N/V-limit for Langevin dynamics in continuum

TL;DR

This paper constructs an infinite-volume Langevin dynamics for particles with velocities in $ ^d$, using a limiting process from finite systems with periodic boundary conditions, applicable to general Ruelle-type potentials.

Contribution

It introduces a method to construct infinite particle Langevin dynamics in continuum, extending previous finite-volume approaches with improved correlation bounds and general interaction potentials.

Findings

Established an improved Ruelle bound for correlation functions.

Proved tightness and identified limits as martingale solutions.

Demonstrated the invariant measure as a tempered grand canonical Gibbs measure.

Abstract

We construct an infinite particle/infinite volume Langevin dynamics on the space of configurations in $\R^d$ 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 $\phi$ of Ruelle type (e.g. the Lennard-Jones potential) and all temperatures, densities and dimensions $d\geq 1$.