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 \centerline{\bfb Ergodicity in Infinite 
 Hamiltonian Systems with Conservative Noise}
 \vskip1.5cm
 \centerline{{\rmb Carlangelo Liverani}\footnote{$^1$}{ II Universit\`a di Roma ``Tor
 Vergata", Dipartimento di Matematica, 00133 Roma, Italy.  E-mail:
 liverani@ccd.utovrm.it .}}
 \vskip.5cm
 \centerline{{\rmb Stefano Olla}\footnote{$^2$}{ 
Centre de Math\'ematiques Appliqu\'ees, Ecole
Polytechnique, 91128 Palaiseau Cedex, France and Politecnico di Torino, 
 Dipartimento di Matematica, corso Duca degli
 Abruzzi 24, 10129 Torino, Italy. E--mail: olla@paris.polytechnique.fr .}}
 \footnote{}{\bfp The authors wish to thank L.Chierchia, J.Fritz, 
 J.L.Lebowitz and G.Tarantello for helpful 
 discussions. In addition, C.Liverani is grateful to the CNR--GNFM for providing 
 travel funds, and S.Olla would like to thank the Courant Institute, New York,
 for the warm hospitality while this work was being completed.  A last thanks goes to
 the referees for pointing out some inadequacies in the early version of the
 paper and for forcing us to work out some unpleasant but very relevant details.}
 \vskip.5cm
 \centerline{\bf Abstract}
  
 {\sl \baselineskip5mm
 We study the stationary measures of an infinite Hamiltonian system of 
 interacting particles in $\RR^3$ subject to a stochastic local perturbation
 conserving energy and momentum. We prove that the translation invariant
 measures that are stationary for the deterministic Hamiltonian dynamics,
 reversible for the stochastic dynamics, and with finite entropy density are convex
combination of ``Gibbs'' states. This result implies
 hydrodynamic behavior for the systems under consideration.}
  
 \vskip 1cm
 {\bf INTRODUCTION}
 \vskip .5cm
  
 The ergodicity problem in Hamiltonian dynamical systems is at the base of 
 equilibrium statistical mechanics. 
 While, beginning with the celebrated Sinai's paper [Si],
 some result are known for finite system (see [LW] for a general 
 approach to ergodicity in Hamiltonian systems), 
 very little is known concerning infinite systems (some 
 results are known for special systems with an arbitrary, but finite, 
 number of particles
 [BLPS]). By ergodicity of an infinite system we mean that  convex 
 combinations of Gibbs measures are the only 
 invariant measures, within a reasonably ``regular'' class.
  
 Furthermore, recent developments in non-equilibrium hydrodynamics 
 (cf. [OVY]) 
 show that the ergodicity of an infinite systems is a main ingredient
 in the rigorous derivation of Euler equations as a macroscopic description
 of the conservation laws for the density, the momentum and the energy 
 (at least in the smooth regime of these equations).
  
 Since no results are present for deterministic systems, it is natural to ask
 if a stochastic perturbation may help in proving ergodicity. 
 The stochastic perturbation should conserve the energy, the
 momentum and the number of particles of the system, while destroying 
 {\it locally} the other possible invariant of the motion.
  
 A stochastic perturbation of this type is introduced in [OVY]: 
 any two particles exchange randomly momentum in such a way as
 to preserve only the total momentum and energy of the two particles.
 The rate of exchange is assumed to decrease when the distance 
 between the two particles increases, but the range is infinite. 
 Accordingly, any particle is interacting stochastically
 with any other and the corresponding diffusion on the momenta space of any
 finite number of particles is elliptic. 
 This permits to characterize the distribution of the momenta 
 of any finite number of particles conditioned to the
 positions: it must be a uniform measure on the corresponding 
 invariant manifold in the momenta space. 
 The equivalence of ensembles implies that the distribution of the
 momenta conditioned on the position is a convex combination of ``Maxwellians''.
 In addition, one can localize the invariance, under the Hamiltonian dynamics, 
 of the distribution, and prove that 
 the distribution of the positions satisfies the DLR equations
 with respect to the corresponding interaction.
  
 The purpose of the present paper is to extend the foregoing argument 
 to {\it finite} range
 stochastic interactions. Two difficulties arise immediately: one of local
 and the other of global type.
  
 Locally, restricting oneself to a finite ``chain'' (or cluster) of particles
 interacting stochastically, the diffusion on the space of momenta
 is no longer elliptic; it becomes then necessary to prove that it is, 
 at least, hypoelliptic.
 This is done quite easily with an inductive argument and in grand generality: 
 only the convexity of the kinetic energy is needed. 
  
 The global obstacle is of a more serious nature. The diffusion on 
 the momenta is hypoelliptic 
 only when restricted to chains of interacting particles.
 But, several clusters of 
 particles, too far apart to interact stochastically, may be present; hence,
 they could be at ``different temperatures".
 We need the help of the deterministic Hamiltonian dynamics 
 to ``connect'' distant clusters. 
  
 Taking commutators between the vector fields
 generating the stochastic dynamics and the Hamiltonian generator one obtains
 a Lie algebra of vector fields large enough 
 to generate all the tangent space to the energy-momentum
 manifold on the phase space (i.e., position and momentum). This means that our 
 system is invariant for the dynamics  generated by these vector fields (that turn out
 to be local) which enable, after some work, to produce ``cluster deformations" that
 connect any cluster with the others. 
 Proceeding in such a way we can obtain, in each sufficiently large 
 finite box, a ``unique cluster" and consequently prove that
 the momenta are uniformly distributed.
  
 A further difficulty arises if the kinetic energy is quadratic (i.e., the
 usual ``Gaussian case''). In fact, in this case all the above 
 mentioned dynamics preserve also the center of mass 
 of any finite cluster of particles. 
 To complete the argument in this case it would be necessary to perform cluster
 deformations that conserve the center of mass, hence substantially complicating the
 above argument.
 We belive that our program could be carried out for the Gaussian case as well but
we stop short of it also in view of the fact that its application to hydrodynamics
is unclear (see point (d) in the following discussion).  
 As in [OVY] we consider only  stationary measures 
 having finite entropy 
 density with respect to a grancanonical Gibbs measure. This condition seems to 
 characterize a nice class of regular measures.  
 To complete our argument, various extra assumptions are necessary:
 \item{(a)} The range of the stochastic interaction is finite but 
 must be strictly 
 larger than the one of the deterministic potential.
 \item{(b)} The invariant measures considered must have sufficiently high
 particles density. More precisely, we need to be sure that, 
 for almost any configuration, any sufficiently large box contains 
 at least two particles interacting stochastically. 
 The bound on the density we ask here is very rough, and we believe
 it can be substantially improved by using a more refined argument.
Alternatively one can assume that the average potential energy is positive,
which will imply that in a box large enough at last two particle interact
deterministically, though stochastically. Unfortunately potential energy
is not a conserved quantity, so usually one does not have any information
about its average value, that is why we prefer a condition on the density,
which is stricter but easier to use. 
 \item{(c)} We assume that the measures considered are separately invariant
 for the deterministic and the stochastic dynamics. Furthermore, they must be
 reversible for the stochastic dynamics alone. 
 The reversibility with respect to the global
 stochastic dynamics is a more general condition than the ones needed to derive
 the hydrodynamic limit: the invariance for each local 
 stochastic dynamics (cf. proposition 2.2) would suffice (cf. [OVY]).
 \item{(d)} In order to apply our results to obtain hydrodynamic 
 limits following [OVY] we consider kinetic energies that are 
 not quadratic, since [OVY] does not apply to the quadratic case. 
 Nonetheless, we must assume a mild restriction on the kinetic energy 
 function: the local dynamics cannot have undesired invariant
 (like the center of mass in the Gaussian case, see lemma 2.5).
 We provide examples of kinetic energy functions that 
 satisfy both our condition and the ones assumed in [OVY] (cf. Appendix 1).
  
 \noindent
 As a consequence of our result, theorem 2.1 of [OVY] 
 is valid for Hamiltonian dynamics with stochastic perturbation 
 of the non-Gaussian type considered in the present
 paper.
  
 For lattice systems the problem of ergodicity is solved in [FFL] in a
 more satisfactory way. In fact, there it is not needed condition (c), i.e.,
 only the invariance with respect to the total dynamics 
 (deterministic + stochastic) is required.
  
 Concerning condition (c), notice that we could have asked the invariance 
 for the finite stochastic dynamic in each finite box. We prove indeed
 that this is equivalent to the global reversibility (cf. proposition 2.2).
 Proposition 2.2 has an interest in itself: 
 it says that if a stochastic dynamics
 on a lattice in finite dimension is hypoelliptic then for the corresponding 
 infinite dynamics all the reversible measures are given by Gibbs measures.
 This generalize a result of M. Zhu (cf.[Z]) 
 to the ``hypoelliptic'' situation.
  
 The next section contains a more precise description of the results outlined
 here, together with the plan of the paper.
  
 \vskip 1cm
  {\bf 1. NOTATIONS AND RESULTS}
 \vskip.5cm
  
 \numsec=1\numfor=1\numtheo=1
  
 \noindent{\bf Sample space}
  
 A point of $\RR^3\times\RR^3$
 will be denoted by $(q,p)$ and the sample space $\Omega$ will consist of
 points $\omega=\{(q_\alpha, p_\alpha)\}$.
 Any bounded region $B$ in $\RR^3$ will contain only a finite number of
 particles, with positions $q_\alpha$, in addition 
 one can think of $p_\alpha$ as tags, and consider
 the corresponding finite configuration in $B \times \RR^3$.
  
 \vskip 5pt
 \noindent{\bf Interaction}
  
 We consider a radial repelling finite range smooth pair potential 
 $V(q_\a-q_\b)$ such that:
 \item{i)}$ \hskip 10pt
 V(x)=0 \hskip 10pt |x|>R_0
 \hskip 10pt$ (finite range)
 \item{ii)} $V$ is superstable (i.e. it satisfies the superstability inequality:
 {\it there exists $B>0$ and $A>0$ such that for any finite 
 box $\Lambda$ and any configuration we have}:
 $$
 \sum_{q_a\in\Lambda}\sum_\b V(q_\a-q_\b)\ 
 \ge {A\over |\Lambda|} |\o_\Lambda|^2- B |\o_\Lambda|
 \Eq (superstability)
 $$
 (see [R])).
 \item{iii)} $\hskip 10pt \langle x,\,\nabla V(x)
 \rangle \ \equiv \sum_i^3 x_i{\partial V\over\partial x_i}(x) \le 0 \hskip10pt
\forall x\in
\RR^3\hskip 10pt$
 (repelling interaction)
  
 The repelling condition (iii) is of a technical nature and it should 
 be possible to remove it by a more accurate analysis. 
 \vskip 5pt
 \noindent{\bf Kinetic Energy}
  
 It is given by a strictly convex function $\phi (p) \in C^\infty(\RR^3)$.
 We consider two cases:
  
 \item{(G)} $\phi (p)$ is a quadratic function of $p$, which is the classical 
 {\it Gaussian} case. 
 \item{(NG)} $\phi (p)=\sum_{i=1}^3 \varphi(p^i)$ with $\varphi$ a 
 strictly convex smooth positive function on $\RR$ 
 with 
 $$
 {1\over 2}{d^2\over dx^2}(\varphi''(x))^2= 
 \varphi'''(x)^2+\varphi^{iv}(x)\varphi''(x)\neq 0
 \Eq (NG)
 $$ 
 apart from, at most, finitely many points. In addition, we require the invariance
 for reflections, i.e. $\varphi(x)=\varphi(-x)$, and that $\varphi$ is not too flat
 near the origin; more precisely, we assume that there exist $m$ such that
 $\varphi^{(m)}(0)\neq 0 $.\nfootnote{This
 last condition will be needed only in the proof of Lemma 3.2}  
 We will refer to this  case as the {\it non-Gaussian} case. 
  
 \noindent
 Notice that, if ${d^2\over dx^2}(\varphi''(x))^2= 0$ for each $x$ the
 condition $\varphi(x)=\varphi(-x)$ implies 
 $\varphi''(x)=$cons\-tant, i.e., we have the Gaussian case. This shows that,
 morally, our conditions cover all the possible cases; yet, it could be
 interesting to carry out a more detailed investigation.
  
 As already mentioned, our main motivation to treat the case (NG) 
 is to apply the present results to the derivation of the hydrodynamic limit.
 To do so, the kinetic energy function must satisfy the conditions:
 $$
 \left\vert {\partial \phi\over\partial p^j}\right\vert \le C',\hskip 10pt
 \left\vert {\partial^2 \phi\over\partial p^j\partial p^i}\right\vert \le 
 C''
 \hskip 20pt \forall p\in\RR^3
\Eq(bv)
 $$
 which are clearly not satisfied by the classical case (G). 
  
 \vskip 5 pt
  
 \noindent{\bf Hamiltonian Dynamics}
  
 The Hamiltonian is defined by the formal expression:
 $$
 {\cal H}(\omega)=\sum_\a\phi(p_\a) + 
 {1\over 2}\sum_\a\sum_{\b\ne\a}V(q_\a-q_\b)
 $$
 and the Liouville operator by
 $$
 L=\sum_\a\sum_{i=1}^3\left[\partial_{p_\a^i}{\cal H}\ \partial_{q_\a^i}-
 \partial_{q_\a^i}{\cal H}\ \partial_{p_\a^i}\right] .
 $$
  
 In this paper, we are not concerned with the existence of the 
 dynamics generated by
 $L$ or its stochastic perturbations. Our aim is simply to characterize 
 the probability
 measures on $\Omega$ that are `formally' invariant (see Th. 1.1). For a 
 more detailed description of the above objects see $[AGGLM,\,\S 2]$.
 \vskip 5pt
  
 \noindent{\bf Stochastic perturbation of the dynamics}
  
 We will use the notation 
 $v_\a^i\equiv\phi_i(p_\a)\equiv\partial_{p_\a^i}\phi$. In the following  
 smooth will mean always differentiable infinitely many times.
  
 For each smooth function $\eta_{\a\b}:\RR^6\to\RR^3$, 
(i.e. $\eta_{\a\b}= \eta_{\a\b} (p_\a,p_\b)$ )
 we define the vector field
 $$
 X(\eta_{\a\b})=\langle \eta_{\a\b},\,D_{\a\b}\rangle \equiv
 \sum_{i=1}^3 \eta_{\a\b}^i D_{\a\b}^i
 $$
 where $D_{\a\b}=\partial_{p_\a}-\partial_{p_\b}$.
  
 We are interested in vector fields with null divergence, i.e.,
 $$
 \hbox{div}(X(\eta_{\a\b}))=\langle \,D_{\a\b},\eta_{\a\b}\rangle = 
 \sum_{i=1}^3  D_{\a\b}^i \eta_{\a\b}^i =0   .
 \Eq (divnull)
 $$
  
 Furthermore, we ask that $X(\eta_{\a\b})$ is tangent to the surfaces, 
 $\RR^3\times\RR^3$,
 $$
 \cases{p_\a^i+p_\b^i=c^i\qquad i=1,2,3\cr 
 \phi(p_\a)+\phi(p_\b)=c^0 , }  
 $$
 that is, the orthogonality relation 
 $$
 \langle \eta_{\a\b},\,D_{\a\b}(\phi(p_\a)+\phi(p_\b))\rangle=0
 \Eq (orthorel)
 $$ 
 (equivalently, $\langle \eta_{\a\b},\, v_\a\rangle =\langle\eta_{\a\b},\,
 v_\b\rangle$), which will imply the conservation of energy and momenta with respect
 to the stochastic dynamics.
  
 Let $X(\eta_{\a\b})^*$ be the adjoint of $X(\eta_{\a\b})$ with 
 respect to the measures
 $$
 e^{\lambda_4(-\phi(p_\a)-\phi(p_\b))+\lambda\cdot(p_\a+p_\b)}\ dp_\a\ dp_\b
 $$
 for any $\lambda_4>0$ and $\lambda=(\lambda_1,\,\lambda_2,\,\lambda_3)$,
  with the restriction that
 $$
 \int\exp(-\lambda_4\phi(p)+\lambda\cdot p)dp<+\infty\ .
 $$
 We have, because the null divergence and the orthogonality property,
 that $X(\eta_{\a\b})^*=-X(\eta_{\a\b})$.
  
