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Lévy anomalous diffusion and fractional Fokker–Planck equation

Author

Listed:
  • Yanovsky, V.V.
  • Chechkin, A.V.
  • Schertzer, D.
  • Tur, A.V.

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 Lévy stable stochastic forcing. A precise determination of this equation is obtained by substituting a Lévy 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 Lévy 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 Lévy stable distributions. Furthermore, with the help of important examples, we show the applicability of the fractional Fokker–Planck equation in physics.

Suggested Citation

  • Yanovsky, V.V. & Chechkin, A.V. & Schertzer, D. & Tur, A.V., 2000. "Lévy anomalous diffusion and fractional Fokker–Planck equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 282(1), pages 13-34.
    • Handle: RePEc:eee:phsmap:v:282:y:2000:i:1:p:13-34
    DOI: 10.1016/S0378-4371(99)00565-8
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    Cited by:

    1. Evangelista, L.R. & Lenzi, E.K. & Barbero, G. & Scarfone, A.M., 2024. "On the Einstein–Smoluchowski relation in the framework of generalized statistical mechanics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 635(C).
    2. Bai, Zhan-Wu & Hu, Meng, 2015. "Escape rate of Lévy particles from truncated confined and unconfined potentials," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 428(C), pages 332-339.
    3. Tawfik, Ashraf M. & Elkamash, I.S., 2022. "On the correlation between Kappa and Lévy stable distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 601(C).
    4. Lubashevsky, Ihor, 2013. "Equivalent continuous and discrete realizations of Lévy flights: A model of one-dimensional motion of an inertial particle," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(10), pages 2323-2346.
    5. Oraby, T. & Suazo, E. & Arrubla, H., 2023. "Probabilistic solutions of fractional differential and partial differential equations and their Monte Carlo simulations," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    6. Tarasov, Vasily E. & Zaslavsky, George M., 2008. "Fokker–Planck equation with fractional coordinate derivatives," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(26), pages 6505-6512.

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