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 \centerline{\bfb Reversibility in Infinite Hamiltonian Systems}
 \centerline{\bfb with Conservative Noise}
 \vskip.5cm
 \centerline{\bf In Memoriam Roland Dobrushin}
 \vskip1cm
 \centerline{{\rmb J\'ozsef Fritz}$^1$,
  {\rmb Carlangelo Liverani}$^2$ and {\rmb Stefano Olla}$^3$}


\footnote{}{$^1$~Department of Probability and Statistics,
 E\"otv\"os Lor\'and University of Sciences,
 H-1088 Budapest, M\'uzeum krt. 6-8, Hungary.
 E-mail: jofri@cs.elte.hu}
 \footnote{}{$^2$~ II Universit\'a di Roma ``Tor Vergata'',
 Dipartimento di Matematica, 00133 Roma, Italy.
 E-mail: liverani@mat.utovrm.it}
 \footnote{}{$^3$~ Universit\'e de Cergy--Pontoise,  D\'epartement de 
 Math\'ematiques, 2 avenue Adolphe Chauvin, Pontoise 95302 Cergy--Pontoise 
 Cedex, France and Centre de Math\'ematiques Appliqu\'ees, Ecole Polytechnique,
 91128 Palaiseau Cedex, France.\hfill\break
 E-Mail: olla@paris.polytechnique.fr}
 \footnote{}{\bf\noindent Work partially supported by grants CIPA-CT92-4016 and
 CHRX-CT94-0460 of
 the Commission of the European Community, and by grant T 16665 of
 the Hungarian NSF. Two of us (J.F. and C.L.) acknowledge hospitality of Ervin
 Schr\"odinger Institute.}

 \vskip.5cm
 \noindent{\bf Abstract:} {\sl The set of stationary measures of an infinite
 Hamiltonian system with noise is investigated. The model consists of
 particles moving in $\RR^3$ with bounded velocities and subject to a
 noise that does not
 violate the classical laws of conservation,  see  [OVY]. Following
 [LO] we assume that the noise has also a finite radius of interaction,
 and prove that translation invariant stationary states of finite
 specific entropy are reversible with respect to the stochastic
 component of the evolution. Therefore the results of { [LO]} imply
 that such invariant measures are superpositions of Gibbs states.}

 \vskip.5cm
 {\bf 0. INTRODUCTION}
 \vskip .25cm
 \numsec=0\numfor=1\numtheo=1

 Let $\O$ denote the space of locally finite
 configurations $\o=(q_\a,p_a)_{\a\in I}$ indexed by a countable set
 $I\,,$ that is $q_\a,p_\a\in\RR^3$ are the position and momentum
 of particle $\a\in I\,;$ the set $\{q_\a\}_{\a\in I}$ has no limit
 points in $\RR^3$ by assumption. The classical
 dynamics of the system is governed by a formal Hamiltonian $\Cal H\,,$
 $$
 {\cal H}(\o)=\sum_{\a\in I}\phi(p_\a)+{1\over2}
 \sum_{\a\in I}\sum_{\b\ne\a}V(q_\a-q_\b)\,,
 $$
 where the kinetic energy $\phi:\RR^3\mapsto\RR$ is strictly convex with
 bounded derivatives, and $V:\RR^3\mapsto\RR$ is a symmetric and superstable
 pair potential of finite range.  The associated Liouville operator will
 be denoted by $L\,,$
 $$
 L\p=\sum_{\a\in I}\quad\bigl\lan{\dd\Cal H\over\dd p_\a},{\dd\p\over\dd
 q_\a}\bigr\ran
 -\bigl\lan{\dd\Cal H\over\dd q_\a},{\dd\p\over\dd p_\a}\bigr\ran\,,
 $$
 where $\lan\cdot,\cdot\ran$ denotes the usual scalar product in
 $\RR^3$. Almost nothing is known on the ergodic properties of such infinite
 systems. In fact, very few results are available even for finite systems
 of this type (e.g., [KSS], [DL], [BLPS], [LW]). To ensure a proper ergodic
 behavior of the system we
 add some noise whereby obtaining stochastic equations of motion; these
 equations read as
 $$
 \eqalign{
 dq_\a&=\phi'(p_\a)\,dt\,,\cr
 dp_\a&=-\sum_{\b\neq\a}V'(q_\a-q_\b)\,dt
 +b_\a(\o)\,dt+\sum_{\t=1}^d\sum_{\b\neq\a}
 \sigma^\t_{\a,\b}(\o)\,dw^\t_{\a,\b} ,}
 \Eq(stochdyn)
 $$
 where $w^\t_{\a,\b}$ is a family of independent one-dimensional Wiener
 processes for $\t=1,2,...,d$ and $\a\ne\b$ such that
 $w^\t_{\a,\b}=-w^\t_{\b,\a}\,;$ $\phi'$ and $V'$ denote the gradient of
 $\phi$ and $V\,,$
 respectively. The coefficients $b_\a,\sigma^\t_{\a,\b}:\O\mapsto\RR^3$
 are smooth local functions to be specified in the next section in such
 a way that total energy and momentum are both preserved by the
 randomized evolution (0.1). In addition, any Gibbs state $\PP$ 
 with energy $\Cal H$ will be a reversible measure for the stochastic 
 part of the evolution:
 $$
 \int \p(\o)\wh L\psi(\o)\,\PP(d\o)=\int\psi(\o)\wh L\p(\o)\,\PP(d\o)
 \Eq(reversibility)$$
 for all smooth local functions $\p,\psi:\O\mapsto\RR\,,$ where
 $$
 \wh L\psi=\sum_{\a\in I} \bigl\lan b_\a,{\dd\psi\over\dd p_\a}\bigr\ran
 +{1\over4}\sum_{\t=1}^d\sum_{\a\in I}\sum_{\b\ne\a}
 \bigl\lan\sigma^\t_{\a,\b},(D^2_{\a,\b}\psi)\sigma^\t_{\a,\b}\bigr\ran\,,
 \Eq(stochgen)$$
 and  $D^2_{\a,\b}\psi$ is the matrix of second derivatives obtained
 by applying $D_{\a,\b}=\dd/\dd_{p_\a}-\dd/\dd_{p_\b}$ twice to
 $\psi\,.$ Since the Liouville operator is antisymmetric with respect to
 Gibbs distributions, the full generator, $\wt L=L+\wh L$ also
 satisfies the stationary Kolmogorov equation, $\smallint \wt
 L\psi\,d\PP=0$ for a wide class of test functions $\psi$ and any Gibbs
 state $\PP\,.$

 The converse problem is much more complex. In our basic reference [LO]
 it is shown that if a translation invariant measure $Q$ with finite
 specific entropy satisfies the stationary Kolmogorov equation and (0.2),
 together with some other technical conditions, then $Q$ enjoys
 the Gibbs property.  Let us remark that finiteness of
 specific entropy is a fairly natural and effective condition in the
 theory of hydrodynamics limits (see [OVY]). On the contrary, 
 condition (0.2) 
 looks rather restrictive and, at least in general, not particularly 
 natural. The main purpose of this paper
 is to show that condition (0.2) of reversibility is superfluous (i.e., it 
 follows from the stationarity of the measure).\nfootnote{Note that in 
 applications to hydrodynamics the reversibility (0.2) is insured by 
 construction (see [OVY] lemma 4.4), hence the present paper does not add to 
 hydrodynamics type problems for which the results in [LO] suffice. The 
 focus here is on the classification of stationary translation invariant 
 measures.}  To obtain such a result we are forced to prove the 
 existence of a semigroup defined by (0.1);
 its regularity (locality) will play a crucial role in the argument. This
 problem may not seem to be a very difficult one since $\phi'(p_\a)\,,$
 the velocity of particle $\a$, is bounded by assumption. However,
 the evolution must be defined for a very large set $\bar\O\subset\O$ of
 initial configurations: we need
 $Q(\bar\O)=1$ for any probability measure $Q$ of finite specific entropy.
 On the other hand, to obtain the necessary regularity properties of the
 dynamics we have to restrict the configuration space by excluding
 extremely high values of particle density.
 We shall see that the desired construction fails unless the dimension
 of the space is less than four, cf. {[FD]} and {[S]}.