 We use the previous vector fields to define an operator of the second order
 that will be the generator of the stochastic perturbation.
 Consider a finite number $K\ge 3$ of vectors $\{\eta^\theta_{\a\b}\}$ 
 with the properties above.
 We define the operator
 $$
 \hat L_{\a\b}=-{1\over2}\sum_{\theta=1}^K\ X(\eta^\theta_{\a\b})^*
 X(\eta^\theta_{\a\b}) \ =\ {1\over2}\sum_\theta\ X(\eta^\theta_{\a\b})^2.
 $$
 Moreover, we require that, at each point, the linear combination of
 $\{\eta_{\a\b}^\theta\}$ spans a two dimensional subspace of $\RR^3$ (the 
maximum compatible with \equ(orthorel),
 eventually apart from a set $\wt{\mit\Sigma}_{\a\b}^s$ consisting of the 
 finite union of codimension--two manifolds.
 Therefore, $\hat L_{\a\b}$ is selfadjoint, 
 and elliptic outside $\wt{\mit\Sigma}_{\a\b}^s$. For later purposes, we
 define 
 $$
 \wt{\mit\Sigma}_{\a\b}=\wt{\mit\Sigma}_{\a\b}^s\cup\{(p_\a,\,p_\b)
 \;|\;v_\a=v_\b\},
 \Eq (mitsigma)
 $$ 
 by convexity follows that $\wt{\mit\Sigma}_{\a\b}$ is the
 finite union of smooth manifold with codimension two  as well.
  
 Let $\sigma(q)$ be a radial smooth function on $\RR^3$, such that 
 $\sigma(q)>0$ for each $\|q\|< R_1$, and $\sigma(q)=0$ for each $\|q\|\ge
 R_1>4R_0$. Then we consider the operator
  
 $$
 \hat L=\sum_{\a,\b} \sigma(q_\a-q_\b)\hat L_{\a\b}.
 $$
 In the following considerations it will be important that $\sigma$ 
 is strictly positive
 for a radius $R_1$ strictly greater than $4R_0$, i.e. that the range of 
 the stochastic interaction is larger than the one of the `deterministic'
 interaction.\nfootnote{The factor 4 is due to technical reasons and 
 plays a role only in section 4.} 
 \vskip 5pt
  
 \noindent{\bf Gibbs Measures}
  
 Let $\Lambda\subset\RR^3$. Each configuration $\omega\in \Omega$ 
 can be written as 
 $\omega=\{\omega_\Lambda,\omega_{\Lambda^c}\}$ where $\omega_\Lambda=
 \{(q_\a,p_\a)\in\omega\;|\; q_\a\in\Lambda\}$.
  
 Let $\PP$ be a probability measure on $\Omega$. If the $\PP$-conditional distribution
 of $\omega_\Lambda$ given the configuration outside 
 $ \omega_{\Lambda^c}$ is proportional
 to
 $$
 {1\over n!}\exp\left[\lambda_0n + \sum_{\a=1}^n\sum_{i=1}^3\lambda_i p_\a^i
 -\lambda_4{\cal H}_{\Lambda,n}(\omega_\Lambda,\omega_{\Lambda^c})\right]
 $$
 then $\PP$ is called Gibbs Measure (or grandcanonical Gibbs measure).
 In the above expression $n$ is the number of 
 particles in $\Lambda$ (that we will denote by $|\omega_\Lambda|$) 
 and the local Hamiltonian is defined by
 $$
 {\cal H}_{\Lambda,n}(\omega_\Lambda,\omega_{\Lambda^c})=
 \sum_{q_\a\in\omega_\Lambda}\left[\phi(p_\a)+{1\over 2}
 \sum_{q_\b\in\omega_\L;\;\alpha\neq\beta}V(q_\a-q_\b)
 +\sum_{q_\b\in\omega_{\L^c}}V(q_\a-q_\b)\right].
 $$ 
  
  
 \vskip 20pt
 \noindent{\bf Statement of the result}
 \vskip 5pt
  
  
 Let $Q$ and $P$ two probability measures on $\Omega$, and let $Q_\Lambda$
 and $P_\Lambda$ their restriction on a finite box $\Lambda$. The relative 
 entropy of $Q_\Lambda$ with respect to $P_\Lambda$ is defined by 
 $$
 H_\Lambda(Q|P)\ =\ \sup_{F\in{\cal F}_\L}\left\{Q(F)-\log P(\exp(F))\right\}
 \Eq (entropy)
 $$
 where ${\cal F}_\Lambda$ are the smooth functions localized in $\Lambda$.
 For the properties of $H_\Lambda$ see, for example, [OVY]. In the following $Q$ will
 be the translation invariant measure under consideration, 
 while $\PP$ will be any
 grancanonical Gibbs measure for the interaction $V$.
  
  
 \proclaim{\Lemma (entropybounds)}.
 If there exists a constant $C$ such that for each box $\Lambda$,  
  $$H_\Lambda(Q|\PP) \le C|\Lambda|$$
 then,
 $$
 \leqalignno{
 Q&\left(|\Lambda|^{-2}
 |\omega_\Lambda|^2\right)\le C_1 < \infty&(i)\cr
 Q&\left(|\Lambda|^{-1}\sum_{q_\a \in \Lambda} \|p_a\|\right)\le C_2  <\infty
 &(ii)\cr
 Q&\left(|\Lambda|^{-1}\sum_{q_\a \in \Lambda} 
 \left[\phi(p_\a)+\sum_{q_\b\in\omega} V((q_\a-q_\b)\right]\right)\le C_3 
 <\infty&(iii)
 }
 $$
 where $C_1,C_2,C_3$ are constants independent on $\Lambda$.
  
  
 \proclaim{Proof}.

 The inequalities (ii) and (iii) are consequences of the 
 following entropy inequality:
 $$
 Q(F)\le{1\over\b}\log \PP \left(\exp(\b F)\right)\ +\ {1\over\b}H(Q|\PP)
 \Eq (e-ineq) 
 $$
 which is valid for any local function $F$ and any constant $\b>0$. It follows
 directly from the definition \equ(entropy).
 Then for sufficiently small $\b$, 
 we have:
 $$
 Q\left(|\Lambda|^{-1}\sum_{q_\a \in \Lambda} \|p_a\|\right)\le
 {1\over\b|\Lambda|}\log P\left(\exp\left(\b \sum_{q_\a \in \Lambda}
  \|p_a\|\right)\right)\  +\ {1\over\b}C \le {1\over\b}C'
 $$
  
 In a similar way one can prove (iii). While (i) follows by the same 
 argument and the superstability inequality \equ(superstability).
 $\qed$
  
 \medskip
 Define 
 $$
 \eqalign{
 \rho(\omega)\ &=\ \lim_{|\Lambda| \to\infty}|\Lambda|^{-1}
 |\omega_\Lambda|\cr
 \pi(\omega)\ &=\ \lim_{|\L| \to\infty}|\L|^{-1}\Pi_\L\ 
 \equiv\ \lim_{|\L|\to\infty} |\L|^{-1}\sum_{q_\a \in \L} p_a\cr
 e(\omega)\ &=\ \lim_{|\L| \to\infty}|\L|^{-1} E_\L\ 
 \equiv\ \lim_{|\L|\to\infty}|\L|^{-1} \sum_{q_\a \in \L} 
 \left[\phi(p_\a)+{1\over 2}\sum_{q_\b\in\L;\; \alpha\neq\beta}
 V(q_\a-q_\b)\right.\cr
 &\qquad\left. +\sum_{q_\b\not\in\L} V(q_\a-q_\b)\right]\cr}
 $$
  
 The above Lemma \equ(entropybounds), and the translation invariance of Q, 
 insures that the limits
 $\rho(\omega)$, $e(\omega)$, $\pi(\omega)$ exist $Q$-almost everywhere.
  
  
 The aim of this paper is to prove the following:
  
 \proclaim {\Theorem (ergo)}.
 Let $Q$ be a translation invariant probability measure on $\Omega$, 
 if
 \item{(i)} There exists a constant $C$ such that for each box $\Lambda$,  
  $H_\Lambda(Q|\PP) \le C|\Lambda|$;
 \item{(ii)} $Q\left(\left\{\omega\in\Omega\;|\;
 \rho(\omega)> \rho_*\right\}\right)\;=\; 1$ where $\rho_*={3\over4R_1^3\pi}$;
 \item{(iii)} $Q$ is invariant w.r.t. the dynamics generated by $L$ (the
deterministic part), 
 in the sense that, for any smooth local function 
 $F_\Lambda(\omega_\Lambda)$,\nfootnote{By $\E^Q$ we mean the expectation with
 respect to the measure $Q$.}
 $$
 \E^Q(LF_\Lambda)\ =\ 0 \; ;
 $$
 \item{(iv)} $Q$ is reversible with respect to $\hat L$ (the stochastic
perturbation), 
 i.e., for any two smooth local functions $\p$ and $\psi$ holds
 $$
 \E^Q(\psi\hat L\p)\ =\ \E^Q(\p\hat L\psi)\ ;
 $$
 \noindent
 then Q is a convex combination of (gran canonical) Gibbs Measures.
  

\proclaim{Remark (1.3)}.

Condition (ii) on the density is a sufficient condition in order to always find
at least two particle interacting stochastically. 
Since the range of the deterministic interaction is smaller than the one 
of the stochastic interaction, this condition
 may be replaced by ensuring that the average potential energy
is strictly positive, i. e. if we define
$$
u(\o)\ =\ \lim_{|\L| \to\infty}|\L|^{-1} U_\L\ \equiv\ 
\lim_{|\L| \to\infty}|\L|^{-1}\sum_{q_\a,q_\b\in\L} V(q_\a-q_\b)
$$
then the condition reads
$$
Q(u(\o)>0)\ =\ 1
$$
This condition is not practical because $u(\o)$ does not correspond to a conserved
quantity. It will be of no use for the application to hydrodynamics (cf.[OVY]),
where we cannot have such information on Q. 
 \par
 The proof of Theorem 1.1 will be carried out in three parts. In the next 
 section  we will construct a multitude of local dynamics that leave
 the finite dimensional restrictions of the measure $Q$ invariant.  Section three
 is dedicated to the characterization of typical configurations for the class of
 measures $Q$ under consideration. In section four we show that 
 the above mentioned dynamics give a local characterization 
 weaker than the one implied by DLR equations, but sufficient to 
 claim that the global 
 distribution of the momenta, conditioned to the positions, 
 is given by a convex
 combination of ``Maxwellian'' (corresponding to the proper $\phi$).
 We conclude the argument in section five, along the line of [OVY], 
 by proving that in the infinite limit the 
 kinetic energy is ``invariant'' for the deterministic dynamic generated 
 by $L$. Thus, each component of the convex combination is 
 invariant for $L$. A classic argument (cf.[GV] 
 and [OV]) shows that invariant 
 distribution for L that have  distribution of the momenta conditioned to 
 the position given by a Maxwellian are canonical Gibbs measures.
  
 \vskip 1cm
  {\bf 2. CLUSTERS AND LOCAL DYNAMICS}
 \vskip.5cm
  
 \numsec=2\numfor=1\numtheo=1
  
 Given a  configuration $\omega$, we call ``connected'' two particles 
 that are sufficiently close to interact stochastically ($\a$ and $\b$
 are connected if $\sigma(q_\a-q_\b)>0$, i.e. $|q_\a-q_\b|<R_1$).
 We call ``cluster'' a set of 
 particles such that any two can be joined by a chain of connected ones.
 We call ``isolated cluster'' a cluster
 such that no other particle is connected to any particle in the cluster.
  
 Any configuration $\omega$ can be grouped in many isolated clusters 
 $\{\Gamma_i(\omega_q)\}$.
 These clusters may be finite or infinite. Also, the restriction 
 of any configuration to a 
 finite region $\omega_\Lambda$ is grouped into finite clusters 
 $\{\Gamma_i^\Lambda(\omega_q)\}$, where we have overlooked the connections 
 with the particles outside $\Lambda$.
  
 To simplify notations, in this section we denote by $\ec$ the expectation of Q 
 conditioned to a configuration of positions $\o_q$.
  
 Consider a cluster $\G_i^\L$, and let $n$ be the number of particles in it.
 Then $\hat L_{\G_i^\L}=\sum\limits_{\a\b\in\Gamma_i^\Lambda}\sigma
 (q_\a-q_\b)\hat L_{\a\b}$ is an operator on $\RR^{3n}$ and it conserves the 
 quantities
 $$
 \sum_{q_\a\in\G_i^\L}p_\a^1,\hskip 15pt \sum_{q_\a\in\G_i^\L}p_\a^2,
 \hskip 15pt
 \sum_{q_\a\in\G_i^\L}p_\a^3,\hskip 15pt
 \sum_{q_\a\in\G_i^\L}\phi(p_\a) .
 $$
  
 Let us consider the corresponding connected hypersurfaces of 
 dimension $3n-4$:
 $$
 {\mit \Sigma}_c\equiv\left\{(p_1,\ldots,p_n)\;\bigg|\;\sum_\a p_\a^1=c^1,
 \sum_\a p_\a^2=c^2,
 \sum_\a p_\a^3=c^3, T=\sum_\a \phi(p_\a)=c^4\right\} ,
 \Eq (hypersurf)
 $$
 and the sets
 $$
 \widetilde{\mit \Sigma}_c\equiv\left\{(p_1,\ldots,p_n)\;\big|\;
 (p_\a,\,p_\b )\in\widetilde{\mit \Sigma}_{\a\b}\ \hbox{ for some } 
 \a,\,\b\in\{1,\ldots,n\}\right\}
 $$
where $\widetilde{\mit \Sigma}_{\alpha\beta}$ has been defined by \equ(mitsigma).
 Note that $\widetilde{\mit\Sigma}_c$ is the finite union of smooth 
 submanifolds in ${\mit\Sigma}_c$ of
 codimension, at least, two; hence
 ${\mit\Sigma}_c\setminus\widetilde{\mit\Sigma}_c$ is connected.
 \vskip 10pt
 \proclaim {Lemma 2.1}.
 $L_{\G_i^\L}$ is an hypoelliptic operator on 
 ${\mit \Sigma}_c\backslash\widetilde{\mit\Sigma}_c$, i.e., the Lie algebra
 generated by 
 $$
 \left\{X^\theta_{\a\b}\;|\; \a,\b\in\G_i\right\}
 $$
 spans the tangent space of ${\mit \Sigma}_c$ at each point in
 ${\mit \Sigma}_c\setminus \widetilde{\mit \Sigma}_c$.
  
 \proclaim Proof.
  
 Let us fix our attention on an arbitrary point in 
 ${\mit \Sigma}_c\backslash \widetilde{\mit \Sigma}_c$.
  
