As a simple model of the Brownian motor, we consider hopping motion of a particle in a periodic asymmetric double-well potential which randomly switches between two states. The potential profiles of the states are identical but shifted by half a period. The current and the efficiency are explicitly calculated as functions of the parameters of the model, including also a load force. Such a flashing ratchet is shown to be particularly efficient, with the efficiency tending to unity when the highest peak of the potential is high enough to suppress the backward motion.

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A system is considered in which transitions between two states occur through two reaction channels. Because of coupling with an external process which consists of cyclic switching between two regimes (each characterized by a certain fixed set of rate constants), the net circulation flux arises in the system even in the absence of an external generalized force. Such a mechanism underlying a catalytic wheel of many biological processes is considered as a Brownian motor. The basic operational motor characteristics are calculated for the regular and random inter-regime switching, being better in the former case and reaching the optimum at equal relaxation-to-lifetime ratios for the two regimes. The general Brownian motor formalism is exemplified by two particular realizations, the electroconformational-coupling model and the flashing-potential model. The former concerns enzymatically catalyzed ligand pumping through a membrane, and the latter describes particle motion under two sets of potential wells and barriers. Because of a unified thread between the two models, their parameters are interrelated, and all of the relevant conclusions are valid for either of them. In the tight coupling limit, the optimal conditions are analyzed, and they imply that a catalytic wheel operated by the Brownian motor works with the maximum output energy (useful work) or with the maximum efficiency tending to unity under certain conditions.