Method of Langevin Equation for Irreversible Processes in Non-Linear Systems

Abstract

A stochastic equations \dotA(t) = α1{A(t)} + R(t) for a macroscopic observable A in a large system is rigorously derived from microscopic equations of motion, where α1(a) is the first derivate moment of transition probability of the corresponding master equation. The correlation function of the random force satisfies ≪R(0)R(t)> = Dδ(t), where the diffusion constant D is given by D ≡ ½ \int α2(a)feq(a)da = - \int α1(a)afeq(a)da by the use of the second derivate moment α2(a) and the equilibrium distribution function feq(a). This expression is a generalization of the Einstein relation. The stochastic equation is stochastically equivalent to the master equation with the aid of averages ≪R(t1)R(t2) … R(tn)>A(0) conditional on the initial value of A and, thus, gives a new method for the study on the irreversible process. On the assumption that the random force is a Gaussian process, the stochastic equation can be considered as a generalization of the Langevin equation in a non-linear system. A fluctuation-dissipation theorem and a general response theory are presented. A brief discussion on an extension of Onsager's relation of reciprocity to a system far away from equilibrium is also presented.