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Steady State, Master Equation, Chaotic Sequence
---------------1212261734374
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{L}ett.}},
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Marra},
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\begin{document}
 
\title{Entropic Chaoticity for the Steady State of a Current Carrying System.}

\author[*]{F. Bonetto}
\author[*]{M. Loss}
\affil[*]{School of Mathematics, Georgia Institute of Technology, Atlanta, GA}

\renewcommand\Authands{ and }

\maketitle


\begin{abstract}
The steady state for a system of $N$ particle under the influence of an external
field and a Gaussian thermostat and colliding with random ``virtual'' scatterers
can be obtained explicitly in the limit of small field. We show the sequence
of steady state distribution, as $N$ varies, forms a chaotic sequence in the
sense that the $k$ particle marginal, in the limit of large $N$, is the $k$-fold
tensor product of the 1 particle marginal. We also show that the
chaoticity properties holds in the stronger form of entropic chaoticity.
\end{abstract}

\section{introduction}
Ever since its introduction by Kac \cite{kac} in 1956, the notion of a chaotic
sequences has become an important concept in studying many body systems. Chaotic
sequences and propagation of chaos are the principal tools for passing from many
body problems to effective equations. The aim of this article is to give yet
another example of the interplay of chaotic sequences and effective equations.

In [BDLR] the authors consider a system consisting of $N$ particles moving in a
2 dimensional billiard and colliding with convex scatterers that form a
dispersing billiard. The particle are subject to an external electric field
$\bE$ and a Gaussian thermostat that keeps the kinetic energy fixed. The
equation of motion between collisions are thus:
\begin{equation}\label{eq2}
\left\{
\begin{array}{l}
   \dot\bq_i = \bv_i  \qquad\qquad i=1,\ldots,N \\
   \dot\bv_i = \bF_i = \bE-\frac{\displaystyle \bE\cdot
\bj}{\displaystyle U}\,\bv_i + \cF_i \end{array}
\right.
\end{equation}
where
\begin{equation}  \label{eq3}
    \bj(\bV)=\sum_{i=1}^N \bv_i,\qquad U(\bV) = \sum_{i=1}^N |\bv_i|^2
\end{equation}
and $\cF_i$ is the force exerted on the $i$th particle by collisions with the
fixed scatterers. Very little is known about billiards with more than one
particle. In particular there is no existence theorem for the SRB measure of
this system.

The authors introduced a stochastic version of the above model in which the
deterministic collisions are replaced by Poisson distributed collision
processes. More precisely, in the time interval between $t$ and $t+dt$, the
$i$-th
particle as a probability $|\bv_i|^\alpha dt$ of suffering a collision. When a
collision happens, the velocity of the particle is randomly updated, i.e. if the
particle's velocity direction before collision is $\omega=\bv/|\bv|$, after
collision it will be distributed as $K(\omega'\cdot \omega)d\omega'$. The
details of the collision kernel $K$ are largely irrelevant. For what follows
it will be enough to assume that $K(x)>0$ for $x$ in a open set
${\cal U}\in[-1,1]$. We
note that this stochastic process make sense for any
dimension $d$.

The master equation for this process is given by
\begin{eqnarray}  \label{dW}
 \frac{\partial W(\bQ,\bV,t;\bE)}{\partial t} = 
&-&\sum_{i=1}^N\bv_i\frac{\partial W(\bQ,\bV,t;\bE)}{\partial \bq_i}
-\sum_{i=1}^N
 \nabla_{\bv_i}\biggl[\Bigl(\bE-\frac{\bE\cdot\bj(\bV)}{U(\bV)}
\bv_i\Bigl)W(\bQ,\bV,t;\bE)
\biggr]\nonumber\\
  &+&\sum_{i=1}^N|\bv_i|^\alpha\int_{\cS^{d-1}(1)} K(\omega_i \cdot \omega'_i)
\bigl[W(\bQ,\bV_i',t;\bE)-W(\bQ,\bV,t;\bE)\bigr]\,d\sigma^{d-1}
(\omega')\nonumber\\
\end{eqnarray}
where $\cS^{m}(R)$ is the $m$-dimensional sphere with radius $R$,
$d \sigma^{m}(\cdot)$ is the uniform surfaces measure on $\cS^{m}(R)$ and $\bj =
\bJ/U$ as in \eqref{eq3}. Moreover
\begin{equation}
\bQ = (\bq_1,\ldots,\bq_N)\quad\text{,}\quad
 \bV = (\bv_1,\ldots,\bv_i,\ldots,\bv_N)
 \quad\text{and}\quad
 \bV_i' = (\bv_1,\ldots,\bv_i',\ldots,\bv_N),
\end{equation}
and $\bv_i'=|\bv_i|\omega_i'$ if $\bv_i=|\bv_i|\omega_i$. Note that
the variable $\bQ$ is not part of the dynamics, i.e., if the initial condition
$W(\bQ,\bV,0)$ is independent of $\bQ$ so will be $W(\bQ,\bV,t)$. Moreover,
if $W(\bQ,\bV,0)$ is concentrated on the surface of given
energy $U_0$, that is if 
\[
 W(\bV,0)=\delta(U(\bV)-U_0)F(\bV,0)
\]
then so will be the solution of .\eqref{dW}:
\[
 W(\bV,t;\bE)=\delta(U(\bV)-U_0)F(\bV,t;\bE).
\]
Finally if $F(\bV,0)$ is a symmetric function so is $F(\bV,t;
\bE)$. Thus we will,
henceforth, only consider symmetric spatially homogeneous solutions
concentrated on the surface of energy $U_0=N$.