 \vskip .5cm
 {\bf 1. NOTATIONS AND RESULTS}
 \vskip.25cm

 \numsec=1\numfor=1\numtheo=1

 Configurations can be interpreted as $\sigma$-finite integer valued
 measures on $\RR^3\times\RR^3\,;$ sometimes we write $\o=(q,p)$ with
 $q=(q_\a)_{\a\in I}$ and $p=(p_\a)_{\a\in I}\,,$ and if
 $\L\subset\RR^3$ then $\o_\L$ denotes the restriction of $\o$ to
 $\L\,,$ i.e. $\o_\L=(q_\a,p_\a)_{q_a\in\L}\,,$ while $|\o_\L|$ is the
 cardinality of this set.  The centered cubic box of side $r>0$ will be
 denoted by $\Lambda_r\,.$ Referring to functionals
 $\o(\p):=\sum_{\a\in I} \p(q_\a,p_\a)\,,$
 where $\p:\RR^3\times\RR^3\mapsto\RR$ is continuous with compact
 support, we equip $\O$ with the associated weak topology and Borel
 structure, and $C_0(\Omega)$ denotes the space of cylinder (local)
 functions
 $\Psi(\o)=f(\o(\p_1),\o(\p_2),...,\o(\p_n))$ such that $f\in
 C(\RR^n)\,.$

 Since all sets $\Sigma(\delta)$, defined for an increasing sequence
 $\delta=(\delta_1,\delta_2,...)$ such that $\delta_1\ge 1$ by
 $$
 \Sigma(\delta):=\bigl\{\o\in\O\,:\,|\o_{\Lambda_n}|\le\delta_n
 \hbox{ and } |p_\a|\le\delta_n\hbox{ if } q_\a\in\Lambda_n\bigr\}
 $$
 are compact, we need not worry too much
 about topology. Indeed, in the forthcoming considerations we always do
 have an a priori bound allowing us to restrict calculations to some
 compact $\Sigma(\delta)\,.$ In this situations all reasonable
 topologies coincide, moreover any continuous function can be uniformly
 approximated by elements of $C_0(\Omega)$ in view of the
 Stone-Weierstrass Theorem.

 \vskip 5pt 
\noindent{\bf Interaction}
\vskip 2pt

 We consider a repelling pair potential $V$ of finite range such that
 $V(x)=V(-x)$ is twice continuously differentiable, $V(0)>0$ but
 $V(x)=0$ if $|x|>R_0\,,$ finally $\lan x,V'(x)\bigr\ran\le0$ for all
 $x\in\RR^3\,.$ These conditions imply that $V$ is superstable, see [R]:
 for each
 cubic box or ball $\L\subset\RR^3$ there exist some constants $A_\L\ge0$
 and $B_\L>0$ such that, for any configuration, we have
 $$
 \sum_{\a:q_\a\in\Lambda}\sum_{\b\ne\a} V(q_\a-q_\b)
 \ge B_\L |\o_\Lambda|^2- A_\L |\o_\Lambda| .
 \Eq (superstability)
 $$

 \vskip 5pt
 \noindent{\bf Kinetic energy}\vskip 2pt

 We assume that $\phi$ has bounded second derivatives, and velocities
 are also bounded, i.e. $|\phi'(y)|\le \bar c<+\infty$ for all
 $y\in\RR^3\,.$ To define Gibbs measures we need a lower bound:
 $\lim\inf_{|y|\to\infty}\phi(y)/|y|\ge \underline c>0\,.$ When results
 of [LO] are applied and extra technical condition on $\phi$ is needed. 
 For simplicity one can consider the case in which $\phi (y)=\sum_{i=1}^3
 \phi_0(y_i)$ for $y=(y_1,y_2,y_3)\,,$ where  $\phi_0\in
 C^{\infty}(\RR)$ is strictly convex and
 $$
 {1\over 2}{d^2\over du^2}(\phi_0''(u))^2=
 \phi_0'''(u)^2+\phi_0^{iv}(u)\phi''(u)\neq 0
 $$
 apart from, at most, finitely many points (see [LO], section two, 
 ``Condition on the Noise" for the general condition that $\phi$ 
 must satisfy).
 Notice that if ${d^2\over du^2}(\phi''_0(u))^2= 0$ for each $u$ and if
 we require the natural condition $\phi_0(u)=\phi_0(-u)$ then
 $\phi_0''$ is a constant, which is the classical case of a quadratic
 kinetic energy function.

 \vskip 5pt
 \noindent{\bf Stochastic perturbation of classical dynamics}
 \vskip 2pt

 There are several ways to select the coefficients of the stochastic
 perturbation, we set
 $$
 b_\a(\o)=\sum_{\b\ne\a}\gamma_{\a,\b}(q)F(p_\a,p_\b)\hbox{ and }
 \sigma^\t_{\a,\b}(\o)=\sqrt{\gamma_{\a\b}(q)}G_\theta(p_\a,\,p_\b)\,,
 \Eq (coeff)$$
 where, as in { [LO]},\nfootnote{In fact, in [LO], the functions
 $\gamma_{\a\b}$
 depend only on the variables $q_\a$ and $q_\b$; yet all is done
 there  applies
 without changes to the situation described here.}
 $\ga_{\a,\b}(q)=\ga_{\b,\a}(q)\ge 0$ is
 continuously differentiable, $\gamma_{\a,\b}(q)>0$ if
 $|q_\a-q_\b|< R_1$ and it is zero for $|q_\a-q_\b|\ge R_1\,,$ i.e.
 $R_1>R_0$ is the radius of stochastic interaction.  The functions
 $F,G_\t:\RR^6\mapsto \RR^3$  are infinitely
 differentiable and bounded together with their derivatives; they
 are chosen in such a way that the stochastic interaction also
 preserves the total momentum and energy of an interacting
 couple of particles. Moreover, $\{G_\theta\}_{\theta=1}^d$ spans, 
 at each point, all $\RR^3$. It is natural to assume that $\ga_{\a,\b}$
 depends only on the interparticle distances, and it does not depend on a
 coordinate $q_\de$ if $|q_\a-q_\de|>R_2$ or $|q_\b-q_\de|>R_2\,,$ where
 $R_2>2R_1$ is a constant. Therefore the stochastic interaction is also
 translation invariant and has a finite range $R_3:=R_1+R_2\,.$ A new
 feature of the present model is that
 $\gamma_{\a,\b}$ vanishes when the number of particles near $q_\a$ or
 $q_\b$ tends to infinity. For convenience, we set
 $\ga_{\a,\b}(q)=\si(q_\a-q_\b)\Te_{\a,\b}(q)\,,$ where
 $$
 \Te_{\a,\b}(q):=\Bigl(1+\sum_{\de\in I}\chi(q_\a-q_\de)
 +\sum_{\de\in I}\chi(q_\b-q_\de)\Bigr)^{-1} .
 \Eq(gamma)$$
 In (1.3) $\sigma,\chi:\RR^3\mapsto[0,1]$ are twice continuously
 differentiable, $\sigma(x)=\sigma(-x)>0$ if $|x|< R_1$ and it is zero
 for $|x|\ge R_1\,.$ Similarly, $\chi(x)=\chi(-x)>0$ if $|x|\le2R_1$ and
 $\chi(x)=0$ if $|x|>R_2$ with some $R_2>2R_1\,.$ A technical condition,
 $|\chi'(x)|\le K\chi(x)^{1-\ka}\,,$ where $0<\ka<1/9\,,$ will be
 exploited in Lemma 2.4.

