Broad class of nonlinear Langevin equations with drift and diffusion cofficients separable in time and space: Generalized n-moment, ergodicity, Einstein relation and fluctuations of the system

TL;DR

This paper analyzes a broad class of nonlinear Langevin equations with separable drift and diffusion coefficients, exploring their ergodic properties, generalized moments, Einstein relation, and fluctuation behavior.

Contribution

It introduces a generalized n-moment framework for nonlinear Langevin equations with separable coefficients, revealing ergodicity and fluctuation characteristics.

Findings

System may exhibit ergodic behavior with space-time-dependent coefficients.

A generalized Einstein relation is derived.

First two moments and variance effectively describe system fluctuations.

Abstract

A wide class of nonlinear Langevin equations with drift and diffusion coefficients separable in time and space driven by the Gaussian white noise is analyzed in terms of a generalized n-moment. We show the system may present ergodic property, a key property in statistical mechanics, for space-time-dependent drift and diffusion coefficients. A generalized Einstein relation is also obtained. Besides, we show that the first two generalized moments and variance are useful to describe the drift and fluctuations of the system.