Abstract
AbstractWe 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 $$1/\epsilon $$
1
/
ϵ
, and we prove the convergence in the fast-reaction limit $$\epsilon \rightarrow 0$$
ϵ
→
0
. We establish a $$\Gamma $$
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 characterises 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 $$\Gamma $$
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.
Funder
Deutsche Forschungsgemeinschaft
Publisher
Springer Science and Business Media LLC
Reference43 articles.
1. Agazzi, A., Dembo, A., Eckmann, J.-P.: Large deviations theory for Markov jump models of chemical reaction networks. Annals of Applied Probability 28(3), 1821–1855 (2018)
2. Ambrosio, L., Gigli, N., Savaré, G.: Gradient flows in metric spaces and in the space of probability measures, 2nd edn. Lectures in Mathematics. ETH Zürich. Birkhauser, Basel, Switzerland (2008)
3. Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Macroscopic fluctuation theory. Rev. Mod. Phys. 87(2) (2015)
4. Baiesi, M., Maes, C., Netočný, K.: Computation of current cumulants for small nonequilibrium systems. Journal of Statistical Physics 135(1), 57–75 (2009)
5. Bothe, D.: Instantaneous limits of reversible chemical reactions in presence of macroscopic convection. Journal of Differential Equations 193(1), 27–48 (2003)
Cited by
7 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献