 We can represent a cluster of $n$ particles by a 
 graph $\widetilde{\cal G}_n$ with $n$ vertices. 
 Each vertex of the graph  corresponds to a particle; two vertices are 
 joined by a bond (edge) if and only if they correspond 
 to particles close enough to interact stochastically. 
 In a cluster of $n$ particles there are at least $n-1$ bonds:
 $$
 \sharp\{(\a,\b)\;|\;\sigma(q_\a-q_\b)< R_1\}\ge\ n-1  .
 $$
 We consider then a minimal acyclical (i.e., the edges do not form any loop)
 connected sub-graph ${\cal G}_n$ of
  $\widetilde{\cal G}_n$. This amounts to choose $n-1$ bonds.
 We choose a set of $2n-2$ linearly independent vector fields 
 $X_{\a\b}^{\theta_i}$, $i\in\{1,2\}$, $(\a\b)$ being a bond in $\Cal G_n$. 
 \par
 Two bonds are said to be contiguous if they have one vertex in common. 
 There are at least
 $n-2$ couples of contiguous bonds. 
 If we compute the commutator between two contiguous
 bonds $\{\a\b\}$ and $\{\gamma\b\}$ we obtain:
 $$
 [X_{\a\b}^{\theta_i},X_{\gamma\b}^{\theta_j}]\ =\langle 
 \eta_{\a\b}^{\theta_i},\,A_{\theta_j}D_{\gamma\b}\rangle-
 \langle\eta_{\gamma\b}^{\theta_j},\,B_{\theta_i}D_{\a\b}\rangle
 $$
 where $A_{\theta_j}$ is the matrix whose element $kl$ is given by
 $D^k_{\a\b}(\eta^{\theta_j}_{\gamma\b})_l$ (that is, the derivative, 
 with respect to
 the $k$ component of $p_\a-p_\b$, of the $l$ component of
 $\eta^{\theta_j}_{\gamma\b}$) and $B_{\theta_i}$ is the matrix 
 $D^k_{\gamma\b}(\eta^{\theta_i}_{\a\b})_l$.
 \par
 Let ${\cal G}_k,\;k<n$ be a sequence of connected sub-graph of $\Cal G_n$ 
 containing $k$ vertices, ${\cal G}_k\subset {\cal G}_{k+1}$, and
 ${\cal A}_k$ the Lie algebra generated by the vector fields 
 $X_{\a\b}^{\theta_i},\;
 \a,\,\b\in{\cal G}_k$. We will show by induction that ${\cal A}_n$ contains
 $3n-4$ linearly independent vector fields.
 \par
 The fact is clear for ${\cal A}_2$. Let us suppose that it is true for 
 ${\cal A}_k$ and let us show that ${\cal A}_{k+1}$ contains three extra 
 vector fields linearly independent from all the previous ones.
 \par
 We start by noticing that ${\cal G}_{k+1}$ is obtained by adding a particle
 (vertex), 
 say $\a$, to $\Cal G_k$ and that such particle
 is connected with the rest of the graph by only one bond.
 Let us call $\b$ the particle to which $\a$ is connected and $\gamma$
 a particle, in $\Cal G_k$, connected to $\b$. 
 ${\cal A}_{k+1}$
 is larger (or equal) than the set of vector fields generated by all 
 the vectors in ${\cal A}_k$, $X_{\a\b}^{\theta_i}$ and 
 $[X_{\a\b}^{\theta_i},\,X_{\gamma\b}^{\theta_j}]$. If our 
 inductive hypothesis is false, then there exist $\lambda_i\in 
 \RR$, $\mu_{ij}\in \RR$ and $Y_{ij}\in{\cal A}_k$ such that
 $$
 0=\lambda_1 X_{\a\b}^{\theta_1}+\lambda_2 X_{\a\b}^{\theta_2}+
 \mu_{ij} [X_{\a\b}^{\theta_i},\,X_{\gamma\b}^{\theta_j}]+ Y_{ij}
 $$
 for each $i,j\in\{1,2\}$.
 To see that this is impossible we apply the above tangent vector to $p_\a$
 $$
 0=\lambda_1\eta_{\a\b}^{\theta_1}+\lambda_2\eta_{\a\b}^{\theta_2}-
 \mu_{ij}\left(B_{\theta_i}\right)^T\eta_{\gamma\b}^{\theta_j} .
 $$
 By multiplying this relation by $D_{\a\b}T$, where $T$ is the kinetic energy
 defined inside \equ(hypersurf), and by 
 using the fact that, for points not in $\widetilde{\mit\Sigma}_c$, 
 $D_{\a\b}T\neq 0$, follows
 $$
 0=\mu_{ij}\langle B_{\theta_i}D_{\a\b}T,\,\eta_{\gamma\b}^{\theta_j}\rangle.
 $$
 To simplify the previous expression we recall that, by definition,
 $\langle\eta_{\a\b}^{\theta_j},\,D_{\a\b}T\rangle=0$ and applying the
 operator $D_{\gamma\b}$ to such an equality yields
 $$
 0=B_{\theta_i}D_{\a\b}T+H_\b\eta_{\a\b}^{\theta_i},
 $$
 where $H_\b$ is the matrix $\partial_{p_\b}\partial_{p_\b} T$. 
 It is essential to 
 notice that the Hessian $H_\b$ is positive definite, given the 
 convexity of $\phi$.
 \par
 By collecting the previous relations follows
 $$
 0=\mu_{ij}\langle H_\b\eta_{\a\b}^{\theta_i},\,\eta_{\gamma\b}^{\theta_j}
 \rangle   .
 $$
 To conclude the proof it is enough to show that at least one of the scalar
 products $\langle H_\b\eta_{\a\b}^{\theta_i},\,\eta_{\gamma\b}^{\theta_j}
 \rangle$ is different from zero.
 Notice that 
 $\{\eta_{\a\b}^{\theta_i}\}$ and $\{\eta_{\gamma\b}^{\theta_j}\}$ 
 span two 
 two-dimensional planes in $\RR^3$. Such planes must intersect, at least, at a 
 line. Let $\zeta\neq 0$ be a vector belonging to such a line. Clearly there exists
 $\tau_i$ $\tau_i'$ such that $\sum_l\tau_l\eta_{\a\b}^{\theta_l}=\zeta$
 and  $\sum_l\tau_l'\eta_{\a\b}^{\theta_l}=\zeta$. 
 This yields to the contradiction
 $0=\langle H_\b\zeta,\,\zeta\rangle $.
 That is, the vectors $X_{\a\b}^{\theta_1}$, $X_{\a\b}^{\theta_2}$,
 $\sum_{ij}\tau_i\tau_j' [X_{\a\b}^{\theta_i},\,X_{\gamma\b}^{\theta_j}]$
 are linearly independent with respect to themselves and with respect 
 to the ones in the algebra $\Cal A_k$.
 $\qed$ 
  
 \proclaim {Proposition 2.2}.
 The condition (iv) of reversibility of the measure 
 Q implies that, for any bond corresponding to connected particles $b=\{\a,\b\}$ in the 
 configuration $\o_q$ and any smooth local function $\phi$, we have
 $$
 \E^Q(X^\theta_{\a\b}\phi\;|\;\omega_q)\equiv
 \ec\left(X^\theta_{\a\b}\phi\right)=0
 $$
 provided that $\hbox{supp}(\phi)
 \cap \widetilde{\mit \Sigma}_c=\emptyset$.
 \vskip 10pt
  
 \proclaim Proof.     
  
 Let $\p$ and $\psi$ be an arbitrary smooth local functions with support in a finite 
 region $\Lambda'$, and $\Lambda$ a region containing $\Lambda'$ and so large
 that particles inside $\Lambda'$ cannot interact with particles 
 outside $\Lambda$. Since the size of the support of a test function can be 
 assumed arbitrarily small, without loss of generality, we will carry out
 a local argument; namely we will assume that the support of $\psi$ is 
 contained in a conveniently small neighborhood of an arbitrary configuration
 not belonging to $\widetilde{\mit \Sigma}_c$ (i.e., the support must be so 
 small as not to intersect $\wt{\mit\Sigma}_c$). 
 If the configuration $\o_q$ does not contain any cluster in $\Lambda$, 
 then the 
 Proposition is obviously true. Next, we will assume that only one 
 isolated cluster\nfootnote{Remember that an ``isolated cluster" in 
 $\L$ can still have interactions with particles outside $\L$.}
 $\Gamma_\L$ is present (the case in which several isolated 
 clusters are present
 can be treated in the same way, as we will remark at the end of the proof). 
 The graph associated to $\Gamma_\L$ is finite and it will 
 contain $M$ bonds.
 The reversibility condition (iv) implies:
 $$
 -\ec\left(\psi \hat L \p\right) = 
 \sum_{b\in\Gamma_\L}\sum_\theta\sigma_b 
 \ec\left(X^\t_b\psi X^\t_b\p\right)
 $$
 where $b=\{\a,\b\}$ is a generic bond in $\Gamma_\L$, i.e., $\sigma_b=
 \sigma(q_\a -q_\b) > 0 $. Note that the operators $X_{\a\b}$ with $q_\a$  
 or $q_\b$ not in $\L$ do not appear in the right hand side of the 
 above equation, although it is possible that $\sigma(q_\a-q_\b)\neq 0$;
 this is due to the fact that, since the test functions depend only on
 the particles in $\L'$, if $q_\a\not\in\L$ and 
 $\sigma(q_\a-q_\b)\neq 0$, then $q_\b\not\in\L'$ which implies
 $X_{\a\b}\psi=0=X_{\a\b}\varphi$.
  
 A technical obstacle to our proof is that, in general, the vector fields 
 $\{X^\t_{b_1},\ldots,X^\t_{b_M}\}$ 
 are neither linearly independent nor their linear combinations 
 span all the Lie algebra that they generate. Typically, only $L\leq KM$
 such vector fields will be linearly independent,\nfootnote{KM is the 
 cardinality of $\{X^\t_{b_1},\ldots,X^\t_{b_M}\}$; remember that $\theta\in
 \{1,\,...,\,K\}$.}
 while the Lie algebra will be $N\ge L$ dimensional.
 To overcome such problem we choose, 
 among $\{X^\t_{b_1},\ldots,X^\t_{b_M}\}$ and their commutators, 
 a subset of linearly independent vector fields 
 $\{X_1,\ldots,X_N\}$ that form a base of the Lie algebra.\nfootnote{This
  is possible provided the support of $\psi$ is sufficiently
 small.} In addition, we require 
 $$
 \{X_1,\ldots,X_L\}\subset  \{X^\t_{b_1},\ldots,X^\t_{b_M}\} .
 $$
  
 Thus, the original $KM$ vector fields can be expressed as linear combinations 
 of the independent vector fields $\{X_1,\ldots,X_L\}$:
 $$
 X_{b_j}^\t=\sum_{i=1}^L \nu_{ji}^\t X_i \qquad\qquad j=1,\ldots,KM .
 $$
  
 Since,
 $$
 [X_{b_j}^\t,\,X_{b_k}^{\t'}]= \sum_{l,\,p}[\nu_{jl}^\t X_l,\,
 \nu_{kp}^{\t'} X_p]=\sum_{l,\,p}\left\{\nu_{jl}^\t(X_l \nu_{kp}^{\t'})
 X_p-\nu_{kp}^{\t'}(X_p \nu_{jl}^\t)X_l+
 \nu_{jl}^\t \nu_{kp}^{\t'} [X_l,\, X_p]\right\}
 $$
 it is clear that  $\{X_1,\ldots,X_L\}$  generates the complete Lie 
 algebra under consideration.
  
 Let $A$ be the $L\times L$ matrix with elements defined by
 $$
 a_{i,k}=\sum_{j=1}^{M}\sum_\t \sigma_{b_j}\nu_{ji}^\t \nu_{jk}^\t
 $$
 then
 $$
 \sum_{b\in\Gamma_\L}\sum_\t\sigma_b \ec\left(X^\t_b\psi X^\t_b\p\right) 
 = \sum_{i,k=1}^L\ec\left(a_{ik} X_i\psi X_k\p\right)     .
 $$
  
 It is easy to check that the matrix A is positive defined, and therefore 
 invertible.
  
 According to Lemma 2.1  $\{X_{1},\ldots,X_{N}\}$ span the tangent
 space of ${\mit \Sigma}_c$ (the surfaces associated to the cluster
 $\Gamma_\L$). Since such surfaces foliate the phase space of the
 particles contained in $\Gamma_\L$,
 we can choose coordinates $(c,\,y)$ such that the
 vector fields $\{Y_i\}_1^N$, associated to the coordinates $\{y_i\}_1^N$, 
 generates the tangent space of ${\mit \Sigma}_c$ (i.e., for each
 $c$, $\{y\}$ is a system of coordinates for ${\mit \Sigma}_c$). 
 This implies, $\forall i,j$,
  
 $$
 [Y_i,Y_j]=0 \quad ;\qquad Y_i^*=-Y_i\quad ; 
 \qquad Y_iy_j=\delta_{ij}\quad.
 $$
  
 In addition, there exists an invertible $N\times N$ matrix $\L$, 
 such that
 $$
 X_i=\sum_{j=1}^N\L_{ij}Y_j \; .
 $$
  
 Let us choose as function $\p$ a coordinate function $y_j$ multiplied 
 by a smooth
 function with value one on the support of $\psi$, which, consequently, 
 can be ignored.
 Applying $\hat L$ we have
 $$
 \eqalign{
 \hat L\ y_j = \sum_{k,i}\ X_k\ a_{ki}\ X_i\ y_j \cr
 =\sum_{k,i}\ X_k\ a_{ki}\ \L_{ij} \cr}
 $$
  
 where we have used
 $$
 X_i\ y_j\ =\ \sum_l\ \L_{il}\ Y_l\ y_j\ =\ \L_{ij}.
 $$
  
  
 The reversibility relation then gives us:
 $$
 -\sum_{k,i}\ec\left(\psi X_k\ \left( a_{ki}\ \L_{ij}\right)\right)\ 
 =\ \sum_{k,i}\ec\left( a_{ki}\ \L_{ij}\ X_k\ \psi\right)
 $$
 which is equivalent to
 $$
 \sum_{k,i}\ec\left( X_k \left(\psi\ a_{ki}\ \L_{ij}\right)\right)\ =\ 0 .
 $$
  
 Let $V=\L^{-1}\RR^L\subset \RR^N$,\nfootnote{By $\RR^L$, here we mean 
 $\{v\in\RR^N\;|\;v_i=0\;\forall i>L\}$.} then $A\L\,:\,V\to\RR^L$  is 
 one to one and onto.
 Which means  that for each $e^k\in\RR^L$, $e^k=(0,\dots,1,\dots,0)$, there
 exists $\alpha^k\in V\subset \RR^N$ such that $A\L\alpha^k=e^k$. Moreover, in some 
 small neighborhood of any configuration, $\alpha^k$ will vary smoothly.
  
 We can make the following $L^2$ different choices of $\psi$
 $$
 \psi_{jh}\ =\ \alpha^h_j\phi
 $$
 where $\phi$ is a function with sufficiently small support around the
 configuration we are considering.
  
 Summing over $j$ we obtain
 $$
 0=\sum_{i,j,k}\ec\left(\ X_k (\alpha^h_j\ \L_{ij}a_{ki}\ 
 \phi)\right)=\sum_k\ec\left(X_k e^h_k\phi\right) ,
 $$
 that is to say 
 $$
 \ec\left(X_h\ \phi\right)\ =\ 0 \qquad\forall\;h\in\{0,\dots,L\}
 $$
 which implies our thesis.
  
 The generalization to the situation where many clusters appear in the region
 $\L$ is straightforward since, in the above argument, the coordinate
 functions $y_j$ are localized on the particular cluster
 we are considering. Hence, the argument simply factors over the different 
 clusters.
 $\qed$
  
 \vskip 20pt
 Up to now we have seen that the measure is invariant with respect to
 vector fields that 
 generate the tangent space to the surfaces of the momenta of the 
 clusters  $\G_i^\L$ with constant kinetic energy and momentum.
 This was done only by using the reversibility of the stochastic dynamics. 
If in $\L$ the cluster was unique (like in the case with infinite range
stochastic interaction), then
this would imply that the measure on the momenta conditioned on 
the position is microcanonical, i.e. we would have directly
lemma 5.1 below. Unfortunately in our case we cannot ignore the existence of
isolated clusters. So what we can conclude at this point is that, conditioned on
the positions, the distribution of the velocities in each cluster is microcanonical.
In order to arrive to the statement of lemma 5.1, we need to somehow exchange 
the particles between clusters. The only way to do this is to generate,
with the help of the Hamiltonian dynamics, other dynamics for which the measure 
is invariant and that permit such exchanges of particles among clusters.
In the rest of the section we will define these dynamics and prove their local 
properties, and in the section four we will use them to move particles
among clusters.

We start by studying the Lie algebra generated by 
 $\{X^\theta_{\a\b};\;[X^\theta_{\a\b},\,L]\}_{q_\a,q_\b\in\L}$.
  
 \proclaim{Lemma 2.3}. 
 For each region $\L$, each local smooth function $\varphi$ localized in 
 $\L$, calling $\Cal A_\L$ the Lie algebra generated by the
 operators
 $\{X^\theta_{\a\b};\;[X^\theta_{\a\b},\,L]\}_{q_\a,q_\b\in\L}$, we have
 $$
 \E^Q\left(X\varphi\;\big|\;|\o_\L|=n;\,\o_{\L^c}\right)=0
 $$
 for each $X\in\Cal A_\L$.
  
 \proclaim{Proof}.
  
 The difficulties arise because $L$ does not conserve the number of particles 
 in a finite region. We need to use here the stationarity of $Q$ 
 with respect to $L$.
  
 Let $\chi_\ve(q)$ be a smooth function equal to one if 
 $q\in\L$, and equal to $0$ if
 the distance between $q$ and $\L$ is larger than $\ve$.
 We can then define $N_\ve\equiv\sum_\a\chi_\ve(q_\a)$ to be an approximation
 of the number of particles in $\L$. Clearly, when $\ve$ goes to
 zero, $N_\ve$ tends to the number of particles contained in the closure of
 $\L$, which, since $Q$ is locally absolutely continuous, 
 equals almost everywhere the number of particles contained in the interior. 
  
 Let $h$ be a smooth function on $\RR^+$ with compact support and $\varphi$
 any smooth local function with support contained in the interior of $\L$;
 in addition, we
 consider arbitrary smooth functions $\psi_{\a\b}:\RR^6\to\RR$ with support in
 $\L\times\L$ and we use them to define the local operators
 $X(\psi)\equiv\sum\limits_{\a\b}\psi_{\a\b}(q_\a,\,q_\b)\sigma(q_\a-q_\b)
 X_{\a\b}$ (clearly all these operators are part of the Lie algebra
 $\Cal A_\L$). Using the previous definitions, since 
 $X(\psi)h(N_\varepsilon)=0$, we have,
 $$
 0=\E([L,\,X(\psi)]\varphi h(N_\ve))=
 \E(h[L,\,X(\psi)]\varphi)-\E(\varphi X(\psi)Lh) .
 $$
 Since $L h(N_\ve)=h'(N_\ve)\sum\limits_{\gamma\in\L^c}
 \langle p_\gamma,\,\nabla\chi_\ve(q_\gamma)\rangle$, we have that
 $X(\psi)L h = 0$. So we conclude that
 $$
 0=\E(h(N_\ve)[L,\,X(\psi)]\varphi) .
 $$
 Letting $\ve\to 0$ proves that it is possible to condition with respect to
 the number of particles in $\L$; a similar computation shows
 that it is possible to condition with respect to the configuration outside
 $\L$ as well.
 $\qed$
  
Lemma 2.3 shows that $\Cal A_\L$ has interesting local properties, these are further
clarified by the following Lemma. Consider configurations with $n$ particles in $\L$
and define $\Pi_\L$, $E_\L$ like in the equations above theorem 1.2.
  
