Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Asymptotic analysis for the generalized Langevin equation with singular potentials

Manh Hong Duong, Hung D. Nguyen

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.

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.
Paper Structure (20 sections, 14 theorems, 275 equations)

This paper contains 20 sections, 14 theorems, 275 equations.

Table of Contents

  1. Introduction
  2. Geometric ergodicity
  3. Small mass limit
  4. Organization of the paper
  5. Assumptions and main results
  6. Main assumptions
  7. Geometric ergodicity
  8. Small mass limit
  9. Geometric Ergodicity
  10. Single-particle GLE
  11. Positive viscous constant $\gamma>0$
  12. Zero viscous constant $\gamma=0$
  13. N-particle GLEs
  14. Proof of Theorem \ref{['thm:ergodicity:N-particle']}
  15. Small mass limit
  16. ...and 5 more sections

Key Result

Proposition 2.7

Under Assumption cond:U and Assumption cond:G, for every initial condition $X_0=(\mathrm{x}(0),\mathrm{v}(0)$, $\mathrm{z}_{1}(0)$,$\dots$,$\mathrm{z}_N(0))\in \mathbf{X}$, system eqn:GLE:N-particle admits a unique strong solution $X_m(t;X_0)=(\mathrm{x}_m(t)$, $\mathrm{v}_m(t)$, $\mathrm{z}_{1,m}(t

Theorems & Definitions (32)

  • Remark 2.2
  • Remark 2.4
  • Remark 2.6
  • Proposition 2.7
  • Theorem 2.8
  • Proposition 2.9
  • Theorem 2.10
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • ...and 22 more