Exact transformation of a Langevin equation to a fluctuating response equation

TL;DR

This paper presents an exact transformation of a Langevin equation into a form that explicitly shows the velocity response to perturbations, aiding in understanding non-equilibrium stochastic systems.

Contribution

It introduces a novel operator-based transformation that links Langevin equations to response equations, enhancing analysis of non-equilibrium stochastic dynamics.

Findings

Provides a new exact transformation method.

Facilitates calculation of system response and force decomposition.

Offers a physical basis for force analysis in Langevin systems.

Abstract

We demonstrate that a Langevin equation that describes the motion of a Brownian particle under non-equilibrium conditions can be exactly transformed to a special equation that explicitly exhibits the response of the velocity to a time dependent perturbation. This transformation is constructed on the basis of an operator formulation originally used in nonlinear perturbation theory for differential equations by extending it to stochastic analysis. We find that the obtained expression is useful for the calculation of fundamental quantities of the system, and that it provides a physical basis for the decomposition of the forces in the Langevin description into effective driving, dissipative, and random forces in a large-scale description.