Deterministic ratchets, circle maps, and current reversals

In this work we transform the deterministic dynamics of an overdamped tilting ratchet into a discrete dynamical map by looking stroboscopically at the continuous motion originally ruled by differential equations. We show that, for the simple and widely used case of periodic dichotomous driving forces, the resulting discrete map belongs to the class of circle homeomorphisms. This approach allows us to apply the well-known properties of such maps to derive the necessary and sufficient conditions that the ratchet potential must satisfy in order to have a vanishing current. Furthermore, as a consequence of the above, we show (i) that there is a class of periodic potentials which do not exhibit the rectification phenomenon in spite of their asymmetry and (ii) that current reversals occur in the deterministic case for a large class of ratchet potentials.