Green's function method in the theory of Brownian motors

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.