Levy anomalous diffusion and fractional Fokker-Planck equation

We demonstrate that the Fokker-Planck equation can be generalized into a 'f ractional Fokker-Planck' equation, i.e., an equation which includes fractio nal space differentiations, in order to encompass the wide class of anomalo us diffusions due to a Levy stable stochastic forcing. A precise determinat ion of this equation is obtained by substituting a Levy stable sourer to th e classical Gaussian one in the Langevin equation. This yields not only the anomalous diffusion coefficient, but a non-trivial fractional operator whi ch corresponds to the possible asymmetry of the Levy stable source. Both of them cannot be obtained by scaling arguments, The (mono-) scaling behavior s of the fractional Fokker-Planck equation and of its solutions are analyse d and a generalization of the Einstein relation for the anomalous diffusion coefficient is obtained. This generalization yields a straightforward phys ical interpretation of the parameters of Levy stable distributions. Further more, with the help of important examples, we show the applicability of the fractional Fokker-Planck equation in physics, (C) 2000 Published by Elsevi er Science B.V. All rights reserved.