A fractional diffusion equation to describe Levy flights

A fractional-derivatives diffusion equation is proposed that generates the Levy statistics. The fractional derivatives are defined by the eigenvector equation partial derivative(x)(alpha)e(ax) = a(alpha)e(ax) and for one dimension the diffusion equation in an isotropic medium reads partial derivative(t)n = (D/2)(partial derivative(x)(alpha) + partial derivative(-x)(alpha))n + upsilon partial derivative(x)n, 1 < alpha less than or equal to 2. The equation is based on a proposed generalization of Fick's law which reads j = -(D/2)(del(r)(alpha-1) - del(-r)(alpha-1))n + nu n. The diffusion equation is also written for an anisotropic medium, and in this case it generates an asymmetric Levy statistics. (C) 1998 Published by Elsevier Science B.V.