Fractional Diffusion Equation Approach to the Anomalous Diffusion on Fractal Lattices

摘要

A generalized fractional diffusion equation (FDE) is presented,which describes the time-evolution of the spatial distribution of a particle performing continuous time random walk (CTRW) on a fractal lattice.For a case corresponding to the CTRW with waiting time distribution that behaves as psi(t) approx t~(-(alpha+1)) the FDE is solved to give analytic expressions for the Green's function and the mean squared displacement (MSD).In agreement with the previous work of Blumen et al.[Phys.Rev.Lett.1984,53,1301],the time-dependence of MSD is found to be given as approx t~(2alpha/d_w),where d_w is the walk dimension of the given fractal.A Monte-Carlo simulation is also performed to evaluate the range of applicability of the proposed FDE.