 \proclaim {Lemma 2.4}.
 The Lie Algebra $\Cal A_\L$
 is tangent to the surface $\Pi_{\L}$=constant, $E_{\L}$=constant,
 and acts only on observables depending on the coordinates of the particles
 inside $\L$.
  
 \proclaim Proof.
  
 Given two particles $\a,\,\b\in\Gamma$  we have
 $$
 \eqalign{
 & X^\theta_{\a\b}\Pi_{\L} =0\cr 
 & X^\theta_{\a\b}E_{\L} =0\cr 
 & [X^\theta_{\a\b},\,L] \Pi_{\L} =X^\theta_{\a\b}\sum_{\gamma}
 {\partial T\over \partial q_\gamma}=0\cr
 & [X^\theta_{\a\b},\,L] E_{\L} =X^\theta_{\a\b}\left[
 \sum_\gamma\langle{\partial \Cal H\over \partial p_\gamma},\,
 {\partial E_{\L}\over \partial q_\gamma}\rangle -
 \langle{\partial \Cal H\over \partial q_\gamma},\,
 {\partial E_{\L}\over \partial p_\gamma}\rangle\right]\cr
 }
 $$
 Letting $\Delta=\Cal H -E_{\L}=\sum_{\gamma\not\in\L} 
 \phi(p_\gamma)+{1\over 2}
 \sum_{q_\gamma\not\in\L} \sum_{q_\delta\not\in
\L;\;\delta\neq\gamma}V(q_\gamma-q_\delta)$,
  and $H_\gamma=\left({\partial^2\phi(p_\gamma)\over\partial p^i_\gamma  
  \partial p^j_\gamma}\right)$, we can rewrite the last equation as
 $$
 \eqalign{
 [X^\theta_{\a\b},\,L] E_{\L} &=X^\theta_{\a\b}\left[
 \sum_\gamma\langle{\partial \Delta\over \partial p_\gamma},\,
 {\partial E_{\L}\over \partial q_\gamma}\rangle -
 \langle{\partial \Delta\over \partial q_\gamma},\,
 {\partial E_{\L}\over \partial p_\gamma}\rangle\right]\cr
 &=-\langle {\partial \Delta\over \partial q_\a},\, H_\a\eta^\theta_{\a\b}
 \rangle
 +\langle {\partial \Delta\over \partial q_\b},\, H_\b\eta^\theta_{\a\b}
 \rangle =0
 }
 $$
 since $ \Delta$ does not depend on $q_\alpha$ or $q_\beta$.
  
 Similarly, a direct computation shows that, if $q_\gamma\not
 \in\L$, then
 $$
 \eqalign{
 [X^\theta_{\a\b},\,L] q_\gamma &= 0\cr
 [X^\theta_{\a\b},\,L] p_\gamma &= 0 \, .\cr
 }
 $$
 $\qed$
  
  
 At this point we have to distinguish between the Gaussian 
 and the non-Gaussian case.
 The difference is that in the gaussian case the center of mass is always 
conserved. Define
$$
\Theta_\L\ =\ \sum_{q_\a\in\L} q_\a
$$
  
 \proclaim {Lemma 2.5}.
 If $\phi$ is quadratic, the Lie Algebra $\Cal A_\L$
 is tangent to the surface $\Pi_{\L}$=constant, $E_{\L}$=constant, 
 and   $\Theta_{\L}$=constant.
  
 \proclaim Proof.
  
 All we need to compute is
 $$
 \eqalign{
 & X^\theta_{\a\b}Q_{\L} =  0\cr
 &[X^\theta_{\a\b},\,L] Q_{\L} = X^\theta_{\a\b}
 \sum_{\gamma\in\L} {\partial\phi\over\partial p_\gamma}=
 (H_\a-H_\b)\eta^\theta_{\a\b}=0\cr
 }
 $$
 since, in the present case, $H_\a=H_\b$=constant.
 $\qed$
 \vskip .5cm
  
 This means that, in the Gaussian case, the vector fields we are considering 
 conserve the center of mass, even if this is not conserved by $L$;
 accordingly, the Lie Algebra generated by $\{X_{\a\b}^\theta,\,
 [X_{\a\b}^\theta,\,L]\}$, for some $\a,\b\in\Gamma$ ($\Gamma$ being 
 some cluster in $\L$), 
 can be at most five dimensional.\nfootnote{Here and in the following for 
dimension of a Lie Algebra we mean the minimal dimension of it when 
restricted to the tangent spaces at different points.} 
We prove that the algebra has the largest
 possible dimension. 
  
 \proclaim {Lemma 2.6}.
 If $\phi$ is quadratic, and $\a,\b\in\Gamma$ are connected, 
 then the Lie Algebra generated by $\{X^\theta_{\a\b};\;
 [X^\theta_{\a\b},\,L]\}$ is five dimensional.
  
 \proclaim Proof.
  
 Applying the vector fields to $q_\a$ we have
 $$
 \eqalign{
 [X^\theta_{\a\b},\,L]q_\a=&H_\a\eta^\theta_{\a\b}\cr
 [X^\theta_{\a\b},\,[X^\theta_{\a\b},\,L]]q_\a=&H_\a 
 D_{\a\b}(\eta^\theta_{\a\b})\eta^\theta_{\a\b}\cr}
 $$
 This vectors span a three dimensional vector space and are linearly 
 independent with respect with the vectors $X_{\a\b}^\t$. To see this, it 
 is sufficient to consider a generic linear combination, equal it to 0, and
 multiply it by $H_\a^{-1}D_{\a\b}E$, then
 $$
 0=\sum_i\mu_i\langle D_{\alpha\beta}E,\, 
 \eta_{\alpha\beta}^{\theta_i}\rangle+
 \nu\langle D_{\alpha\beta}E,\,D_{\alpha\beta}(\eta_{\alpha\beta}^{\theta_1})
 \eta_{\alpha\beta}^{\theta_1} \rangle  .
 $$
 Next, remember that $\langle D_{\alpha\beta}E,\,
 \eta_{\alpha\beta}^{\theta}\rangle=0$,
 differentiating such an expression by $D_{\alpha\beta}$ one gets
 $$
 (H_\alpha+H_\beta)\eta_{\alpha\beta}^{\theta}+
 D_{\alpha\beta}(\eta_{\alpha\beta}^{\theta})^T D_{\alpha\beta}E=0
 $$
 and, multiplying it by $\eta_{\alpha\beta}^{\theta} $,
 $$
 \langle \eta_{\alpha\beta}^{\theta},\,(H_\alpha+H_\beta)
 \eta_{\alpha\beta}^{\theta}\rangle=
 -\langle D_{\alpha\beta}E,\,D_{\alpha\beta}(\eta_{\alpha\beta}^{\theta})
 \eta_{\alpha\beta}^{\theta}\rangle.
 $$
 Using the above equalities we obtain
 $$
 \nu\langle\eta_{\alpha\beta}^{\theta_1},\,(H_\alpha+H_\beta)
 \eta_{\alpha\beta}^{\theta_1}\rangle=0
 $$
 that is $\nu=0$. From this follows $\mu_i=0$.
 $\qed$
\vskip 0.5 cm
  
 In the non-Gaussian case the center of mass is not conserved 
 by the vector fields we are considering, 
 and we have no other obvious conserved quantity. 
 We impose a condition on the noise to make sure that 
 there are no conserved quantities, beside those considered in Lemma 2.4.
 More precisely we require the following:
  
 \proclaim Condition on the Noise.
 For each two particles $\a,\,\b$, interacting stochastically,
 we require that the Lie algebra generate by the vectors
 $X_{\a\b}^\theta$ and $[X^\theta_{\a\b},\,L]$ is eighth dimensional
 at each point of every surface with fixed total energy and total momentum
 except, at most, for the finite union of smooth manifolds of codimension two
 $\wt{\mit\Sigma}_{\a\b}$.
  
 In Appendix 1 we will show that if $\phi$ satisfy (NG) then the above 
 condition is satisfied.
  
 We introduce two family of surfaces in $\RR^{6n}$,
 $$
 \eqalign{
 \Xi(n,\,\Pi,\,E,\,\omega_c)&=\left\{(q,\,p)\in\RR^{6n}\;\bigg|\;
 \sum_\a p_\a=\Pi;\;\sum_\a\phi(p_\a)+{1\over 2}
 \sum_{\a,\b} V(q_\a-q_\b)\right.\cr
 &\ \ +\left.\sum_\a\sum_{\b\in\omega_c}V(q_\a-q_\b)=E\right\}\cr
 \Xi(n,\,\Theta,\,\Pi,\,E,\,\omega_c)&=\left\{(q,\,p)\in
 \Xi(n,\,\Pi,\,E,\,\omega_c)\;\big |\;\sum_\a q_\a=\Theta\right\}\cr
 }
 $$
 and let $\widetilde\Xi$ be the union of the sets for which $(p_\a,\,p_\b)\in 
 \wt{\mit\Sigma}_{(\a,\b)}$, for some $\a\neq \b$.
 By hypotheses $\widetilde \Xi$ has at least codimension two in $\Xi$, in
 additions it has zero Lebesgue measure.
  
 \proclaim{Lemma 2.7}.
 For all $n\in\NN$, for almost all $\Pi,\, E,\,\Theta$, 
 for each $X\in\Cal A_\L$, 
 and any local function $\varphi$ with support 
 contained in $\L$ and disjoint from $\wt\Xi$
 $$
 \E(X\varphi\;|\;\o_\L\in\Xi,\,\o_{\L^c})=0.
 $$
 In addition, if $\L$ contains a unique cluster, then 
 the Lie algebra $\Cal A_\L$ contains all the
 tangent space of $\Xi$ at each point of $\Xi\backslash\widetilde\Xi$.
  
 \proclaim {Proof}.
  
  
 The first condition follows from lemma 2.3 and lemma 2.4 (or lemma 2.5 for
 the Gaussian case). To address the second part of the lemma we
 start an induction argument similar to the one used in lemma 2.1. 
 We want to generate a 6n-7 dimensional
 Lie algebra in the Gaussian case and a 6n-4 dimensional Lie algebra in the 
 non-Gaussian case. In both cases we need at each step of the induction 
 argument, i.e., for every particle $\a$ that
 we add to a cluster, six new independent vector fields.
  
 From the proof of Lemma 2.1 we have already three independent 
 vector fields generated by 
 $\{X^{\theta^i}_{\a\b},X^{\theta^j}_{\b\gamma}\}$. All these are acting 
 only in the direction of the momenta, so all we need is to look at the action 
 of the new vector
 fields on the positions direction to establish their linear independence.
  
 Define
 $$
 \widetilde L_{\a\b}^{\theta_k} = [X_{\a\b}^{\theta_k},\,L]
 $$
  
 $$
 \overline{L}_{\a\b\gamma}^{\theta_k\theta_l} =  [X_{\b\gamma}^{\theta_l},
 \widetilde L_{\a\b}^{\theta_k}].
 $$
  
 Applying these vector fields to $q_\a$ we have:
 $$
 \eqalign{
 \widetilde L_{\a\b}^{\theta_k}q_\a &= X_{\a\b}^{\theta_k} L q_\a = 
 H_\a \eta^{\theta_k}_{\a\b}\cr
 \overline{L}_{\a\b\gamma}^{\theta_k\theta_l} q_\a &=
 X_{\b\gamma}^{\theta_l} H_\a \eta^{\theta_k}_{\a\b} = 
 H_\a\left( D_\b\eta^{\theta'}_{\a\b}\right)^T\eta^\theta_{\b\gamma} .\cr}
 $$
  
 It is then enough to prove that the vectors 
 $X_{\a\b}^{\theta_i}$, $[X_{\a\b}^{\theta_i},\,X_{\gamma\b}^{\theta_j}]$, 
 $\widetilde L_{\a\b}^{\theta_i}$,
 $\sum_{ij} \xi_{ij}\overline{L}_{\a\b\gamma}^{\theta_i\theta_j}$, for
 some choice of $\xi_{ij}$, and $Y$ (where $Y$ belongs to the lie algebra
 generated by the vectors already considered during the induction
 procedure) are linearly
 independent. Again we assume that it is not so, i.e.,
  
 $$
 0=\sum_{k=1}^2\L_k X_{\a\b}^{\theta_k}+
 \mu_{ij} [X_{\a\b}^{\theta_i},\,X_{\gamma\b}^{\theta_j}]+ 
 \sum_{k=1}^2\nu_k \widetilde L_{\a\b}^{\theta_k}
 +\tau\sum_{kl}\xi_{kl}\overline{L}_{\a\b\gamma}^{\theta_k\theta_l}+Y,
 $$
 for some $\L_i,\,\mu_{ij},\,\nu_i,\,\tau,\,Y$. 
 We apply the previous expression to, $q_\a$ and obtain
 $$
 0=\sum_{k=1}^2\nu_k H_\a\eta^{\theta_k}_{\a\b}+\tau\sum_{kl} \xi_{kl} H_\a
 \left(D_{\b}\eta^{\theta_l}_{\a\b}\right)^T\eta^{\theta_k}_{\gamma\b}.
 $$
 If we multiply by $H_\a^{-1}D_{\a\b}E$, recalling the properties of $\eta$,
 we obtain
 $$
 0=\tau\sum_{kl}\xi_{kl}\langle 
 H_\a\eta^{\theta_k}_{\a\b},\,\eta^{\theta_l}_{\gamma\b}\rangle .
 $$
 Which shows that, out of $\mit{\widetilde\Sigma}$, it is always possible to 
 choose $\xi_{kl}$ such that the sum is different from zero. This implies
 $\tau=0$ and allows us to conclude the proof in complete analogy with 
 lemma 2.1.
 $\qed$
 
 As promised, we have found a bundle of local dynamics preserving the measure $Q$
(or, more precisely, its local conditional measures), i.e. the dynamics 
generated by the vector fields in the Lie algebra
$\Cal A_\L$.


\vskip 1cm {\bf 3. CONDITIONING TO TYPICAL CONFIGURATIONS }.
 \vskip.5cm
 
 \numsec=3\numfor=1\numtheo=1

Using the entropy bound and large deviations estimates, we will show here
that certain configurations have probability 0 for any measure $Q$
satisfying our hypotheses. We will need to exclude these configurations from the
considerations of the next section.  
 
First of all, we want to disregard configurations with 
locally big barriers of potential, so we are going to analyse those
configurations with local high density.
 
 \proclaim{Lemma 3.1}.
 Let $\L\subset\RR^3$ and $\Delta\subset\L$ be a
box of size $R_1$, consider the following configurations
 $$
 \Omega^\ve_\L=\{\omega \;|\;
 \exists \Delta\subset\L\,:\, |\omega_\Delta|\ge\ve^{-1}|\L|^{1\over 2}\}.
\Eq (bad1)
 $$ 
 If $Q$ satisfies condition (i) of theorem 1.2 (entropy bound), then there 
 exists $C>0$ such that:
 $$ 
 Q(\Omega_\L^\ve)\leq C\ve^2.
 $$
 

\proclaim{Proof}.

 By the entropy inequality
 $$ Q(\Omega^\ve_\L) \le {\log 2 + H_\L (Q|\PP) \over 
 \log\left(1+ \PP(\Omega^\ve_\L)^{-1}\right)}
 $$ 
 (which is a consequence of \equ(e-ineq) ), and condition (i) of theorem 1.2,
 we need only to prove that for a given grancanonical measure $\PP$
 $$
 \PP(\Omega^\ve_\L) \le C_2|\L|\exp(-C_1|\L|\ve^{-2})
 $$ for some constants $C_1,C_2>0$ independent from $\ve$ and $\L$.
  
 Since the measure $\PP$ is translation invariant
 $$
 \PP(\Omega_\L^\ve) \le {C_1\over R_1}|\L|
 \PP(\{|\omega_{\Delta}|>\ve^{-1}|\L|^{1\over2}\}).
 $$
  Accordingly, (setting
 $\Gamma=\int_{\RR^3}e^{-\l_4\phi(p)+\sum_{i=1}^3\l_i p_i}dp$)
 $$
 \eqalign{
 \PP(|\omega_\Delta|>\ve^{-1}|\L|^{1/2})=  
 &Z_\Delta^{-1}\sum_{n\ge
 \ve^{-1}|\L|^{1/2}}^\infty  {e^{\L_0 n}\Gamma^n\over n!}
 \int_{\Delta^n}  e^{-\l_4 V_{\Delta}}\cr
 \leq &Z_\Delta^{-1}\sum_{n\ge \ve^{-1}|\L|^{1/2}}^\infty{e^{\l_0
 n}\Gamma^n\over n!} \exp\left[-\l_4 A {\ve^{-2}|\L|\over\Delta} +
 \l_4 B
 \ve^{-1}|\L|^{1/2}\right] |\Delta|^n\cr
 \leq& C_2 e^{- C_3 \ve^{-2}|\L|}\cr}
 $$
 
 where we have used the explicit form of the grand canonical measures,  the positivity
 and the superstability of the potential.
 $\qed$
\medskip

 Another information needed in the following arguments is a bound on the total kinetic
 energy shared by a large number of particles.
 
