L evy flights in external force fields: from models to equations

We consider different generalizations of the Fokker–Planck equation (FPE) devised to describe L evy processes in potential force fields. We show that such generalizations can proceed along different lines. On one hand, L evy statistics can emerge from the fractal temporal nature of the underlying process, i.e., a high variability in the rate of microscopic events. On the other hand, they may be a direct consequence of the scale-free spatial structure on which the process evolves. Although both forms considered lead to Boltzmann equilibrium, the relaxation patterns are quite different. As an example, generalized diffusion in a double-well potential is considered. 2002 Elsevier Science B.V. All rights reserved.