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.