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On the Geometry of Chemical Reaction Networks: Lyapunov Function and Large Deviations

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Abstract

In an earlier paper, we proved the validity of large deviations theory for the particle approximation of quite general chemical reaction networks. In this paper, we extend its scope and present a more geometric insight into the mechanism of that proof, exploiting the notion of spherical image of the reaction polytope. This allows to view the asymptotic behavior of the vector field describing the mass-action dynamics of chemical reactions as the result of an interaction between the faces of this polytope in different dimensions. We also illustrate some local aspects of the problem in a discussion of Wentzell–Freidlin theory, together with some examples.

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Notes

  1. It can happen that several reactions maximize the scalar product. This will be discussed later.

  2. Where the angle between hyperplanes \(B,\,B' \subset \mathbb {R}^{d_\mathcal {P}}\) passing through the origin is \(\angle (B,\,B') := \arccos (\max \{\min \{|\langle b,\,b' \rangle |:b \in B,\, \Vert b\Vert _2 = 1\}:b' \in B',\, \Vert b'\Vert _2 = 1\})\).

  3. Assuming that the set \({\mathcal {D}}\) respects [1, Assumption A.3].

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Acknowledgements

AA’s and JPE’s Research was partially supported by ERC Advanced Grant 290843 (BRIDGES). AA also acknowledges the SNSF Grant 161866. AD’s research was partially supported by NSF Grant DMS-1613091.

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Authors and Affiliations

  1. Département de Physique Théorique, Université de Genève, 1211, Geneva 4, Switzerland

    A. Agazzi & J.-P. Eckmann

  2. Statistics Department, Stanford University, Stanford, CA, 94305, USA

    A. Agazzi & A. Dembo

  3. Section de Mathématiques, Université de Genève, 1211, Geneva 4, Switzerland

    J.-P. Eckmann

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  1. A. Agazzi
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  2. A. Dembo
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  3. J.-P. Eckmann
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Correspondence to J.-P. Eckmann.

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To Herbert, Jürg, and Tom, friends and inspirations.

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Agazzi, A., Dembo, A. & Eckmann, JP. On the Geometry of Chemical Reaction Networks: Lyapunov Function and Large Deviations. J Stat Phys 172, 321–352 (2018). https://doi.org/10.1007/s10955-018-2035-8

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