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

Abstract

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