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 the Kramers–Smoluchowski equation (KS). This equation governs the time evolution of the probability density of a particle performing a Brownian motion under the influence of a chemical potential $H/\varepsilon$. We choose $H$ having two wells corresponding to two chemical states $A$ and $B$. We prove that after a suitable rescaling the solution to KS converges, in the limit of high activation energy ($\varepsilon\to0$), to the solution of a simple system modeling the diffusion of $A$ and $B$, and the reaction $A\rightleftharpoons B$. The aim of this paper is to give a rigorous proof of Kramers's formal derivation and to embed chemical reactions and diffusion processes in a common variational framework which allows one 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 analyzed by means of $\Gamma$-convergence in the space of finite Borel measures endowed with the weak-$*$ topology.
Original languageEnglish
Pages (from-to)1805-1825
JournalSIAM Journal on Mathematical Analysis
Volume42
Issue number4
DOIs
Publication statusPublished - 2010

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