Fractional Fokker-Planck equation for ultraslow kinetics

• Published in: Europhysics letters Vol. 63; no. 3; pp. 326 - 332
• Main Authors: Chechkin, A. V, Klafter, J, Sokolov, I. M
• Format: Journal Article
• Language: English
• Published: Les Ulis IOP Publishing 01.08.2003; EDP Sciences; EDP sciences
• Subjects: 02.50.Ey; 05.40.-a; Exact sciences and technology; Fluctuation phenomena, random processes, noise, and brownian motion; Mathematical methods in physics; Physics; Probability theory, stochastic processes, and statistics; Statistical physics, thermodynamics, and nonlinear dynamical systems; Stochastic processes; Probability distribution; Fokker-Planck equation; Diffusion; Power law; Weight function; Random walk
• ISSN: 0295-5075; 1286-4854
• DOI: 10.1209/epl/i2003-00539-0

Several classes of physical systems exhibit ultraslow diffusion for which the mean-squared displacement at long times grows as a power of the logarithm of time (“strong anomaly”) and share the interesting property that the probability distribution of particle's position at long times is a double-sided exponential. We show that such behaviors can be adequately described by a distributed-order fractional Fokker-Planck equations with a power law weighting function. We discuss the equations and the properties of their solutions, and connect this description with a scheme based on continuous-time random walks.