Space- and time-fractional diffusion and wave equations, fractional Fokker–Planck equations, and physical motivation Ralf Metzler a,b, * , Theo F. Nonnenmacher c a Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA b NORDITA, Blegdamsvej 17, DK-2100 København Ø, Denmark c Department of Mathematical Physics, University of Ulm, Albert-Einstein-Allee 11, 89069 Ulm/Donau, Germany Received 29 October 2001 Abstract We investigate the physical background and implications of a space- and time-fractional diffusion equation which corresponds to a random walker which combines competing long waiting times and L evy flight properties. Explicit solutions are examined, and the corresponding fractional Fokker–Planck–Smoluchowski equation is presented. The framework of fractional kinetic equations which control the systems relaxation to either Boltzmann–Gibbs equilibrium, or a far from equilibrium L evy form is explored, putting the fractional approach in some perspective from the standard non-equilibrium dynamics point of view, and its generalisation. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 05.40.Fb; 05.60.Cd; 02.50.Ey Keywords: Fractional diffusion equation; Fractional Fokker–Planck equation; Anomalous diffusion; L evy flights; L evy walks 1. Introduction A cornerstone in the development of modern physics was the introduction of the concept of probability into atomistic physics in Maxwell’s theory of gas kinetics [1], and in Boltzmann’s transport equation [2]. In these theories, particles are usually viewed, from a stochastic standpoint, in a bath of equivalent particles, giving rise to collisions, and eventually to systems equilibration, uniquely towards the Maxwell–Boltzmann distribution W ðvÞ¼ð2pk B T =mÞ 1=2 e mv 2 =ð2k B T Þ ð1Þ of velocities. Chemical Physics 284 (2002) 67–90 www.elsevier.com/locate/chemphys * Corresponding author. Tel.: +45-353-25507; fax: +45-353-89157. E-mail addresses: metz@nordita.dk (R. Metzler), theo.nonnenmacher@physik.uni-ulm.de (T.F. Nonnenmacher). 0301-0104/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII:S0301-0104(02)00537-2