 \proclaim{Lemma 3.2}. Let $a>0$, and $\ve>0$ sufficiently small,
 $\L\subset\RR^3$ and consider the following configurations
 $$
 \widetilde\Omega^\ve_\L=\left\{\omega \;\bigg|\;\exists
 \{\a_i\}_{i=1}^{a|\L|}\,:\,q_{\a_i}\in \L,\ 
\sum_{i=1}^{a|\L|}\left[\phi(p_{\a_i})-\phi
\left({\sum_{i=1}^{a|\L|}p_{\a_i}\over
 a|\L|}\right)\right]<\ve a|\L|\right\},
\Eq (bad2) 
$$ then there exists $C>0$: 
 $$ \lim_{|\L|\to\infty} Q(\widetilde\Omega^\ve_\L)\leq
 {C\over a\ln(\ve^{-1})}.
 $$
 
 \proclaim{Proof}.
 
 We will use the same entropy bound as in the previous lemma. 
In order to simplify
 notations, we choose a grancanonical measure $\PP$ corresponding to the parameters
 $ \l_1=\l_2=\l_3=0$ and $\l_4$ such that 
 $\Gamma_\L = \int e^{-\l_4\phi(p)} dp = 1$. 
 
 Let us define
 $$  Y_m = {1\over m}\sum_{i=1}^m \phi(p_i) - \phi \left({1\over m}\sum_{i=1}^m
 p_i\right)
 $$ and observe that since $\phi$ is convex $Y_m$ is non--negative. Then we have
 $$
 \PP(\widetilde\Omega^\ve_\L) = Z_\L^{-1}\sum_{n\ge a|\L|}
 {e^{n\l_0}\over n!}{n\choose [a|\L|]} J_{[a|\L|]}\int e^{-\l_4
 V_\L} = e^{c|\L|}J_{[a|\L|]}\PP\left(\{\omega ||\omega_\L|\ge 
 [a|\L|]\}\right)
 $$ 
 where $c$ is some constant depending on $\PP$, and
 $$ J_m = \int_{Y_m<\ve} e^{-\l_4 \sum_{i=1}^m \phi(p_i)}\;d^m p
 $$ and in the following $m=[a|\L|]$. By exponential Chebicheff inequality, for
 any
 $\beta>0$
 $$ J_m \le e^{\beta\ve m} \int e^{-m\beta Y_m} e^{-\l_4 \sum_{i=1}^m
 \phi(p_i)}\;d^m p  .
 $$ 
 By large deviation asymptotic (cf.[V])
 $$
 \lim_{m\to\infty}{1\over m}\log 
 \int e^{-m\beta Y_m} e^{-\l_4 \sum_{i=1}^m \phi(p_i)}\;d^m p =
 \sup_{\mu}\left\{\beta\left[\phi(\bar\mu) - \widehat\phi(\mu)\right] - 
 I(\mu)\right\},
 $$ 
 where $\mu(p)$ are probability densities on $\RR^3$  (with respect to
 $e^{-\l_4\phi}dp$), 
 $$
 \bar\mu = \int p \mu(p) e^{-\l_4\phi}dp,\qquad
 \widehat\phi(\mu) = \int \phi(p) \mu(p) e^{-\l_4\phi}dp
 $$ 
 and
 $$ I(\mu) = \int \mu(p)\log\left(\mu(p)\right)
  e^{-\l_4\phi(p)}dp\ .
 $$ 
 Since $\phi(x)=\phi(-x)$ the variational problem can be explicitly
 solved and the maximizing $\mu$ is given by
 $$ 
 {e^{-\beta\phi(p)}\over \int e^{-(\beta+\l_4)\phi(p')}dp'} .
 $$ 
 We can then compute
 $$
 \sup_{\mu}\left\{\beta\left[\phi(\bar\mu) - \widehat\phi(\mu)\right] - I(\mu)\right\}
 =\log \int e^{-(\beta+\l_4)\phi(p)} dp .
 $$ 
 Optimizing on $\beta$ we obtain
 $$
 \lim_{m\to\infty}{1\over m}\log J_m \le \inf_{\beta>0}
 \left[\beta\ve + \log \int e^{-(\beta+\l_4)\phi(p)} dp \right].
 $$ 
 By hypothesis there exists $k\in\NN^+$ :
 $$
 \eqalign{
 \phi(p)&\ge c_1\|p\|^k\quad\forall \|p\|\leq 1\cr
 \phi(p)&\ge c_2\|p\|\quad\forall\|p\|>1,}
 $$
 it follows
 $$
 \int e^{-\nu\phi(p)}dp\leq\int_{\|p\|\leq 1}e^{-\nu c_1\|p\|^k}
 +\int_{\|p\|\ge 1} e^{-\nu c_2\|p\|}\leq c_3\nu^{-{1\over k}},
 $$
 for each $\nu>1$. Using the above estimates and minimizing over $\beta$ the 
 lemma follows.
 $\qed$ 
 


\vskip 1cm {\bf 4. CLUSTERING }.
 \vskip.5cm
 
\numsec=4\numfor=1\numtheo=1
   
Before getting into the technicalities of the clusters deformations,
let us pause here to explain our strategy. 

As we already mentioned in section 2, from lemma 2.1 and 2.2 follow that
the measure $Q$ on a box $\L_0$ conditioned on the positions,
on the total momentum and on total kinetic energy:
$$
Q_{\L_0} \left(dp_1,\dots,dp_n\Big|q_1,\dots,q_n ; \sum_{\a =1}^n p_\a, 
\sum_{\a =1}^n \phi(p_\a) \right)
\Eq (ponq)
$$
is Microcanonical only for the $p$'s corresponding to
the particles in the same cluster, in particular this measure is symmetric for 
exchange of momentum between particles of the same cluster (by 
``exchange of momentum'' we mean any transfer of momentum between two particles that
conserves the total kinetic energy). 
If we could show that a measure is symmetric  for exchange of momentum between
clusters, it would follow that such a measure is Microcanonical, i.e. lemma 5.1
below (see appendix II for details). One way to achieve this could be to find a transformation on the phase
space, for which the measure $Q$ is invariant, that brings a particle $\a$ from a
cluster
$\Gamma_1$ in ``contact'' to another cluster $\Gamma_2$, then exchanges the momenta 
with a particle $\b$ of $\Gamma_2$, then brings back $\a$ to the initial position in
the cluster $\Gamma_1$. We cannot do exactly this, but we will exchange momenta
between the clusters performing more complicated transformations for which our measure
$Q$ is still invariant.

Given a box $\L_0$ and
 a configuration $\o\in\Omega$, let $T_{\a,\b}\o$
the configuration obtained exchanging  momenta between the particle $\a$
and particle $\b$, where $\a$ and $\b$ are two particles with
position in $\L_0$ (fix any amount $\eta\in\RR^3$ of momenta to be exchanged
compatible with the conservation of the total kinetic energy of the two particles).
Observe that only momenta is exchanged  while positions are unchanged.
 Furthermore such
operation does not change the total momenta in $\Pi_{\L_0}$, nor the total kinetic
energy $K_{\L_0}$ in the region $\L_0$. All we have to prove is that
$$
\int\sum_{q_\a,q_\b \in \L_0} \left[ F(T_{\a,\b}\omega) -\ F(\omega) \right]\; 
dQ(\omega )\ =\ 0
\Eq (exbis)
$$
for any local smooth function $F(\omega)$. 

It is very easy to see why \equ(exbis) implies the symmetry of the measure on the momenta \equ(ponq). 
Choose $F(\omega)= F_1(p_{\L_0}) F_2(q_{\L_0},\Pi_{\L_0},
 K_{\L_0} )$. Since $T_{\a,\b}$
leaves invariant $F_2$, one can condition the relation \equ(exbis) on the quantities 
on which $F_2$ depends and obtain
$$
\int\left[ F_1(T_{\a,\b}p_{\L_0}) -\ F_1(p_{\L_0}) \right]\; 
dQ( p_{\L_0} \big| q_{\L_0}, \Pi_{\L_0}, K_{\L_0})\ =\ 0
\Eq (exmom)
$$
i.e. that the measure defined by \equ(ponq) is invariant for exchange of
momenta between particles.

What we already know is that \equ(exmom) is true if $\a$ and $\b$ 
are in the same cluster (defined by the configuration $q_{\L_0}$ on which
we have conditioned).

By condition (ii)\nfootnote{If, as noted in remark 1.3, the condition is on the
potential energy, just substitute the definition of $\widehat\Omega^{\L,\ve}$ with
$$
\widehat\Omega^{\L,\ve} = \left\{ \o:U_{\L_1}>0, U_\L>0\right\}
\cap (\Omega^\ve_\L)^c \cap (\widetilde\Omega^\ve_\L)^c.
$$
and the rest of the argument of this section will remain essentially 
unchanged.} 
of our main theorem, we can choose $a > 0$ such that 
$\rho(\o) >\rho_* + 2a$ with $Q$--probability 1. 
 For any $\ve > 0$ small enough,
and $\L \supset \L_0$ large enough, with linear size $L$, 
define the set of good configurations 
$$
\widehat\Omega^{\L,\ve} = \left\{ \o: \Big| {|\o_{\L_1}|\over|\L_1|}-\rho(\o)| \Big|
\le a\ ;\ \Big| {|\o_{\L}|\over |\L|} -\rho(\o)| \Big| \le a \right\}
\cap (\Omega^\ve_\L)^c \cap (\widetilde\Omega^\ve_\L)^c ,
$$
where $\L_1$ is a box concentric to $\L$ of linear size $L/2$.
Then by lemma 3.1 and 3.2
$$
\lim_{\ve\to 0} \lim_{|\L|\to\infty} 
Q\left((\widehat\Omega^{\L,\ve})^c \right)\ =\ 0 \ .
$$

So it is enough to show that, for any $\ve>0$ we can find $\L$  large enough 
such that
$$
\int_{\widehat\Omega^{\L,\ve}}\sum_{q_\a,q_\b \in \L_0} \left[ F(T_{\a,\b}\omega) 
-\ F(\omega) \right]\; 
dQ(\omega )\ =\ 0
\Eq (vexmom)
$$
for any bounded function F localized in $\L_0$.

Let $\Xi_\L (n,\,\Pi,\,E,\,\omega_c)\subset \RR^{6n}$  be the surface 
on which $n$ particles have positions in $\L$, total momentum $\Pi$, and
total energy $E$ (note that  the total
energy inside $\L$ is affected by $\omega_c$).
Because of the boundaries $\o_c$, this surface may have many different connected
components  $\Xi^j_\L (n,\,\Pi,\,E,\,\omega_c)\subset \RR^{6n}$.
 
 \proclaim {Proposition 4.1}. For any $\ve>0$ there exists $\L$ large enough
such that the measure $Q$ restricted to
 $\Xi^j_\L (n,\,\Pi,\,E,\,\omega_c)
\cap {\widehat\Omega^{\L,\ve}}$, is proportional to the Microcanonical 
measure\nfootnote{\rm To define the
 Microcanonical measure consider that
 $(\L\times \RR^3)^n$ is foliated by the surfaces $\Xi(E,\,\Pi)$ 
when varying  $E$ and $\Pi$.
 Accordingly it is possible to define the conditioning of the Lebesgue 
 measure on
 $(\L\times \RR^3)^n$ to almost all the above mentioned surfaces. 
 Such a conditional measure is
 exactly the Microcanonical measure on $\Xi$.
This Microcanonical measure is also the only one invariant for the action of 
every element of the tangent space.} for almost all $\Pi$, $E$ and $\omega_c$.
  
It is easy to see that \equ (exbis) follows from proposition 4.1. In fact, 
$\o$ and $T_{\a,\b}\o$ belong always to the same connected component (connected
components can be distinguished only by the positions $q$'s), and Microcanonical
measures are invariant for exchanges of momenta between the particles.

The rest of the section will be then dedicated to the proof of proposition 4.1.
We will fix now the box $\L$, and we will drop the index $\L$ when this 
will not create confusion; moreover, in the 
rest of the paragraph we will drop the index $j$ and $\Xi$ will  refer to 
a fixed connected component.


What we have proven in the previous section is that our Lie algebra 
$\cal A$ generates the
 tangent space of $\Xi (n,\,\Pi,\,E,\,\omega_c)$ only at those points corresponding 
to a unique cluster.
  
 Let us call $d\mu_{n,\,\Pi,\,E,\,\omega_c}(q,\,p)$ 
the  measure Q conditioned on surface $\Xi (n,\,\Pi,\,E,\,\omega_c)$ i.e.
 $$
 \eqalign{
 \int_{\Xi (n,\,\Pi,\,E,\,\omega_c)}&
 f(q,\,p)d\mu_{n,\,\Pi,\,E,\,\omega_c}(q,\,p) \cr
 &=\E^Q\left(f(\o_{\L})\;\big|\;|\o_\L|=n,\,
 \Pi_\L(\o)=\Pi,\,E_\L(\o)=E,\,\o_{c}\right).
 }
 $$
 
 Since all the quantities we
 have conditioned on, in the definition of $\mu$, are  conserved by the vector fields
 of the Lie subalgebra generated only by the particles in $\L$, the  conditional
 measure
 $d\mu$ is invariant for such a subalgebra (Lemma 2.7); moreover, the subalgebra is
 composed by null  divergence vector fields. This implies that, in a sufficiently small
 neighborhood $B$ of a point corresponding  to a configuration with a unique cluster,
 the measure
 $d\mu_n$ is proportional to the Microcanonical measure.
 More precisely consider an open set
 $B\subset\Xi$ with a constant cluster structure and let $\chi$ be the characteristic
 function of such a set. If all the configurations in $B$ have a {\bf unique}
 cluster and
 $v_i\not=v_j$ for every $i,\,j$, it follows
 $$
 \eqalign{&
 \int_{\Xi(n,\,\Pi,\,E,\,\omega_c)}\chi(q,\,p) F_\L(q,\,p)
 d\mu_{n,\,\Pi,\,E,\,\omega_c}(q,\,p)\cr &= Z(n,\,\Pi,\,E,\,\omega_c)
 \int_{\Xi(n,\,\Pi,\,E,\,\omega_c)} \chi(q,\,p)F_\L(q,\,p) dM(q,\,p) }
 $$ 
 where $dM$ is the Microcanonical measure on $\Xi$ and $Z$ is a  normalization
 constant. 
 In fact, the Microcanonical measure is invariant with respect to $\Cal A$. 
 Moreover, there exists vector fields
 $\{Y_i\}_{i=1}^m$ from $\Cal A$ that span all the tangent space of $\Xi$ at
 each point of $B$ (provided $B$ is chosen small enough). Hence, $d\mu$ must be an
 invariant measure for the elliptic operator $\sum_{i=1}^m Y_i^*Y_i$. The claim follows
 since it is well known that such an elliptic operator has a unique invariant measure.
 \par  
 If in the configurations in $B$ are present several not interacting clusters
 $\{\Gamma_i\}=\widetilde\Gamma$, then from section 2 follows that the Lie algebra
 $\Cal A$, restricted to $\Xi$, does not necessarily span  all the tangent
 space. Yet, for each
 $\Gamma_i\in\wt\Gamma$, we can consider the surface $\Xi(\Gamma_i)$ obtained by
 fixing the positions of the particles not in $\Gamma_i$.\nfootnote{To be more
 precise, suppose that $\Gamma_i$ consists of $m$ particles. Fix the position and
 velocities of all the particles in
 $\L$ not belonging to $\Gamma_i$ and call their total energy $E_1$ and  their
 total momentum $\Pi_1$. Then, $\Xi(\Gamma_i)$ is the surface in $\RR^m$ defined by
 $\sum_{\a\in\Gamma_i}p_\a=\Pi-\Pi_1\equiv\Pi'$ and
 $\sum_{\a\in\Gamma_i}\phi(p_\a)+{1\over 2}\sum_{\a,\,\b\in\Gamma_i}V(q_\a-q_\b)+
 \sum_{\a\in\Gamma_i,\,\b\not\in\Gamma_i}V(q_\a-q_\b)=E-E_1\equiv E'$. Notice that we
 are not writing explicitly the dependence on $E'$ and $\Pi'$, since this does not
 create ambiguities.} From section 2 follows then that the Lie
 Algebra
 $\Cal A _\L$, restricted to the surface
 $\Xi(\Gamma_i)$ spans all its tangent space. Thus, the  simple application of the
 invariance with respect to the available  vector fields yields the weaker result
 $$
 \eqalign{&
 \int_{\Xi_j(n,\,\Pi,\,E,\,\omega_c)}\chi(q,\,p) F_\L(q,\,p)
 d\mu_{n,\,\Pi,\,E,\,j,\,\omega_c}(q,\,p)\cr &= Z(n,\,\Pi,\,E,\,j,\,\omega_c)
 \int_{\Xi_j(n,\,\Pi,\,E,\,\omega_c)} \chi(q,\,p)F_\L(q,\,p)
 dM_{\widetilde\Gamma}(q,\,p)  }
 $$ where
 $$ M_{\widetilde\Gamma}(q,\,p)(\cdot\;|\;(q_j,\,p_j)\not\in\Gamma_i)= 
 M_{\Gamma_i}((q,\,p)\in\Gamma_i)
 $$
 $M_{\Gamma_i}$ being the  Microcanonical measure for the particles belonging to
 $\Gamma_i$.
 \par 
 Yet, it is possible to use the dynamics generated by the vector fields in order
 to get a better result. We will show that one can  construct maps,
 connected to cluster deformations, with the property of  preserving both the measures
 $d\mu$ and $dM$. To be more concrete we need to define precisely what is meant by
 deforming a cluster. 

