Stochastic Transport in a Disordered Solid. I. Theory

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 $\stackrel{\mathrm{´}}{\mathrm{t}}$o hopping conduction, an exact expression for the conductivity $\sigma \left(\omega \right)$ for the CTRW process is derived. The frequency dependence of $\sigma \left(\omega \right)$ is determined by the Fourier transform of the zeroth and second spatial moments of the function $\psi (\overrightarrow{\mathrm{s}},t)$, which is equal to the probability per unit time that the displacement and time between hops is $\overrightarrow{\mathrm{s}}$, $t$. 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.