A model of diffusion in a spatially disordered lattice

An approximate method for the study of one-particle diffusion in a three-dimensional disordered lattice is proposed. The method is based on the locator expansion of a generalized discrete version of the diffusion equation. Approximations are performed through a convenient interpretation of the resulting equations in terms of known quantities that characterize a discrete-time random walk. The method is applied to a model of a disordered lattice in which allowed sites are randomly distributed in a continuum at a given concentration n and hopping is allowed between sites separated by a distance not greater than a specified fixed value a 0 . The results are in good agreement with the expected physical situation, showing the existence of two regions in the parameter space (n,a 0 ), one of which is characterized by the existence of normal diffusion and the other by the vanishing of the diffusion constant, with the random walker confined in a cluster of finite size. The two regions are separated by a critical curve, along which the diffusion is shown to be anomalous. The three different regimes are characterized by a single parameter, the average number of nearest neighbours. A connection with percolation theory is made, the formalism yielding values for the exponents gamma and nu . The results gamma =2 and nu =1 are obtained in the 3D case. For dimensions greater than four it is shown that the predicted critical exponents agree with the mean field values gamma =1 and nu = 1 / 2 .