From diffusion to reaction via $\Gamma$-convergence

M.A. Peletier, G. Savaré, M. Veneroni

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Abstract

We study the limit of high activation energy of a special Fokker-Planck equation, known as Kramers-Smoluchowski (K-S) equation. This equation governs the time evolution of the probability density of a particle performing a Brownian motion under the inuence of a chemical potential H/e. We choose H having two wells corresponding to two chemical states A and B. We prove that after a suitable rescaling the solution to (K-S) converges, in the limit of high activation energy (e->0), to the solution of a simple system modeling the diffusion of A and B, and the reaction AB. The aim of this paper is to give a rigorous proof of Kramer's formal derivation and to embed chemical reactions and diffusion processes in a common variational framework which allows to derive the former as a singular limit of the latter, thus establishing a connection between two worlds often regarded as separate. The singular limit is analysed by means of Gamma-convergence in the space of finite Borel measures endowed with the weak-*topology.
Original languageEnglish
Publishers.n.
Number of pages18
Publication statusPublished - 2009

Publication series

NamearXiv.org [math.AP]
Volume0912.5077

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    Peletier, M. A., Savaré, G., & Veneroni, M. (2009). From diffusion to reaction via $\Gamma$-convergence. (arXiv.org [math.AP]; Vol. 0912.5077). s.n.