Lévy Ratchet in a Weak Noise Limit: Theory and Simulation

Abstract

We study the motion of a particle embedded in a time independent periodic potential with broken mirror symmetry and subjected to a Lévy noise possessing Lévy stable probability law (Lévy ratchet). We develop analytical approach to the problem based on the asymptotic probabilistic method of decomposition proposed by P. Imkeller and I. Pavlyukevich [J. Phys. A {\bf39}, L237 (2006); Stoch. Proc. Appl. {\bf116}, 611 (2006)]. We derive analytical expressions for the quantities characterizing the particle motion, namely the splitting probabilities of first escape from a single well, the transition probabilities and the particle current. A particular attention is devoted to the interplay between the asymmetry of the ratchet potential and the asymmetry (skewness) of the Lévy noise. Intensive numerical simulations demonstrate a good agreement with the analytical predictions for sufficiently small intensities of the Lévy noise driving the particle.

14 pages, 11 figures, 63 references

References in corpus (9)

• The scaling laws of human travel
• Levy Flights, Non-local Search and Simulated Annealing
• Escape driven by $\alpha$-stable white noises
• Stationary states in Langevin dynamics under asymmetric Lévy noises
• Cooling down Levy flights
• Transport in a Levy ratchet: Group velocity and distribution spread
• Fluctuation-driven directed transport in the presence of Levy flights
• Stationary states in single-well potentials under symmetric Levy noises
• Bimodality and hysteresis in systems driven by confined Lévy flights

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• Stationary States in Bistable System Driven by Lévy Noise
• Levy ratchets with dichotomic random flashing
• Underdamped, anomalous kinetics in double-well potentials
• Drifted escape from the finite interval
• Stochastic kinetics under combined action of two noise sources
• How to simulate Lévy flights in a steep potential: An explicit splitting numerical scheme