The Ursell-function structure of the memory function

Abstract

The integrodifferential equation which defines a memory function f (t) in terms of the time−correlation function g (t) generates ordered Ursell functions having the cluster property. Thus f (t) has an expansion f (2) (t) + f (3) (t) + ... in which f (n) (t) is an n−1 fold time integral whose integrand vanishes when the interval between any two successive times is much greater than a certain correlation time of the system. An interaction representation is used in deriving these results and it is found that f (n) (t) depends, in a certain sense, upon the nth power of the interaction term H 1 of the Hamiltonian. The integrand of f (n) (t) is closely related to the ’’cumulant averaged’’ Liouville operators introduced by Kubo and Tomita and developed further by Kubo, Freed, and van Kampen. Thus, in the Markoffian limit f (2) (t) is simply related to the Redfield relaxation matrix. However, the peculiar time−ordering problems of the cumulant expansion theory do not appear here. Except in very simple cases, all of these results depend upon identifying g (t) as a correlation matrix (Kubo) and the f (t) is the corresponding memory matrix. As a first application the memory function for the EPRrelaxation of aqueous Ni++ is calculated in terms of the spin parameters. It is assumed that fluctuating zero−field splitting causes the relaxation, but it is not assumed that the EPRrelaxation obeys Bloch’s equations.