On Effective Conductivity on ${\mathbb Z}^d$ Lattice

TL;DR

This paper derives high-order series expansions for the effective conductivity of a random resistor network on a lattice, comparing these with classical approximations and revealing their accuracy limits.

Contribution

It provides exact expressions for the effective conductivity expansion up to fifth order and a sixth order in 2D, enhancing understanding of disorder effects in lattice conductance.

Findings

Exact fifth-order expansion for $\sigma_e$ in $d extgreater= 2$

Sixth-order expansion in 2D using duality symmetry

Comparison shows Bruggeman approximation matches up to 4th order in 2D

Abstract

We study the effective conductivity $\sigma_e$ for a random wire problem on the $d$-dimensional cubic lattice ${\mathbb Z}^d, d \geq 2$ in the case when random conductivities on bonds are independent identically distributed random variables. We give exact expressions for the expansion of the effective conductivity in terms of the moments of the disorder parameter up to the 5th order. In the 2D case using the duality symmetry we also derive the 6th order expansion. We compare our results with the Bruggeman approximation and show that in the 2D case it coincides with the exact solution up to the terms of 4th order but deviates from it for the higher order terms.