Recall that the $k$-particle marginal $f^{(k)}_N(\bv_1, \dots, \bv_k)$  of a
distribution $F_N(\bV)$ is defined by the equation
\[
\int_{\cS^{dN-1}(\sqrt{N})} \varphi(\bv_1, \dots, \bv_k) F_N(\bV) d
\sigma^{dN-1}(\bV) =
\int_{\mathbb{R}^{dk}} \varphi(\bv_1, \dots, \bv_k)
f^{(k)}_N(\bv_1, \dots, \bv_k) d \bv_1 \cdots d \bv_k  \ ,
\]
where $\varphi(\bv_1, \dots, \bv_k)$ is any bounded continuous function on
$\mathbb{R}^{dk}$.
%$\cS^{m}(R)$ is the $m$-dimensional sphere with radius $R$
%and $d \sigma^{m}(\cdot)$ is the uniform surfaces measure on $\cS^{m}(R)$.
Simple computations show that
\begin{equation} \label{kmarginal}
f^{(k)}_N(\bV_k) = \sqrt{\frac{N}{N - |\bV_k|}}
\int_{\cS^{d(N-k)-1}\left(\sqrt{N - |\bV_k|^2}\right)}
F_N(\bV_k, \bV^k) d \sigma^{d(N-k)-1}(\bV^k) \ ,
\end{equation}
where $\bV_k = (\bv_{1} , \dots, \bv_k)$ and $\bV^k = (\bv_{k+1} , \dots,
\bv_N)$. A sequence of densities
$\{F_N\}_{N=1}^\infty $ form a {\bf chaotic sequence with marginal $f $} if for
any bounded continuous function $\varphi$
\begin{equation}\label{weak}
\lim_{N \to \infty} \int_{\cS^{dN-1}(\sqrt{N})} \varphi(\bV_k)
F_N(\bV) d \sigma^{dN-1}(\bV) =
\int_{\mathbb{R}^{dk}} \varphi(\bV_k)
\prod_{j=1}^k f(\bv_j)  d \bv_1 \cdots d \bv_k \
\end{equation}

It was shown in \cite{BCELM} that for finite time $t$ the master
equations \eqref{dW} propagates chaos, i.e.,  the solution of the master
equation \eqref{dW} forms a chaotic sequence  if the initial condition does.
More precisely, for any bounded continuous function on $\mathbb{R}^{dk}$,
$\varphi(\bv_1, \dots, \bv_k)$, we have
\[
\lim_{N \to \infty} \int_{\cS^{dN-1} } \varphi(\bV_k) F(\bV, t;
\bE) d\sigma(\bV) =
\int_{\mathbb{R}^{dk}}  \varphi(\bV_k) \prod_{j=1}^k f  (\bv_j, t;
\bE) d\bv_1 \cdots d\bv_k
\]
where
\[
  f (\bv_1,t;\bE)=\lim_{N\to\infty}  f^{(1)}_N(\bv_1,t;\bE) \ .
\]



It was shown in \cite{BCELM} that $  f(\bv_1,t;\bE)$ satisfies
the Boltzmann equation
\begin{equation}\label{BE}
 \frac{f(\bv,t;\bE)}{dt}+\frac{\partial}{\partial
\bv}\left[\bigl(\bE-(\bE\cdot\hat\bj(t,E)) \bv\bigl)f
\right]=|\bv|^\alpha\int_{\cS^{d-1}(1)} K(\omega \cdot \omega')
  \bigl[f(\bv',t;\bE)-f(\bv,t;\bE)\bigr]\,d\sigma^{d-1}(\omega')
\end{equation}
where $\hat \bj(t,E)$ is given by the self consistent condition
\[
 \hat \bj(t,E)=\frac{1}{u}\int \bv f(\bv,t;\bE) d\bv
\]
with
\[
 u=\int |\bv|^2 f(\bv,t;\bE) d\bv.
\]
The initial condition is given by $f(\bv) = \lim_{N \to \infty} f^{(1)}_N(\bv,
0)$. It is easy to see that $u$ is independent of time and since we have chosen
$U_0=N$, $u=1$.

Concerning steady states, the situation is far from clear. In \cite{BDLR} and
\cite{BCKLs} it
was shown that a steady state $F_{\rm ss} (\bV;\bE)$ exists for the Kac master
equation \eqref{dW} provided that  $\bE \not= 0$. If $\bE = 0$ any density
$F(\bV)$ that depends only on the magnitude of the velocities furnishes a
stationary state. It is, however, an open question whether $F_{\rm ss}
(\bV;\bE)$ tends to a limiting distribution as $\bE \to 0$. Interestingly,
assuming the {\it existence} of the limiting distribution, it can be computed
exactly and it is given by
\begin{equation}\label{exN}
F_{\rm ss}(\bV;0)=\frac{1}{\widetilde{Z}_N}\delta(U(\bV)-N)F(\bV)
\end{equation}
with 
\begin{equation} \label{eff}
 F_N(\bV)=\frac{1}{\left(\sum_{i=1}^N
|\bv_i|^{2+\alpha}\right)^{\frac{dN-1}{2+\alpha}} }
\end{equation}
where $\widetilde{Z}_N$ is the normalization constant
\begin{equation} \label{normalization}
\widetilde{Z}_N= \int_{\cS^{dN-1}(\sqrt{N})}F_N(\bV)d\sigma^{Nd-1} (\bV) \ .
\end{equation}
For details, the reader should consult \cite{BDLR} and \cite{BCKLs}. Thus, the
electric field
`selects' the right steady state as the electric field tends to zero.



