Minimal Brownian Ratchet: An Exactly Solvable Model

TL;DR

This paper introduces an exactly solvable three-state discrete-time Brownian ratchet model, analyzing steady-state probabilities, detailed balance, and how external noise influences directional motion.

Contribution

It presents a minimal, exactly solvable model of a Brownian ratchet with asymmetric transition probabilities and explores noise effects on directional motion.

Findings

Steady-state solutions do not generally satisfy detailed balance.

A null-curve exists where net current vanishes and detailed balance holds.

External noise can induce directional motion even with zero net driving force.

Abstract

We develop an exactly-solvable three-state discrete-time minimal Brownian ratchet (MBR), where the transition probabilities between states are asymmetric. By solving the master equations we obtain the steady-state probabilities. Generally the steady-state solution does not display detailed balance, giving rise to an induced directional motion in the MBR. For a reduced two-dimensional parameter space we find the null-curve on which the net current vanishes and detailed balance holds. A system on this curve is said to be balanced. On the null-curve, an additional source of external random noise is introduced to show that a directional motion can be induced under the zero overall driving force. We also indicate the off-balance behavior with biased random noise.