Ratchet effect in inhomogeneous inertial systems: II. The square-wave   drive case
  The underdamped Langevin equation of motion of a particle, in a symmetric periodic potential and subjected to a symmetric periodic forcing with mean zero over a period, with nonuniform friction, is solved numerically. The particle is shown to acquire a steady state mean velocity at asymptotically large time scales. This net particle velocity or the ratchet current is obtained in a range of forcing amplitudes $F_0$ and peaks at some value of $F_0$ within the range depending on the value of the average friction coefficient and temperature of the medium. At these large time scales the position dispersion grows proportionally with time, $t$, allowing for calculating the steady state diffusion coefficient $D$ which, interestingly, shows a peaking behaviour around the same $F_0$. The ratchet current, however, turns out to be largely coherent. At intermediate time scales, which bridge the small time scale behaviour of dispersion$\sim t^2$ to the large time one, the system shows, in some cases, periodic oscillation between dispersionless and steeply growing dispersion depending on the frequency of the forcing. The contribution of these different dispersion regimes to ratchet current is analysed. 
S.Saikia and Mangal C. Mahato