A similar problem exists on the level of the Boltzmann equation. Again it is
possible to show that the steady state $f_{\rm ss}(\bv;\bE)$ for the Boltzmann
equation \eqref{BE} exists and is unique if $E\not=0$. This clearly implies the
existence of a steady state current $\hat \bj_{\rm ss}(E)$. In \cite{BL} it
was shown that, assuming that $f(v)=\lim_{E\to 0} f_{\rm ss}(v,E)$ exists and
that $\hat \bj_{\rm ss}(E)=O(E)$, one has
\begin{equation}\label{exin}
 f(\bv)=\frac{\mu^{\frac{d}{2}}}{c}e^{-(\sqrt{\mu} |\bv|)^{2+\alpha}}
\end{equation}
where $c$ and $\mu$ are uniquely determined by normalization and the
condition $u=1$. One easily get
\begin{equation} \label{constants}
\mu=\frac{\Gamma\left(\frac{d+2}{2+\alpha}\right)}{\Gamma\left(\frac{d}{
2+\alpha
} \right)}\qquad\qquad
c=\frac{|\cS^{d-1}(1)|\Gamma\left(\frac{d}{2+\alpha
} \right) }{(2+\alpha)}
\end{equation}
which for $\alpha=1$ and $d=2$ gives
\[
\mu=\frac{\Gamma\left(\frac{4}{3}\right)}{\Gamma\left(\frac{2}{3}\right)}
\approx 0.65948\qquad\qquad
c=\frac{2\pi}{3}\Gamma\left(\frac{2}{3}\right)
 \approx 2.83605
\]
For details the reader may consult \cite{BL} where the existence of the small
field limit of the steady state distribution is proved for $d=1$.

It is now natural to ask whether the distribution \eqref{exN} is chaotic with
marginal $f$ given by  \eqref{exin}. For the reasons mentioned above, this
cannot be deduced from the previous results on propagation of chaos since these
results do not hold uniform in time.
A more serious impediment is the fact that
the small field limits of the steady states are not known to exist. 
As explained before, the limit as $\bE \to 0$
selects a steady state for the Kac master equation as well as for its Boltzmann version.
It is far from clear that the selection mechanism is such as to preserve
chaoticity. In this note
we prove that, nevertheless, the distribution \eqref{exN} is chaotic with
marginal \eqref{exin}.


\medskip
\begin{thm} \label{chaoticss}
 Let $f_N(\bv)$ be the one particle marginal of $F_N(\bV)$ i.e., defined by
 \begin{equation}\label{NN}
  f^{(1)}_N(\bv_1)=\frac{\sqrt{N}}{\sqrt{N-
 |\bv_1|^2}} \frac{1}{\widetilde{Z}_N}\int_{\cS^{d{N-1}-1}(\sqrt{N-|\bv_1|^2})}
F_N(\bV)d\sigma(\bV^1)
\end{equation}
and set
\begin{equation}
f(\bv)=\frac{\mu^{\frac{d}{2}}}{c}e^{-(\sqrt{\mu} |\bv|)^{2+\alpha}}
\end{equation} 
with the constants given by \eqref{constants}.  Then for any bounded continuous function $\varphi(\bv)$ 
\begin{equation}\label{prove}
 \lim_{N\to\infty} \int_{\mathbb{R}^d} \varphi(\bv) f_N^{(1)}(\bv) d \bv =\int_{\mathbb{R}^d} \varphi(\bv) f(\bv) d \bv
\end{equation}
and for every $k$,
the $k$ particle marginal $f^{(k)}_N(\bv_1,\ldots,\bv_k)$ of $F(\bV)$  satisfies
\begin{equation}\label{prove1}
 \lim_{N\to\infty}\int_{\mathbb{R}^{kd}} \varphi(\bv_1,\ldots,\bv_k)  f^{(k)}_N(\bv_1,\ldots,\bv_k) d \bv_1 \cdots d \bv_k =
 \int_{\mathbb{R}^{kd}} \varphi(\bv_1,\ldots,\bv_k)  \prod_{i=1}^k f(\bv_k)  d \bv_1 \cdots d \bv_k
\end{equation}
where, again, $\varphi$ is any bounded continuous function on $\mathbb{R}^{kd}$. Thus $F_N(\bV)$
from a chaotic sequence with marginal $f$.
% Finally the limit \eqref{prove} and
% \eqref{prove1} are also true uniformly on $\mathbb{R}^d$ and in
% $L^1(\mathbb{R}^d)$
\end{thm}
\medskip



\section{Proof of Theorem \ref{chaoticss}.}

The following elementary lemma sets the stage for the proof. It will be
expressed in terms of the probability distribution 
\[
g(\bw) := \frac{e^{-|\bw|^{2+\alpha}}}{\int_{\mathbb{R}^d} e^{- |\bw|^{2+\alpha}} d\bw } \ .
\]
\begin{lem} \label{basic} The following formulas hold for $F_N(\bV)$:
\begin{equation} \label{formulaeff}
F_N(\bV) =  \frac{2+\alpha}{\Gamma(\frac{dN-1}{2 +\alpha})} \frac{1}{Z_N}
\int_0^\infty   t^{dN-1}   \prod_{j=1}^N g( \bv_j t ) \frac{dt}{t} \ ,
\end{equation} 
\begin{equation} \label{formulazee}
 Z_N=\frac{(2+\alpha)}{\Gamma\left(\frac{dN-1}{2+\alpha}\right)}\int_{
\mathbb{R}^{dN}}\frac{
\prod_{i=1}^Ng(\bw_i)}{|\bW|}\,d\bW
\end{equation}
and
\begin{align} \label{formulakay}
f^{(k)}_N(\bV_k) :&= \sqrt{\frac{N}{N - |\bV_k|}}
\int_{\cS^{d(N-k)-1}(\sqrt{N - |\bV_k|^2})}
F_N(\bV_k, \bV^k) d \sigma^{d(N-k)-1}(\bV^k)  \\
&=  \frac{2+\alpha}{\Gamma(\frac{dN-1}{2 +\alpha})Z_N }  \sqrt{\frac{N}{(N -
|\bV_k|)^{dk+1}}}\times\nonumber\\
&\phantom{=\frac{2+\alpha}{\Gamma(\frac{dN-1}{2 +\alpha})Z_N
}}\int_{\mathbb{R}^{d(N-k)}}    \prod_{j=1}^k g\left(\frac{\bv_j
|\bW^k|
}{\sqrt{N- |\bV_k|^2}}\right) |\bW^k|^{dk-1}
\prod_{j=k+1}^N g(\bw_j)  d\bW^k \ .\nonumber
\end{align}

