Ergodicity and asymptotic limits for Langevin interacting systems with singular forces and multiplicative noises

TL;DR

This paper investigates ergodicity and asymptotic behaviors of classical and relativistic Langevin systems with singular forces and multiplicative noises, establishing convergence rates and limits for both models.

Contribution

It proves ergodicity and limits for Langevin systems with singular forces, including classical and relativistic models, using Lyapunov functions.

Findings

Exponential convergence to Boltzmann-Gibbs distribution in classical model

Algebraic mixing rate to Maxwell-Jüttner distribution in relativistic model

Recovery of overdamped and Newtonian limits in respective models

Abstract

In this paper, we study systems of $N$ interacting particles described by the classical and relativistic Langevin dynamics with singular forces and multiplicative noises. For the classical model, we prove the ergodicity, obtaining an exponential rate of convergence to the invariant Boltzmann-Gibbs distribution, and the small-mass limit, recovering the $N$-particle interacting overdamped Langevin dynamics. For the relativistic model, we establish the ergodicity, obtaining an algebraic mixing rate of any order to the Maxwell-J\"uttner distribution, and the Newtonian limit (that is when the speed of light tends to infinity), approximating a system of underdamped Langevin dynamics. The proofs rely on the construction of Lyapunov functions that account for irregular potentials and multiplicative noises.