H. T. Yau
Logarithmic Sobolev Inequality for Lattice Gases
with Mixing Conditions
(385K, ps)
ABSTRACT. Let $\bar \mu$ be a
prLet $\mu^{gc}_{\L_L, \l}$ denote the grand canonical Gibbs measure of a lattice gas in a
cube of size $L$ with the chemical potential $\l$ and a fixed boundary condition.
Let $\mu^c_{\L_L, n}$ be the corresponding canonical measure defined by conditioning
$\mu^{gc}_{\L_L, \l}$ on $\sum_{x \in \L} \eta_x = n$.
Consider the lattice gas
dynamics for which each particle performs random walk
with rates depending on near-by particles. The rates are chosen
such that, for every $n$ and $L$ fixed, $\mu^c_{\L_L, n}$ is a
reversible measure. Suppose that the Dobrushin-Shlosman mixing conditions
holds for $\mu_{L, \l}$ for {\it all} chemical potentials $\l \in \RR$.
We prove that
$\int f \log f d\mu^c_{\L_L, n} \le \hbox{const.} L^2 D(\sqrt f)$
for any probability density $f$ with respect to $\mu^c_{\L_L, n}$; here
the constant is independent of $n$ or $L$ and $D$ denotes the Dirichlet form of
the dynamics. The dependence on $L$ is optimal.