论文标题
通过$γ$ - 通量率功能的快速反应限制
Fast reaction limits via $Γ$-convergence of the Flux Rate Functional
论文作者
论文摘要
我们研究了图形节点上支持的度量的演化方程序列的收敛性。进化方程本身可以解释为马尔可夫跳跃过程的前向kolmogorov方程，也可以等效地为线性反应网络中浓度的方程。跳高率或反应率分为两类； “缓慢”的速率是恒定的，并且“快速”速率缩放为〜$ 1/ε$，我们证明了快速反应限制$ε\ to0 $中的收敛性。 我们以每个节点的浓度和每个边缘的通量（级别2.5速率函数）的浓度来建立速率功能的$γ$合并结果。限制系统再次由功能描述，并表征系统中的快速和慢速通量。 这种证明方法具有三个优点。首先，不需要详细的余额条件。其次，在浓度和通量方面的配方导致了$γ$ - 融合的简短证明；付款的价格是一种更涉及的紧凑性证明。最后，证明方法涉及近似解决方案，该解决方案的功能不是零，而是较小，没有任何更改。
We study the convergence of a sequence of evolution equations for measures supported on the nodes of a graph. The evolution equations themselves can be interpreted as the forward Kolmogorov equations of Markov jump processes, or equivalently as the equations for the concentrations in a network of linear reactions. The jump rates or reaction rates are divided in two classes; `slow' rates are constant, and `fast' rates are scaled as~$1/ε$, and we prove the convergence in the fast-reaction limit $ε\to0$. We establish a $Γ$-convergence result for the rate functional in terms of both the concentration at each node and the flux over each edge (the level-2.5 rate function). The limiting system is again described by a functional, and characterizes both fast and slow fluxes in the system. This method of proof has three advantages. First, no condition of detailed balance is required. Secondly, the formulation in terms of concentration and flux leads to a short and simple proof of the $Γ$-convergence; the price to pay is a more involved compactness proof. Finally, the method of proof deals with approximate solutions, for which the functional is not zero but small, without any changes.
