Random walks on cubic lattices with bond disorder

We consider diffusive systems with static disorder, such as Lorentz gases, lattice percolation, ants in a labyrinth, termite problems, random resistor networks, etc. In the case of diluted randomness we can apply the methods of kinetic theory to obtain systematic expansions of dc and ac transport properties in powers of the impurity concentrationc. The method is applied to a hopping model on ad-dimensional cubic lattice having two types of bonds with conductivityσ andσ0=1, with concentrationsc and 1−c, respectively. For the square lattice we explicitly calculate the diffusion coefficientD(c,σ) as a function ofc, to O(c2) terms included for different ratios of the bond conductivityσ. The probability of return at long times is given byP0(t)≈ [4πD(c,σ)t]−d/2, which is determined by the diffusion coefficient of the disordered system.

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