The entropy method for reaction-diffusion systems without detailed balance: first order chemical reaction networks

TL;DR

This paper extends the entropy method to analyze the convergence to equilibrium in reaction-diffusion systems from first order chemical reaction networks, including cases without detailed balance, showing exponential decay and stability.

Contribution

It introduces an entropy structure for weakly reversible networks without detailed balance and proves exponential convergence to equilibrium, broadening the applicability of the entropy method.

Findings

Solutions converge exponentially fast to equilibrium in weakly reversible networks.

Species in source or transmission components decay exponentially, while target species stabilize.

Results apply even when all but one diffusion coefficient are zero.

Abstract

In this paper, the applicability of the entropy method for the trend towards equilibrium for reaction-diffusion systems arising from first order chemical reaction networks is studied. In particular, we present a suitable entropy structure for weakly reversible reaction networks without detail balance condition. We show by deriving an entropy-entropy dissipation estimate that for any weakly reversible network each solution trajectory converges exponentially fast to the unique positive equilibrium with computable rates. This convergence is shown to be true even in cases when the diffusion coefficients all but one species are zero. For non-weakly reversible networks consisting of source, transmission and target components, it is shown that species belonging to a source or transmission component decay to zero exponentially fast while species belonging to a target component converge to the corresponding positive equilibria, which are determined by the dynamics of the target component and the mass injected from other components. The results of this work, in some sense, complete the picture of trend to equilibrium for first order chemical reaction networks.