Influence of superdiffusion on directional motion

The directional motion of a particle in an unbound potential in the presence of either non-Ohmic memory friction or Lévy flight is considered. Time-dependent probability passing over the saddle point of a parabolic potential is derived exactly. The results show that the Lévy flights break directional behavior of the system in the long time limit. The minimal initial velocity of the particle needed appears in superdiffusion with anomalous exponent δ∼1.6, however, a strong superdiffusion (δ closing to 2.0) hinders directional motion. It is found that the steady current of the particle in a flashing ratchet shows a nonmonotonic behavior as a function of the exponent μ of Lévy flights. A giant enhancement of the current related to normal diffusion can be realized by the rectification of superdiffusion induced by non-Ohmic friction.