NDLI: Chemical Reactions as $\Gamma$-Limit of Diffusion

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Chemical Reactions as $\Gamma$-Limit of Diffusion

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Peletier, Mark A. Veneroni, Marco Savar, Giuseppe
Copyright Year	2012
Abstract	We study the limit of high activation energy of a special FokkerPlanck equation known as the KramersSmoluchowski 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 the KS converges, in the limit of high activation energy ($\varepsilon\to0$), to the solution of a simpler system modeling the spatial diffusion of A and B combined with the reaction $A\rightleftharpoons B$. With this result we give a rigorous proof of Kramers's formal derivation, and we show how chemical reactions and diffusion processes can be embedded in a common framework. This allows one to derive a chemical reaction as a singular limit of a diffusion process, thus establishing a connection between two worlds often regarded as separate. The proof rests on two main ingredients. One is the formulation of the two disparate equations as evolution equations for measures. The second is a variational formulation of both equations that allows us to use the tools of variational calculus and, specifically, $\Gamma$-convergence.
Starting Page	327
Ending Page	352
Page Count	26
File Format	PDF
ISSN	00361445
DOI	10.1137/110858781
e-ISSN	10957200
Journal	SIAM Review (SIREAD)
Issue Number	2
Volume Number	54
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2012-05-08
Access Restriction	Subscribed
Subject Keyword	Mosco-convergence unification variational evolution equations Methods involving semicontinuity and convergence; relaxation Chemical kinetics scale-bridging activation energy Dirichlet forms high-energy limit Fokker-Planck equations Reaction-diffusion equations upscaling Variational principles of physics
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Theoretical Computer Science Computational Mathematics

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