EDP-convergence for nonlinear fast-slow reaction systems with detailed balance

TL;DR

This paper studies the limiting gradient structure of nonlinear reaction systems with fast and slow reactions satisfying detailed balance, using EDP convergence to rigorously derive effective thermodynamic and kinetic descriptions.

Contribution

It introduces a rigorous EDP convergence framework for nonlinear reaction systems with detailed balance, revealing the structure of the limiting gradient flow.

Findings

Effective gradient structure can be derived via EDP convergence.

The limiting entropy may differ from Boltzmann form.

Reactions may deviate from mass-action kinetics in the limit.

Abstract

We consider nonlinear reaction systems satisfying mass-action kinetics with slow and fast reactions. It is known that the fast-reaction-rate limit can be described by an ODE with Lagrange multipliers and a set of nonlinear constraints that ask the fast reactions to be in equilibrium. Our aim is to study the limiting gradient structure which is available if the reaction system satisfies the detailed-balance condition. The gradient structure on the set of concentration vectors is given in terms of the relative Boltzmann entropy and a cosh-type dissipation potential. We show that a limiting or effective gradient structure can be rigorously derived via EDP convergence, i.e. convergence in the sense of the Energy-Dissipation Principle for gradient flows. In general, the effective entropy will no longer be of Boltzmann type and the reactions will no longer satisfy mass-action kinetics.