# Geometric ergodicity of Langevin dynamics with Coulomb interactions

## TL;DR

This paper proves that Langevin dynamics for Coulomb gases in multi-dimensional space converge exponentially to a unique equilibrium, using a novel Lyapunov function to analyze long-term behavior.

## Contribution

It introduces a new Lyapunov function approach to establish exponential ergodicity for Coulomb gas Langevin dynamics.

## Key findings

- Exponential convergence to the Boltzmann-Gibbs measure.
- Construction of a novel Lyapunov function for Coulomb systems.
- Proof of ergodicity in multi-dimensional Coulomb gases.

## Abstract

This paper is concerned with the long time behavior of Langevin dynamics of {\em Coulomb gases} in $\mathbf{R}^d$ with $d\geq 2$, that is a second order system of Brownian particles driven by an external force and a pairwise repulsive Coulomb force. We prove that the system converges exponentially to the unique Boltzmann-Gibbs invariant measure under a weighted total variation distance. The proof relies on a novel construction of Lyapunov function for the Coulomb system.