Directed current due to broken time-space symmetry

We consider the classical dynamics of a particle in a one-dimensional space -periodic potential U(X) = U(X + 2 pi) under the influence of a time-period ic space-homogeneous external field E(t) = E(t + T). If E(t) is neither a s ymmetric function of t nor antisymmetric under time shifts E(t +/- T/2) not equal -E(t), an ensemble of trajectories with zero current at t = 0 yields a nonzero finite current as t --> 0. We explain this effect using symmetry considerations and perturbation theory. Finally we add dissipation (fricti on) and demonstrate that the resulting set of attractors keeps the broken s ymmetry property in the basins of attraction and leads to directed currents as well.