Universal time dependent dispersion properties for diffusion in a one-dimensional critically tilted potential

We consider the time dependent dispersion properties of overdamped tracer particles diffusing in a one dimensional periodic potential under the influence of an additional constant tilting force $F$. The system is studied in the region where the force is close to the critical value $F_c$ at which the barriers separating neighboring potential wells disappear. We show that, when $F$ crosses the critical value, the shape of the Mean-Square Displacement (MSD) curves is strongly modified. We identify a diffusive regime at intermediate time scales, with an effective diffusion coefficient which is much larger than the late time diffusion coefficient for $F>F_c$, whereas for $F<F_c$ the late time and intermediate time diffusive regimes are indistinguishable. Explicit asymptotic regimes for the MSD curves are identified at all time scales.