Some properties on the reversibility and the linear response theory of Langevin dynamics

TL;DR

This paper rigorously justifies linear response theory for Langevin dynamics, providing conditions for smoothness, convergence, and response behavior, with applications to Green-Kubo relations and generalized Langevin systems.

Contribution

It offers new mathematical characterizations and conditions for the validity of linear response theory in Langevin dynamics, including both overdamped and underdamped cases.

Findings

Established equivalent characterizations for reversible Langevin dynamics.

Provided sufficient conditions for smoothness and exponential convergence to invariant measures.

Proved asymptotic dependence of response functions and validated Green-Kubo relations.

Abstract

Linear response theory is a fundamental framework studying the macroscopic response of a physical system to an external perturbation. This paper focuses on the rigorous mathematical justification of linear response theory for Langevin dynamics. We give some equivalent characterizations for reversible overdamped/underdamped Langevin dynamics, which is the unperturbed reference system. Then we clarify sufficient conditions for the smoothness and exponential convergence to the invariant measure for the overdamped case. We also clarify those sufficient conditions for the underdamped case, which corresponds to hypoellipticity and hypocoercivity. Based on these, the asymptotic dependence of the response function on the small perturbation is proved in both finite and infinite time horizons. As applications, Green-Kubo relations and linear response theory for a generalized Langevin dynamics are also proved in a rigorous fashion.