Gauge Theory for the Rate Equations: Electrodynamics on a Network

TL;DR

This paper develops a gauge theory framework analogous to electrodynamics to analyze the conservation laws and structure of rate equations on networks, revealing insights into their degrees of freedom and connectivity.

Contribution

It introduces a novel gauge theory approach to rate equations, providing a new perspective on their conservation laws and network properties.

Findings

Network is maximally connected with respect to electromagnetic fields.

Electric and magnetic fields have equal degrees of freedom.

Framework offers insights into classical abelian gauge theory beyond rate equations.

Abstract

Systems of coupled rate equations are ubiquitous in many areas of science, for example in the description of electronic transport through quantum dots and molecules. They can be understood as a continuity equation expressing the conservation of probability. It is shown that this conservation law can be implemented by constructing a gauge theory akin to classical electrodynamics on the network of possible states described by the rate equations. The properties of this gauge theory are analyzed. It turns out that the network is maximally connected with respect to the electromagnetic fields even if the allowed transitions form a sparse network. It is found that the numbers of degrees of freedom of the electric and magnetic fields are equal. The results shed light on the structure of classical abelian gauge theory beyond the particular motivation in terms of rate equations.