A homogenization theorem for Langevin systems with an application to Hamiltonian dynamics

TL;DR

This paper presents a new homogenization theorem for Langevin systems, providing explicit limits for stochastic differential equations with applications to Hamiltonian dynamics under detailed balance conditions.

Contribution

It introduces a sufficient condition based on the solvability of a PDE for homogenization limits in Langevin systems, extending previous results.

Findings

Derived explicit homogenization limits for Langevin systems

Established a new theorem under generalized detailed balance conditions

Applied the theorem to systems with position-dependent temperature

Abstract

This paper studies homogenization of stochastic differential systems. The standard example of this phenomenon is the small mass limit of Hamiltonian systems. We consider this case first from the heuristic point of view, stressing the role of detailed balance and presenting the heuristics based on a multiscale expansion. This is used to propose a physical interpretation of recent results by the authors, as well as to motivate a new theorem proven here. Its main content is a sufficient condition, expressed in terms of solvability of an associated partial differential equation ("the cell problem"), under which the homogenization limit of an SDE is calculated explicitly. The general theorem is applied to a class of systems, satisfying a generalized detailed balance condition with a position-dependent temperature.