Understanding the L?vy ratchets in terms of L?vy jumps

We investigate the overdamped dynamics of a particle in a spatially periodic potential with broken reflection symmetry and subject to the action of a symmetric white Levy noise. The system (referred to as the Levy ratchet) has been previously studied using both Langevin and fractional Fokker-Planck formalisms, with the main find being the existence of a preferred direction of motion toward the steepest slope of the potential, producing a non-vanishing current. In this contribution we develop a semi-analytical study combining the Fokker-Planck and Langevin formalisms to explore the role of Levy flights on the system dynamics. We analyze the departure positions of Levy jumps that take the particle out of a potential well as well as the rates and lengths of such jumps, and we study the way in which long jumps determine the non-vanishing current. We also discuss the essential difference from the Gaussian-noise case (producing no current). Finally we study the current for different potential shapes as a function of the amplitude of the potential barrier. In particular, we show that standard Levy ratchets produce a non-vanishing current in the infinite-barrier limit. This latter counterintuitive result can be easily understood in terms of the long Levy jumps and analytically demonstrated.