Rectification of current in ac-driven nonlinear systems and symmetry   properties of the Boltzmann equation
  We study rectification of a current of particles moving in a spatially periodic potential under the influence of time-periodic forces with zero mean value. If certain time-space symmetries are broken a non-zero directed current of particles is possible. We investigate this phenomenon in the framework of the kinetic Boltzmann equation. We find that the attractor of the Boltzmann equation completely reflects the symmetries of the original one-particle equation of motion. Especially, we analyse the limits of weak and strong relaxation. The dc current increases by several orders of magnitude with decreasing dissipation. 
O. Yevtushenko, S. Flach, Y. Zolotaryuk and A. A. Ovchinnikov