An effective medium treatment of the random bias problem

We consider here a general model for the incoherent transport of an electron or a one-dimensional disordered lattice. No special symmetry is assumed to hold between the hopping rates. This model is applicable then to charge transport in highly anisotropic disordered systems in the presence of a static electric field. We show how an effective medium theory for the diffusion constant and the velocity of an initially localised particle can be formulated directly from the equations of motion. The theory is shown to predict accurately the velocity for all values of the applied static field including the observed vanishing of the velocity at low values of the static field. The results for the diffusion constant are shown to be in agreement with known results for intermediate to high values of the applied field. We also discuss how the effective medium can easily be constructed in higher dimensions.