We present the main results of the theory of Brownian motors obtained using the authors’ approach, in which a Brownian particle moving in a slightly fluctuating potential profile is considered. By using the Green’s function method, the perturbation theory in small fluctuations of potential energy is constructed. This approach allows obtaining an analytic expression for the mean particle velocity that is valid for two main types of Brownian motors (flashing and rocking ratchets) and any time dependence (stochastic or deterministic) of the fluctuations. The advantage of the proposed approach lies in the compactness of the description and, at the same time, in the variety of motor systems analyzed with its help: the overwhelming majority of known analytic results in the theory of Brownian motors follow from this expression. The mathematical derivations and analysis of those results are the main subject of these methodological notes.
We revisit two known models of deterministically driven ratchets, which exhibit high energetic efficiency, with the goal to uncover similarities and differences in the principles of their operation. Both the models rely on adiabaticity of the potential change process, however, the adiabaticity that we deal with in the two cases is of different types, slow and fast. It is shown that in the former (latter) case the drift velocity is an even (odd) functional of the potential, with the notable consequence that for the adiabatically slow driven ratchet the necessary symmetry breaking occurs only due to time-dependent parametric perturbations, while the spatial asymmetry of the potential is a mandatory condition for the adiabatically fast driven ratchet to operate. To treat energetic characteristics, the models are restated in terms of traveling potential ratchets. With such an approach, we find that in these cases (i) the conditions of high energetic efficiency to be reached are similar, and (ii) the symmetry properties of the kinetic coefficients are different. Based on our results, a strategy for designing efficient Brownian motors is suggested.
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