Equilibration problem for the generalized Langevin equation

TL;DR

This paper investigates the conditions under which a single oscillator governed by the generalized Langevin equation reaches equilibrium, revealing that nonlinearity can prevent equilibration due to localized excitations.

Contribution

It provides a detailed analysis of equilibration conditions for harmonic potentials and demonstrates how nonlinearity can hinder equilibration through localized modes.

Findings

Harmonic systems can equilibrate under certain conditions.

Nonlinearities can prevent equilibration even when harmonic systems do.

Localized excitations like discrete breathers are responsible for non-equilibration.

Abstract

We consider the problem of equilibration of a single oscillator system with dynamics given by the generalized Langevin equation. It is well-known that this dynamics can be obtained if one considers a model where the single oscillator is coupled to an infinite bath of harmonic oscillators which are initially in equilibrium. Using this equivalence we first determine the conditions necessary for equilibration for the case when the system potential is harmonic. We then give an example with a particular bath where we show that, even for parameter values where the harmonic case always equilibrates, with any finite amount of nonlinearity the system does not equilibrate for arbitrary initial conditions. We understand this as a consequence of the formation of nonlinear localized excitations similar to the discrete breather modes in nonlinear lattices.