Dispersionless motion and ratchet effect in a square-wave-driven inertial periodic potential system

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 timescales. At these large timescales the position dispersion grows proportionally with time, t, allowing for calculating the steady state diffusion coefficient D. Interestingly, D shows a peaking behaviour around the same F-0 where the net current peaks. The net (ratchet) current, however, turns out to be largely coherent. At an intermediate timescale, which bridges the small timescale behaviour of dispersion similar to t(2) to the large time one, the system shows periodic oscillation between dispersionless and steeply growing dispersion depending on the amplitude and frequency of the forcing. The contribution of these different dispersion regimes to ratchet current is analysed.