Coarse-graining via EDP-convergence for linear fast-slow reaction systems

TL;DR

This paper studies the limiting behavior of linear reaction systems with fast and slow reactions, showing how to rigorously derive a coarse-grained model with preserved gradient structure using EDP-convergence.

Contribution

It introduces a rigorous EDP-convergence framework for coarse-graining linear reaction systems with detailed balance, preserving gradient structures in the limit.

Findings

Effective limit gradient structure derived for coarse-grained models

EDP-convergence with tilting established for reaction systems

Coarse-grained equations retain cosh-type gradient structure

Abstract

We consider linear reaction systems with slow and fast reactions, which can be interpreted as master equations or Kolmogorov forward equations for Markov processes on a finite state space. We investigate their limit behavior if the fast reaction rates tend to infinity, which leads to a coarse-grained model where the fast reactions create microscopically equilibrated clusters, while the exchange mass between the clusters occurs on the slow time scale. Assuming detailed balance the reaction system can be written as a gradient flow with respect to the relative entropy. Focusing on the physically relevant cosh-type gradient structure we show how an effective limit gradient structure can be rigorously derived and that the coarse-grained equation again has a cosh-type gradient structure. We obtain the strongest version of convergence in the sense of the Energy-Dissipation Principle (EDP), namely EDP-convergence with tilting.