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

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

2001-11-30

arxiv.org/abs/math-ph/0111048v3

J. Stat. Phys., {\bf 108}, N 5/6, 1015 -- 1031 (2002)

Physics

Mathematical Physics

17 pages

Scientific paper

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.

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