\end{lem}
\begin{proof}
Formula \eqref{formulaeff} follows from  \eqref{eff} and 
\begin{equation}\label{exp}
A^{- \gamma} = \frac{1}{\Gamma(\gamma)} \int_0^\infty s^\gamma e^{-As} \frac{ds}{s} \ ,
\end{equation}
valid for all $A>0$ and $\gamma >0$ by setting
\[
A = \left(\sum_{i=1}^N
|\bv_i|^{2+\alpha}\right) \  ,  \ \gamma = dN-1
\]
and substituting $s =t^{2+\alpha}$. The normalization constant $\widetilde{Z}_N$
given in \eqref{normalization} is then given as 
\[
\frac{2+\alpha}{ \Gamma(\frac{dN-1}{2 +\alpha})} \int_{\cS^{Nd-1}(\sqrt N)}
\int_0^\infty   t^{dN-1}  \prod_{j=1}^N
e^{-\left(|\bv_j| t \right)^{2+\alpha}} \frac{dt}{t} d \sigma^{Nd-1} (\bV)
\]
 which, using Fubini's theorem and changing variables $\bv_j = \sqrt N \bw_j$
equals 
 \[
 \frac{(2+\alpha)N^{\frac{Nd-1}{2}}}{\Gamma(\frac{dN-1}{2 +\alpha})}
\int_0^\infty  \int_{\cS^{Nd-1}(1)}    \prod_{j=1}^N e^{-( |\bw_j|t\sqrt N)
^{2+\alpha}}  d \sigma^{Nd-1} (\bW)  t^{dN-1}  \frac{dt}{t}  \ . 
  \]
One more variable change $r = \sqrt N t $  yields
\[
 \frac{(2+\alpha)}{\Gamma(\frac{dN-1}{2 +\alpha})}   \int_0^\infty
\int_{\cS^{Nd-1}(1)}    \prod_{j=1}^N e^{-( |\bw_j|r ) ^{2+\alpha}}  d
\sigma^{Nd-1} (\bW)   r^{dN-1} \frac{dr}{r} \ .  
\]
Hence
\[
Z_N = \frac{(2+\alpha)}{\Gamma(\frac{dN-1}{2 +\alpha})}   \int_0^\infty
\int_{\cS^{Nd-1}(1)}    \prod_{j=1}^N g ( \bw_jr)  d \sigma^{Nd-1} (\bW)
r^{dN-1} \frac{dr}{r} \ .  
\]
This integral is  \eqref{formulazee} written in terms of polar coordinates. To
see \eqref{formulakay} we start with \eqref{kmarginal} and find
\[
f_N^{(k)}(\bV_k) =
\sqrt{\frac{N}{N -|\bV_k|}}
\frac{2+\alpha}{\Gamma(\frac{dN-1}{2 +\alpha})} \frac{1}{Z_N}
\int_{\cS^{d(N-k)-1}(\sqrt{N -|\bV_k|^2})} \int_0^\infty
t^{dN-1}   \prod_{j=1}^N g (\bv_j t) \frac{dt}{t}  d \sigma^{d(N-k)-1}(\bV^k) \
.
\]
Once more, using Fubini's theorem and changing variables $\bv_j = \sqrt{N -
|\bV_k|^2} \bw_j, j=k+1, \dots, N$ yields
\begin{align}
\sqrt{\frac{N}{N -|\bV_k|^2}}&
\frac{2+\alpha}{\Gamma(\frac{dN-1}{2 +\alpha})} \frac{1}{Z_N}
(N- |\bV_k|^2)^{\frac{d(N-k)-1}{2}} \times\nonumber\\
&\int_0^\infty   t^{dN-1}    \prod_{j=1}^k g(\bv_j t)
\int_{\cS^{d(N-k)-1}(1)}
\prod_{j=k+1}^N g\left(\sqrt{N - |\bV_k|^2} \bw_j t \right) d
\sigma^{d(N-k)-1}(\bW^k) \frac{dt}{t} \ . \nonumber
\end{align}
Changing variables $r = \sqrt{N - |\bV_k|^2} t$ yields the
expression
\begin{align*}
&\sqrt{\frac{N}{N - |\bV_k|^2}}
\frac{2+\alpha}{\Gamma(\frac{dN-1}{2 +\alpha})} \frac{1}{Z_N}
\frac1{(N-  |\bV_k|^2)^{\frac{dk}{2}}} \times\nonumber\\\
&\qquad \int_0^\infty   r^{d(N-k)-1}    \prod_{j=1}^k g\left(\frac{\bv_j
r}{\sqrt{N-
|\bV_k|^2}} \right) r^{dk-1}
\int_{\cS^{d(N-k)-1}(1)}
\prod_{j=k+1}^N g\left(\bw_j r \right)   d \sigma^{d(N-k)-1}(\bW^k) dr
\nonumber\ .
 \end{align*}
 which equals
 \[
 \frac{2+\alpha}{\Gamma(\frac{dN-1}{2 +\alpha})Z_N}  \sqrt{\frac{N}{(N -
 |\bV_k|^2)^{dk+1}}}
  \int_{\mathbb{R}^{d(N-k)}}    \prod_{j=1}^k g\left(\frac{\bv_j |\bW^k|
}{\sqrt{N-|\bV_k|^2}} \right)|\bW^k|^{dk-1}     \prod_{j=k+1}^N
g\left(\bw_j \right)  d\bW^k \ . 
 \]
\end{proof}


\begin{lem}  \label{limitnorm}
We have the following limit
\[
 \lim_{N \to \infty} \int_{\mathbb{R}^{Nd}} \frac{\sqrt N}{|\bW|} \prod_{j=1}^N
g(\bw_j) d\bW  = \frac{1}{\sqrt \mu} 
\]
where
\[
 \mu := \int_{\mathbb{R}^d} |\bw|^2 g(\bw) d \bw \ ,
\]
so that
\[
\lim_{N \to \infty} \sqrt N \Gamma\left(\frac{dN-1}{2+\alpha}\right)Z_N
=\frac{(2+\alpha)}{\sqrt \mu}
\ . 
\]


