2000/01/18 by V. V. Yanovsky, Vladimir Yanovsky, Aleksei V. Chechkin +5 · 3 citations
Mathematics · Physics and Astronomy · #Anomalous diffusion #Differential equation #Diffusion #Diffusion equation #Fokker–Planck equation #Fractional Differential Equations Solutions #Fractional calculus #Generalization #Langevin equation #Lévy flight #Mathematical analysis #Mathematical physics #Mathematics #Physics #Quantum mechanics #Random walk #Scaling #Statistical Mechanics and Entropy #Statistical physics #nlin.CD #stochastic dynamics and bifurcation
paper · pdf · doi:10.1016/s0378-4371(99)00565-8
22 pages; To Appear in Physica A
arxiv created 2000/01/18 · openalex publication_date 2000/07/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.