Effective medium treatments of random simple square and simple cubic conductance networks

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Abstract
A simple treatment of networks containing conductances gA and gB gives an exact formula for the conductivity in terms of gA, gB, the fraction b of conductances gB and a quantity lambda . Setting lambda =d-1 in d dimensions 1 - 2/pi in two dimensions and 0.210 in three dimensions yields effective medium results for the bond, two-dimensional 'non-overlapping' site and three-dimensional 'non-overlapping' site problems respectively. This is shown to be a good approximation for all values of gB/gA and b. For the ordinary site problem the lambda values 1 - 2/pi and 0.210 yield good approximations only when b to 0 or gB=0 and lambda increases by approximately 50% as b to 1 with gB not=0.