WIAS Preprint No. 2643, (2019)

Coarse-graining via EDP-convergence for linear fast-slow reaction systems


Authors

  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888
  • Stephan, Artur
    ORCID: 0000-0001-9871-3946

2010 Mathematics Subject Classification

  • 49S05 47D07 47J30 92E20 60J20

Keywords

  • Markov process with detailed balance, linear reaction system, entropic gradient structure, energy-dissipation balance, EDP-convergence, microscopic equilibrium, coarse graining, reconstruction operators

DOI

10.20347/WIAS.PREPRINT.2643

Abstract

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.

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