On dispersionless transport in washboard potentials

TL;DR

This paper clarifies that Brownian particles in tilted periodic potentials do not exhibit a true dispersionless regime, but rather normal diffusion follows an initial superdiffusive stage, with the duration of the apparent dispersionless phase increasing exponentially at low temperatures.

Contribution

The study provides a detailed quantitative analysis of the transition from superdiffusion to normal diffusion in washboard potentials, challenging previous claims of dispersionless transport.

Findings

Particles exhibit normal diffusion immediately after transitioning to the running state.

The flat dispersion segment is due to initial broad spatial distribution, not dispersionless transport.

Superdiffusion exponent is $oldsymbol{ ext{α=3}}$, with the dispersionless regime duration diverging exponentially as temperature approaches zero.

Abstract

We reassess the "dispersionless transport regime" of Brownian particles in tilted periodic potentials. We show that the particles exhibit normal diffusive motion right after transitioning into the running state dragged by the constant bias force. No special transient dynamics appears, contrary to conjectures in the previous studies. The observed flat segment in the dispersion evolution curve is solely due to the broad spatial distribution of particles formed in the early superdiffusion stage. We quantitatively describe the whole evolution of the distribution function during superdiffusion and the transition to the normal diffusion that follows, in the framework of the 2-well potential in the velocity space model. We show that the superdiffusion exponent is $\alpha=3$. Estimate of the duration of the ostensible "dispersionless regime" is provided. It is shown to diverge exponentially as the temperature decreases to zero.