A fractional diffusion equation to describe Lévy flights

doi:10.1016/S0375-9601(97)00947-X

Physics Letters A

Volume 239, Issues 1–2, 23 February 1998, Pages 13-16

https://doi.org/10.1016/S0375-9601(97)00947-X Get rights and content

Abstract

A fractional-derivatives diffusion equation is proposed that generates the Lévy statistics. The fractional derivatives are defined by the eigenvector equation ∂_x^αe^ax = a^αe^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 < α ≤ 2. 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.

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^☆: This work was supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico.

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A fractional diffusion equation to describe Lévy flights☆

Abstract

Phys. Rev. Lett.

Phys. Rep.

Phys. Rev. E

Phys. Rev. E

J. Stat. Phys.

SIAM Rev.