INSTRUCTIONS The text between the lines BODY and ENDBODY is made of 2716 lines and 111871 bytes (not counting or ) In the following table this count is broken down by ASCII code; immediately following the code is the corresponding character. 71023 lowercase letters 3430 uppercase letters 1640 digits 15535 ASCII characters 32 16 ASCII characters 33 ! 16 ASCII characters 34 " 139 ASCII characters 35 # 2638 ASCII characters 36 $ 249 ASCII characters 37 % 91 ASCII characters 38 & 132 ASCII characters 39 ' 842 ASCII characters 40 ( 842 ASCII characters 41 ) 19 ASCII characters 42 * 116 ASCII characters 43 + 1565 ASCII characters 44 , 255 ASCII characters 45 - 920 ASCII characters 46 . 19 ASCII characters 47 / 65 ASCII characters 58 : 123 ASCII characters 59 ; 15 ASCII characters 60 < 394 ASCII characters 61 = 32 ASCII characters 62 > 4 ASCII characters 63 ? 2 ASCII characters 64 @ 133 ASCII characters 91 [ 5904 ASCII characters 92 \ 133 ASCII characters 93 ] 653 ASCII characters 94 ^ 1937 ASCII characters 95 _ 63 ASCII characters 96 ` 1350 ASCII characters 123 { 225 ASCII characters 124 | 1350 ASCII characters 125 } 1 ASCII characters 126 ~ BODY \magnification = 1200 \def\a {\alpha} \def\b {\beta} \def\G {\Gamma} \def\L {\Lambda} \def\l {\lambda} \def\o {\omega} \def\p {\varphi} \def\t {\theta} \def\E {I\!\! 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E' possibile citare anche %%% formule o figure che appaiono in altri file, purche' sia presente il %%% corrispondente file .aux; basta includere all'inizio l'istruzione %%% \include{nomefile} %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \global\newcount\numsec \global\newcount\numfor \global\newcount\numtheo \global\advance\numtheo by 1 \global\newcount\numfig \gdef\profonditastruttura{\dp\strutbox} \def\senondefinito#1{\expandafter\ifx\csname#1\endcsname\relax} \def\SIA #1,#2,#3 {\senondefinito{#1#2} \expandafter\xdef\csname #1#2\endcsname{#3}\else \write16{???? ma #1,#2 e' gia' stato definito !!!!} \fi} \def\etichetta(#1){(\veroparagrafo.\veraformula) \SIA e,#1,(\veroparagrafo.\veraformula) \global\advance\numfor by 1 \write15{\string\FU (#1){\equ(#1)}} \write16{ EQ \equ(#1) == #1 }} \def\letichetta(#1){\veroparagrafo.\verotheo \SIA e,#1,{\veroparagrafo.\verotheo} \global\advance\numtheo by 1 \write15{\string\FU (#1){\equ(#1)}} \write16{ Sta \equ(#1) == #1}} \def\tetichetta(#1){\veroparagrafo.\veraformula %%%%copy four lines \SIA e,#1,{(\veroparagrafo.\veraformula)} \global\advance\numfor by 1 \write15{\string\FU (#1){\equ(#1)}} \write16{ tag \equ(#1) == #1}} \def \FU(#1)#2{\SIA fu,#1,#2 } \def\etichettaa(#1){(A\veroparagrafo.\veraformula) \SIA e,#1,(A.\veroparagrafo.\veraformula) \global\advance\numfor by 1 \write15{\string\FU (#1){\equ(#1)}} \write16{ EQ \equ(#1) == #1 }} \def\getichetta(#1){Fig. \verafigura \SIA e,#1,{\verafigura} \global\advance\numfig by 1 \write15{\string\FU (#1){\equ(#1)}} \write16{ Fig. \equ(#1) ha simbolo #1 }} \newdimen\gwidth \def\BOZZA{ \def\alato(##1){ {\vtop to \profonditastruttura{\baselineskip \profonditastruttura\vss \rlap{\kern-\hsize\kern-1.2truecm{$\scriptstyle##1$}}}}} \def\galato(##1){ \gwidth=\hsize \divide\gwidth by 2 {\vtop to \profonditastruttura{\baselineskip \profonditastruttura\vss \rlap{\kern-\gwidth\kern-1.2truecm{$\scriptstyle##1$}}}}} } \def\alato(#1){} \def\galato(#1){} \def\veroparagrafo{\number\numsec}\def\veraformula{\number\numfor} \def\verotheo{\number\numtheo} \def\verafigura{\number\numfig} \def\geq(#1){\getichetta(#1)\galato(#1)} \def\Eq(#1){\eqno{\etichetta(#1)\alato(#1)}} \def\teq(#1){\tag{\tetichetta(#1)\hskip-1.6truemm\alato(#1)}} %%%%%this line \def\eq(#1){\etichetta(#1)\alato(#1)} \def\Eqa(#1){\eqno{\etichettaa(#1)\alato(#1)}} \def\eqa(#1){\etichettaa(#1)\alato(#1)} \def\eqv(#1){\senondefinito{fu#1}$\clubsuit$#1\else\csname fu#1\endcsname\fi} \def\equ(#1){\senondefinito{e#1}\eqv(#1)\else\csname e#1\endcsname\fi} %next six lines by paf (no responsibilities taken) \def\Lemma(#1){Lemma \letichetta(#1)\alato(#1)\hskip-1.6truemm} \def\Theorem(#1){{Theorem \letichetta(#1)\alato(#1)}\hskip-1.6truemm} \def\Proposition(#1){{Proposition \letichetta(#1)\alato(#1)}\hskip-1.6truemm} \def\Corollary(#1){{Corollary \letichetta(#1)\alato(#1)}\hskip-1.6truemm.} \def\Remark(#1){{\noindent{\bf Remark \letichetta(#1)\alato(#1)\hskip-1.6truemm.}}} \let\ppclaim=\plainproclaim \let\EQS=\Eq\let\EQ=\Eq \let\eqs=\eq \let\Eqas=\Eqa \let\eqas=\eqa %%%%%%%%%%%%%%%%%% Numerazione verso il futuro ed eventuali paragrafi %%%%%%% precedenti non inseriti nel file da compilare \def\include#1{ \openin13=#1.aux \ifeof13 \relax \else \input #1.aux \closein13 \fi} \openin14=\jobname.aux \ifeof14 \relax \else \input \jobname.aux \closein14 \fi \openout15=\jobname.aux %%%%%%%%%%%%%%%%%%%%%%%%%%%% % \BOZZA \hfuzz 16pt \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| 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 $r=\sup_{\alpha,\beta:\;|q_\alpha-q_\beta|\ve^{-8}$) 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. 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