\end{lem}
\begin{proof}  The proof follows with a slight modification from the law of
large numbers. We denote  
\[
\mathbb{P}(A) :=\int_A \prod_{j=1}^N g(\bw_j) d\bW \ , \
\ {\rm and} \  \sigma^2 := \int_{\mathbb{R}^d} (|\bw|^2 - \mu)^2 g(\bw) d \bw \ .
\]
If
\[
A_\varepsilon =\left\{ \bW \in \mathbb{R}^{Nd} : \left| \frac{|\bW|^2}{N} -
\mu\right| > \varepsilon \right\}
\]
then it is a standard estimate that
\begin{equation} \label{tcheb}
\mathbb{P}(A_\varepsilon) \le \frac{\sigma^2}{\varepsilon^2 N} \ .
\end{equation}
From this it follows readily that
\begin{equation}
\frac{1}{\sqrt{\mu+\varepsilon}} \mathbb{P}(A^c_\varepsilon) \le
\int_{A^c_\varepsilon} \frac{\sqrt N}{|\bW|} \Pi_{j=1}^N g(\bw_j) d \bW  
\le \frac{1}{\sqrt{\mu-\varepsilon}} \mathbb{P}(A^c_\varepsilon) \ .
\end{equation}
The set $A_\varepsilon$ can be written as $A_\varepsilon = A^< _\varepsilon \cup
A^> _\varepsilon$ where 
\[
A^< _\varepsilon = \{ \bW \in \mathbb{R}^{Nd} :  \frac{|\bW|^2}{N} <  \mu - \varepsilon \} \ {\rm and} \ 
 A^>_\varepsilon = \{ \bW \in \mathbb{R}^{Nd} :  \frac{|\bW|^2}{N} >  \mu + \varepsilon \} \ .
\]
Clearly
\[
 \int_{A^>_\varepsilon} \frac{\sqrt N}{|\bW|} \prod_{j=1}^N g(\bw_j) d \bW  \le
\frac{1}{\sqrt{\mu+\varepsilon}} \mathbb{P}(A^>_\varepsilon) \ , \] 
 however, to estimate the corresponding integral in the set $A^<_\varepsilon$ is
a little bit trickier, on account of the singularity of $\frac{1}{|\bW|}$. Fix
some number $a < \sqrt{\frac{\mu - \varepsilon}{d}}$ independent of $N$  so that
for $N$ sufficiently large
 \[
 \int_{C^d} \frac{g(\bw)}{|\bw|^{\frac{1}{N}}} d \bw <  \frac{1}{2}
 \]
 where $C^d =[-a,a]^d$. 
 Note that the cube $C^{Nd} =[-a,a]^{Nd} \subset A^<_\varepsilon$. By the
arithmetic-geometric mean inequality 
 \begin{equation}\label{agm}
 \sqrt{\frac{N}{|\bW|^2}} \le \prod_{j=1}^N |\bw_j|^{-1/N}
 \end{equation}
 so that
\[
\int_{C^{Nd}} \frac{\sqrt N}{|\bW|} \prod_{j=1}^N g(\bw_j) d \bW \le
\left(\int_{C^d} \frac{g(\bw)}{|\bw|^{\frac{1}{N}}} d \bw \right)^N \le 
\frac{1}{2^N} \ .
\]
It remains to estimate the integral
\[
 \int_{A^<_\varepsilon \setminus  C^{Nd}} \frac{\sqrt N}{|\bW|} \Pi_{j=1}^N g(\bw_j) d \bW   \ .  
\]
Here we note that the ball centered at the origin of radius $a$ is a subset of
$C^{Nd}$ and hence 
\[
\sqrt{\frac{N}{|\bW|^2}} \ge \frac{\sqrt N}{a} 
\]
and this leads to the estimate
\[
 \int_{A^<_\varepsilon\setminus  C^{Nd}} \frac{\sqrt N}{|\bW|} \prod_{j=1}^N
g(\bw_j) d \bW   \le
  \frac{\sqrt N}{a} \mathbb{P}(A^<_\varepsilon \setminus  C^{Nd}) \ .
\]
Collecting these bounds yields
\begin{equation}\label{Ae}
\int_{A_\varepsilon} \frac{\sqrt N}{|\bW|} \prod_{j=1}^N
g(\bw_j) d \bW   \le\frac{\sqrt N}{a}
\mathbb{P}(A^<_\varepsilon \setminus C^{Nd}) +\frac{1}{2^N} +
\frac{1}{\sqrt{\mu+\varepsilon}} \mathbb{P}(A^>_\varepsilon)
\end{equation}
and finally
\begin{eqnarray}
 \frac{1}{\sqrt{\mu+\varepsilon}} - \frac{1}{\sqrt{\mu+\varepsilon}}
\mathbb{P} (A_\varepsilon)
  &\le&  \int_{\mathbb{R}^{Nd}} \frac{\sqrt N}{|\bW|} \prod_{j=1}^N g(\bw_j) d
\bW
\nonumber \\ &\le& \frac{1}{\sqrt{\mu-\varepsilon}} + \frac{\sqrt N}{a}
\mathbb{P}(A^<_\varepsilon \setminus C^{Nd}) +\frac{1}{2^N} +
\frac{1}{\sqrt{\mu+\varepsilon}} \mathbb{P}(A^>_\varepsilon)       
\end{eqnarray}
and using the estimate \eqref{tcheb} we find that
\[
\frac{1}{\sqrt{\mu+\varepsilon}}(1  - \frac{\sigma^2}{\varepsilon^2 N})
\le  \int_{\mathbb{R}^{Nd}} \frac{\sqrt N}{|\bW|} \Pi_{j=1}^N g(\bw_j) d \bW  \le 
 \frac{1}{\sqrt{\mu-\varepsilon}}(1 + \frac{\sigma^2}{\varepsilon^2 N} ) +
\frac{\sqrt N}{a}  \frac{\sigma^2}{\varepsilon^2 N} +\frac{1}{2^N}     
\]
By choosing $\varepsilon = N^{-1/8}$ the lemma follows.
\end{proof}

