From diffusion to reaction via Gamma-convergence

TL;DR

This paper rigorously derives a reaction-diffusion system from a high-activation-energy limit of the Kramers-Smoluchowski equation, connecting chemical reactions and diffusion within a unified variational framework using Gamma-convergence.

Contribution

It provides a rigorous proof of Kramer's formal derivation and embeds chemical reactions and diffusion processes in a common variational framework via Gamma-convergence.

Findings

Convergence of the Kramers-Smoluchowski equation to a reaction-diffusion system as activation energy increases.

Establishment of a variational framework linking diffusion and chemical reactions.

Application of Gamma-convergence in the space of finite Borel measures to analyze the singular limit.

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 influence 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 A <-> B. 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.