 Recall that 
 $\widetilde\Xi=\{(q,\,p)\in\Xi\;|\; (p_i,\,p_j)\in\wt\Sigma_{ij}\hbox{ for 
 some } i,\,j\}$. 
 Moreover, given a partition $\P$ of the particles (i.e., $\cup_{P\in\Cal P}
 P=\{1,\,...,\,n\}$) we will say that  a measure is Microcanonical with respect to the
 partition $\P$ if for each $P\in\P$ conditioning the measure to all the particles not
 in $P$ one obtains the Microcanonical measure for the particles in $P$.  (From now
 on, with an evident abuse of notations, we will use $M_{\Cal P}$ to designate any
 measure which is  Microcanonical with respect to $\Cal P$.)
 
 Furthermore, by $\Cal A_{\delta,\,\P}$ we will mean the Lie algebra 
 generated by the vector fields associated to bonds
 in which the particles are closer than $R_1-\delta$, for some fixed 
 $\delta$ smaller than $R_1-R_0$, and belongs to the 
 same element of the partition $\P$; finally, by $\Cal A_{\delta,\,\P}(\xi)$  
 we designate the restriction of
 $\Cal A_{\delta,\,\P}$ at $\Cal T_\xi\Xi$.\nfootnote{Clearly 
 $\Cal A_{\delta,\,\P}(\xi)$ is a linear subspace of $\Cal T_\xi\Xi$.} 
 
 \proclaim {Definition 4.2}. By ``allowed deformation" with respect to a 
 partition $\P$ and a tolerance $\delta$, we mean a  piecewise smooth 
 curve $\gamma:[0,\,1]\to \Xi\backslash\widetilde\Xi$ with the property that, 
 for some $\delta\in \RR^+$ and for each $s\in[0,\,1]$, $\gamma'(s)\in 
 \Cal A_{\delta,\,\P}(\gamma(s))$.
 
 Note that, in a given configuration, the clusters form a partition.
 
 \proclaim{Definition 4.3}. Given a set $B\in\Xi$ we call ``$\Cal P(B)$" the
 coarsest partition of $\{1,\,...,\,n\}$ finer than the partitions produced 
 by the isolated clusters of each $\xi$ in $B$.
 
 
 \proclaim{Proposition 4.4}. Given a configuration 
 $\xi\equiv (q,\,p)\in\Xi\backslash\wt\Xi$, 
 let $r=\sup_{\alpha,\beta:\;|q_\alpha-q_\beta|<R_1}|q_\alpha-q_\beta|$ and
 $\delta\leq R_1-r$, suppose that there exists a neighborhood $B$ of $\xi$   
 in which the measure $\mu$ is Microcanonical with respect to some partition 
 $\P$ (coarser than $\P(\{\xi\})$) and an allowed 
 deformation with respect to $\P$ and $\delta$ connecting 
 $\xi$ to some other configuration $\xi_1\in\Xi\backslash\widetilde\Xi$, 
 then there exists a
 neighborhood $B_1\subset B$ of $\xi$ such that
 $$
 \mu|_{B_1}=M_{\P\wedge\P(\{\xi_1\})}|_{B_1} .
 $$ 
 
 The proof of proposition 4.4 is the content of subsection 4.1. 


 Proposition 4.4. shows that we can have the
 wanted result, provided we can generate enough allowed deformation. In
 sub--section 4.2, we will prove  the existence of enough cluster deformations, more
 precisely we will prove proposition 4.1.
  
 
 \vskip.5cm {\bf 4.1 Cluster deformation}.
 \vskip.25cm
 
 
 \proclaim {Proof of Proposition 4.4}.

The following is a useful auxiliary Lemma.
  
\proclaim{Lemma 4.5}. If the measure is Microcanonical
 with respect to a partition $\P_1$ and, at the same time, with respect to a
 partition $\P_2$, then it is Microcanonical with respect to the partition
 $\P_1\wedge\P_2$.
 
\proclaim{Proof}.

Start with the following observation: let $P$ and $G$ two element respectively
of $\P_1$ and $\P_2$ such that $P\cap Q\ne \emptyset$. Then since the
measure is invariant for exchanges of momenta between particles inside $P$ 
and inside $Q$, then it is invariant for exchanges in $P\cup Q$ 
(cf. appendix II). 
This implies that it
is Microcanonical in $P\cup Q$. By definition of
$\P_1\wedge\P_2$, it follows that it is microcanonical
with respect to this coarser partition. 
$\qed$

 
 We start by constructing explicitly a map $\Upsilon$ that leaves invariant
 $\mu$ and such that $\Upsilon(\xi)=\xi_1$. The idea
 is to define a one parameter family of vector fields in a neighborhood of the allowed
 deformation
 $\gamma$; the wanted map will be obtained by integrating  such a family
 in a  neighborhood of the curve $\gamma$. 
 \par 
 We parameterize $\gamma$ by arc-length. By definition, 
 $\gamma\subset\Xi\backslash\wt\Xi$. Let $L=\hbox{length}(\gamma)$ then
 $\gamma(0)=\xi$  and $\gamma(L)=\xi_1$. Moreover, for each $s\in[0,L]$, 
 $\gamma'(s)$ can be written as $\sum_{i=1}^{m}a_i(s) X_i$ where $a_i(s)$ 
 are smooth functions depending only on $s$ and
 $X_i$ are vector fields, in the Lie algebra 
 $\Cal A_{\delta,\,\P}(\gamma(s))$,  
 that leave $Q$ invariant;\nfootnote{The fact that the $a_i$ can be chosen 
 smooth follows from our requirement that $\gamma'\in\Cal A_\delta(\gamma)$, 
 and is indeed the reason of such a requirement.} here $m$ is some 
 integer depending only on
 $\gamma$.  Then we define a time dependent vector
 field by
 $$ 
 V(q,\,p,\,s)=\sum_{i=1}^m a_i(s) X_i(q,\,p)
 $$ 
 and the corresponding flow $\Phi$ by\nfootnote{Here we are abusing 
notations and using the same symbol to designate both the vector field and 
its coordinates with respect to the basis $\{\partial_q,\,\partial_p\}$.}
 $$
 \eqalign{ &{d\over ds}\Phi(q,\,p,\,s)=V(\Phi(q,\,p,\,s),\,s)\cr
 &\Phi(q,\,p,\,0)=(q,\,p) . \cr }
 $$ 
 By continuity there exists $\varepsilon$ such that for each 
 $s\in[0,\,1]$, setting $\gamma(s)=(q(s),\,p(s))$, if
 $\widetilde q\in\L^n$, $\widetilde p\in\RR^{3n}$, $\|\widetilde q_i-q_i(s)
 \|<\varepsilon$, then $V(\widetilde q,\,\widetilde p,\,s)\in
 \Cal A_{0,\,\P}((\widetilde q,\,\widetilde p))$.
 By choosing the initial conditions in a sufficiently small neighborhood 
 $B_0$ of $\xi$, 
 we can insure that the solutions of the differential equations, with 
 initial conditions in $B_0$, are closer than
 ${\varepsilon \over 2}$ to the curve $\gamma$. 
 
 We define $\Upsilon$ on $B_0$ by:
 $$
 \Upsilon(\xi)=\Phi(\xi,\,L)  .
 $$ 
 Since $V$ is always tangent to $\Xi$ it follows that $\Upsilon$ is a well defined
 function from $B_0\subset \Xi\backslash\widetilde\Xi$ to 
 $\Xi\backslash\widetilde\Xi$. 
 \par 
 $\Upsilon(\xi)=\xi_1$ follows immediately from the construction ($\gamma(s)$ is a
 solution of the differential equation). In addition, for each smooth 
function $F$, with $\hbox{supp} F\subset B_0$, $\E^Q(F\circ\phi^{-1}_s)=\E^Q(F)$; 
that is $\mu|_{B_0}$ is invariant with respect to $\Upsilon$. 
 \par 
 Next, it is easy to check that the vector fields $X_i$ have zero
 divergence and are tangent to the surfaces of constant momentum and energy of the
 various elements in $\P$. Accordingly, any measure that is 
 Microcanonical with respect to a partition coarser than $\P$ will be left 
 invariant by the flow; that is $M_\P$ is invariant with respect to 
 $\Upsilon$. 
 
 Next choose an neighborhood $B_1\subset B_0$ of $\xi$ such that 
 $\P(\Upsilon B_1)$ is equal to $\P(\xi_1)$.
 Let $\chi$ be the 
 characteristic function of $\Upsilon B_1$ and $F_\L$ a smooth local 
 function with support disjoint from $\widetilde \Xi$. 
 $$
 \eqalign{&
 \int_{\Xi(n,\,\Pi,\,E,\,\omega_c)}
 \chi(q,\,p) F_\L(q,\,p)d\mu_{n,\,\Pi,\,E,\,\omega_c}(q,\,p) \cr  
 &=\int_{\Xi(n,\,\Pi,\,E,\,\omega_c)}
 \chi\circ \Upsilon(q,\,p) F_\L\circ
 \Upsilon(q,\,p)d\mu_{n,\,\Pi,\,E,\,\omega_c}(q,\,p)\cr
 &=\int_{\Xi(n,\,\Pi,\,E,\,\omega_c)}
 \chi\circ\Upsilon(q,\,p)F_\L\circ\Upsilon(q,\,p)dM_{\P}(q,\,p)\cr
 &=\int_{\Xi(n,\,\Pi,\,E,\,\omega_c)} \chi (q,\,p)F_\L(q,\,p)
 dM_{\P}(q,\,p) }
 $$ 
 Since $M_{\P}(\widetilde\Xi)=0$, for each partition $\P$,  and
 $\mu_{n,\Pi,E,\o_c}$ is absolutely continuous with respect to $M$ for almost all
 $\Pi$, $E$, $\o_c$, the  equality holds true for each $F_\L$. Note that we have
 used the invariance, with respect to the map $\Upsilon$, both of the 
 measure $\mu$ and of the Microcanonical measures $M_{\P}$.
 
 The consequence of the above chain of equalities is that $\mu$, 
 restricted to $\Upsilon B_1$, must be
 Microcanonical with respect to $\Cal P$. But we know from section 2 
 that it must be Microcanonical with respect to $\P(\{\xi_1\})$ as well. 
 Hence, by Lemma 4.5, it must be Microcanonical with respect to $\P\wedge\P
 (\{\xi_1\})$ in $\Upsilon B_1$. Using again the invariance of the 
 Microcanonical measure, 
 with respect to partitions coarser than $\P$, under the map 
 $\Upsilon$, the result follows.
 $\qed$
 \smallskip

 Our condition on the density insures that in the box $\L$ there is, 
 at least, a
 two particle cluster. If all the clusters in a given 
configuration could be connected,
 via an allowed deformation, to each nearby cluster, the argument would be
 easily concluded. Unfortunately, this is not always possible.
 The reason for this is that in order
 to extract two particles from a cluster may be needed
 more energy than it is available in the cluster itself.\nfootnote{E.g., think of a
 cluster at a corner with a concave shape and zero kinetic energy; in  addition the
 boundary condition may prevent the cluster from sliding along  a side of the box.}
 Such clusters are ``locked" and, in principle, it may be impossible 
 to remove them. 
 
 It is then painfully clear that our argument can be concluded only via a discussion
 of such a pathological behavior; this the task of the next section.
 
 
 \vskip.5cm {\bf 4.2 Killing locked clusters}.
 \vskip.25cm
 
 
 This section is dedicated to the proof of Proposition 4.1.
 
 Let $\xi_0\in(\Xi\backslash\wt\Xi)\bigcap \widehat\Omega_{\L,\,\ve}$, consider a
 neighborhood $B_0\subset\Xi\backslash\wt\Xi$, contained in a ball of sufficiently
 small radius $\delta$ such that $\P_0=\P(\{\xi_0\})$ is equal to 
 $\P(B_0)$. Our task will be to show that there is an allowed deformation that,
 together with Proposition 4.4, can be used to show that $\mu$ is Microcanonical 
 in a neighborhood of $\xi_0$ contained in $B_0$.
 
 Given a partition
 $\Cal P$ let us define
 $$
 \Pi(\xi,\,\Cal P)=\left\{(q',\,p')\in\Xi\backslash\hat\Xi\;\bigg|\;q'=q;\;\forall
 P\in\Cal P\;\;\sum_{j\in P}p_j=\sum_{j\in P}p_j';\;\sum_{j\in P}\phi(p_j)=\sum_{j\in
 P}\phi(p_j')\right\}.
 $$
 We will call $\Cal P$-complete a configuration $\xi$ that can be connected 
 via an allowed deformation with respect to $\P$ to each configuration in 
 $\Pi(\xi,\,\Cal P)$.\nfootnote{The reason why we are
 interested in this concept is that, later on in the argument, we will need to be
 able to allocate the kinetic energy, of any element of the partition, 
 among its particles in an arbitrary fashion.} 
 By definition, $\xi_0$ is $\Cal P_0$-complete.
 
We intend to produce a sequence of configurations $\xi_i$ and coarser and 
coarser partitions $\P_i$, where $\xi$ 
is connected to $\xi_{i+1}$ by an allowed deformation with respect to $\
P_{i+1}$, such that $\mu=M_{\P_i}$ in some neighborhood of 
$\xi_i$. From this it will follow that $\mu=M_{\P_i}$ 
also in some neighborhood of $\xi_0$.
 
 We will insure that the configurations
 $\xi_i$ that we are going to construct are all $\P_i$-complete and that 
 $\xi_i\in\widehat\Omega_{\L,\,\ve}$.
 
In order to start we need that a cluster  of two or more particles exists in $\L$.
But in the configurations under consideration we cannot have all the particles isolated
due to the bounds on the density.

We start dealing with clusters that do not interact with particles
outside $\L$ (if they exist).  Thanks to our hypothesis that the potential is
repelling, it is clear that any such cluster will have two particles that can be
``extracted"\nfootnote{By ``extracted" we mean that the particles can be moved out of
the region defined by the convex hall of the support of the deterministic potential
of the particles belonging to the cluster.} from the cluster always decreasing the
energy along the deformation; that is: clusters that do not touch the 
boundary cannot be locked. We choose any cluster that is away from the boundary of
$\L$ and perform the following allowed transformations: first select two 
particles, interacting between themselves,
with the above mentioned property and extract them from the cluster as a
two--particle cluster (the
excess of energy so create is stored in extra kinetic energy of the other particles in
the cluster) and  move the two particles to touch a neighbor cluster (without
interacting deterministically). Clearly, this deformation involves only the 
vector fields relative to the two particles, hence the deformation is 
allowed relative to $\P_0$. From proposition 4.4 follows that there exists 
a neighborhood $B_1\subset B_0$ of $\xi_1\equiv \xi_0$ such that 
$\mu|_{B_1}=M_{\P_1}|_{B_1}$, where $\P_1$ is the partition obtained by
$\P_0$ by joining the element from which the two particles have been 
extracted with the element with which they have been brought into contact.

It remains to show that $\xi_1$ is $\P_1$ complete. This is done by 
exhibiting allowed deformations relative to $\P_1$ that connect $\xi_1$ to 
any other configuration in $\Pi(\xi_1,\,\P_1)$. Such deformations are 
constructed as follows: transfer to the two particles mentioned before 
any needed amount of energy and momentum compatible with the conservation of 
total energy and momentum in the cluster, after extract them from the cluster 
(the excess of energy so create is stored in extra kinetic energy of the other particles in
the cluster) and  move the two particles to touch the same neighbor cluster 
as before (without interacting deterministically), at this point energy and 
momentum can be exchanged, finally move back the two particles in their 
original position.  In this way we can show that $\xi_1$ is 
complete with respect to the partition $\P_1$. 
Although the fact seems quite intuitive the 
proof is not immediate (due to the limitations on the momentum that can be 
exchanged conserving the kinetic energy) and can be found in
Appendix II. 

We can proceed in a similar fashion  producing coarser and coarser partitions $\P_i$.
The process will stop only if we obtain a partition $\P_i$ with all the elements
extending till the boundary of $\L$ or consisting of only 1 particles. 
Note that up
to now we are considering always the same reference configuration 
$\xi_i\equiv\xi_0$. 
 
