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.

中文翻译：

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线性快慢反应系统通过 EDP 收敛的粗粒度

我们考虑具有慢速和快速反应的线性反应系统，可以将其解释为有限状态空间上马尔可夫过程的主方程或 Kolmogorov 前向方程。如果快速反应速率趋于无穷大，我们将研究它们的极限行为，这会导致粗粒度模型，其中快速反应产生微观平衡的簇，而簇之间的交换质量发生在慢时间尺度上。假设详细平衡，反应系统可以写成相对于相对熵的梯度流。着眼于物理相关的 cosh 型梯度结构，我们展示了如何严格推导有效的极限梯度结构，并且粗粒度方程再次具有 cosh 型梯度结构。