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arXiv · 2305.03637

Asymptotic analysis for the generalized Langevin equation with singular potentials

Manh Hong Duong·Hung D. Nguyen
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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.

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Manh Hong Duong, Hung D. Nguyen. 2024-03-13. Asymptotic analysis for the generalized Langevin equation with singular potentials. https://arxiv.org/abs/2305.03637

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