Rectification of asymmetric surface vibrations with dry friction: an exactly solvable model

TL;DR

This paper presents an exactly solvable stochastic model for the directed motion of an object on a surface with asymmetric vibrations and dry friction, revealing complex stationary velocity features and non-monotonic transport behavior.

Contribution

It introduces a novel exactly solvable model incorporating dry friction and asymmetric vibrations, providing analytical insights into velocity distributions and transport phenomena.

Findings

Stationary velocity density shows discontinuities and delta peaks.

Mean velocity varies non-monotonically with dry friction strength.

Transport can improve with increased dissipation.

Abstract

We consider a stochastic model for the directed motion of a solid object due to the rectification of asymmetric surface vibrations with Poissonian shot-noise statistics. The friction between the object and the surface is given by a piecewise-linear friction force. This models the combined effect of dynamic friction and singular dry friction. We derive an exact solution of the stationary Kolmogorov-Feller (KF) equation in the case of two-sided exponentially distributed amplitudes. The stationary density of the velocity exhibits singular features such as a discontinuity and a delta-peak singularity at zero velocity, and also contains contributions from non-integrable solutions of the KF equation. The mean velocity in our model generally varies non-monotonically as the strength of the dry friction is increased, indicating that transport improves for increased dissipation.