Renormalized entropy production for optimal transport in jump processes: Make conservative forces optimal again

TL;DR

This paper introduces the concept of renormalized entropy production to identify conservative forces that optimize jump processes, highlighting differences from traditional entropy minimization in continuous systems.

Contribution

It defines and analyzes renormalized entropy production, providing a new framework for understanding force optimization in jump processes on discrete spaces.

Findings

Conservative forces are characterized as minimizers of renormalized entropy production.

Renormalized entropy production shares properties with entropy but differs in key aspects.

Numerical examples illustrate the theoretical concepts.

Abstract

For continuous-space diffusion processes, there is a strong connection between conservative forces and entropy production. For a given time evolution of the system's state, the entropy production is minimized when the system is driven by a unique conservative force. However, this relation does not extend to jump processes on a discrete state space. In this case, the forces that minimize the entropy production are generally nonconservative, this effect is more pronounced far from equilibrium in the presence of high energy barriers. Here we show that, while conservative forces do not minimize the entropy production for a given time evolution, they are nevertheless uniquely characterized as the minimizer of a quantity we dub the renormalized entropy production. This work explores the properties this quantity shares with entropy production as well as crucial differences between them. We also discuss the conceptual and physical differences between the corresponding optimization problems in finite time. Our theoretical calculations are illustrated with explicit numerical examples.