Scaling of heterogeneous distributions of conductances: Renormalization versus exact results

Abstract

In this work we are concerned with the calculation of effective conductance of heterogeneous media. We are interested in determining the conductance when the system becomes macroscopically homogeneous, and the disorder length $({\ensuremath{\xi}}_{D})$, i.e., the length scale necessary to reach this effective value. Numerical computation of conductances in two-dimensional lattices is done using an exact numerical method and two different renormalization procedures. The conductance values are extracted from (a) power-law, (b) log-normal, or (c) Weibull probability distributions in the interval $[0,1]$. A parameter ${\ensuremath{\mu}}^{\ensuremath{-}1}$ is used to measure the degree of heterogeneity of all three distributions. For the power-law distribution, ${\ensuremath{\xi}}_{D}$ diverges as $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\mu}}0$ with the exponent $\ensuremath{\nu}$ of the percolation correlation length on the same geometrical support. The log-normal and Weibull distribution reduce to a percolation distribution function, $P(g)=p\ensuremath{\delta}(g\ensuremath{-}1)+(1\ensuremath{-}p)\ensuremath{\delta}(g)$, in the limit $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\mu}}0$. The disorder length remains finite or diverges depending on whether the effective occupation probability $p$ is above the percolation threshold or not. The analysis carried out here may be generalized to a large number of long-tailed distributions, for which percolation ideas apply.