Adiabatic Ratchet Effect in Systems with Discrete Variables

An adiabatic theory of the ratchet effect in various systems described by random transitions between discrete states has been developed. The theory is based on the development of a discrete analog of the Parrondo lemma, which is one of the fundamental relations of the theory of ratchet systems allowing the calculation of integral fluxes in time intervals from the initial to final (equilibrium) distributions. Correspondence between the transition probabilities and the parameters of the potential profile, in which the hopping motion is described by the developed theory and is a low-temperature limit of continuous motion, has been proposed. The rate of the ratchet effect calculated by the proposed theory is well confirmed by a numerical simulation. The developed approach makes it possible to study the characteristics of the operation of Brownian motors of various natures by simple methods.