On the markoffian approximation in the description of relaxation in open systems

Abstract

Most traditional discussions of the Markoffian Born approximation of relaxation phenomena (e.g., Bloch equations) are based on the assumption of widely separated time scales in the non-Markoffian weak coupling generalized master equation. By exploiting the physically reasonable Gaussian factorization property of multitime correlation functions which enter through the system reservoir interaction, recent work has shed significant new light on the relationship between the Markoffian-Born approximation and the exact generalized master equation (GME) describing the evolution of the relaxing subsystem. Two limits for which the GME becomes the Born approximation relaxation equation arc discussed: (1) the Van Hove (λ2t) limit rigorously discussed for open systems by Davies, (2) the zero memory (white noise) suggested by Middleton—Schieve and recently discussed for open systems by Gorini—Kossakawsky and Palmer. We have outlined the proof of this limit. We then discuss the physical meaning of these two exact limits in the context of the general theory of kinetic equations of Resibois. The dominant pole contributions are isolated to all orders by means of an expansion in which the terms are in norm proportional to powers of the ratio of the time scales τc/τB where τc is the two point correlation memory time and τB the relaxation time in the Born approximation. Each of the above exact limits corresponds to the vanishing of this ratio. Comments are made on nonexponential decay.