96-215 H. T. Yau

Logarithmic Sobolev Inequality for Lattice Gases with Mixing Conditions (385K, ps) May 21, 96

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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.

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