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\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. 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
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\vskip.5cm
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\bye
ENDBODY