Asymptotic analysis for the generalized Langevin equation with singular potentials

2023/05/05 by Duong, Manh Hong, Nguyen, Hung D. · 1 citation
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2305.03637

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.

Cited by

• Error Analysis of Generalized Langevin Equations with Approximated Memory Kernels

Related

• The Generalized Langevin Equation with a power-law memory in a nonlinear potential well
• Ergodicity and Lyapunov functions for Langevin dynamics with singular potentials
• Exponential ergodicity of Lévy driven Langevin dynamics with singular potentials
• Gibbsian dynamics and the generalized Langevin equation
• Trend to equilibrium and Newtonian limit for the relativistic Langevin equation with singular potentials