We now turn our attention to $f_N^{(k)}$. We first need the following 
elementary Lemma.

\begin{lem}\label{change}
Let $\bY=(\by_1,\ldots,\by_N)$ be defined by

\[\begin{cases}
 \by_i=\frac{|\bV^k|\bv_i}{\sqrt{N-|\bV_k|^2}}& 1\leq
i\leq k\\
\by_i=\bv_i& k+1\leq
i\leq N\ .
\end{cases}
\]
Then
\begin{equation}\label{Jac}
 \bv_i=\by_i\sqrt{\frac{N}{|\bY|^2}}\qquad\qquad 0\leq i \leq k\ .
\end{equation}
Moreover the Jacobian determinant is given by
\[
 \left|\frac{\partial (\bv_1,\ldots,\bv_k)}{\partial
(\by_1,\ldots,\by_k)}\right|=\frac{|\bY^k|^2}{N}\left(\frac{N}{|\bY|^2}
\right)^{\frac
{dk}2+1}=\frac{|\bV^k|^2}{N}\left(\frac{N^2-|\bV_k|^2}{|\bV^k|^2}\right)^{
\frac {dk}2+1}
\]
\end{lem}
\begin{proof} We have
\[
|\bY_k|^2=\frac{|\bV^k|^2|\bV_k|^2}{N-|\bV_k|^2}
\]
so that
\[
 N-|\bV_k|^2=\frac{N|\bY^k|^2}{|\bY|^2}
\]
from which \eqref{Jac} follows immediately. To conclude the proof of the
Lemma it is enough to observe that
\[
 \frac{\partial \bv_i}{\partial\by_j}=
 \sqrt{\frac{N}{|\bY|^2}}
\left(\delta_{i,j } - \frac{y_iy_j}{|\bY|^2}\right)
\]

 
\end{proof}

Let now $\varphi(\bV_k)$ be a continuous function on
$\mathbb{R}^{dk}$ such that
\[
 \sup_{\bV_k\in\mathbb{R}^{dk}}\varphi(\bV_k)<K\ .
\]
Using Lemma \ref{change} and \ref{formulaeff} we get
\begin{eqnarray}\label{inte}
&&\int d\bV_k\varphi(\bV_k)
f_N^{(k)}(\bV_k)=\nonumber\\
&&\qquad\frac{2+\alpha}{\Gamma(\frac{dN-1}{2
+\alpha})Z_N }\int_{\mathbb{R}^{dk}} d\bY_k\prod_{i=1}^kg(\by_i)
\int_{\mathbb{R}^{d(N-k)}}d\bY^k
\frac{\prod_{j=k+1}^N g(\by_j)}{|\bY|}%\frac{|\bY^k|}{|\bY|}
\varphi\left(\by_1\frac{\sqrt{N}}{|\bY|} , \ldots
, \by_k\frac {
\sqrt{N}}{|\bY|^2}\right):=\nonumber\\
&&\qquad\int_{\mathbb{R}^{dk}} d\by_1\cdots d\by_k\prod_{i=1}^kg(\by_i)
H_N(\by_1,\ldots,\by_k)
\end{eqnarray}
Since
\[
\frac{|\bY^k|^2}{|\bY^k|^2+|\bY_k|^2}<1
\]
we get
\[
 |H_N(\by_1,\ldots,\by_k)|\leq
K\frac{\Gamma\left(\frac{d(N-k)-1}{2+\alpha}\right)Z_{N-k}}{\Gamma\left(\frac{
dN-1}{2+\alpha}\right)Z_N}\leq K'
\]
for a suitable constant $K'$. For the last inequality we have used Lemma
\ref{limitnorm}. The proof can then be completed by applying the Dominated
Convergence Theorem if we can show that
\[
 \lim_{N\to\infty}H_N(\by_1,\ldots,\by_N)=\varphi\left(\frac{\by_1}{\mu},\ldots,
\frac{\by_N}{\mu}\right)
\]
pointwise in $\mathbb{R}^{dk}$. The proof of this fact follows closely that of
Lemma \ref{limitnorm}. From \eqref{Ae} it follows that
\begin{eqnarray*}
&& \int_{A_\varepsilon}d\bY^k
\prod_{j=k+1}^N
g(\by_j)\frac{\sqrt{N}}{|\bY^k|}\nonumber\sqrt{\frac{|\bY^k|^2}{|\bY^k|^2+
|\bY_k|^2}}
\left|\varphi\left(\by_1\frac{\sqrt{N}}{|\bY|} , \ldots ,
\by_k\frac{\sqrt{N}}{|\bY|}\right)-\varphi\left(\frac{\by_1}{\mu }
,\ldots,
\frac{\by_N}{\mu}\right)\right|\leq\nonumber\\
&&\qquad\qquad2K\left(\frac{\sqrt N}{a}
\mathbb{P}(A^\varepsilon \setminus C^{Nd}) +\frac{1}{2^N} +
\frac{1}{\sqrt{\mu+\varepsilon}} \mathbb{P}(A^>_\varepsilon)\right)
\end{eqnarray*}
while
\begin{eqnarray*}
&& \int_{\mathbb{R}^{Nd}\setminus A_\varepsilon}d\bY^k
\prod_{j=k+1}^N
g(\by_j)\frac{\sqrt{N}}{|\bY^k|}\nonumber\sqrt{\frac{|\bY^k|^2}{|\bY^k|^2+
|\bY_k|^2}}\left|\varphi\left(\by_1\frac{\sqrt{N}}{|\bY|} ,
\ldots , \by_k\frac{\sqrt{N}}{|\bY|}\right)-
\varphi\left(\frac{\by_1}{\mu }
,\ldots,
\frac{\by_N}{\mu}\right)
\right|\leq\nonumber\\
&&\qquad\qquad\frac{1}{\sqrt{\mu-\varepsilon}}\sup_{\lambda\in
I_N}|\varphi(\lambda\by_i,\ldots,\lambda\by_k)-\varphi(\by_i,\ldots,
\by_k)|
\end{eqnarray*}
where $I_N$ is the interval
\[
 I_N=\left[\sqrt{\frac{N}{N(\mu+\varepsilon)+|\bY_k|^2}},
 \sqrt{\frac{N}{N(\mu-\varepsilon)+|\bY_k|^2}}\right]\ .
\]
Collecting these estimates we get
\begin{eqnarray}
\biggl|H_N(\by_i,\ldots,\by_k)&-&\varphi\left(\frac{\by_1}{\mu},\ldots,
\frac{\by_N}{\mu}\right)\biggr|\leq\nonumber\\
&&\frac{2+\alpha}{\Gamma(\frac{dN-1}{2
+\alpha})\sqrt{N}Z_N }\left[ \frac{1}{\sqrt{\mu-\varepsilon}}\sup_{\lambda\in
I_N}|\varphi(\lambda\by_i,\ldots,\lambda\by_k)-\varphi\left(\frac{\by_i}{\mu},
\ldots ,
\frac{\by_k}{\mu}\right)|+\right.\nonumber\\
&&\left.2K\left(\frac{\sqrt N}{a}
\mathbb{P}(A^\varepsilon \setminus C^{Nd}) +\frac{1}{2^N} +
\frac{1}{\sqrt{\mu+\varepsilon}} \mathbb{P}(A^>_\varepsilon)\right)\right]
\end{eqnarray}
from which Theorem \ref{chaoticss} follows by choosing $\varepsilon=N^{-\frac
18}$.

