Stochastic Transport in a Disordered Solid. I. Theory

Abstract

A general theory of stochastic transport in disordered systems has been developed. The theory is based on a generalization of the Montroll-Weiss continuous-time random walk (CTRW) on a lattice. Starting from a general mobility formalism, specialized $\stackrel{\mathrm{\ifmmode\acute\else\textasciiacute\fi{}}}{\mathrm{t}}$o hopping conduction, an exact expression for the conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ for the CTRW process is derived. The frequency dependence of $\ensuremath{\sigma}(\ensuremath{\omega})$ is determined by the Fourier transform of the zeroth and second spatial moments of the function $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{s}},t)$, which is equal to the probability per unit time that the displacement and time between hops is $\stackrel{\ensuremath{\rightarrow}}{\mathrm{s}}$, $t$. The conductivity corresponding to characteristically different types of hopping distributions is discussed, as well as the basic approximation in adopting a CTRW on a lattice to transport in disordered solids.