fs(\(0\))774 1649 y Fr(2)808 1670 y Fs(=)860 1602 y Fh(Z)918 1670 y Fj(dx)971 1677 y Fk(i)993 1670 y Fj(x)1021 1649 y Fr(2)1021 1682 y Fk(i)1049 1602 y Fh(Z)1107 1670 y Fj(P)1138 1677 y Fk(x)1158 1682 y Fe(i)1172 1677 y Fk(;x)1202 1682 y Fe(i)1217 1670 y Fs(\()p Fj(ds)1284 1677 y Fk(i)1298 1670 y Fs(\))p Fj(e)1340 1649 y Fg(\000)p Fk(A)1393 1638 y Fb(\003)1412 1649 y Fk(s)1428 1638 y Fd(2)1428 1660 y Fe(i)1447 1670 y Fj(:)281 b Fs(\()p Fj(A:)p 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2410 y(Y)l(o)q(o)h(w)o(as)h(supp)q(orted)g(b)o(y)f (gran)o(t)g(of)h(P)o(ost-Do)q(c.)22 b(Program)15 b(from)f(Kyungp)q(o)q (ok)j(National)e(Univ)o(ersit)o(y)-35 2485 y(\(1998\).)-35 2678 y Ft([AH-K)23 b Fs(])17 b(S.)f(Alb)q(ev)o(erio)f(and)j(R.)e (H\034egh-Krohn,)h(Homogeneous)g(random)f(\014elds)g(and)i(statistical) 87 2752 y(mec)o(hanics,)13 b Fm(J.)k(F)l(unct.)24 b(A)o(nal.)e Ft(19)p Fs(,)16 b(242-272)j(\(1975\).)-35 2878 y Ft([BR1)k Fs(])17 b(O.)f(Bratteli)g(and)i(D.)e(W.)h(Robinson,)h Fm(Op)n(er)n(ator)f(algebr)n(as)h(and)h(quantum)g(statistic)n(al)g(me-) 87 2953 y(chanics,)f(1)e Fs(\(Springer-V)l(erlag,)f(New)h(Y)l (ork/Heidelb)q(erg/Berlin,)d(1979\).)p eop %%Page: 21 21 21 20 bop -35 14 a Fn(Statistical)15 b(mec)o(hanics)f(for)j(un)o(b)q (ounded)g(spin)f(systems)783 b Fs(21)-35 168 y Ft([BR2)23 b Fs(])17 b(O.)f(Bratteli)g(and)i(D.)e(W.)h(Robinson,)h Fm(Op)n(er)n(ator)f(algebr)n(as)h(and)h(quantum)g(statistic)n(al)g(me-) 87 243 y(chanics,)f(2)e Fs(\(Springer-V)l(erlag,)f(New)h(Y)l 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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--