Optimal protocol for a collective flashing ratchet

TL;DR

This paper derives an optimal control protocol for a system of Brownian particles in a flashing ratchet to maximize long-term center of mass speed, using Bellman's principle, and demonstrates its effectiveness through numerical solutions and simulations.

Contribution

It introduces a method to find the optimal flashing protocol for any number of particles using Bellman's principle, improving upon existing open-loop strategies.

Findings

Optimal protocol outperforms open-loop strategies for 2 particles.

Numerical solutions validate the effectiveness of the derived protocol.

Simulations confirm improved long-term speed in the system.

Abstract

We study a system of independent Brownian particles in a flashing ratchet potential that can be turned on or off depending on the position of the particles, with the aim of maximising the speed of the center of mass in the long run. First, an explanation on how to find the optimal protocol using Bellman's principle of optimality for any number of particles is given. Then the problem is numerically solved for a system of 2 particles. Simulations show that the optimal protocol performs better than the maximisation of instantaneous center-of-mass speed, a protocol known to give better results for 2 particles than any open-loop protocol.