Levy Anomalous Diffusion and Fractional Fokker--Planck Equation

Abstract : We demonstrate that the Fokker-Planck equation can be generalized into a \'Fractional Fokker-Planck\' equation, i.e. an equation which includes fractional space differentiations, in order to encompass the wide class of anomalous diffusions due to a Levy stable stochastic forcing. A precise determination of this equation is obtained by substituting a Levy stable source to the classical gaussian one in the Langevin equation. This yields not only the anomalous diffusion coefficient, but a non trivial fractional operator which corresponds to the possible asymmetry of the Levy stable source. Both of them cannot be obtained by scaling arguments. The (mono-) scaling behaviors of the Fractional Fokker-Planck equation and of its solutions are analysed and a generalization of the Einstein relation for the anomalous diffusion coefficient is obtained. This generalization yields a straightforward physical interpretation of the parameters of Levy stable distributions. Furthermore, with the help of important examples, we show the applicability of the Fractional Fokker-Planck equation in physics.
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V. V. Yanovsky, A. V. Chechkin, D Schertzer, A. V. Tour. Levy Anomalous Diffusion and Fractional Fokker--Planck Equation. Physica A, Elsevier, 2000, 282, pp.13-34. ⟨hal-00008415⟩



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