Understanding the Lévy ratchets in terms of Lévy jumps

Abstract

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 Lévy noise. The system (referred to as the Lévy 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 Lévy flights on the system dynamics. We analyze the departure positions of Lévy 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 Lévy ratchets produce a non-vanishing current in the infinite-barrier limit. This latter counterintuitive result can be easily understood in terms of the long Lévy jumps and analytically demonstrated.