Stochastic Equations for Nonequilibrium Processes

Abstract

A system of weakly interacting particles is described by a time-dependent joint probability distribution in the occupation numbers of the individual particle states. The "master" equation for the distribution is obtained by considering the time evolution of the system as a Markoff process with transition probabilities per unit time given by first-order quantum mechanical perturbation theory. This is done for particles obeying classical and quantum statistics. The resulting equations include the usual rate equations for the average occupation numbers as special cases; but they also yield all higher moments and correlations in the occupation numbers. The general solution and its properties are discussed for the case in which a relaxing subsystem interacts via binary collisions with a larger system having a fixed but not necessarily thermal distribution. The explicit solution for the joint distribution in occupation numbers for all time is constructed for the case of identical harmonic oscillators which have an arbitrary initial distribution. These interact via binary collisions with a reservoir of similar oscillators, the coupling being linear in each oscillator coordinate. This model is also generalized and solved for a case in which the number of interacting particles is not conserved.