\section{Extension and Remarks.}

It is easy to extend the results of the previous section in a couple of
interesting directions. We first observe that one can give a stronger
definition of chaoticity by requiring that given a sequence of functions
$F_N(\bV)$ on $\cS^{dN-1}(\sqrt{N})$, the entropy per particle of this sequence
converge to the entropy of the 1 particle marginal. More precisely if
\[
 S_N=\int_{\cS^{dN-1}(\sqrt{N})}F_N(\bV)\log
F_N(\bV)d\sigma(\bV
\]
is the entropy of the $N$ particles system than
\[
 \lim_{N\to \infty}\frac{S_N}{N}=\int_{\mathbb{R}^d} f(\bv)\log
f(\bv)d\bv
\]
where, as before
\[
 f(\bv)=\lim_{N\to\infty} f^{(1)}_N(\bv).
\]
If this is true we say that the sequence $F_N$ is {\bf entropically chaotic},
see \cite{CCLRV}.

\begin{cor}
The sequence $F_N(\bV)$ given by \eqref{eff} is entropically chaotic and
\begin{equation}\label{SSN}
\lim_{N\to \infty} N^{-1}\int_{\cS^{dN-1}(\sqrt{N})}F_N(\bV)\log
F_N(\bV)d\sigma(\bV)=\log\left(\frac{\mu^{\frac{d}{2}}}{c}\right)-\frac{d}{
2+\alpha}=\int_{\mathbb{R}^d} f(\bv)\log f(\bv)d\bv
\end{equation}
\end{cor}

\begin{proof}
We will just report here the minor modification to the proof of Lemma
\ref{limitnorm} needed to prove the corollary. We observe that
\[
 x\log x=\lim_{\delta\to 0} \frac{x^{1+\delta}-x}{\delta}
\]
Applying this to \eqref{exp} we get
\[
 A^{-\gamma}\log A^{-\gamma}=\frac{1}{\Gamma(\gamma)}\int_0^\infty
s^{\gamma}\log s^{\gamma}e^{-As}ds-\gamma\psi(\gamma)A^{-\gamma}
\]
where $\psi(x)=\Gamma'(x)/\Gamma(x)$ is the digamma
function. Following the proof of Lemma \ref{basic}, we find
\begin{eqnarray}\label{SN}
\frac{S_N}{N}&=& -\frac{\log\tilde
Z_N}{N}-\frac{dN-1}{(2+\alpha)N}\psi\left(\frac{dN-1}{2+\alpha}
\right)+\nonumber\\
&&\frac{dN-1}{N}\frac{(2+\alpha)}{\Gamma\left(\frac{dN-1}{2+\alpha}\right)Z_N}
\int_ {\mathbb{R}^{dN}}\log\left(\frac{|\bW|}{\sqrt{N}}\right)\frac{
\prod_{i=1}^Ng(\bw_i)}{|\bW|}\,d\bW
\end{eqnarray}
where $\tilde Z_N=c^{N}Z_N$. Using Stirling formula we get that
\[
 \lim_{N\to\infty}\left(\frac{\log\tilde
Z_N}{N}-\frac{dN-1}{(2+\alpha)N}\psi\left(\frac{dN-1}{2+\alpha}
\right)\right)=-\log c-\frac{d}{2+\alpha}
\]
Finally we need to compute the integral in the last term of \eqref{SN}.
It is very easy to adapt the proof of Lemma \ref{limitnorm} to this expression.
Indeed we have that
\[
 \log\left(\frac{|\bW|}{\sqrt{N}}\right)\frac{\sqrt{N}}{|\bW|}
<\max\left(e , \frac{\log\sqrt{\mu+\epsilon}}{\sqrt{\mu+\epsilon}}\right)
\]
for $\bW\in A_\varepsilon^>$. On the other hand, the function $|(\log x)/x|$ is
increasing for $x<1$ so that from \eqref{agm} we get
\[
\left|\log\left(\frac{|\bW|}{\sqrt{N}}\right)\frac{\sqrt{N}}{|\bW|}\right|\leq
\frac{\sum_{i=1}^N\left|\log|\bw_i|\right|}{N}\prod
\frac{1}{|\bw_i|^{\frac{1}{N}}}\qquad\qquad\mathrm{for\
}\frac{|\bW|}{\sqrt{N}}<1
\]
One can now proceed like in Lemma \ref{limitnorm} with the only difference that
$a$ must be chosen such that
\[
\int_{C^d} \frac{\left|\log|\bw_i|\right|}{|\bw|^{\frac{1}{N}}} g(\bw)d \bw <
\frac{1}{2}
\]
We finally obtain that
\[
 \lim_{N\to\infty}\int_
{\mathbb{R}^{dN}}\log\left(\frac{|\bW|}{\sqrt{N}}\right)\frac{
\sqrt{N}}{|\bW|}\prod_{i=1}^Ng(\bw_i)\,d\bW=\frac{\log\sqrt{\mu}}{\sqrt{\mu}}
\] 
Collecting the above computation we get the first equality in \eqref{SSN}. The
second equality is immediate.

