Logarithmic Sobolev inequality for lattice gases with mixing conditions

Let mu(Lambda L,lambda)(gc) denote the grand canonical Gibbs measure of a lattice gas in a cube of size L with the chemical potential lambda and a fixed boundary condition. Let mu(Lambda L,n)(c) be the corresponding canonical measure defined by conditioning mu(Lambda L,lambda)(gc) on Sigma(x is an element of Lambda)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(Lambda L,n)(c) is a reversible measure. Suppose that the Dobrushin-Shlosman mixing conditions holds for mu(L,lambda)(gc) for all chemical potentials lambda is an element of R. We prove that integral f log f d mu(Lambda L,n)(c) less than or equal to const. L(2)D(root f) for any probability density f with respect to mu(Lambda L,n)(c); here the constant is independent of n or L and D denotes the Dirichlet form of the dynamics. The dependence on L is optimal.