Ergodicity and Lyapunov functions for Langevin dynamics with singular potentials

TL;DR

This paper proves exponential convergence to equilibrium for Langevin dynamics with singular potentials, like Lennard-Jones, using Lyapunov functions, and improves understanding of the system's stability from various initial conditions.

Contribution

It introduces a novel Lyapunov function construction that establishes geometric convergence for Langevin systems with singular potentials, extending previous results.

Findings

Exponential convergence to the Gibbs measure is proven.

The convergence holds from a broad class of initial distributions.

A new Lyapunov function approach is developed.

Abstract

We study Langevin dynamics of $N$ particles on $R^d$ interacting through a singular repulsive potential, e.g.~the well-known Lennard-Jones type, and show that the system converges to the unique invariant Gibbs measure exponentially fast in a weighted total variation distance. The proof of the main result relies on an explicit construction of a Lyapunov function. In contrast to previous results for such systems, our result implies geometric convergence to equilibrium starting from an essentially optimal family of initial distributions.