Gibbsian dynamics and the generalized Langevin equation

TL;DR

This paper investigates the invariant structures of the nonlinear generalized Langevin equation with power-law memory kernels, constructing solutions via a Gibbsian approach and establishing conditions for unique steady states.

Contribution

It introduces a Gibbsian framework for solving the GLE with broad memory kernels and generalizes conditions for steady state uniqueness beyond exponential kernels.

Findings

Constructed solutions for GLE using Gibbsian methods

Established decay conditions for memory kernels

Generalized steady state uniqueness results

Abstract

We study the statistically invariant structures of the nonlinear generalized Langevin equation (GLE) with a power-law memory kernel. For a broad class of memory kernels, including those in the subdiffusive regime, we construct solutions of the GLE using a Gibbsian framework, which does not rely on existing Markovian approximations. Moreover, we provide conditions on the decay of the memory to ensure uniqueness of statistically steady states, generalizing previous known results for the GLE under particular kernels as a sum of exponentials.