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 ei genvector equation partial derivative(x)(alpha)e(ax) = a(alpha)e(ax) a nd for one dimension the diffusion equation in an isotropic medium rea ds partial derivative(t)n = (D/2)(partial derivative(x)(alpha) + parti al derivative(-x)(alpha))n + upsilon partial derivative(x)n, 1 < alpha less than or equal to 2. The equation is based on a proposed generali zation 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 aniso tropic medium, and in this case it generates an asymmetric Levy statis tics. (C) 1998 Published by Elsevier Science B.V.