Logarithmic Sobolev inequality for lattice gases with mixing conditions

Abstract

Let μgcΛL,λμΛL,λgc denote the grand canonical Gibbs measure of a lattice gas in a cube of sizeL with the chemical potential γ and a fixed boundary condition. Let μcΛL,nμΛL,nc be the corresponding canonical measure defined by conditioning μgcΛL,λμΛL,λgc on Σx∈Ληx=nΣx∈Ληx=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 everyn andL fixed, μcΛL,nμΛL,nc is a reversible measure. Suppose that the Dobrushin-Shlosman mixing conditions holds for μgcL,λμL,λgc forall chemical potentials λ ∈ γ ∈ ℝ. We prove that ∫flogfdμcΛL,n≦const.L2D(f√)∫flog⁡fdμΛL,nc≦const.L2D(f) for any probability densityf with respect to μcΛL,nμΛL,nc ; here the constant is independent ofn orL andD denotes the Dirichlet form of the dynamics. The dependence onL is optimal.