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A fractional diffusion equation to describe Lévy flights. (English) Zbl 1026.82524

Summary: A fractional-derivatives diffusion equation is proposed that generates the Lévy statistics. The fractional derivatives are defined by the eigenvector equation x α e ax =a α e ax and for one dimension the diffusion equation in an isotropic medium reads

t n=(D/2)( x α + -x α )n+v x n,1<α2·

The equation is based on a proposed generalization of Fick’s law which reads j=-(D/2)( r α-1 - -r α-1 )n+vn. The diffusion equation is also written for an anisotropic medium, and in this case it generates an asymmetric Lévy statistics.

82C31Stochastic methods in time-dependent statistical mechanics