Ratchet effect in inhomogeneous inertial systems: I. Adiabatic case
  Risken's matrix continued fraction method is used to solve the Fokker-Planck equation to calculate particle current in an inertial symmetric (sinusoidal) periodic potential under the action of a constant force. The particle moves in a medium with friction coefficient also varying periodically in space as the potential but with a finite phase difference $\phi (\ne n\pi, n=0,1,2, ...)$. The algebraic sum of current with applied forces $\pm|F|$ gives the ratchet current in the adiabatic limit. Even though, the effect of frictional inhomogeneity is weak, the ratchet current shows very rich qualitative characteristics. The effects of variation of $F$, the temperature $T$, and the average friction coefficient $\gamma_0$ on the performance characteristics of the ratchet are presented. 
W. L. Reenbohn and Mangal C. Mahato