Fluctuation-driven directed transport in the presence of Lévy flights

The role of Lévy flights on fluctuation-driven transport in time independent periodic potentials with broken spatial symmetry is studied. Two complementary approaches are followed. The first one is based on a generalized Langevin model describing overdamped dynamics in a ratchet type external potential driven by Lévy white noise with stability index α in the range 1<α<2. The second approach is based on the space fractional Fokker–Planck equation describing the corresponding probability density function (PDF) of particle displacements. It is observed that, even in the absence of an external tilting force or a bias in the noise, the Lévy flights drive the system out of the thermodynamic equilibrium and generate an up-hill current (i.e., a current in the direction of the steeper side of the asymmetric potential). For small values of the noise intensity there is an optimal value of α yielding the maximum current. The direction and magnitude of the current can be manipulated by changing the Lévy noise asymmetry and the potential asymmetry. For a sharply localized initial condition, the PDF of staying at the minimum of the potential exhibits scaling behavior in time with an exponent bigger than the −1/α exponent corresponding to the force free case.