Gradient and Generic systems in the space of fluxes, applied to reacting particle systems

TL;DR

This paper develops a framework linking microscopic particle flux large deviations to macroscopic flux gradient systems, extending previous work on concentration-based systems, with applications to reacting particles and diffusion.

Contribution

It introduces a novel approach to derive flux-based gradient and generic systems from microscopic large deviations, expanding the scope beyond concentration-based models.

Findings

Flux large deviations induce generalized gradient systems.

The framework connects flux systems to concentration systems via continuity equations.

Applications include reacting particle systems with spatial diffusion.

Abstract

In a previous work we devised a framework to derive generalised gradient systems for an evolution equation from the large deviations of an underlying microscopic system, in the spirit of the Onsager-Machlup relations. Of particular interest is the case where the microscopic system consists of random particles, and the macroscopic quantity is the empirical measure or concentration. In this work we take the particle flux as the macroscopic quantity, which is related to the concentration via a continuity equation. By a similar argument the large deviations can induce a generalised gradient or Generic system in the space of fluxes. In a general setting we study how flux gradient or generic systems are related to gradient systems of concentrations. The arguments are explained by the example of reacting particle systems, which is later expanded to include spatial diffusion as well.