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Moment Closure—A Brief Review

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Control of Self-Organizing Nonlinear Systems

Part of the book series: Understanding Complex Systems ((UCS))

Abstract

Moment closure methods appear in myriad scientific disciplines in the modelling of complex systems. The goal is to achieve a closed form of a large, usually even infinite, set of coupled differential (or difference) equations. Each equation describes the evolution of one “moment”, a suitable coarse-grained quantity computable from the full state space. If the system is too large for analytical and/or numerical methods, then one aims to reduce it by finding a moment closure relation expressing “higher-order moments” in terms of “lower-order moments”. In this brief review, we focus on highlighting how moment closure methods occur in different contexts. We also conjecture via a geometric explanation why it has been difficult to rigorously justify many moment closure approximations although they work very well in practice.

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Acknowledgments

I would like to thank the Austrian Academy of Science (ÖAW) for support via an APART Fellowship and the EU/REA for support via a Marie-Curie Integration Re-Integration Grant. Support by the Collaborative Research Center 910 of the German Science Foundation (DFG) to attend the “International Conference on Control of Self-Organizing Nonlinear Systems” in 2014 is also gratefully acknowledged. Furthermore, I would like to thank Thomas Christen, Thilo Gross, Thomas House and an anonymous referee for very helpful feedback on various preprint versions of this chapter.

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  1. Institute for Analysis and Scientific Computing, Vienna University of Technology, 1040, Vienna, Austria

    Christian Kuehn

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  1. Christian Kuehn
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Correspondence to Christian Kuehn .

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  1. Inst für Theoretische Physik, Technische Universität Berlin, Berlin, Berlin, Germany

    Eckehard Schöll

  2. Institut für Theoretische Physik, Technische Universität Berlin, Berlin, Germany

    Sabine H. L. Klapp

  3. Institut für Theoretische Physik, Technische Universität Berlin, Berlin, Germany

    Philipp Hövel

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Kuehn, C. (2016). Moment Closure—A Brief Review. In: Schöll, E., Klapp, S., Hövel, P. (eds) Control of Self-Organizing Nonlinear Systems. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-28028-8_13

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