An expansion in orthogonal polynomials of binomial-random variables for inhomogeneous transport in disordered media

Abstract

A general method is presented for expanding random functions in orthogonal polynomials of binomial-random variables for transport in randomly inhomogeneous media. The expansion projects the stochastic equation onto the statistically orthogonal polynomials of binomial variables. It generates an infinite set of coupled equations for the determination of kernels in the expansion where randomness is removed at the outset. The expansion in orthogonal polynomials is applied to inhomogeneous transport in bond-disordered resistor networks (bond model). The expression for the effective conductivity is obtained, to order c2 (c being the fraction of broken bonds), by truncating the infinite set of coupled equations for kernels after the third term. It is found that the expression agrees with that derived from the two-bond approximation. The expansion in orthogonal polynomials is also applied to the clumped-bond model and the continuum model. Truncated equations are derived to govern the kernels in the expansions.