Asymptotic Analysis for the Generalized Langevin Equation with Singular Potentials

Manh Hong Duong, Dang Hung 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.
Original languageEnglish
Article number62
JournalJournal of Nonlinear Science
Volume34
Issue number4
DOIs
Publication statusPublished - 14 May 2024

Keywords

  • 60H10
  • Lyapunov functions
  • Singular potentials
  • Small mass limits

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