From Diffusion to Reaction via $\Gamma$-Convergence

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 o...