N/V-limit for Langevin dynamics in continuum

Abstract

We construct an infinite particle/infinite volume Langevin dynamics on the space of configurations in ${\mathbb{R}}^d$ having velocities as marks. The construction is done via a limiting procedure using $N$-particle dynamics in cubes $(-\lambda ,\lambda {]}^d$ with periodic boundary conditions. A main step to this result is to derive an (improved) Ruelle bound for the canonical correlation functions of $N$-particle systems in $(-\lambda ,\lambda {]}^d$ with periodic boundary conditions. After proving tightness of the laws of finite particle dynamics, the identification of accumulation points as martingale solutions of the Langevin equation is based on a general study of properties of measures on configuration space (and their weak limit) fulfilling a uniform Ruelle bound. Additionally, we prove that the initial/invariant distribution of the constructed dynamics is a tempered grand canonical Gibbs measure. All proofs work for general repulsive interaction potentials $\varphi$ of Ruelle type (e.g. the Lennard-Jones potential) and all temperatures, densities and dimensions $d\ge 1$.

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