 Since $w^\t_{\a,\b}=-w^\t_{\b,\a}\,,$ the condition
 $F(p_\a,p_\b)=-F(p_\b,p_\a)$ of antisymmetry of $F$ clearly implies
 the conservation of total momentum. For convenience, we choose
 $$
 F(p_\a,\,p_\b):={1\over 2}\sum_{\t=1}^d\bigl\lan G_\t(p_\a,\,p_\b),
 D_{\a,\b}\bigr\ran G_\t(p_\a,\,p_\b) \hbox{ and }
 X^\t_{\a,\b}\p:={1\over \sqrt 2}\bigl\lan
 G_\t(p_\a,\,p_\b),D_{\a,\b}\p\bigr\ran\,,
 $$
 then the formal generator $\wh L$ of the random component of our
 process becomes\nfootnote{The future requirements (1.5) and (1.6) imply
 that ${X^\t_{\a,\b}}^*=-X^\t_{\a,\b}$, 
 where the adjoint is taken with respect to the measure defined by the 
 kinetic energy, (see[LO]).}
 $$
 \wh L\p={1\over 2}\sum_{\t=1}^d\sum_{\a\in I}\sum_{\b\ne\a}
 \gamma_{\a,\b}(q) X^\t_{\a,\b}\bigl(X^\t_{\a,\b}\p\bigr).
 \Eq (genoise)
 $$
 In this case the orthogonality relations
 $$
 \bigl\langle G_\t(p_\a,\,p_\b),\phi'(p_\a)
 -\phi'(p_\b)\bigr\rangle=0
 \Eq (orthorel)
 $$
 imply the formal conservation of energy, see {[LO]}. To have
 conservation of phase volume it is also assumed that
 $$
 \bigl\lan D_{\a,\b},G_\t(p_\a,p_\b)\bigr\ran=0\,,
 \Eq (divnull)
 $$
 i.e. the operators $\gamma_{\a,\b}X^\t_{\a,\b}X^\t_{\a,\b}$ are
 symmetric with respect to Lebesgue measure $dp_\a dp_\b\,.$ As a
 consequence we shall see that
 the conservation laws imply the reversibility of Gibbs states with
 respect to $\wh L\,.$ For an explicit example of $F$ and $G_\t$ see
 the Appendix of [LO].

 \vskip 5pt
 \noindent{\bf Gibbs measures}\vskip 2pt

 Let $\l=(\l_0,\l_1,\l_2,\l_3,\l_4)$ be a set of real parameters with
 $\l_4>0$ and $\l^2_1+\l^2_2+\l^2_3<\underline c^2\,,$ and denote by $\Pi$
 the distribution of a Poisson process of unit intensity in
 $\RR^3$ . A probability measure $\PP$ on $\O$ is called
 a Gibbs state for $\Cal H$ with parameters $\l$ if its conditional
 distributions given the configuration outside of any cubic box
 $\L\subset\RR^3$ can be represented as
 $$
 \PP[d\o_\L|\o_{\L^c}]=
 {1\over Z_\L}\exp\left[\lambda_0|\o_\L| +
 \sum_{\a=1}^n\sum_{i=1}^3\lambda_i p_\a^i
 -\lambda_4{\cal H}_{\Lambda}(\omega_\Lambda,\omega_{\Lambda^c})\right]
 \,\Pi(dq_\L)\,dp_\L\,,
 $$
 where $Z_\L$ is the normalization, and a natural decomposition
 $\o_\L=(q_\L,p_\L)$ is used, see [D]. The local Hamiltonian, $\Cal
 H_\L$ is defined as
 $$
 {\cal H}_{\Lambda}(\omega_\Lambda,\omega_{\Lambda^c})=
 \sum_{q_\a\in\omega_\Lambda}\left[\phi(p_\a)+{1\over 2}
 \sum_{q_\b\in\omega_\L;\;\alpha\neq\beta}V(q_\a-q_\b)
 +\sum_{q_\b\in\omega_{\L^c}}V(q_\a-q_\b)\right]\,,
 $$
 the set of such measures will be denoted by $\Cal P_\l\,,$ see {[R]}
 for the existence of Gibbs states for superstable interactions.

 \vskip 5pt
 \noindent{\bf Relative entropy}
 \vskip 2pt

 Let $Q$ and $P$ be probability measures on $\Omega$, and for any
 $\L\subset\RR^3$ denote $\Cal F_\L$ the set of continuous and bounded
 functions $\psi:\O\mapsto\RR$ such that $\psi(\o)=\psi(\o_\L)$ for all
 $\o\in\O\,.$ The entropy of $Q$ in $\L\,,$ relative to $P\,,$ is
 defined by
 $$
 H_\Lambda(Q|P)\ =\ \sup_{\psi\in{\cal F}_\L}\left\{\E^Q(\psi)
 -\log\E^P(e^\psi)\right\}
 \Eq (entropy)
 $$
 where $\E^Q$ denotes the expectation with respect to the probability
 measure $Q\,.$ If $\L=\RR^3$ then the subscript $\L$ of $H_\L$ will be
 omitted, for properties of $H_\L$ see, for example, [OVY]. As a
 reference measure a distinguished, translation invariant, Gibbs state
 $P=\PP$ will be chosen. We say that $Q$ has finite specific entropy if
 there exists a constant $C$ such that $H_\L(Q|\PP)\le C(1+|\L|)$ for any
 cubic box $\L\,.$ If $Q$ is translation invariant with finite specific
 entropy, then the particle density $\rho=\rho(\o)$ is $Q$-a.s. defined
 as the following limit taken along any increasing sequence of cubic
 boxes, see [LO],
 $$
 \rho(\o)=\lim_{|\Lambda|\to\infty}|\Lambda|^{-1}|\omega_\Lambda| .
 $$

 Main results of [LO] for the system under considerations can be
 summarized as follows.

 \proclaim {\Theorem (old)}.
 Suppose that $Q$ is a translation invariant probability measure on
 $\Omega$ with finite specific entropy, and let $\rho_c:=3/(4\pi
 R^3_1)\,.$ If
 \item{(i)} $Q[\rho(\omega)>\rho']\;=\; 1$ for some $\rho'>\rho_c\,,$
 \item{(ii)} $Q$ is invariant with respect to $\wt L=L+\wh L$
 in the sense that, for any smooth local function $\psi$ we have
 $\E^Q(\wt L\psi)=0\,,$
 \item{(iii)} $Q$ is reversible with respect to $\wh L\,,$
 i.e.  $\E^Q(\psi\wh L\p)\ =\ \E^Q(\p\wh L\psi)\;$ for any pair
 $\p,\psi$ of smooth local functions,
 
 \vskip-\medskipamount
 \noindent{\sl then Q is a convex combination of Gibbs states.}
 \medskip


 \vskip 5pt
 \noindent{\bf Statement of the result}
 \vskip 2pt

 Notice that the theorem above is stated without any reference to the
 existence of the infinite dynamics, properties (ii) and (iii)  of
 invariance are purely formal.  However, the extraction of local
 information
 as reversibility is usually based on a method of Liapunov functions,
 namely entropy and its rate of change are compared, so the first step
 of our argument is intrinsically related to the evolution.

 \proclaim {\Theorem (dynamics)}. Under the conditions on the
 stochastic dynamics listed above, there exists an explicitly defined
 set $\bar\O\subset\O$ such that  $Q(\bar\O)=1$ for each $Q$ with
 finite specific entropy. Moreover, for each $\o_0\in\bar\O$ we have
 a unique strong solution $\o(t)\,,\,t\ge0$ to (0.1) such that
 $\o(0)=\o_0$ and $\o(t)\in\bar\O$ a.s. The solution is a measurable
 function of the initial configuration, and every Gibbs state
 $\PP\in\Cal P_\l$ with $\l_4>0$ and $\l_1=\l_2=\l_3=0$ is a stationary
 measure for the random evolution.

 This theorem is proven in the next section, solutions are defined by a
 limiting procedure starting  from finite systems. The restriction on
 the parameters of a Gibbs measure in the last statement could have been
 removed by elaborating some technical details, but we do not need such a
 general assertion. 

Having constructed the infinite evolution we can 
consider stationary measures instead of simply measures formally invariant 
as in theorem 1.1 (ii).\nfootnote{Notice that, since the infinite dynamics 
satisfies equations (0.1), if $Q$ is stationary, then it satisfies (ii) of 
theorem 1.1.}

 \proclaim {\Theorem (reversible)}. Every translation invariant
 stationary measure with finite specific entropy is reversible with
 respect to the stochastic part $\wh L$ of the generator; that is, 
 condition (iii) in Theorem 1.1 holds.

 The proof of Theorem \equ (reversible) is the content of Section 3.
 Combining the above results we get the final result of the paper:

 \proclaim {\Theorem (main)}. Let $Q$ be a translation invariant
 stationary
 measure with finite specific entropy, then condition (i) of Theorem 1.1
 implies that $Q$ is a superposition of Gibbs states.

