# Logarithmic Sobolev inequality for generalized simple exclusion processes

DOI: 10.1007/s004400050140

- Cite this article as:
- Yau, HT. Probab Theory Relat Fields (1997) 109: 507. doi:10.1007/s004400050140

- 8 Citations
- 92 Downloads

## Summary.

Let \(\) be a probability measure on the set {0,1, . . .,*R*} for some *R*∈ℕ and Λ_{L} a cube of width *L* in *ℤ*^{d}. Denote by μ^{gc}_{ΛL} the (grand canonical) product measure on the configuration space on Λ_{L} with \(\) as the marginal measure; here the superscript indicates the grand canonical ensemble. The canonical ensemble, denoted by μ^{c}_{ΛL,n}, is defined by conditioning μ^{gc}_{ΛL} 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 μ^{c}_{ΛL,n} is reversible. We prove the logarithmic Sobolev inequality in the sense that ∫*f*log*fd*μ^{c}_{ΛL,n}≤\(\) for any probability density *f* with respect to μ^{c}_{ΛL,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.