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We study a singular-limit problem arising in the modelling of chemicalreactions. At finite {\epsilon} > 0, the system is described by a Fokker-Planckconvection-diffusion equation with a double-well convection potential. Thispotential is scaled by 1/{\epsilon}, and in the limit {\epsilon} -> 0, thesolution concentrates onto the two wells, resulting into a limiting system thatis a pair of ordinary differential equations for the density at the two wells.This convergence has been proved in Peletier, Savar\'e, and Veneroni, SIAMJournal on Mathematical Analysis, 42(4):1805-1825, 2010, using the linearstructure of the equation. In this paper we re-prove the result by using solelythe Wasserstein gradient-flow structure of the system. In particular we make nouse of the linearity, nor of the fact that it is a second-order system. Thefirst key step in this approach is a reformulation of the equation as theminimization of an action functional that captures the property of being acurve of maximal slope in an integrated form. The second important step is arescaling of space. Using only the Wasserstein gradient-flow structure, weprove that the sequence of rescaled solutions is pre-compact in an appropriatetopology. We then prove a Gamma-convergence result for the functional in thistopology, and we identify the limiting functional and the differential equationthat it represents. A consequence of these results is that solutions of the{\epsilon}-problem converge to a solution of the limiting problem.

Passing to the Limit in a Wasserstein Gradient Flow: From Diffusion to Reaction