Analytical Solution to Transport in Brownian Ratchets via Gambler's Ruin Model

TL;DR

This paper develops an analytical approach to understanding transport in Brownian ratchets by modeling it as a Gambler's Ruin problem, deriving explicit solutions for current and conditions for current reversal.

Contribution

It introduces a novel analogy between Brownian ratchets and Gambler's Ruin, providing explicit formulas for current and transition times at any temperature.

Findings

Explicit current solutions for arbitrary temperature

Conditions for zero current and current reversal identified

Numerical verification via Langevin simulations

Abstract

We present an analogy between the classic Gambler's Ruin problem and the thermally-activated dynamics in periodic Brownian ratchets. By considering each periodic unit of the ratchet as a site chain, we calculated the transition probabilities and mean first passage time for transitions between energy minima of adjacent units. We consider the specific case of Brownian ratchets driven by Markov dichotomous noise. The explicit solution for the current is derived for any arbitrary temperature, and is verified numerically by Langevin simulations. The conditions for vanishing current and current reversal in the ratchet are obtained and discussed.