 \vskip .5cm
 {\bf 2. INFINITE DYNAMICS}
 \vskip.25cm

 \numsec=2\numfor=1\numtheo=1

 We start this section by describing the set of  allowed
 initial configurations. Although the definition is a bit technical our 
choice boils down to configurations for which the energy in a box does not 
grow too fast with respect to the size of the box. The exact meaning 
of this construction will become
 more clear later on when the desired a priori bounds for a family of
 partial dynamics and the requirements for the existence of a unique
 limiting dynamics are discussed.

 \vskip 5pt
 \noindent{\bf Initial conditions}\vskip 2pt

 Let $\Cal H_m(\o,r)$ denote the total energy of $\o\in\Omega$ in a ball
 $B_m(r)\subset
 \RR^3$ of center $m$ and radius $r\le\infty$, i.e. $\Cal H_m(\o,r)
 \equiv \Cal H(\o_{B_m(r)})$; the number of points of $q$ in
 $B_{q_\a}(r)$ will be denoted as $N_\a(q,r)\,.$ For $\kappa\in
 (0,1/9)\,,$ see \equ(gamma), and $r\ge R_3 = R_1+R_2$, define
 $$
 \bar \Cal H_{\kappa,r}(\o) := \sup_{|m|\le r} {\Cal H_m(\o,R_3)\over
 1+|m|^{3+2\kappa}}\,,\qquad
 \bar\Omega_{\kappa,r} (h) = \{\o\in\Omega \;:\; \ \bar \Cal
 H_{\kappa,r}(\o)\le h \}.
 $$
 Let $\Cal Q_r(k)$ be the set of Borel probability measures on
 $\bar\Omega_{\kappa,r} (h)$ such that $Q(\Cal H_m(\o,R_3)) \le k$
 for all $|m|\le r$.

 Now the set of all allowed configurations is defined as
 $$
 \bar\Omega_{\kappa,\infty}  = \bigcup_{h>0}
 \bar\Omega_{\kappa,\infty}(h)
 = \{ \o\in\Omega\;:\;\bar\Cal H_{\kappa,\infty}(\o) <\infty \}.
 $$
 Remember that the level sets $\bar\Omega_{\kappa,\infty} (h)$ are
 compact in the weak topology of $\Omega$ and, in view of
 \equ(superstability), $N_\a(q,R_3)=O(\sqrt h L^{{3/2}+\ka})$ for
 $q_\a\in B_m(L-R_3)$ and $(q,p)\in\bar\O_{\ka,L}(h)\,.$ We shall
 see that the initial condition for the existence and uniqueness of the
 limiting dynamics could have been formulated in terms of $N_\a$ only,
 but a preservation of bounds on kinetic energy will be needed when we
 prove locality of the dynamics.

 \proclaim{\Lemma(initial)}. If $\kappa>0$ then for any fixed $k>0$ we
 have
 $$
 \lim_{h\to\infty} \inf_{r\ge R_3} \inf_{Q\in \Cal Q_r(k)}
 Q(\bar\Omega_{\kappa,r} (h) ) = 1 ,
 $$
 that is $Q(\bar\O_{\ka,\infty}) =1$ if $Q\in \Cal Q_\infty:=
 \bigcup_{k>0}\Cal Q_\infty(k)$.

 \smallskip\noi{\bf Proof:}
 This statement is a direct consequence of the Markov inequality. In
 fact we have some universal $v>0$ such that (by $v \ZZ^3$ we denote the 
 tridimensional cubic lattice of size $v$)
 $$
 \eqalign{
 Q(\bar\O_{\ka,r}(h)^c)&\le \sum_{m\in B_0(r)\cap v\Z^3}
 Q[\Cal H_m(\o,\,R_3)>vh(1+|m|^{3+2\ka})\cr
 &\le \sum_{m\in B_0(r)\cap v\Z^3}{Q(\Cal H_m)\over h(1+|m|^{3+2\ka})}\le
 {k\over vh}\sum_{m\in v\Z^3}{1\over (1+|m|^{3+2\ka})}
 }
 $$
 which proves the statement for any $\kappa>0\,.$ $\qed$
 \smallskip

 Observe that the entropy condition $H_\L(Q|\PP)\le C (1+|\L|)$
 implies $Q\in \Cal Q_\infty$ via (1.7) and (1.1), see Lemma 3.1 of
 [LO].

 \vskip 5pt
 \noindent{\bf Local dynamics}\vskip 2pt

 There are several ways to define a family of partial dynamics, the
 advantage of the following construction consists in its direct relation
 to Gibbs states. Let $a:\RR^3\mapsto[0,1]$ be twice continuously
 differentiable with compact support, we assume also $|a'(x)|\le 1$
 for all $x\in\RR^3\,.$ We interpret $a$ as a smooth
 version of the indicator function of a ball, its concrete shape is not
 very important. For every such cutoff $a$ and inverse temperature
 $\l_4>0$ we consider a system of stochastic differential equations,
 $$
 \eqalign{
 dq_\a=&-{1\over\l_4}e^{\l_4{\cal H}_\L(\o_\L,\o_{\L^c})}
 {\partial\over\partial p_\a}\left(a(q_\a)e^{-\l_4
 {\cal H}_\L(\o_\L,\o_{\L^c})}\right) dt\cr
 dp_\a=& {1\over\lambda_4} e^{\lambda_4
 {\cal H}_\L(\o_\L,\o_{\L^c})}{\partial\over\partial q_\a}\left(a(q_\a)
 e^{-\lambda_4 {\cal H}_\L(\o_\L,\o_{\L^c})}\right)\,dt\cr
 &+a(q_\a)\sum_{\b\ne\a}\gamma_{\a,\b}(q)a(q_\b)F(p_\a,p_\b)\,dt\cr
 &+\sqrt{a(q_\a)}\sum_{\t=1}^d\sum_{\b\ne\a}
 \sqrt{a(q_\b)\gamma_{\a,\b}(q)}G_\t(p_\a,p_\b)\,dw_{\a,\b}^\t\,,
 }
 \Eq(dynloc)$$
 where it is assumed that $\L\subset\RR^3$ is bounded and contains the
 support of $a$ in its interior; in such a situation the equations above
 do not depend on the particular choice of $\L\,.$ Notice that in a
 region where $a=1$ our particles follow the original equations of
 motion, while they are frozen outside of the support of $a\,,$ i.e.
 $\dot q_\a=\dot p_\a=0\,.$ Particles approaching the boundary of the
 support of $a$ slow down, thus we have a smooth transition between
 moving and frozen particles, see [F1] for a similar construction. This
 means that we essentially have a finite dimensional diffusion, let
 $P^t_{\lambda_4,a}$ denote the Markov semigroup induced by  partial
 dynamics (2.1), i.e. $P^t_{\l_4,a}\psi(\o):=\E_w(\psi(\o(t))\,,$ where
 $\o(t)$ is the solution with initial condition $\o(0)=\o\,,$
 $\psi:\O\mapsto\RR$ is continuous and bounded, while $\E_w$
 denotes the expectation with respect to the joint distribution of our
 Wiener processes.

 By a direct application of the Ito lemma we see
 that the (formal) generator of $P^t_{\l_4,a}$ decomposes as
 $\wt L_{\l_4,a}=L_{\l_4,a}+\wh L_{a}\,,$ where
 $$\eqalign{
 L_{\l_4,a}\psi&=-{1\over\l_4}\sum_{\a\in I}
 e^{\l_4{\cal H}_\L(\o_\L,\o_{\L^c})}{\partial\over\partial p_\a}
 \left(a(q_\a)e^{-\l_4 {\cal H}_\L(\o_\L,\o_{\Lambda^c})}\right)
 {\partial\psi\over\partial q_\a}\cr
 &+{1\over\l_4}\sum_{\a\in I} e^{\lambda_4{\cal H}_\L
 (\o_\L,\o_{\L^c})}{\partial\over\partial q_\a}
 \left(a(q_\a)e^{-\l_4 {\cal H}_\L(\o_\L,\o_{\L^c})}\right)
 {\partial\psi\over\partial p_\a} ,\cr
 \wh L_{a}\psi &={1\over 2}\sum_{\t=1}^d\sum_{\a\in I}\sum_{\b\ne\a}
 \gamma_{\a,\b}(q)a(q_\a)a(q_\b)X^\t_{\a,\b}(X^\t_{\a,\b}\psi).
 }
 \Eq(pargen)$$
 Since the coefficients of \equ(dynloc) are bounded smooth functions,
 we have a differentiable dependence of solutions on initial
 values. Therefore a class $\Cal D_a$ of twice continuously
 differentiable functions forms a common core of $L_{\l_4,a}$ and
 $\wh L_a\,,$ e.g. in the space
 of continuous and bounded functions. An extension to the
 space $L^2(\PP_\l)$ of square integrable functions with respect to a
 distinguished Gibbs state $\PP_\l$ follows by  the next Lemma.

