Lévy flights in external force fields: Langevin and fractional Fokker-Planck equations and their solutions

Abstract

We consider Levy flights subject to external force fields. This anomalous\ntransport process is described by two approaches, a Langevin equation with Levy\nnoise and the corresponding generalized Fokker-Planck equation containing a\nfractional derivative in space. The cases of free flights, constant force and\nlinear Hookean force are analyzed in detail, and we corroborate our findings\nwith results from numerical simulations. We discuss the non-Gibbsian character\nof the stationary solution for the case of the Hookean force, i.e. the\ndeviation from Boltzmann equilibrium for long times. The possible connection to\nTsallis's q-statistics is studied.\n