Directed transport in a spatially periodic potential under periodic non-biased forcing

TL;DR

This paper investigates how a particle in a spatially periodic potential exhibits directed transport under a slow, unbiased periodic force, revealing long-term ratchet effects and deriving formulas for average velocity.

Contribution

It introduces a theoretical analysis of ratchet transport in a nonlinear system under general periodic forcing, extending understanding of directed motion in such setups.

Findings

Directed transport occurs in the chaotic regime over long times.

Formulas for average velocity are derived for small and large force amplitudes.

Results are applicable to non-harmonic periodic potentials.

Abstract

Transport of a particle in a spatially periodic harmonic potential under the influence of a slowly time-dependent unbiased periodic external force is studied. The equations of motion are the same as in the problem of a slowly forced nonlinear pendulum. Using methods of the adiabatic perturbation theory we show that for a periodic external force of general kind the system demonstrates directed (ratchet) transport in the chaotic domain on very long time intervals and obtain a formula for the average velocity of this transport. Two cases are studied: the case of the external force of small amplitude, and the case of the external force with amplitude of order one. The obtained formulas can also be used in case of a non-harmonic periodic potential.