Deriving fractional Fokker-Planck equations from a generalised master equation

¹ School of Chemistry, Tel Aviv University - 69978 Tel Aviv, Israel
² Department of Chemistry and Center for Materials Science and Engineering Massachusetts Institute of Technology 77 Massachusetts Avenue, Building 6-230, Cambridge, MA 02139, USA

Received: 12 February 1999
Accepted: 13 March 1999

Abstract

A generalised master equation is constructed from a non-homogeneous random walk scheme. It is shown how fractional Fokker-Planck equations for the description of anomalous diffusion in external fields, recently proposed in the literature, can be derived from this framework. Long-tailed waiting time distributions which cause slowly decaying memory effects, are demonstrated to give rise to a time-fractional Fokker-Planck equation that describes systems close to thermal equilibrium. An extension to include also Lévy flights leads to a generalised Laplacian in the corresponding fractional Fokker-Planck equation.