On the effective conductivity of flat random two-phase models

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An approximate equation for the effective conductivity sigmaₑff of systems with a finite maximal scale of inhomogeneities is deduced. An exact solution of this equation is found and its physical meaning is discussed. A two-phase randomly inhomogeneous model is constructed by a hierarchical method and its effective conductivity at arbitrary phase concentrations is found in the mean-field-like approximation. These expressions satisfy all the necessary symmetries, reproduce the known formulas for sigmaₑff in the weakly inhomogeneous case and coincide with two recently found partial solutions of the duality relation. It means that sigmaₑff even of two-phase randomly inhomogeneous system may be a nonuniversal function and can depend on some details of the structure of the inhomogeneous regions. The percolation limit is briefly discussed.