About the entropic structure of detailed balanced multi-species cross-diffusion equations
Esther S. Daus, Laurent Desvillettes, Helge Dietert
TL;DR
The paper establishes a formal and then rigorous link between the entropy structure of the SKT multi-species cross-diffusion system under the detailed balance condition and the entropy of a reversible microscopic Markov process on a discretised space. Beginning with a mean-field limit on a fixed grid, the authors obtain a quadratic master equation whose entropy matches the macroscopic relative entropy when chaos is assumed; they then refine the spatial discretisation and prove convergence to the SKT system as the grid is refined. The key result is a robust, entropy-preserving discretisation approach that yields global weak solutions for the SKT model with an arbitrary number of species under detailed balance, providing a new strategy for proving existence via discrete entropy dissipation and compactness. The methodology highlights how microscopic reversibility and entropy production can drive rigorous macroscopic limits in cross-diffusion systems and suggests extensions to more general or reaction-type terms.
Abstract
This paper links at the formal level the entropy structure of a multi-species cross-diffusion system of Shigesada-Kawasaki-Teramoto (SKT) type satisfying the detailed balance condition with the entropy structure of a reversible microscopic many-particle Markov process on a discretised space. The link is established by first performing a mean-field limit to a master equation over discretised space. Then the spatial discretisation limit is performed in a completely rigorous way. This by itself provides a novel strategy for proving global existence of weak solutions to a class of cross-diffusion systems.