Self-consistent mode-coupling theory of electrical conductivity and incoherent excitation transport in disordered media

We present a new method for calculating transport properties of an excitation such as an electron, exciton, or vibron in a disordered medium, in which the dynamics are governed by a Pauli master equation. A hierarchy of self-consistent mode-coupling equations for the master equation propagator is developed, and the first member of the hierarchy is explicitly constructed. This equation can be used to calculate the propagator for an excitation transfer rate with any dependence on the intermolecular separation. The resulting propagator is exactly correct to lowest order in the density and in the ordered lattice limit. Extensive calculations are presented for the site percolation problem, for which a crossover in time to fractal dynamics is observed. The spectral dimension of a percolation cluster in three dimensions is predicted to be 4/3, in agreement with the conjecture of Alexander and Orbach and with recent simulations.