  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.
