Trend to equilibrium for reaction-diffusion systems arising from complex balanced chemical reaction networks

TL;DR

This paper proves exponential convergence to equilibrium in reaction-diffusion systems from complex balanced chemical networks using entropy methods, including explicit rate estimates and stability analysis of boundary equilibria.

Contribution

It provides a rigorous entropy-based framework for showing exponential convergence in complex balanced systems, with explicit rates and stability results for boundary equilibria.

Findings

Exponential convergence to equilibrium for complex balanced systems without boundary equilibria.

Explicit rate estimates for cyclic reaction systems.

Boundary equilibria are shown to be unstable, ensuring convergence to positive equilibria.

Abstract

The quantitative convergence to equilibrium for reaction-diffusion systems arising from complex balanced chemical reaction networks with mass action kinetics is studied by using the so-called entropy method. In the first part of the paper, by deriving explicitly the entropy dissipation, we show that for complex balanced systems without boundary equilibria, each trajectory converges exponentially fast to the unique complex balance equilibrium. Moreover, a constructive proof is proposed to explicitly estimate the rate of convergence in the special case of a cyclic reaction. In the second part of the paper, complex balanced systems with boundary equilibria are considered. We investigate two specific cases featuring two and three chemical substances respectively. In these cases, the boundary equilibria are shown to be unstable in some sense, so that exponential convergence to the unique strictly positive equilibrium can also be proven.