Understanding the Lévy ratchets in terms of Lévy jumps

IBÁÑEZ, S.A.; RISAU-GUSMAN, S.; BOUZAT, S.

JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT

IOP PUBLISHING LTD

Año: 2013 vol. 2013

1742-5468

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. © 2013 IOP Publishing Ltd and SISSA Medialab srl.