Communications in Mathematical Physics

, Volume 181, Issue 2, pp 367–408 | Cite as

Logarithmic Sobolev inequality for lattice gases with mixing conditions

  • Horng-Tzer Yau


Let\(\mu _{\Lambda _L ,\lambda }^{gc} \) 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\(\mu _{\Lambda _L ,n}^c \) be the corresponding canonical measure defined by conditioning\(\mu _{\Lambda _L ,\lambda }^{gc} \) on\(\Sigma _{x \in \Lambda } \eta _x = n\). 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 everyn andL fixed,\(\mu _{\Lambda _L ,n}^c \) is a reversible measure. Suppose that the Dobrushin-Shlosman mixing conditions holds for\(\mu _{L,\lambda }^{gc} \) forall chemical potentials λ ∈ γ ∈ ℝ. We prove that\(\smallint f\log fd\mu _{\Lambda _L ,n}^c \leqq const.L^2 D(\sqrt f )\) for any probability densityf with respect to\(\mu _{\Lambda _L ,n}^c \); here the constant is independent ofn orL andD denotes the Dirichlet form of the dynamics. The dependence onL is optimal.


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© Springer-Verlag 1996

Authors and Affiliations

  • Horng-Tzer Yau
    • 1
  1. 1.Courant InstituteNew York UniversityNew YorkUSA

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