  We report a theoretical study of an overdamped Brownian particle dynamics in
the presence of both a spatially modulated one-dimensional periodic potential
and a periodic alternating force (AF). As the periodic potential 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 of the particle coordinate is derived and the nonlinear dependence
of the macroscopic velocity on the frequency and the amplitude of AF is found.
In particular, our theory allows to go beyond the adiabatic limit and to
explain the peculiar reversal of the velocity sign found previously in the
numerical analysis.
