A broad class of nonlinear Langevin equations with drift and diffusion coefficients separable in time and space: generalized n-moment, ergodicity, Einstein relation, and fluctuations of the system

Abstract

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