DOI: 10.1209/epl/i2003-00539-0
Europhys. Lett., 63 (3) , pp. 326-332 (2003)

Fractional Fokker-Planck equation for ultraslow kinetics

A. V. Chechkin¹, J. Klafter^{2, 1} and I. M. Sokolov<sup>3

1</sup>  Institute for Theoretical Physics, N SC KIPT Akademicheskaya st. 1, 61108 Kharkov, Ukraine
²  School of Chemistry, Sackler Faculty of Exact Sciences Tel Aviv University - Tel Aviv 69978, Israel
³  Institut für Physik, Humboldt-Universität zu Berlin Invalidenstrasse 110, D-10115 Berlin, Germany

achechkin@kipt.kharkov.ua
klafter@post.tau.ac.il
igor.sokolov@physik.hu-berlin.de

(Received 29 January 2003; accepted in final form 2 June 2003)

Abstract
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.

PACS
02.50.Ey - Stochastic processes.
05.40.-a - Fluctuation phenomena, random processes, noise, and Brownian motion.

© EDP Sciences 2003