Glauber dynamics of continuous particle systems

Abstract

This paper is devoted to the construction and study of an equilibrium Glauber-type dynamics of infinite continuous particle systems. This dynamics is a special case of a spatial birth and death process. On the space $\Gamma$ of all locally finite subsets (configurations) in ${\mathbb{R}}^d$, we fix a Gibbs measure $\mu$ corresponding to a general pair potential $\varphi$ and activity $z>0$. We consider a Dirichlet form $\mathcal{E}$ on $L^2(\Gamma ,\mu )$ which corresponds to the generator $H$ of the Glauber dynamics. We prove the existence of a Markov process $\mathbf{M}$ on $\Gamma$ that is properly associated with $\mathcal{E}$. In the case of a positive potential $\varphi$ which satisfies $\delta :={\int }_{{\mathbb{R}}^d}(1-e^{-\varphi (x)})zdx<1$, we also prove that the generator $H$ has a spectral gap $\ge 1-\delta$. Furthermore, for any pure Gibbs state $\mu$, we derive a Poincaré inequality. The results about the spectral gap and the Poincaré inequality are a generalization and a refinement of a recent result by L. Bertini, N. Cancrini, and F. Cesi.

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