Dispersionless Transport in a Washboard Potential

We study and characterize a new dynamical regime of underdamped particles in a tilted washboard potential. We find that for small friction in a finite range of forces the particles move essentially nondispersively, that is, coherently, over long intervals of time. The associated distribution of the particle positions moves at an essentially constant velocity and is far from Gaussian-like. This new regime is complementary to, and entirely different from, well-known nonlinear response and large dispersion regimes observed for other values of the external force.

Figure 1

Mean velocity $v$ (left axis, open circles) and mobility $\mu$ (right axis, solid circles) vs the external force. Inset: diffusion coefficient $D$ vs external force. The “empty” portion of the diffusion coefficient curve indicates that we did not achieve a stationary state up to the longest simulation times [$\tau \sim O({10}^7)$]. This is the regime of persistent dispersionless transport (see text).

Figure 2

Dispersion vs time for different forces. Note that the curves for small ($F=0.04$) and large ($F=4.0$) forces quickly cross over from the short-time superdiffusive regime typical of low friction to the asymptotic diffusive regime. Intermediate forces exhibit a long dispersionless regime, eventually followed by normal dispersion.

Figure 3

Unnormalized distribution of particle positions at different times for $F=0.12$. Top left: early times regime. Values of $\tau$: 500 (dot-dashed); 1000 (solid); 2000 (dotted); 4000 (dashed). Bottom left: dispersionless time regime for $\tau =7000$ (solid), 30 000 (dotted). Top right: longtime dispersive regime for $\tau =2\times {10}^5$ (solid), ${10}^6$ (dotted). Bottom right: Log plot of bottom left distribution to display the exponential tail. The straight line is the exponential form (2) with $l_0=(F/\gamma ){\tau }_0=3600$.

Figure 4

Bottom panel: Two individual $z$ trajectories for $F=0.12$. The two trajectories emerge from the initial potential well at slightly different times of order ${10}^3$ and move essentially nondispersively for a long time, until dispersive motion eventually sets in. Upper panel: associated dispersion curve.