A MODEL OF DIFFUSION IN A SPATIALLY DISORDERED LATTICE

An approximate method for the study of one-particle diffusion in a thr ee-dimensional (3D) disordered lattice is proposed. The method is base d on the locator expansion of a generalized discrete version of the di ffusion equation. Approximations are performed through a convenient in terpretation of the resulting equations in terms of known quantities t hat characterize a discrete-time random walk. The method is applied to a model of a disordered lattice in which allowed sites are randomly d istributed in a continuum at a given concentration n and hopping is al lowed between sites separated by a distance not greater than a specifi ed fixed value a0. The results are in good agreement with the expected physical situation, showing the existence of two regions in the param eter space (n, a0), one of which is characterized by the existence of normal diffusion and the other by the vanishing of the diffusion const ant, with the random walker confined in a cluster of finite size. The two regions are separated by a critical curve, along which the diffusi on is shown to be anomalous. The three different regimes are character ized by a single parameter, the average number of nearest neighbours. A connection with percolation theory is made, the formalism yielding v alues 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 s hown that the predicted critical exponents agree with the mean field v alues gamma = 1 and nu = 1/2.