A New Expansion of the Master Equation

Abstract

A new expansion of the master equation is derived from the statistical-mechanical standpoint. The expansion parameter is a slowness parameter which controls the rate of change with time of macroscopic state variables, and the lowest-order terms lead to the generalized Fokker-Planck equation of the Green-Zwanzig type. In this expansion a fluctuation-dissipation relation between generalized diffusion coefficients and dissipative drift coefficients holds even in higher-order terms, thus ensuring the existence of a stationary solution in each order. As a simple application a nonlinear Langevin equation and the corresponding master equation are derived for the Brownian motion of a heavy particle floating in a liquid.