Universal Form of Stochastic Evolution for Slow Variables in Equilibrium Systems

TL;DR

This paper derives a universal form of stochastic evolution equations for slow variables in equilibrium systems, clarifying their structure and relation to previous formulations through a path integral approach.

Contribution

It introduces a universal nonlinear Langevin equation for slow variables, using a novel discretization scheme in path integrals to unify different existing expressions.

Findings

Derivation of a universal nonlinear Langevin equation

Clarification of the relation between path integral formulations

Application to Brownian motion with nonlinear friction

Abstract

Nonlinear, multiplicative Langevin equations for a complete set of slow variables in equilibrium systems are generally derived on the basis of the separation of time scales. The form of the equations is universal and equivalent to that obtained by Green. An equation with a nonlinear friction term for Brownian motion turns out to be an example of the general results. A key method in our derivation is to use different discretization schemes in a path integral formulation and the corresponding Langevin equation, which also leads to a consistent understanding of apparently different expressions for the path integral in previous studies.