N/V-LIMIT FOR LANGEVIN DYNAMICS IN CONTINUUM

Abstract

We construct an infinite particle/infinite volume Langevin dynamics on the space of simple configurations in ℝd having velocities as marks. The construction is done via a limiting procedure using N-particle dynamics in cubes (-λ, λ]d with periodic boundary condition. A main step to this result is to derive an (improved) Ruelle bound for the canonical correlation functions of N-particle systems in (-λ, λ]d with periodic boundary condition. After proving tightness of the laws of the 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 fulfilling a uniform Ruelle bound (and their weak limits). Additionally, we prove that the initial/invariant distribution of the constructed dynamics is a tempered grand canonical Gibbs measure. All proofs work for a wide class of repulsive interaction potentials ϕ (including, e.g., the Lennard–Jones potential) and all temperatures, densities and dimensions d ≥ 1.