Asymptotic analysis for the generalized Langevin equation with singular potentials
Abstract
We consider a system of interacting particles governed by the generalized Langevin equation (GLE) in the presence of external confining potentials, singular repulsive forces, as well as memory kernels. Using a Mori-Zwanzig approach, we represent the system by a class of Markovian dynamics. Under a general set of conditions on the nonlinearities, we study the large-time asymptotics of the multi-particle Markovian GLEs. We show that the system is always exponentially attractive toward the unique invariant Gibbs probability measure. The proof relies on a novel construction of Lyapunov functions. We then establish the validity of the small mass approximation for the solutions by an appropriate equation on any finite-time window. Important examples of singular potentials in our results include the Lennard-Jones and Coulomb functions.
Cite
@article{arxiv.2305.03637,
title = {Asymptotic analysis for the generalized Langevin equation with singular potentials},
author = {Manh Hong Duong and Hung D. Nguyen},
journal= {arXiv preprint arXiv:2305.03637},
year = {2024}
}