### 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 language | English |
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Publisher | s.n. |

Number of pages | 18 |

Publication status | Published - 2009 |

### Publication series

Name | arXiv.org [math.AP] |
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Volume | 0912.5077 |

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## Cite this

Peletier, M. A., Savaré, G., & Veneroni, M. (2009).

*From diffusion to reaction via $\Gamma$-convergence*. (arXiv.org [math.AP]; Vol. 0912.5077). s.n.