On the bulk velocity of Brownian ratchets

TL;DR

This paper analyzes the unidirectional transport in Brownian ratchets modeled by Fokker-Planck equations, providing explicit formulas and proving the non-zero velocity conjecture for stochastic Stokes' drift.

Contribution

It establishes a linear relation between transport velocity and bi-periodic solutions, deriving asymptotic formulas and confirming the non-zero velocity conjecture.

Findings

Transport velocity is explicitly characterized in various limits.

The non-zero velocity conjecture for stochastic Stokes' drift is proven.

Qualitative results on transport direction are obtained.

Abstract

In this paper we study the unidirectional transport effect for Brownian ratchets modeled by Fokker-Planck-type equations. In particular, we consider the adiabatic and semiadiabatic limits for tilting ratchets, generic ratchets with small diffusion, and the multi-state chemical ratchets. Having established a linear relation between the bulk transport velocity and the bi-periodic solution, and using relative entropy estimates and new functional inequalities, we obtain explicit asymptotic formulas for the transport velocity and qualitative results concerning the direction of transport. In particular, we prove the conjecture by Blanchet, Dolbeault and Kowalczyk that the bulk velocity of the stochastic Stokes' drift is non-zero for every non-constant potential.