Geometric ergodicity of Langevin dynamics with Coulomb interactions

Abstract

This paper is concerned with the long time behavior of Langevin dynamics of {\em Coulomb gases} in ${\mathbf{R}}^d$ with $d\ge 2$, 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.