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 to hopping conduction, an exact expression for the conductivity σ(ω) for the CTRW process is derived. The frequency dependence of σ(ω) is determined by the Fourier transform of the zeroth and second spatial moments of the function ψ(→s,t), which is equal to the probability per unit time that the displacement and time between hops is →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.