 \proclaim{\Lemma(pargibbs)}. Let $\l_1=\l_2=\l_3=0$ while $\l_4>0\,,$
 then every Gibbs state $\PP\in\Cal P_\l$ satisfies
 $$
 \E^\PP(\psi_1 L_{\l_4,a}\psi_2)=-\E^\PP(\psi_2 L_{\l_4,a}\psi_1)\hbox{ and }
 \E^\PP(\psi_1 \wh L_{a}\psi_2)=\E^\PP(\psi_2 \wh L_{a}\psi_1)
 $$
 for $\psi_1,\psi_2\in\Cal D_a\,,$ consequently $\PP$ is a stationary
 measure of the process $P_{\l_4,a}^t$ for each cutoff $a$.

 \smallskip\noi{\bf Proof:}
 Both symmetry relations follow from the definition of $\PP$ by
 integrating by parts. The property of reversibility
 is a direct consequence of (1.6). Integration
 by parts with respect to positions is possible because of the presence
 of the cutoff $a\,.$ $\qed$\smallskip

 Since (2.1) violates the law of momentum conservation in regions where
 $a$ is not a constant, Lemma 2.2 is not true for general Gibbs
 measures.

 \vskip 5pt
 \noindent{\bf Construction of the infinite dynamics}\vskip 2pt

 First we derive an a priori bound for local dynamics; we show that the
 initial condition is preserved for all $t>0$ and the related bound does
 not depend on the particular choice of the cutoff function $a\,.$

 \proclaim{\Lemma(apriori)}. There exists a constant $c_1$ depending
 only on $\l_4$ and on the parameters of the infinite system (0.1) such
 that
 $$
 \E_w\left(\Cal H_m(\o_a(t),R_3)\right)\le (c_1+c_1t)(1+\Cal H_m(\o_a(0),R_3+\bar ct)
 $$
 for all $m,t$ and $a\,,$ where $\o_a(t)$ is any solution to (2.1).

 \smallskip\noi{\bf Proof:} Since all velocities are bounded by
 $\bar c\,,$ we have $|q_\a(t)-q_\a(0)|\le \bar ct\,,$ whence
 $N_\a(q(t),r)\le N_\a(q(0),r+\bar ct)\,,$ which yields an explicit
 deterministic bound for the potential energy via superstability (1.1).
 On the other hand,
 $$
 |b_\a(\o)|+\sum_{\t=1}^d\sum_{\b\ne\a}|\sigma^\t_{\a,\b}(\o)|^2
 \le c_1' N_{\a}(q,R_1)\,,
 $$
 and the same bound holds true for the corresponding coefficients of
 (2.1), from the stochastic equations by the Schwarz inequality we get
 $$
 \E_w\left(|p_\a(t)-p_\a(0)|^2\right)\le c_1''t(1+t) N_{\a}(q(0),R_1+\bar ct)^2\,,
 \Eq(papr)$$
 which completes the proof. Indeed, as $\phi(y)\le\phi(0)+\bar c|y|\,,$
 taking the square root of both sides and summing for $\a\in I$ such
 that $|q_\a(0)-m|\le R_3+\bar ct\,,$ we get a bound for
 $\Cal H_m(\o(t),R_3)\,;$ the square of the number of points at time
 zero is estimated again by superstability. $\qed$\smallskip

 To prove the existence of limiting solutions when the cutoff is
 removed we have to compare different partial solutions. Let $A_L$
 denote
 the set of twice continuously differentiable $a:\RR^3\mapsto [0,1]$
 with compact support such that $|a'(x)|\le1$ for all $x$ and $a(x)=1$
 if $|x|\le L\,.$ For $a,\bar a\in A_L$ let $\o(t)=(p(t),q(t))$
 and $\bar\o(t)=(\bar q(t),\bar p(t))$ denote the corresponding
 solutions to \equ(dynloc) with a common initial value
 $\o(0)=\bar\o(0)=(\xi,\eta)\,.$ Supposing $|x|\le L-2\bar ct$, for
 $|x-\xi_\a|\le R_0+2\bar ct$ we get
 $\dd_t|q_\a-\bar q_\a|\le K_0|p_\a-\bar p_\a|\,,$ whence
 $$
 |q_\a(t)-\bar q_\a(t)|^2\le
 2K_0\int_0^t |q_\a(s)-\bar q_\a(s)||p_\a(s)-\bar p_\a(s)|\,ds\,;
 \Eq(qb)$$
 here and in what follows, $K_0,K_1,K'_1...$ denote constants depending
 only on the coefficients of the infinite system. The case of the
 momentum variables is more complex. If $a=\bar a=1$ can be assumed
 as before, then by Ito's formula we get
 $$
 \E_w\left(|p_\a(t)-\bar
p_\a(t)|^2\right)=\E_w\left(\int_0^t\bigl(J_{\a,1}(s)+J_{\a,2}(s)
 +J_{\a,3}(s)\bigr)\,ds\right)\,,
 $$
 where
 $$
 \eqalign{
 J_{\a,1}&=-2\sum_{\a\ne\b}\bigl\lan p_\a-\bar p_\a,
 V'(q_\a-q_\b)-V'(\bar q_\a-\bar q_\b)\bigr\ran\,,\cr
 J_{\a,2}&=2\sum_{\b\ne \a}\bigl\lan p_\a-\bar p_\a,
 \gamma_{\a,\b}(q)F(p_\a,p_\b)-\gamma_{\a,\b}(\bar q)
 F(\bar p_\a,\bar p_\b)\bigr\ran\cr
 J_{\a,3}&=\sum_{\t=1}^d\sum_{\b\ne \a}
 \bigl|\sqrt{\gamma_{\a,\b}(q)}G_\t(p_\a,p_\b)
 -\sqrt{\gamma_{\a\b}(\bar q)}G_\theta(\bar p_\a,\bar p_\b)\bigr|^2 .
 }
 $$

 Introduce now  $\Ga_i$ for $i=1,2$ and $r,t\ge0$ by
 $$
 \Ga_1(\xi,\eta,a,\bar a;r,t):=\sum_{\a:|\xi_\a|\le r}
 |q_\a(t)-\bar q_a(t)|^2\,;
 $$
 in the definition of $\Ga_2$ the variables $q_\a$ and $\bar q_a$ should
 be replaced by $p_\a$ and $\bar p_\a\,,$ respectively. Our main tool is
 the following:

 \proclaim{\Lemma(converg)}. Suppose that $\ka<1/9\,,$
 $2\bar cT<R_1\,$.   For all
 $r\ge R_3\,,\,t\le T\,,\,h>0$ and $i=1,2$ we have
 $$
 \lim_{L\to\infty}\sup_{a,\bar a\in A_L}\;\sup_{(\xi,\eta)\in\bar
 \O_{\ka,L}(h)}\;\E_w\left(\Ga_i(\xi,\eta,a,\bar a;r,t)\right)=0\,,
 $$
 and the convergence is uniform on the time interval $[0,\,T]$.

 \smallskip\noi{\bf Proof:} The idea of the proof is to define and to
 estimate a ``distance" (based on $\Gamma_i$)
among different partial dynamics in boxes of radius $r<L$. This will lead 
us to equation (2.9) 
in which such a distance in a given box is related to the distance in a larger 
box, the result will easily follow. 

