L\'evy 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 L\'evy noise possessing L\'evy stable probability law (L\'evy 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 L\'evy noise. Intensive numerical simulations demonstrate a good agreement with the analytical predictions for sufficiently small intensities of the L\'evy noise driving the particle.