Space- and time-fractional diffusion and wave equations, fractional Fokker-Planck equations, and physical motivation

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 Levy 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 Levy form is explored, putting the fractional approach in some perspective from the standard non-equilibrium dynamics point of view, and its generalisation. (C) 2002 Elsevier Science B.V. All rights reserved.