# From diffusion to reaction via Gamma-convergence - Mathematics > Analysis of PDEs

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Abstract: We study the limit of high activation energy of a special Fokker-Planckequation, known as Kramers-Smoluchowski K-S equation. This equation governsthe time evolution of the probability density of a particle performing aBrownian motion under the influence of a chemical potential H-e. We choose Hhaving two wells corresponding to two chemical states A and B. We prove thatafter a suitable rescaling the solution to K-S converges, in the limit ofhigh activation energy e -> 0, to the solution of a simple system modelingthe diffusion of A and B, and the reaction A <-> B.The aim of this paper is to give a rigorous proof of Kramer-s formalderivation and to embed chemical reactions and diffusion processes in a commonvariational framework which allows to derive the former as a singular limit ofthe latter, thus establishing a connection between two worlds often regarded asseparate.The singular limit is analysed by means of Gamma-convergence in the space offinite Borel measures endowed with the weak-* topology.

Author: ** Mark A. Peletier, Giuseppe Savaré, Marco Veneroni**

Source: https://arxiv.org/