\end{proof}

A last interesting extension regards the first order correction in $\bE$. In
\cite{BCKLs}, under the assumption that the limit $|\bE|\to0$ exists, it was
shown that
\[
 F_{\rm ss}(\bV,\bE)=\frac{1}{\widetilde{Z}_N}\left(F_N(\bV)+
\sum_{i=1}^N \bE\cdot\bc(\omega_i)|\bv_i|H_N(\bV)+o(|\bE|)\right)
\]
where $\bv_i=|\bv_i|\omega_i$ and 
\[
 H_N(\bV)=\frac{dN-1}{\left(\sum_{i=1}^N
|\bv_i|^{2+\alpha}\right)^{\frac{dN-1}{2+\alpha}+1}
}=\frac{1}{|\bv_1|^{2+\alpha}}\bv_1\cdot\nabla_{\bv_1}F_N(\bV),
\]
$\bc(\omega)$ is the unique solution of
\[
 \left[(\mathrm{Id}-\cK)\bc\right](\omega)=\omega
\]
where $\cK$ is the convolution operator generated by $K$, that is
\begin{equation*}
 \left(\cK\bc\right)(\omega)=\int_{\cS^{d-1}(1)}
K(\omega\cdot\omega')\bc(\omega')d\sigma^ { d-1 }
(\omega').
\end{equation*}
Since $-\bc(-\omega)$ is also a solution if $\bc(\omega)$ is, we have, by
uniqueness, that $\bc(\omega)=-\bc(-\omega)$. As a consequence
\[
 \int_{\cS^{d-1}(1)}\bc(\omega')d\sigma^{d-1}(\omega')=0.
\]

Calling $h_N^{(k)}$ the marginal of $H_N$, we easily get that
\[
 h_N^{(k)}(\bv_1,\ldots,\bv_k)=\frac{1}{|\bv_1|^{2+\alpha}}
\bv_1\cdot\nabla_{\bv_1}f_N^{(k)}(\bv_1,\ldots,\bv_k)
\]
It is easy to see, from \eqref{inte}, that we can take the limit for
$N\to\infty$ on both side and obtain
\[
 \lim_{N\to\infty} h_N^{(k)}(\bv_1,\ldots,\bv_k)=h(\bv_1)\prod_{i=2}^k f(\bv_k)
\]
where
\[
 h(\bv)=\frac{1}{|\bv_1|^{2+\alpha}}
\bv\cdot\nabla_{\bv}f(\bv)=(2+\alpha)\mu^{\frac{2+\alpha}{2}}f(\bv)
\]
Combining the above results we get that the $k$ particle marginal of $F_{\rm
ss}$ is 
\begin{equation}\label{1oN}
 \lim_{N\to\infty} f_{\rm
ss}^{(k)}(\bv_1,\ldots,\bv_k;\bE)=\left(1+(2+\alpha)\mu^{\frac{2+\alpha}{2}}
\sum_{i=1}^k \bE\cdot\bc(\omega_i)|\bv_i|\right)\prod_{i=1}^k f(\bv_k)
\end{equation}

This is consistent with the results on the Boltzmann equation \eqref{BE}. To
solve the steady state equation of \eqref{BE} one as to make an assumption on
the form of $\hat\bj_{\rm ss}(E)$ for small $|E$. It is natural to assume that
\begin{equation}\label{cond}
\hat\bj_{\rm ss}(E)=\tau\underline\kappa E+o(|E|)
\end{equation}
where $\underline\kappa$ is the conductivity tensor for the system with 1
particle and energy 1, that is
\[
 \underline\kappa=\frac{1}{|\cS^{d-1}(1)|}\int_{\cS^{d-1}(1)}
\bc(\omega)\otimes\omega\,d\sigma^{d-1}(\omega).
\]
Under this assumption one finds that
\begin{equation}\label{1oinfty}
 f_{\rm
ss}(\bv,\bE)=\left(1+(2+\alpha)\nu^{\frac{2+\alpha}{2}}
\bE\cdot\bc(\omega)|\bv|\right)\tilde f(\bv)+o(|E|)
\end{equation}
where
\[
 \tilde f(\bv)=\frac{\nu^{\frac{d}{2}}}{b}e^{-(\sqrt{\nu} |\bv|)^{2+\alpha}}
\]
with $\nu$ and $b$ uniquely determined by normalization and \eqref{cond}. One
easily see that the average energy of this solution is
\[
u= \int_{\mathbb{R}^d}
|v^2|\tilde f(\bv)\,dv=\left(\frac{\nu}{\mu}\right)^{\frac{2+\alpha}{2}}
\]
so that, requiring $u=1$ we get back the large $N$ limit of the one particle
marginal of $F_{\rm ss}$.
Clearly the first order in $E$ of the $k$-fold tensor product of
\eqref{1oinfty} give us back \eqref{1oN}. We finally notice that the above
results tell us that the current per particle at small field for a large
system is $(2+\alpha)\mu^{\frac{2+\alpha}{2}}$ times the current of the one
particle system, if the energy per particle is 1.

\subsection*{Acknowledgment} We are indebted to Joel Lebowitz, Ovidiu Costin
and Eric Carlen form many enlightening discussions. 

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