Logarithmic Sobolev inequality for generalized simple exclusion processes

Summary. Let be a probability measure on the set {0,1, . . .,R} for some R∈ℕ and ΛL a cube of width L in ℤd. Denote by μgcΛL the (grand canonical) product measure on the configuration space on ΛL with as the marginal measure; here the superscript indicates the grand canonical ensemble. The canonical ensemble, denoted by μcΛL,n, is defined by conditioning μgcΛL given the total number of particles to be n. Consider the exclusion dynamics where each particle performs random walk with rates depending only on the number of particles at the same site. The rates are chosen such that, for every n and L fixed, the measure μcΛL,n is reversible. We prove the logarithmic Sobolev inequality in the sense that ∫flogfdμcΛL,n≤ for any probability density f with respect to μcΛL,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.