Stochastic theory of multistate diffusion in perfect and defective systems. I. Mathematical formalism

Abstract

A stochastic theory of multistate transport on ideal and defective lattices is presented. A continuous-time random-walk formalism with the inclusion of internal states is used in the derivation of a matrix (whose dimensions are the number of internal states) probability propagator. The propagator describing motion on an ideal lattice is modified owing to the presence of a periodic arrangement of defects. The expression for the modified probability propagator greatly simplifies in the long-time (diffusion) limit. In this limit the presence of defects renormalizes the ideal-lattice propagator through the inclusion of a self-energy-type term which depends upon the concentration of defect sites and the differences in transition rates of the propagating species associated with these sites and ideal ones. The formalism enables the study of complex diffusion mechanisms, as illustrated in the following paper, and allows the calculation of observables such as positional moments, diffusion coefficients, and occupation probability of states.