  We perform a fast-reaction limit for a linear reaction-diffusion system
consisting of two diffusion equations coupled by a linear reaction. We
understand the linear reaction-diffusion system as a gradient flow of the free
energy in the space of probability measure equipped with a geometric structure,
which contains the Wasserstein metric for the diffusion part and cosh-type
functions for the reaction part. The fast-reaction limit is done on the level
of the gradient structure by proving EDP-convergence with tilting. The limit
gradient system induces a diffusion system with Lagrange multipliers on the
linear slow-manifold. Moreover, the limit gradient system can be equivalently
described by a coarse-grained gradient system, which induces a diffusion
equation with a mixed diffusion constant for the coarse-grained slow variable.
