Resistance distribution in the hopping percolation model

TL;DR

This paper investigates the distribution of effective resistance in hopping percolation models, revealing it follows a log-normal distribution dependent on system size and disorder strength, with implications for understanding transport in disordered materials.

Contribution

It introduces a universal description of resistance distribution in hopping percolation models, highlighting the dependence on system size and disorder, and providing a log-normal approximation.

Findings

Resistance distribution is log-normal in both strong and extreme disorder regimes.

Distribution depends only on the ratio L/kappa^nu.

The dispersion of the distribution scales with kappa^nu/L.

Abstract

We study the distribution function, P(rho), of the effective resistance, rho, in two and three-dimensional random resistor network of linear size L in the hopping percolation model. In this model each bond has a conductivity taken from an exponential form \sigma ~ exp(-kappa r), where kappa is a measure of disorder, and r is a random number, 0< r < 1. We find that in both the usual strong disorder regime L/kappa^{nu} > 1 (not sensitive to removal of any single bond) and the extreme disorder regime L/kappa^{nu} < 1 (very sensitive to such a removal) the distribution depends only on L/kappa^{nu} and can be well approximated by a log-normal function with dispersion b kappa^nu/L, where b is a coefficient which depends on the type of the lattice