Generalized Langevin Equations

Abstract

A derivation is presented for a generalized Langevin equation of motion for a dynamical variable φ(R(t), P(t)) where R and P are the position and momentum of a single heavy particle in a bath of light particles. A detailed analysis is given for the conditions required for the validity of the equation. A stochastic Fokker–Planck equation is derived for the quantity D(t) = δ(R(t) − R̂)δ(P(t) − P̂) which may be used to obtain both the generalized Langevin equation for φ and a Fokker–Planck equation for the heavy particle distribution function. The correlation function for D(t) is computed and it is shown how this quantity may be used to obtain all other correlation functions of interest.