A proof by graphical construction of the no-pumping theorem of stochastic pumps

TL;DR

This paper presents a graphical construction method to derive the no-pumping theorem for stochastic pumps, clarifying conditions for directed motion in Markov models of mesoscopic systems.

Contribution

It introduces a simple graphical approach to derive the no-pumping theorem, offering a clearer understanding of the conditions for current generation in stochastic pumps.

Findings

Graphical construction simplifies the derivation of the no-pumping theorem.

Identifies minimal conditions for directed motion in stochastic systems.

Provides a new perspective on controlling mesoscopic stochastic processes.

Abstract

A stochastic pump is a Markov model of a mesoscopic system evolving under the control of externally varied parameters. In the model, the system makes random transitions among a network of states. For such models, a "no-pumping theorem" has been obtained, which identifies minimal conditions for generating directed motion or currents. We provide a derivation of this result using a simple graphical construction on the network of states.