We derive a general effective-medium theory for describing biased diffusion on a bond-disordered lattice in the presence of an external driving field. In our theory, the effective medium associated with a disordered d-dimensional lattice is characterized, for each value of the applied field, by 2d independent parameters describing, respectively, the net drift velocity vv and the diffusion constant Dvv describing the spread of a carrier packet about its mean value, for each of the d crystal axes. The theory correctly predicts the velocity transition occurring in an exactly soluble model studied by Derrida and, in contrast to other recent theories, correctly reproduces the critical velocity at which this transition occurs.
P. E. Parris and B. D. Bookout, "Effective-Medium Theory for the Electric-Field Dependence of the Hopping Conductivity of Disordered Solids," Physical Review B (Condensed Matter), vol. 53, no. 2, pp. 629-635, American Physical Society (APS), Jan 1996.
The definitive version is available at https://doi.org/10.1103/PhysRevB.53.629
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