Directed Current due to Broken Time-Space Symmetry

We consider the classical dynamics of a particle in a one-dimensional space-periodic potential $U\left(X\right)=U(X+2\mathrm{}\pi )$ under the influence of a time-periodic space-homogeneous external field $E\left(t\right)=E(t+T)$. If $E\left(t\right)$ is neither a symmetric function of $t$ nor antisymmetric under time shifts $E(t\pm T/2)\ne -E\left(t\right)$, an ensemble of trajectories with zero current at $t=0\mathrm{}$ yields a nonzero finite current as $t\to \infty$. We explain this effect using symmetry considerations and perturbation theory. Finally we add dissipation (friction) and demonstrate that the resulting set of attractors keeps the broken symmetry property in the basins of attraction and leads to directed currents as well.