 The result of the above discussion can be conveniently summarized by the 
 following. 
 \proclaim{Lemma 4.6}. For each partition $\P_i$, 
 constructed as before $\xi_0$ is $\P_i$-complete.


 If at this point we have obtained a trivial partition we are done. But sometimes
elements may be separated by big clusters with little kinetic energy. At this point
we need to use the properties of the good configurations in 
$\widehat\Omega_{\L,\,\ve}$. The idea is that we will do further transformations
that will produce somewhere a high energy two--particle cluster, i.e. a 
two particle cluster with enough kinetic energy to go anywhere in $\L$.

 
 \proclaim{Definition 4.7}. We will call ``free two particle cluster'' any two
 particles at a distance less than $R_1$ such that their ``available 
 kinetic energy"\nfootnote{By 
``available energy" of a group of $m$ particles, with total momentum $\Pi$, 
we mean the maximal amount of 
kinetic energy that can be liberated and transferred to two 
particles $\alpha,\beta$, 
i.e. $\sum_{\gamma}\phi(p_\gamma)
-m\phi({1\over m}\Pi)$.} is larger than
 $\ve^{-1}|\L|^{{1\over 2}}V_0$.

 The justification of the name lies in the following:
 
 \proclaim{Lemma 4.8}. If a configuration $\xi\in \widehat\Omega_{\L,\,\ve}$
 contains a free two  particles cluster, then, in a neighborhood of $\xi$ 
 $\mu$ is Microcanonical.

 \proclaim{Proof}.

Since $\xi\in \widehat\Omega_{\L,\,\ve}$ it follows that
a free two particles cluster has enough available energy to be moved anywhere in the
box $\L$. Accordingly we can produce allowed deformations that bring the free
two particle cluster in contact with any given particle in $\L$ and the 
result follows from the repeated use of Proposition 4.4. 
$\qed$
\medskip

 Consider a sequence of concentrical boxes $\L_k$, $k\ge 1$, such that
 $\L_{k+1}\supset\L_k$ and
 $B R_1<$dist$(\partial\L_k,\,\partial\L_{k+1})< (B+1) R_1$ for some fixed 
 $B$ large enough. 
 Without loss of generality, we can assume that there exists a constant 
 $C$ such that $\L_{CL}=\L$.
 Given a partition $\Cal P$, we will call an element ``large" if it
 contains at least one particle belonging to $\L_{CL}\backslash\L_{CL-1}$.
 
As a result of our previous construction we are now in a situation in which all the
particles belonging to $\L_1$ (in the configuration $\xi_0$) must either be part of a
large element of the final partition $\P_i$ or constitute a one particle cluster.
To see this suppose that there is an element, not large, containing more 
than two particles, with one particle in $\L_1$. 
Since it is not large it does not touch the boundary so it is possible to 
extract a two particle cluster from its convex hull. Since we have already 
played this game it must be that there are no particles outside the convex 
hull of the element, otherwise one could obtain a coarser partition by 
using the above mentioned strategy. Accordingly, any other particle must be 
contain in the convex hull of such element. Suppose now that there exists 
another element containing more than one particle, consider the cluster 
belonging to such an element closer to the element under consideration, 
such cluster can clearly be translated to touch our element, forming in 
such a way a new coarser partition with respect to which $\xi_0$ will be 
certainly complete. The only alternative is that all the other clusters 
consist of only one particle, but our bound on the density would imply that 
our element contains at least $a({L\over 2})^3$ particles, and therefore 
enough available kinetic energy to form a free two particle cluster (see 
lemma 3.2).

Therefore, all the elements containing more than one particle of whose at 
least one in $\L_1$ must be large in the final $\P_i$ obtained with the 
previous construction. Hence, at least $a({L\over 2})^3$ of the 
particles belong to large elements (the
other $\rho_*({L\over 2})^3$ of $\L_1$ may be isolated one particles clusters).

Another interesting property of large elements is that they must contain at 
least one particle in each region $\Delta_k\equiv\L_{k}\backslash\L_{k-1}$. 
Otherwise, one can extract a two particle cluster from the element at move 
the two particles into the region $\Delta_k$, since the partition is 
already the coarser obtainable with such deformations it is not possible 
that the two particles can be brought in contact with a new particle, i.e. 
no particle can be contained in $\Delta_k$. This implies immediately that no 
other cluster with more than two particles can be present in $\L_{k_1}$ 
(again, if this would not be the case one could extract two particles from 
the other element and move them to the element into consideration), so the 
element would have enough energy to create a free two particle cluster.

It is not obvious that one can continue to extract particles form the cluster to join
different elements once the partition has no longer clusters that do not extend to the
boundary. In the following we will show that something can still be done.

 \proclaim{Definition 4.9}. We call a large element ``fat" if, for each
region $\Delta_k=\L_k\backslash\L_{k-1}$, $k\in\{1,\,...,\,CL\}$, the
cluster has
 at least $L^{3\over 4}$ particles in $\Delta_k$. A non-fat large element will be
called ``thin."  

If at least $a({L\over 2})^3$ belong to fat elements then they will have at their
disposal at least $\ve a({L\over 2})^3$ available energy.
On the other hand a fat element must contain at least 
$({L\over 2})^{7\over 4}\over 10 R_1$
 particles, so there can be at most $10 R_1 ({L\over 2})^{5\over 4}(\rho_*+a)$ fat
elements.
 Hence, at least one fat element will enjoy 
${a\ve \over 16 R_1 (\rho_* +a)} ({L\over 2})^{7\over 4}$ kinetic energy.
 Taking $L$ large enough ($L>\ve^{-8}$) we can 
produce a free two particle cluster.

If this is not the case then the total number of particles belonging to thin elements
is bigger than $a ({L\over 2})^3$, 
then the available energy at their disposal is, at least, 
$\ve a ({L\over 2})^3$. But a thin
element is large, hence it contains at least $L\over B R_1$ particles. 
This implies that there are, at most, ${n B R_1\over L}$ thin elements.
 Hence, at least one of them will have more than
 ${\ve a L^4\over 2n B R_1}> {a\over 2(\rho + a)B R_1}L\ve$
available energy at its disposal.
 Accordingly, we have enough available kinetic energy so that one can construct
a deformation that extracts
 two particles from the element in the
 region $\Delta_k$ containing less than $L^{3\over 4}$
 particles. In fact, the element in  $\L_k\backslash\L_{k_1}$ can 
 invade at most a volume ${4\pi\over 3}R_1^3L^{3\over 4}$ while the 
 available volume is at least $30 R_1 L^2$, so there are both room and 
 energy to extract two particles. We proceed to such an extraction in any direction
and we call  $\xi_{i+1}$ the configuration so obtained. It is important to notice that
$\xi_{i+1}$ has a lover local density that $\xi_0$ and more available energy, moreover
the densities both in $\L_1$ and $\L$ are not changed, so $\xi_{i+1}$ is still in 
$\hat\Omega_{\ve,\,\L}$. It is easy to convince oneself that $\xi_{i+1}$ is still
$\P_i$ complete. 
 If such two particles cannot touch any other element while moving in the
 region $\Delta_k$, this means that the element under
 consideration is the only large element, then the energy at its disposal
 is $\ve{a}({L\over 2})^3$, sufficient to create a free two 
 particle cluster. 
Otherwise we obtain a new partition $\P_{i+1}$ where $\xi_{i+1}$ is
complete (see appendix II again). This shows that it is possible to eliminate
progressively the thin elements until we reach a configuration in which their
density is sufficiently low and the fat elements contain more than $a({L/ 2})^3$
particles.

 
 And this conclude the proof of proposition 4.1. It suffices to apply the 
the previous discussion to each point in $\Xi\cap{\widehat \Omega}^{\L,\ve}$,
accordingly in the neighborhood of each point $\mu$ is proportional to the 
Microcanonical measure. This implies also that conditioning on the 
positions and applying the argument illustrated at the beginning of section 4, 
follows that the conditional measure is constant on the surfaces of 
constant total momenta and energy. 
 
  
 \vskip 1cm
 {\bf 5. PROOF OF THEOREM 1}
  
 \vskip.5cm
  
  
The conclusion of the previous section is summarized by the following lemma:

 \proclaim{Lemma 5.1}. For almost every configuration of the positions $\o_q$
 and any  $\L_0$, 
 the conditional measure Q on $p_{\L_0}$ given $\o_q$ and
 $$
 \eqalign{
 \sum_{q_j\in\L}\phi(p_j)&=\hbox{const}\cr
 \sum_{q_j\in\L} p_j&=\hbox{const}\cr
 }
 $$
 is the Microcanonical measure on the corresponding surface.
  
  
 At this point we are in the same situation as in [OVY] 
 (after lemma (4.5) there). In fact, as a consequence of 
 the previous lemma, the distribution of the
 momentum conditioned on the positions is given by a convex combination of
 measures of the form
 $$
 \pi (d p \, | \,  \L) = \exp \big[\sum_{i=1}^3 \sum_\alpha
 \l_i p^i_\alpha -\l_4 \sum_\alpha \phi(p_\alpha)\big] \, / \,
 \hbox{Normalization.}
 $$
  
 \proclaim Lemma 5.2. For any configuration $\omega =\{ (q_\alpha,p_\alpha)\}$,
 let $\vec z(\omega)$ be the density, momenta and kinetic energy
 associated with the configuration defined by
 $$
 \eqalign{
 z^0 (\omega) &=\lim_{\delta\to 0} z^\mu _{\chi,\delta}(\omega)
 = \lim_{\delta \to 0} \delta^3\sum_{\alpha=1}^N \chi (\delta q_\alpha)\cr
 z^\mu (\omega) &=\lim_{\delta\to 0} z^\mu _{\chi,\delta}(\omega)
 = \lim_{\delta \to 0} \delta^3\sum_{\alpha=1}^N \chi (\delta q_\alpha)
 p^\mu_\alpha(\omega) \ , \,\, \mu=1,2,3\cr
 z^4(\omega) &= \lim_{\delta\to 0} z^\mu_{\chi,\delta}(\omega)
 = \lim_{\delta \to 0} \delta^3\sum_{\alpha=1}^N \chi(\delta q_\alpha)
 \, \phi(p_\alpha)\ . \cr
 }
 $$
  Here $\chi$
 is a cutoff function of total integral one, 
 $\vec z^{\mu}(\omega)$  exist almost 
 everywhere and are independent of the cutoff $\chi$. Furthermore, 
 $\vec z(\omega)$ are constants of the motion for $L$ in the 
 sense that
 $$
 \int h(\vec z(\omega)\, ) \, LF(\omega) \, dQ= 0 ,
 $$
 for all local smooth functions $F$  and all smooth functions 
 $h$ with compact support.
  
 \proclaim Proof.
  
 This was proven in [OVY] for bounded $\phi'$. For completeness, we present 
 here the proof for unbounded $\phi'$.
  
 By the same argument used immediately after lemma (1.1)
 these limits clearly exist and are independent of the cutoff $\chi$.
  
 By condition (iii) in theorem (1.1)
 $$
 \eqalign{
 0 &=\int L(Fh (z^\mu_{\chi,\delta}(\omega)\, )\, ) \, dQ\cr
 &= \int (LF)\, h(z^\mu_{\chi,\delta} (\omega)\, )\, dQ +
 \int F \, L \, h(z^\mu _{\chi,\delta}(\omega)\, ) \, dQ\ . \cr
 }
 $$
 The first term converges to $\int h(z^\mu_\chi(\omega) ) LF \, dQ$
 as $\delta\to 0$. We only have to show that the second term converges to
 zero as $\delta\to 0$. Clearly, it suffices to show that as $\delta \to 0$
 $$
 \int |Lz ^\mu_{\chi,\delta}|\, dQ \to 0 \ , \qquad \mu =0,\ldots, 4\ .
 \eqno{(5.1)}
 $$
 This is easy to show for $\mu=0,1,2,3$ (as in [OVY] pag. 544).
  
 For $\mu=4$ we have
 $$
 \eqalign{
 \E^Q\left(|Lz^4_{\chi,\delta}|\right) 
 =& \E^Q \left(|\delta\, \delta^3 \sum_{i,\alpha} \chi_i
 (\delta q_\alpha) \, \phi_i(p_\alpha) \, \phi(p_\alpha)|\right)\cr
 &+ \E^Q \left(|\delta ^3\sum_{\alpha\ne \beta} \, \sum_i \chi (\delta
 q_\alpha)\, \phi_i(p_\alpha) \, V_i(q_\alpha-q_\beta)|\right)  .\cr
 }
 $$
 Only the second term of the right end side present difficulties.
 Let $w_i(\vec z)$ and $\sigma_i(\vec z)$
 denote the expectation and variance of $\phi_i(p_\alpha)$ with respect to
 $Q$ conditioned on $\vec z$. These can be computed explicitly  by 
 using the characterization of the conditional measure given $\o_q$ and
 $\vec z$.
 We can bound the second term of the RHS of the above expression by
 $$
 \eqalign{
 \E^Q &\left(|\delta^3 \sum_{i} \sum_{\alpha\ne\beta} \chi(\delta q_\alpha)\,
 \phi_i (p_\alpha)\, V_i(q_\alpha -q_\beta)|\right)\cr
 =& \E^Q\left(|\delta^3\sum_i\sum_\a \chi(\delta q_\alpha)
 \big[ \phi_i(p_\alpha)-
 w_i\big] \sum_{\beta\ne \alpha} V_i(q_\alpha -q_\beta)|\right) \cr
 &+ \E^Q\left(|\delta^3 \sum_{\alpha\ne \beta} \, \sum_i \chi(\delta
 q_\alpha)\, V_i(q_\alpha -q_\beta)\, w_i|\right) . \cr
 }
 $$
 The second term of the RHS (third line above) can be bounded as before. Using the
Schwarz
 inequality the first term can be bounded by
 $$
 \eqalign{
 &\sum_i \E^Q\left(\E^Q\left (\left[\delta^3\sum_\a \chi(\delta q_\alpha)
 \big[ \phi_i(p_\alpha)-
 w_i\big] \sum_{\beta\ne \alpha} V_i(q_\alpha -q_\beta)\right]^2\bigg|\;
 \vec z\;\right)^{1/2}\right) \cr 
 &= \sum_i \E^Q \left( \sqrt{\sigma_i(\vec z)}\delta^3 \, 
 \E^Q\left( \sum_\alpha
 \chi(\delta q_\alpha)^2 \, \big(\sum_{\beta} V_i(q_\alpha-q_\beta)\, \big)^2
 \bigg| \; \vec z\;\right)^{1/2}\right)  \cr
 &\qquad\le \sum_i \E^Q(\sigma_i(\vec z))^{1/2} \delta^3 \E^Q\left(
 \sum_\alpha \chi (\delta q_\alpha)^2 \big( \sum_{\beta \ne \alpha} V_i
 (q_\alpha -q_\beta)\, \big)^2\right)^{1/2}  .\cr
 }
 $$
 By the condition on $\phi$ and the entropy argument we have that
 $\E^Q(\sigma_i(\vec z))$ is finite.
 To bound the second expectation, let us divide the set $\big\{ x\, | \, |x|
 \le 2\delta^{-1}\big\}$ into boxes of size $2R_0$  ($R_0$ is
 the range of $V$). Let $\sigma$ index the boxes and let $N_\sigma$ be the
 number of particles in the $\sigma$ box.
 $$
 \delta ^3\sum_i \E^Q \left( \sum_\alpha \chi(\delta q_\alpha)^2 
 \left[\sum_\beta
 V_i (q_\alpha -q_\beta)\, \right]^2 \right)^{1/2} 
 \le \hbox{const. } \delta^3\,\E^Q\left(\sum_\sigma N^3_\sigma \right)^{1/2}
 $$
 By convexity and the inequality
 $(\sum_\sigma N_\sigma^3 )^{1/3}\le (\sum_\sigma N^2_\sigma)
 ^{1/2}$ we see that the above expression is bounded by
 $$
 \hbox{const. } \delta ^3 \E^Q\left( \left[ \sum_\sigma N^2_\sigma\right]
 ^{3/4}\right) \le \hbox{ const. } \delta^3\left[ \E^Q\left( \sum_\sigma
 N^2_\sigma\right)\right]^{3/4}  .
 $$
 By lemma (1.2)(i) and the translation invariance, 
 $\E^Q(\sum_\sigma N^2_\sigma)$ is bounded by $\delta^{-3}$; hence, 
 the quantity under consideration
 is bounded by const.~$\delta^{3/4}$. This concludes the proof 
 of the lemma 4.2. $\qed$
  
 By the previous lemma $Q$ conditioned on $\vec z(\omega)$ is  
 still invariant for $L$. Since we assume that Q is translation invariant,
 we can apply lemma 4.10 in [OVY] and obtain that these conditioned 
 distributions
 are given by grancanonical Gibbs measures, concluding our proof.
  
