Multi-bond expansions for the effective conductivity in bond-disordered resistor networks

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Abstract
A perturbation expansion is given to derive the effective (or macroscopic) conductivity of disordered square and simple cubic resistor networks in which bonds are broken at random (bond model). Successive terms of the multi-bond expansion involve averages over successively larger groups of bonds of the lattice and are given by the partial summations of multi-bond terms in the diagrammatic expansion series, making use of the effective-medium method for multi-bonds. The cluster approximations (multi-bond EMA) which go beyond the two-bond effective-medium approximation (two-bond EMA) can be obtained systematically from the perturbation expansion. It is found that in the non-selfconsistent treatment, the expansion corresponds to the expression of the effective conductivity as an integral power series in the fraction of broken bonds. It is also shown that the multi-bond expansion agrees formally with the group expansion for the continuum model given by Jeffrey (1973).

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