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 \(\partial_x^{\alpha}e^{ax}=a^{\alpha}e^{ax}\) and for one dimension the diffusion equation in an isotropic medium reads \[ \partial_tn=(D/2)(\partial_x^{\alpha}+\partial_{-x}^{\alpha})n+v\partial_xn,\quad 1<\alpha\leq 2. \] The equation is based on a proposed generalization of Fick’s law which reads \(j=-(D/2)(\nabla_r^{\alpha-1}-\nabla_{-r}^{\alpha-1})n+vn\). The diffusion equation is also written for an anisotropic medium, and in this case it generates an asymmetric Lévy statistics.


82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
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