Let $\bar N=\bar N_{L,T}
 :=\max N_\a(\xi,R_3+2\bar cT)$ for all $\a\in I$ such that
 $|\xi_a|+R_3+2\bar cT\le L\,.$ Suppose $r+R_3+2\bar cT<L\,,$
 $|\xi_\a|\le r\,,$ and remember that $|q_\de(t)-\xi_\de|\le \bar ct$
 is always true. The uniform Lipschitz
 continuity of $V'\,,\,\si\,,\,F$ and $G_\t$ shall also be used without
 any further reference in the following calculations. Let
 $\ti\ga=\ti\ga_{\a,\b}(t)$ denote any matrix such that
 $0\le \ti\ga_{\a,\b}(t)\le 1$ if $t\le T\,,$ moreover
 $\ti\ga_{\a,\b}(t)=0$ whenever
 $|\xi_\a-\xi_\b|\ge R_3+2\bar ct\,.$ For $J_1$ we get
 $$
 J_{\a,1}(t)\le K_1|p_\a-\bar p_\a|\sum_{\b\in I}\ti\ga_{\a,\b}(t)
 \bigl(|q_\a-\bar q_\a|+|q_\b-\bar q_\b|\bigr)=:K_1\ti J_{\a,1}(t) .
 \Eq(pb1)$$
 In the case of $J_2$  the
 pattern $|ax-by|\le \min\{|a|,|b|\}|x-y|+|a-b|\max\{|x|,|y|\}$ is used
 several times to derive
 $$
 \eqalign{
 J_{\a,2}(t)&\le K_2\ti J_{\a,1}(t)
 + K_2|p_\a-\bar p_\a|\sum_{\b\ne\a}\ga_{\a,\b}(q)
 \bigl(|p_\a-\bar p_\a|+|p_\b-\bar p_\b|\bigr)\cr
 &+K_2|p_\a-\bar p_\a|\si(q_\a-q_\b)\sum_{\b\ne\a}\sum_{\de\in I}
 |\dd_\de \Te_{\a,\b}(\ti q^{\a,\b})||q_\de-\bar q_\de|
 }
 $$
 where $\dd_\de:=\dd/\dd q_\de$ and $\ti q^{\a,\b}$ is an intermediate
 configuration on the line segment connecting $q$ and $\bar q\,.$
 Observe that by H\"older's inequality
 $$
 \eqalign{
 \sum_{\de\in I}|\dd_\de\Te_{\a,\b}(\ti q)|&\le K\Te^2_{\a,\b}(\ti q)
 \sum_{\de\in I}\bigl(\chi(\ti q_\a-\ti q_\de)^{1-\ka}+
 \chi(\ti q_\b-\ti q_\de)^{1-\ka}\bigr)\cr
 &\le K\Te_{\a,\b}(\ti q)\bigl(N_\a(\ti q,R_2)^\ka
 +N_\b(\ti q,R_2)^\ka\bigr) .
 }
 \Eq(dga)
 $$
 On the other hand, $\si(q_\a-q_\b)>0$ implies $|\ti q^{\a,\b}_{\a}
 -\ti q^{\a,\b}_{\b}|\le R_1+2\bar c t\le 2R_1\,,$ i.e. $N_\a(q,R_1)
 \le N_\a(\ti q^{\a,\b},2R_1)\,.$ This means that
 $N_\a(q,R_1)\le 1/\Te_{\a,\b}(\ti q^{\a,\b})\,,$ consequently
 $$
 \eqalign{
 J_{\a,2}(t)&\le K_2'\bar N^\ka_{L,T}\ti J_{\a,1}(t)
 +K'_2\ti J_{\a,2}(t)\,;\cr
 \ti J_{\a,2}(t)&:=\sum_{\b\ne\a}\ga_{\a,\b}(q)
 \bigl(|p_\a-\bar p_\a|^2+|p_\b-\bar p_\b|^2\bigr).
 }
 \Eq(pb2)
 $$
 In a similar way we get
 $$
 \eqalign{
 J_{\a,3}(t)&\le K_3\ti J_{\a,2}(t)+K_3\ti J_{\a,3}(t)\cr
 &+K_3\sum_{\b\ne\a}{\si(q_\a-q_\b)\over\Te_{\a,\b}(\ti q^{\a,\b})}
 \Bigl(\sum_{\de\in I}|\dd_\de \Te_{\a,\b}(\ti q^{\a,\b})|
 |q_\de-\bar q_\de|\Bigr)^2\,;\cr
 &\ti J_{\a,3}(t):=\sum_{\de\in I}\ti\ga_{\a,\b}(t)
 \bigl(|q_\a-\bar q_\a|^2+|p_\de-\bar p_\de|^2\bigr)
 }
 $$
 whence by the Cauchy inequality and \equ(dga)
 $$
 J_{\a,3}(t)\le K'_3\ti J_{\a,2}+K'_3\bar N^{2\ka}_{L,T}\ti J_{\a,3}(t) .
 \Eq(pb3)
 $$
 Introduce now $d(r,t):=\E_w\left(\Ga_2(\xi,\eta,a,\bar a;r,t)\right)
 +\bar N_{L,T}\E_w\left(\Ga_1(\xi,\eta,a,\bar a;r,t)\right)$ for $t<T\,.$
 Comparing \equ(qb), \equ(pb1)-\equ(pb3) and using the elementary
 inequality
 $$
 2|p_\a-\bar p_\a||q_\de-\bar q_\de|\le \bar N^{-1/2}|p_\a-\bar p_\a|^2
 +\bar N^{1/2}|q_\de-\bar q_\de|^2\,,
 $$
 we obtain, by a direct calculation, 
 $$
 d(r,t)\le K_4\bar N_{L,T}^{{1/2}+\ka}
 \int_0^t d(r+R_3+2\bar cT,s)\,ds
 \Eq(iter)
 $$
 which completes the proof by a standard iteration procedure. Indeed,
 we get
 $$
 d(r,t)\le {T^{\ell+1}\over\ell!}
 \bigl(K_4\bar N^{{1/2}+\ka}_{L,T}\bigr)^\ell\, \sup_{t<T}\,d(L,t)\,,
 \Eq(expl)$$
 where $\ell\,,$ the number of allowed iterations is at least $c_TL$
 with $c_T>0$ depending only on $R_3$ and $T$, while $\bar N_{L,T}=
 O(\sqrt hL^{3/2+\ka})\,.$ Using $|q_\a(t)-\xi_\a|\le \bar ct$ and the
 second a priori bound \equ(papr), we see that the right hand side of
 \equ(expl) vanishes as $L\to+\infty$ because
 $\ell!=O\bigl((\ell/e)^\ell\bigr)$ and $({1/2}+\ka)({3/2}+\ka)<1$ by 
 hypothesis.
 $\qed$\smallskip

 Now we are in a position to prove the existence and uniqueness of
 limiting solutions to (0.1). Let us consider a sequence of partial
 solutions $\o_n=\o_n(t)\,,\,n\in\NN$ of \equ(dynloc) with a common
 initial value $\o_n(0)=(\xi,\eta)\in\bar\O_{\ka,\infty}\,;$ the
 corresponding cutoff $a_n:\RR^3\mapsto\RR$ is assumed to be a 
 decreasing smooth
 function of $|x|$ such that $a_n(x)=1$ if $|x|\le n$ and $a_n(x)=0$ if
 $|x|>n+1\,.$ In view of Lemma 2.4 $\o_n$ converges in
 probability to some limit $\o(t)$ for each $t<T=R_1/2\bar c\,.$
 It is easy to verify that the limit satisfies the infinite system
 (0.1); the uniqueness of limiting solutions follows again by Lemma 2.4.
 Since $T$ does not depend on the initial configuration $(\xi,\eta)
 \in\bar\O_{\ka,\infty}\,,$ the construction extends to all times.

