Symmetry broken motion of a periodically driven Brownian particle: nonadiabatic regime.

We report a theoretical study of an overdamped Brownian particle dynamics in the presence of both a spatially modulated one-dimensional periodic potential U(x) and a periodic alternating force (AF). As the periodic potential U(x) has a low symmetry (a ratchet potential) the Brownian particle displays a broken symmetry motion with a nonzero time average velocity. By making use of the Green function method and a mapping to the theory of Brillouin bands the probability distribution P(x,t) of the particle coordinate x is derived and the nonlinear dependence of the macroscopic velocity on the frequency omega and the amplitude eta of the AF is found. In particular, our theory allows to go beyond the adiabatic limit (omega=0) and to explain the peculiar reversal of the velocity sign found previously in the numerical analysis.