Scaling limits for the generalized Langevin equation

TL;DR

This paper analyzes the diffusive behavior of solutions to the generalized Langevin equation in a periodic potential, deriving asymptotic limits for the effective diffusion coefficient and validating results with spectral numerical methods.

Contribution

It provides new asymptotic results for the GLE's diffusion coefficient in various regimes and employs spectral numerical methods for validation.

Findings

Sharp equilibration estimates for GLE using hypocoercivity

Asymptotic formulas for diffusion coefficient in three regimes

Numerical confirmation of theoretical asymptotics

Abstract

In this paper, we study the diffusive limit of solutions to the generalized Langevin equation (GLE) in a periodic potential. Under the assumption of quasi-Markovianity, we obtain sharp longtime equilibration estimates for the GLE using techniques from the theory of hypocoercivity. We then prove asymptotic results for the effective diffusion coefficient in three limiting regimes: the short memory, the overdamped and the underdamped limits. Finally, we employ a recently developed spectral numerical method in order to calculate the effective diffusion coefficient for a wide range of (effective) friction coefficients, confirming our asymptotic results.