 \vskip 5pt
 \noindent{\bf Properties of the infinite dynamics}\vskip 2pt

 Let $P^t_n$ denote the Markov semigroup induced by the partial
 dynamics $\o_n\,.$ Since $\o_n(t)$ is a continuous function
 of the initial data, it is well defined by $P^t_n\psi(\xi,\eta)
 :=\E_w(\psi(\o_n(t)))$ if $\o_n(0)=(\xi,\eta)$ for any measurable and
 bounded $\psi:\bar\O_{\ka,\infty}\mapsto\RR\,.$ As a limit of
 measurable functions, the limiting solution $\o(t)$ is again a jointly
 measurable function of $(\xi,\eta)$ and the random element representing
 the Wiener processes $w^\t_{\a,\b}\,,$ the limiting semigroup,
 $P^t$ can be defined in the same way. If the initial configuration
 is distributed by $Q\in\Cal Q_{\infty}$ then $QP^t_n$ and $QP^t$
 denote the evolved measure at time $t>0\,.$ In view of Lemma 2.1
 and Lemma 2.3 we know that $QP^t\in\Cal Q_{\infty}\,,$ too. While
 $P^t_n$ has fairly good regularity properties, semigroup theory
 does not apply directly to the limiting case. Nevertheless, all we
 need in the next section is summarized as follows.

 \proclaim{\Lemma(locality)}. Suppose that $\psi:\bar\O_{\ka,\infty}
 \mapsto\RR$ is a continuous and bounded local function, i.e. $\psi(\o)
 \equiv\psi(\o_{B_0(r)})$ for some $r>0\,,$ then
 $$
 \lim_{\ell\to\infty}\sup_{n>\ell+r}\sup_{Q\in\Cal Q_n(k)}
 |QP^t_n\psi-QP^t\psi|=0
 $$
 for all $t,k>0\,,$ and the convergence is uniform on compact time intervals.

 \smallskip\noi{\bf Proof:} The a priori bound of Lemma 2.3 extends
 immediately to the limiting dynamics, thus for any $\e,T>0$ we have
 some $\bar k>k$ and $h>\bar k$ such that
 $Q(\bar\O_{\ka,r}(h))\ge1-\e\,,$
 $QP^t_n(\bar\O_{\ka,r}(h))\ge1-\e\,,$
 and $QP^t(\bar\O_{\ka,r}(h))\ge1-\e$ whenever $t<T\,,$
 $n>r+R_3+2\bar cT$ and $Q\in\Cal Q_n(k)\,.$ Since
 $\bar\O_{\ka,r}(\bar k)$ is compact, there exists also an $\e'>0$ such
 that $|q_\a-\bar q_\a|+|p_\a-\bar p_\a|<\e'$, for all $\a\in I$ with
 $|q_\a|,|\bar q_\a|\le r$, implies $|\psi(\o)-\psi(\bar\o)|\le\e$ for
 $\o,\bar \o\in\bar\O_{\ka,r}(\bar k)\,.$ Therefore the statement follows
 from Lemma 2.4 and Chebishev inequality by a $3\e$ argument. $\qed$\smallskip

 The final statement of Theorem \equ (dynamics) on stationarity of certain Gibbs
 states is now a direct consequence of Lemma 2.2.

 \vskip .5cm
 {\bf 3. AN ENTROPY ARGUMENT}\vskip.25cm

 \numsec=3\numfor=1\numtheo=1


 In this section we extend a familiar argument by Holley [H]
 to the present more complex situation.

 For a probability measure $Q$ on $\O$, let $H(Q|\PP_\lambda)$
 denote the entropy relative to a distinguished Gibbs state
 $\PP_\lambda$ with $\l_1=\l_2=\l_3=0$, as defined by \equ(entropy)
 with $\Lambda = \RR^3$. The family of partial dynamics \equ(dynloc)
 has been chosen such that $\PP_\l$ is a common stationary measure of
 each local dynamics $P_n^t=P_{\l_4,a_n}$ introduced in Section 2.
 Therefore $P_n^t$ is a strongly
 continuous semigroup in $L^2(\PP_\lambda)\,,$ and  smooth cylinder
 functions form a core for its generator $\wt L_n = L_n +\wh L_n\,,$
 see \equ(pargen). Remember that  $L_n\equiv L_{\l_4,a_n}$, the
 Hamiltonian part, is antisymmetric in $L^2(\PP_\l)$ while the symmetric
 (reversible) component is just $\wh L_n\equiv \wh L_{a_n}\,.$

 If $\G$ is a generator in $L^2(\PP_\l)$ then the corresponding
 Donsker-Varadhan rate function is defined as
 $$
 D(Q|\G) \ =\ \sup_\psi\Bigl\{ - \int {\G \psi\over \psi} dQ \;:\;
   \psi\in \hbox{Dom }\G,\,\inf \psi > 0\Bigr\} .
 $$
 If $\G$ is self-adjoint and $\G<0$, then we can apply the following
 result due to Donsker and Varadhan (cf. [DV], Theorem 5)

 \proclaim{\Theorem (donsker-var)}.
 $D (Q|\G )<+\infty$ if and only if $Q<<\PP_\l$ and
 $g:=\sqrt{dQ/d\PP_\l}\in\hbox{Dom }\sqrt{-\G}$; moreover
 $$
 D(Q|\G) \ =\ \int \left(\sqrt{-\G} g\right)^2 d\PP_\l .
 \Eq(dirichlet)
 $$

 Our main tool consists of the following entropy inequality.

 \proclaim{\PProposition (H-th)}. Let $\bar Q_n^t := (1/t) \int_0^t
  QP_n^s ds$. If $H(Q|\PP_\l) <\infty$ then
  $$
  H(QP_n^t|\PP_\l) + 2t D(\bar Q_n^t|\hat L_n) \le H(Q|\PP_\l) .
  \Eq(H-ineq)
  $$

 \smallskip\noi{\bf Proof:}
 Let $P^{*t}_n$ be the adjoint semigroup with respect to $\PP_\l$,
 which is again a diffusion with formal generator $\tilde L_n^* =
 -L_n+\hat L_n$. Both forward and backward diffusion are essentially
 finite dimensional with smooth coefficients, thus twice continuously
 differentiable functions form a common core $\Cal D_n$ of $L_n$ and
 $L^*_n\,.$ This suffice to justify the following computations.
 Observe first that, as an easy consequence of Jensen's inequality, we
 have
 $$
 H(Q'P_n^{\tau}|Q''P_n^{\tau}) \le H(Q'|Q'')
 \Eq (H-increase)
 $$
 for any two measures $Q',\,Q''$.
 For any strictly positive $\psi\in\Cal D_n$ with $\PP_\l(\psi)=1$
 define $Q''$ by $dQ''=\psi d\PP_\l\,.$ Since
 $$
 {dQ''P_n^{\tau}\over d\PP_\l} = P^{*{\tau}}_n \psi
 $$
 we have
 $$
 H(Q'|Q'') = H(Q'|\PP_\l) - Q'(\log \psi)
 $$
 and
 $$
 H( Q' P_n^{\tau} | Q''P_n^{\tau}) = H(Q'P_n^{\tau}|\PP_\l)
 - Q'P_n^{\tau}(\log P^{*{\tau}}_n\psi).
 $$
 Accordingly, by \equ(H-increase),
 $$
 H(Q'|\PP_\l) - H(Q'P_n^{\tau}|\PP_\l) \ge Q'(\log \psi)
 - Q'P_n^{\tau}(\log P^{*{\tau}}_n\psi),
 $$
 whence, by the concavity of the logarithm and the inequality
 $\log(x+1) \le x$,
 $$
 H(Q'|\PP_\l) - H(Q'P_n^{\tau}|\PP_\l) \ge Q'(\log \psi) -
 Q'(\log P_n^{\tau} P^{*{\tau}}_n\psi)\ge
 \int {\psi - P_n^{\tau} P^{*{\tau}}_n\psi\over \psi} dQ' .
 $$
 Remembering that $P_n^t$ and $P^{*t}_n$ are both Feller semigroup and $\psi$
 belongs to the common core of $\tilde L_n$ and $\tilde L_n^*$,
 we have, for small $\tau$,
 $$
 \psi -  P_n^{\tau} P^{*\tau}_n\psi
 = \psi - P_n^{\tau} \psi+ \psi - P^{*\tau}_n\psi
 + P_n^{\tau}\left(\psi - P^{*\tau}_n\psi\right)-
 \left(\psi - P^{*\tau}_n\psi\right)
 = -2\tau \hat L_n \psi+o(\tau).
 $$
 Therefore, by dividing the given interval $[0,t]$ into $m$ small pieces,
 with $\tau=t/m$ and $Q'=QP_n^{{it\over m}}$, we get
 $$
 \eqalign{H(Q|\PP_\l) - H(QP_n^t|\PP_\l)&= \lim_{m\to\infty}
 {1\over m}\sum_{i=0}^{m-1}\left[H(QP_n^{{i\over m}t}|\PP_\l)
 - H(QP_n^{{i+1\over m}t}|\PP_\l)\right]\cr
 &\ge  -2\int_0^t ds \int {\hat L_n \psi \over \psi}\ dQP_n^s .}
 $$
 By taking the supremum over all $\psi$ considered we conclude the
 proof. $\qed$\smallskip

