Directed transport induced byα-stable Lévy noises in weakly asymmetric periodic potentials

Abstract

We study the motion of a particle in spatially periodic potentials with broken mirror symmetry under the influence of white α-stable Lévy noises. We consider both time-independent and fluctuating potentials. We focus on cases in which the spatial asymmetry of the potential is due not to a difference between the distances from an absolute minimum to the absolute maximum on its left and to the absolute maximum on its right but only to the curvatures of the potential profiles. The analysis is performed using the fractional Fokker-Planck formalism. Consistent results from Langevin simulations are also presented. We analyze the influence of the symmetry properties of the potentials in combination with the fluctuating characteristics of the system in the determination of the current. We find different situations in which both the absolute value and the direction of the current depend not only on the properties of the potential but also on the parameters characterizing the α-stable Lévy noise. Among other features, we analyze the case of supersymmetric potentials. In particular, we show that a static supersymmetric potential produces no current, and we analyze the conditions for observing a nonvanishing current when the potential fluctuates between different supersymmetric profiles.

From Markovian to non-Markovian persistence exponents

“…Stable subordinators are an extreme case of α-stable processes. These processes play an important role in line with the generalized central limit theorem [33]: so-called Lévy flights, also known as Lévy-Brownian motions [34], have emerged as powerful tools to model non Gaussian phenomena [35][36][37][38][39]. They are continuous time Markov processes that can be described in Langevin formalism as generalized Brownian motion, where the driving term, instead of a simple Gaussian white noise is a general Lévy noise ζ, that induces independent increments identically distributed according to an α-stable law [34]:ẋ(t) = ζ(t).…”

Directed transport induced by spatially modulated Lévy flights

“…Later on, a semi-analytical study of the minimal ratchet model was performed by using the fractional Fokker-Planck formalism [45]. Also, more involved models of the Lévy ratchets have been considered by means of intensive numerical simulations based on the Langevin and/or fractional Fokker-Planck equations, such as Lévy ratchet effects in underdamped systems [46], with potential fluctuating in time [47,48], or driven by truncated Lévy flights [49].…”