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

概要

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A stochastic equation A^^・(t)=α_1{A(t)}+R(t)for a macroscopic observable A in a large system is rigorously derived from microscopic equations of motion, where α_1(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≡1/2∫α_2(a)f^<eq>(a)da=-∫_1(a)af^<eq>(a)da by the use of the second derivate moment α_2 and the equilibrium distribution function f^<eq>(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(t_1)R(t_2)…R(t_n)>_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.
理論物理学刊行会の論文
1972-10-25