 Observe now that if $D (Q|\hat L_n)< \infty$, then by theorem
 \equ(donsker-var)
 it can be written as a sum, namely, if $g = \sqrt{dQ/d\PP_\l}$,
 \nfootnote{To see this, since $g\in \hbox{Dom }\sqrt{-\hat L_n}$
 (hence $g\in
 \hbox{Dom }\sqrt{a(q_\a)a(q_\b)\gamma_{\a\b}(q)}X_{\a\b})$),
 one can approximate it by local smooth
 functions, then use the closability of the Dirichlet form $D$.}
 $$
 D (Q|\hat L_n ) = {1\over 2}
 \int \sum_{\t, \a,\b} a_n(q_\a) a_n(q_\b) \gamma_{\a\b}(q) (X^\t_{\a\b} g)^2 \
 d\PP_\l  .
 $$
 Let $a_{n,1}(x), a_{n,2}(x),\dots, a_{n,j}(x)$ be smooth non-negative functions
 with compact support, and assume that their supports are disjoint.
 Furthermore, assume that $a_n(x) \ge a_{n,1}(x) + \dots + a_{n,j}(x)$,
 then
 $$
 D (Q|\hat L_n ) \ge D (Q|\hat L_{a_{n,1}} ) + \dots + D (Q|\hat L_{a_{n,j}} ).
 $$
 Therefore, from \equ(H-ineq), we have
 $$
 H(QP_n^t|\PP_\l) + 2t \sum_{i=1}^jD(\bar Q_n^t|\hat L_{a_{n,i}}) \le
 H(Q|\PP_\l).
 $$
 Thus we can choose strictly positive
 and smooth functions $\psi_0, \psi_1,\dots,\psi_j$ such that
 $$
 QP_n^t (\psi_0) - \log \PP_\l(e^{\psi_0})
 - 2t \sum_{i=1}^j \bar Q_n^t \left({\hat L_{a_{n,i}}\psi_i\over \psi_i}\right)
 \le H(Q|\PP_\l)   .
 $$
 This inequality extends by continuity to the infinite dynamics
 (cf. lemma \equ(locality) and note that $Q\in \Cal Q_\infty$)
 $$
 QP^t (\psi_0) - \log \PP_\l(e^{\psi_0})
 - 2t \sum_{i=1}^j \bar Q^t \left({\hat L_{a_{n,i}}\psi_i\over \psi_i}\right)
 \le H(Q|\PP_\l) .
 \Eq(ready)
 $$

 Now we are in a position to take the thermodynamic limit and conclude
 the main result of this section.

 \proclaim{\PProposition (loc-sta)}.
 If $Q_*$ is a translation invariant stationary measure of the
 infinite system \equ(stochdyn), and $Q_*$ has finite specific
 entropy
 with respect to $\PP_\l$, then $D( Q_*|\wh L_{\bar a}) = 0$ for all smooth
 functions $\bar a\le 1$ of compact support.

 \smallskip\noi{\bf Proof}:
 We are going use \equ(ready) with $Q=Q_{*n}$ where $Q_{*n}$
 is defined by
 $$
 Q_{*n}(\psi)\ =\ \int \PP_\l(\psi|\Cal F_{\L_n})\ dQ_*
 $$
 and $\L_n$ denotes the centered cubic box of size $n$.
 Of course, $H(Q_{*n}|\PP_\l) = H_{\L_n}(Q_*|\PP_\l)\,,$ thus
 $$
 \bar H(Q_*|\PP_\l):= \lim_{n\to \infty}{H(Q_{*n}|\PP_\l)\over |\L_n|}
 = \sup_{\psi}\left( Q_*(\psi) - \bar F(\psi)\right)\ ,
 \Eq(H-density)
 $$
 where $\psi$ are the local, bounded and continuous functions; in addition,
 $$
 \bar F(\psi) := \lim_{n\to \infty} {1\over |\L_n|}\log \int \exp
 \left(\sum_{k\in \L_n\cap \Z^3} \vec s^{k} \psi\right) d\PP_\l ,
 \Eq(free)
 $$
 and $\vec s^{k}$ denotes the shift in $\RR^3$ by $k \in
 \RR^3$, i.e. $\vec s^{k}\psi(p,q) = \psi( p,\vec s^{k}q)$ and
 $\vec s^{k}q = \{q_\a +k\}$ if $q = \{q_\a\}$.
 The proof of the existence of \equ(H-density) and \equ(free)
 can be found in [OVY].

 Now we set
 $$
 \psi_0 = \sum_{k\in \L_n\cap \Z^3}\vec s^{k} \psi
 $$
 for some local bounded continuous function $\psi$.

 Without loss of generality we can suppose $n$ so large that $\L_n$ contains the
 support of $\bar a$, and
 define $a_{n,i}(x) = \bar a (x+k_i)$,
 $k_i \in J_n$, and $J_n$ is a discrete subset of $\L_n$
 such that the $a_{n,i}$
 have the disjoint supports contained in $\L_n$, and ${|J_n|\over n^3 }\ge
\bar J_0$, for some fixed constant $\bar J_0$.\nfootnote{This can be
 done as to ensure that $a_n\geq\sum_{i=1}^ja_{n,i}$.}
 Correspondingly we choose $\psi_i =\vec s^{k_i} \bar\psi,\ \vec
 k_i\in J_n$,
 for a local bounded continuous function $\bar\psi$.

 Substituting in \equ(ready) and dividing by $|\L_n|$,
 it remains to prove that
 $$
 \lim_{n\to \infty} {1\over |\L_n|}
 \sum_{k\in \L_n\cap \Z^3} Q_{*n} P^t(\vec s^{k} \psi) =
 Q_*(\psi)
 \Eq(unif1)
 $$
 and
 $$
 \lim_{n\to \infty} {1\over |J_n|} \sum_{k_i\in J_n}
 \bar Q_{*n}^{t} \left(\vec s^{k_i}{\wh L_{\bar a}\bar\psi
 \over\bar \psi}\right)
 = Q_* \left({\wh L_{\bar a}\bar\psi\over \bar\psi} \right) .
 \Eq(unif2)
 $$
 Indeed, then \equ(ready), \equ(H-density),
 \equ(free), \equ(unif1) and \equ(unif2)
imply
 $$
 Q_* (\psi) - \bar F(\psi)
 - 2t J_0 Q_* \left({\hat L_{\bar a}\bar\psi\over\bar\psi}\right)
 \le \bar H(Q_*|\PP_\l),
 $$
 and taking the supremum over all $\psi$ and $\bar\psi$ considered
 we obtain the wanted result.

 To prove \equ(unif1), observe first that the rate of
 convergence in Lemma 2.5 depends only on the magnitude and the
 modulus of continuity of the underlying function. In the present
 situation all functions are translates of each other, thus the convergence
 is uniform on such functions. Therefore, for $k\in\L_{n-\sqrt n}$ we
 approximate $P^t$ with the local dynamics $\vec s^kP^t_{\sqrt n}$
 in the ball $B_k(\sqrt n)\,,$ otherwise we use simply the uniform bound
 of $\psi\,.$ The proof of \equ(unif2) is similar.$\qed$

 As it is well known, see [DV], $D(Q_*|\wh L_{\bar a})=0$ implies the
 reversibility of $Q_*$ with respect to $\wh L_{\bar a}\,,$ which
 completes the proof of Theorem 1.3, whereby proving Theorem 1.4 as well, 
 by a direct argument.

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\bye
ENDBODY