A graphical representation of the memory function for spin relaxation

Abstract

The perturbation expansion of the memory function can be written as a series of graphs on white and black circles which are rather analogous to the white and black circle graphs for spatial correlation functions of equilibrium systems developed by Morita and Hiroike, De Dominices, and Stell. As in the equilibrium theory, the graphical representation of the memory function is especially convenient for generating resummed (or ’’renormalized’’) series based on summations of subclasses of infinite numbers of graphs. The resummed series are useful approximations to the memory function even when the perturbation that makes the spins relax is strong. Our most general results enable one to express the memory function as a generalized continued fraction or to calculate the time-correlation function as the solution to an integrodifferential matrix equation. The graphical representation of the latter result is surprisingly simple: the graphs are isomorphous with the graphs representing the perturbation expansion of the time derivative of the time-correlation function, but the bonds and faces which join the circles have different meanings.