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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 \(\partial_x^{\alpha}e^{ax}=a^{\alpha}e^{ax}\) 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.

MSC:

82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
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[1] Lévy, P., (Theéorie de l’Addition des Variables Aliétoires (1937), Guthier-Villars: Guthier-Villars Paris)
[2] Montroll, E. W.; West, B. J., (Montroll, E. W.; Lebowitz, J. L., Fluctuation Phenomena (1979), North-Holland: North-Holland Amsterdam)
[3] Mandelbrot, B. B., (The Fractal Geometry of Nature (1983), Freeman: Freeman New York) · Zbl 1194.30028
[4] Shlesinger, M. F.; West, B. J.; Klafter, J., Phys. Rev. Lett., 58, 1100 (1987)
[5] Bouchaud, J. P.; Georges, A., Phys. Rep., 195, 127 (1991)
[6] Klafter, J.; Zumofen, G., Phys. Rev. E, 49, 4873 (1994)
[7] Bardou, F.; Bouchaud, J. P.; Emile, O.; Aspect, A.; Cohen-Tannoudji, C., Phys. Rev. Lett., 72, 203 (1994)
[8] Gelfand, I. M.; Shilov, G. E., (Generalized Functions, Vol. 1 (1994), Academic Press: Academic Press New York)
[9] West, B. J.; Seshadri, V.; West, B. J., Physica, 113 A, 203 (1982)
[10] Allegrini, P.; Grigolini, P.; West, B. J., Phys. Rev. E, 54, 4760 (1996)
[11] Tsallis, C.; Levy, S. V.F.; Souza, A. M.C.; Maynard, R., Phys. Rev. Lett., 75, 3589 (1995) · Zbl 1020.82593
[12] Tsallis, C., J. Stat. Phys., 52, 479 (1988) · Zbl 1082.82501
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