 \vskip 1cm
  {\bf APPENDIX 1.} 
  
 \vskip.5cm
  
 To show that our condition on the noise and the kinetic energy in 
 the non-Gaussian
 case are far from empty, we give here an example 
 of stochastic perturbation that
 satisfies such condition. This is the only point where we use our requirement 
 on the form of the kinetic energy $\phi$.
  
 \proclaim  Lemma {A.1}.
 If $\{\eta^\theta_{\a\b}\}=\{e_1\wedge D_{\a\b}E,\,e_2\wedge D_{\a\b}E,\,
 e_3\wedge D_{\a\b}E\}$ and $\phi(p_a)=\sum_{i=1}^3\varphi(p^i_\a)$,
 with $(\varphi''')^2+\varphi^{iv}\varphi''=0$ 
 at most at finitely many points, then  condition 
 on the noise is satisfied.
  
 \proclaim Proof.
  
 A simple computation shows
 $$
 \eqalign{   
 [X^i_{\a\b},\,L]q_\a=&H_\a\eta^i_{\a\b}\cr
 [X^i_{\a\b},\,L]q_\b=&-H_\b\eta^i_{\a\b}\cr
 [X^j_{\a\b},\,[X^i_{\a\b},\,L]]q_\a^l=&
 (\eta^j_{\a\b})_l(\eta^i_{\a\b})_l H_{\a ll}'+ 
 H_{\a ll} \left(e^i\wedge(H_\a+H_\b)\eta^j_{\a\b}\right)_l\cr
 [X^i_{\a\b},\,[X^j_{\a\b},\,L]]q_\b^l=&
 (\eta^j_{\a\b})_l(\eta^i_{\a\b})_l H_{\b ll}'-
 H_{\b ll} \left(e^i\wedge(H_\a+H_\b)\eta^j_{\a\b}\right)_l\cr
 }
 $$
 where $H_{\a ll}$ stand for the element $ll$ of the diagonal matrix $H_\a$.
 The matrix $H_\a'$ is the derivative of the matrix $H_\a$; $(\cdot)_l$
 stands for the $l$--th component of the corresponding vectors.
 Now, let us take the six vectors obtained by letting $i,\,j$ vary only 
 in $\{1,\,2\}$.
 We define the vectors $w^{ij}$ by
 $$
 w_l^{ij}=(\eta_{\a\b}^i)_l (\eta_{\a\b}^j)_l  .
 $$
 Let us consider
 $$
 \sum_{i=1}^2\mu_i [X^i_{\a\b},\,L]  +\sum_{i,j=1}^2\nu_{ij}
 [X^j_{\a\b},\,[X^i_{\a\b},\,L]]=0
 $$
 Applying the above vector fields to $q_\a$, $q_\b$, we have
 $$
 \eqalign{   
 0=&\sum_{i=1}^2\mu_i H_\a\eta^i_{\a\b} +\sum_{i,j=1}^2\nu_{ij}
 \left[H_\a' w^{ij}+H_\a\left(e^i\wedge(H_\a+H_\b)\eta^j_{\a\b}\right)
 \right]\cr
 0=&-\sum_{i=1}^2\mu_i H_\b\eta^i_{\a\b} +\sum_{i,j=1}^2\nu_{ij}\left[
 H_\b' w^{ij}-H_\b\left(e^i\wedge(H_\a+H_\b)\eta^j_{\a\b}\right)\right]\cr
 }
 $$
 If we multiply the first by $(H_\a)^{-1}$, the second by
 $(H_\b)^{-1}$, and add one to the other, then we get
 $$
 0=\sum_{i,j=1}^2\nu_{ij}Aw^{ij} 
 $$
 where $A=1/2\{H_\a'H_\a^{-1}+H_\b'H_\b^{-1}\}$. Notice that $A$ is
 invertible out of a set of codimension 1 (see later for more details), 
 consequently
 $$
 \eqalign{
 0=&\sum_{i,j=1}^2\nu_{ij}w^{ij}\cr
 0=&\sum_{i=1}^2\mu_i\eta^i_{\a\b}+\sum_{i,j=1}^2\nu_{ij}
 e^i\wedge(H_\a+H_\b)\eta^j_{\a\b}\cr
 }
 $$
 To conclude we need an explicit representations of the vectors involved
 in the previous equations. Let $D^i_{\a\b}E=\zeta_i$, $h_i=H_{\a ii}+
 H_{\b ii}$, then a direct computation yields
 $$
 \eqalign{
 &\eta^1=(0,\,-\zeta_3,\,\zeta_2)\cr 
 &\eta^2=(\zeta_3,\,0,\,-\zeta_1)\cr 
 &w^{11}=(0,\,\zeta^2_3,\,\zeta^2_2)\cr    
 &w^{12}=w^{21}=(0,\,0,\,-\zeta_1\zeta_2)\cr
 &w^{22}=(\zeta^2_3,\,0,\,\zeta^2_1)\cr    
 &e^1\wedge(H_\a+H_\b)\eta^1_{\a\b}=(0,\,-\zeta_2 h_3,\,-\zeta_3 h_2)\cr
 &e^1\wedge(H_\a+H_\b)\eta^2_{\a\b}=(0,\,\zeta_1 h_3,\,0)\cr
 &e^2\wedge(H_\a+H_\b)\eta^1_{\a\b}=(\zeta_2 h_3,\,0,\,0)\cr
 &e^2\wedge(H_\a+H_\b)\eta^2_{\a\b}=(-\zeta_1 h_3,\,0,\,-\zeta_3 h_1) .\cr
 }
 $$
 Immediately follows $\nu_{22}=\nu_{11}=0$ and $\nu_{12}=-\nu_{21}$, which, 
 substituted in the remaining equations, yields
 $$
 \Omega\left(\eqalign{&\mu_1\cr&\mu_2\cr&\nu_{12}\cr}\right)=0
 $$
 For some matrix $\Omega$ with det$(\Omega)=\zeta_1\zeta_2\zeta_3(h_2+h_3)$.
 Since the determinant is equal zero on a set of codimension one, we
 have that the vector are linearly independent, 
 out of a set of codimension one.
  
 This set of codimension one consists of $\cup_i\{p\;|\;
 \varphi'''(p_\a^i)\varphi''(p_\a^i)=-\varphi'''(p_\b^i)\varphi''(p_\b^i) \}$, 
 where the matrix $A$ is not invertible\nfootnote{The condition of the 
 hypothesis insure that such set is a smooth codimension one manifold unless
 $\varphi'''(p^i_\a)^2+\varphi^{iv}(p_\a^i)\varphi''(p_\a^i)=
 \varphi'''(p^i_\b)^2+\varphi^{iv}(p_\b^i)\varphi''(p_\b^i)=0$, which can
 happen only on a set of codimension two.}, and
 $\cup_i\{p^i_\a=p^i_\b\}$, where the matrix $\Omega$ is not invertible.
 To get codimension two we have to analyze all the different cases one by one,
 since they are treated all in the same way we will consider only the points
 on the set $\{p^1_\a=p^1_\b\}$, and we will leave the rest to the skeptical
 reader.
 We can clearly ignore points of the above set that also belong to some
 other singular set: they belong to a set of codimension two. For points
 in the set under consider we will have $\zeta_1=0$, while all the other
 components will be different from zero. This implies that
 $w^{12}=w^{21}=e^1\wedge(H_\a+H_\b)\eta^2_{\a\b}=0$, we need then to produce
 more vectors, i.e., compute more commutators.
 It turns out to be sufficient to compute
 $$
 \eqalign{
 [X^2,\,[X^1_{\a\b},\,[X^2_{\a\b},\,L]]]q_\a&= H_\a v_1+ H_\a'v_2\cr
 [X^2,\,[X^1_{\a\b},\,[X^2_{\a\b},\,L]]]q_\b&=-H_\b v_1+ H_\b'v_2\cr
 [X^2,\,[X^2_{\a\b},\,[X^1_{\a\b},\,L]]]q_\a&= H_\a'v_2\cr
 [X^2,\,[X^2_{\a\b},\,[X^1_{\a\b},\,L]]]q_\b&= H_\b'v_2\cr
 }
 $$
 where $v_1=(0,\,\zeta_3h_1h_3,\,0)$, and 
 $v_2=(0,\,0,\,-\zeta_2\zeta_1 h_1)$.
 We have then to study the linear combination
 $$
 \eqalign{
 \sum_{i=1}^2\mu_i [X^i_{\a\b},\,L]  +\sum_{i,j=1}^2\nu_{ij}
 [X^j_{\a\b},\,[X^i_{\a\b},\,L]]+&
 \varepsilon_1[X^2,\,[X^1_{\a\b},\,[X^2_{\a\b},\,L]]]\cr 
 +&\varepsilon_2[X^2,\,[X^2_{\a\b},\,[X^1_{\a\b},\,L]]] =0\cr
 }
 $$
 where $\nu_{12},\,\nu_{21}$ are taken to be zero since the 
 corresponding commutators,
 when restricted to the $q_\a,\,q_\b$ space, would not contribute anything of
 interest. As before, we apply the vectors to the coordinates $q_\a,\,q_\b$,
 we multiply by $H_\a^{-1}$   and $H_\b^{-1}$ and add the corresponding
 equations, in so doing we obtain
 $$
 \sum_{i,j=1}^2\nu_{ij} w^{ij}+(\varepsilon_1+\varepsilon_2) v_2=0
 $$
 from this follows immediately $\nu_{11}=\nu_{22}=0$, 
 $\varepsilon_1=-\varepsilon_2$.
 Substituting in the original equation we get
 $$
 0=\sum_{i=1}^2\mu_i H_\a\eta^i_{\a\b} +\varepsilon_1 H_\a v_1
 $$
 which implies $\mu_i=\varepsilon_i=0$ on a set of codimension two.
 $\qed$
 
\vskip1cm
{\bf APPENDIX II}
\vskip 1cm


We will prove here that if $\xi$ is $\P$--complete, and two particle can be
extracted from an element $P_1$ to join $P_2$ (or viceversa), then
$\xi$ is complete for the partition $\P_*$ obtained from $\P$ joining $P_1$ and
$P_2$.

 Choose $\eta\in\Pi(\xi,\,\P_*)$. Call $\a,\,\b$ the two particles that 
 are allowed to move along $\gamma$.
 
 The rough idea is to transfer energy and
 momentum between the elements\nfootnote{Note that $\{\a,\,\b\}\subset P_1$ and that in
 the configuration $\xi$ $P_1$ still form an element.} $P_1$ and $P_2$ by using the
 particles $\a,\b$. Unfortunately, there are limits to how much momentum or energy we
 can transfer to a particles, due to the necessity to conserve the total energy and
 momentum of the clusters. To overcome this we will show that each
 $\eta\in\Pi(\xi,\,\P_*)$ can be deformed into the special configuration $\zeta
 \in\Pi(\xi,\,\P_*)$ defined by,\nfootnote{For each 
 $P\subset\{1,\,...,\,n\}$, by $\pi(\xi,\,P)$ 
 and $K(\xi,\,P)$ we mean, respectively, the total momentum and 
kinetic energy, in the
 configuration $\xi$, of the particles belonging to $P$; by $\# P$ we mean the
 cardinality of the set $P$.} 
 $$
 \eqalign{
 p_\sigma&={\pi(\xi,\,P_1\cup P_2)\over\# (P_1\cup P_2)}\quad \forall
 \sigma\not\in\{\a,\,\b\}\cr
 p_\a&={\pi(\xi,\,P_1\cup P_2)\over\# (P_1\cup P_2)}+\l v\cr
 p_\b&={\pi(\xi,\,P_1\cup P_2)\over\# (P_1\cup P_2)}-\l v ,
 }
 $$
 with some fixed $v\in\RR^3$, $\|v\|=1$, and $\l$ determined by$^20$
 $$
 K(\xi,\,P_1\cup P_2)=[\# (P_1\cup P_2)-2]
 \phi\left({\pi(\xi,\,P_1\cup P_2)\over\# (P_1\cup P_2)}\right)
 +\phi(p_\a)+\phi(p_\b) .
 $$
 The desired allowed transformation will then be obtained by deforming $\xi$ into
 $\zeta$ and then by running backward the allowed transformation that connects $\eta$
 to $\zeta$ (since the reverse of an allowed transformation it is still an allowed
 transformation).
 
 Since, by convexity, $K(\xi,\,P_1\cup P_2)\ge \# (P_1\cup P_2)
 \phi\left({\pi(\xi,\,P_1\cup P_2)\over\# (P_1\cup P_2)}\right)$, if $\l=0$ then
 $\Pi(\xi,\,\P_*)$, restricted to the particles in $P_1\cup P_2$ consists of only
 the point $\xi$ and we have nothing to prove. Otherwise we proceed as follows: we
 make an allowed deformation that set all the moments in $P_1\backslash\{\a,\b\}$
 equal to ${1\over\# P_1}\pi(\xi,\,P_1)$ while $p_\a={1\over\#
 P_1}\pi(\xi,\,P_1)+\nu_1v$ and 
 $p_\b={1\over\# P_1}\pi(\xi,\,P_1)+\nu_1v$, and $\nu_1$ is
 determined by the conservation of
 $K(\xi, P_1)$. Then we move the coordinates of the particles $\a,\b$ accordingly
 to $\gamma$ but without changing their momenta. Once they get in touch with $P_2$
 we change the momenta of the particles in $P_2$ to 
 $$
 p_*={1\over\# P_2 +2}(\pi(\xi, P_2)+{2\over \# P_1}\pi(\xi,\,P_1)),
 $$
 apart from $p_\a=p_*+\nu_2 v$ and $p_\b=p_*-\nu_2 v$, again $\nu_2$ is determined
 by the conservation of the kinetic energy of the new cluster $P_2\cup\{\a,\b\}$.
 Finally, we move back the particles $\a,\,\b$ to their original position in the
 configuration $\xi$ and share again their momentum among all the particles in $P_1$
 has we have done at the beginning. Let us call $\xi_{1,1}$ the configuration
 reached in such a way. Calling
 $\delta_0={1\over\# P_1}\pi(\xi,\,P_1)- {1\over\# P_2}\pi(\xi,\,P_2)$
 and $\delta_1={1\over\# P_1}\pi(\xi_{1,1},\,P_1)- {1\over\# P_2}\pi(\xi_{1,1},\,P_2)$
 a direct computation shows that
 $$
 \delta_1=\left(1-{2\# (P_1\cup P_2)\over\# P_1(\# P_2+2)}\right)\delta_0 .
 $$
 
 If we iterate further the procedure just described we see that the difference 
 between the average momentum in $P_1$ and $P_2$ goes to zero, this shows that
 we are getting closer and closer to the configuration $\zeta$; unfortunately 
 only asymptotically. Nevertheless, after a finite number of iterations we will
 get to a configuration $\zeta_0$ for which
 $$
 2\phi\left({\pi(\zeta_0,\,P_1)\over 2}-{(\# P_1- 2)
 \pi(\xi,\,P_1\cup P_2)\over 2\#(P_1\cup P_2)}\right)
 +(\# P_1-2)\phi\left({\pi(\xi,\,P_1\cup P_2)\over
 \#(P_1\cup P_2)}\right)< K(\zeta_0,\,P_1).
\eqno{(A2.1)}
 $$

 Let $p_\sigma(\eta)$ be the momentum of the particle $\sigma$ in the configuration 
 $\eta$.
 We deform $\zeta_0$ into $\zeta_1$ defined by
 $$
 \eqalign{
 p_\sigma(\zeta_1)=&{\pi(\xi,\,P_1\cup P_2)\over\#(P_1\cup P_2)} 
 \quad \hbox{for }\sigma\in P_1\backslash\{\a,\,\b\}\cr
 p_\a(\zeta_1)=&{\pi(\xi,\,P_1\cup P_2)\over\#(P_1\cup P_2)}+{1\over 2}
 \left[\pi(\zeta_0,\,P_1)-\# P_1 {\pi(\xi,\,P_1\cup P_2)\over\#(P_1\cup P_2)}
 \right]+\nu v\cr
 p_\b(\zeta_1)=&p_\a(\zeta_1)-2\nu v,
 }
 $$
 where $\nu$ is defined by $K(\zeta_1,\,P_1)=K(\zeta_0,\,P_1)$. All this is possible
 provided (A2.1) is satisfied; in fact, (A2.1)
 express simply that there is
 sufficient energy to deform the momenta of the particles in $P_1$ to the
 above values. After achieving the configuration $\zeta_1$, to obtain the 
 configuration $\zeta$ it suffices to
 take the particles $\a,\,\b$ to $P_2$, adjust the momenta of the particles of
 $P_2$ to the value ${\pi(\xi,\,P_1\cup P_2)\over\#(P_1\cup P_2)}$, which 
 will make all the momenta agree with the ones in the configuration $\zeta$
 and take $\{\a,\,\b\}$ back to their original position in $\xi$.


 
 \vskip 1cm
  {\bf REFERENCES.}
 \vskip.5cm
  
 \frenchspacing
  
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 \bye
ENDBODY