由复杂平衡化学反应网络产生的反应扩散系统的平衡趋向
Trend to equilibrium for reaction-diffusion systems arising from complex balanced chemical reaction networks
摘要：
利用熵方法研究了具有质量作用动力学的复杂平衡化学反应网络的反应扩散系统向平衡态的定量收敛性。在本文的第一部分中，通过显式地导出熵耗散，我们证明了对于没有边界平衡点的复杂平衡系统，每个轨迹都指数快速地收敛到唯一的复杂平衡点。此外，一个建设性的证明，提出了明确估计的收敛速率的特殊情况下的循环反应。第二部分研究了具有边界平衡点的复杂平衡系统。我们调查两个具体的情况下，分别具有两个和三个化学物质。在这些情况下，边界平衡点在某种意义上是不稳定的，因此也可以证明指数收敛到唯一的严格正平衡点。
英文摘要：
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.
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