Theoretical approaches to inhomogeneous transport in disordered media

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Abstract
Continuum theories for electrical conduction in inhomogeneous materials are discussed in connection with the problem of electron localization in disordered systems. The validity of various approximate solutions, including the effective-medium approximation and the cumulant expansion method, is analysed by means of a diagrammatic representation of the perturbation series. In particular, the effective-medium approximation is derived from several different points of view in order to clarify the meaning and applicability of the method. It is confirmed that, at the present stage, the cumulant theory is the best approximation for a three-dimensional material, while the effective-medium theory is the best for a two-dimensional material. In the three-dimensional case, the cumulant approximation holds excellently even near the critical percolation concentration where clustering effects play an essential role and where the effective-medium theory fails to work.