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Weak Chaos, Infinite Ergodic Theory, and Anomalous Dynamics

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From Hamiltonian Chaos to Complex Systems

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 5))

Abstract

This book chapter introduces to the concept of weak chaos, aspects of its ergodic theory description, and properties of the anomalous dynamics associated with it. In the first half of the chapter we study simple one-dimensional deterministic maps, in the second half basic stochastic models, and eventually an experiment. We start by reminding the reader of fundamental chaos quantities and their relation to each other, exemplified by the paradigmatic Bernoulli shift. Using the intermittent Pomeau–Manneville map the problem of weak chaos and infinite ergodic theory is outlined, defining a very recent mathematical field of research. Considering a spatially extended version of the Pomeau–Manneville map leads us to the phenomenon of anomalous diffusion. This problem will be discussed by applying stochastic continuous time random walk theory and by deriving a fractional diffusion equation. Another important topic within modern nonequilibrium statistical physics are fluctuation relations, which we investigate for anomalous dynamics. The chapter concludes by showing the importance of anomalous dynamics for understanding experimental results on biological cell migration.

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Notes

  1. 1.

    This condition could be eliminated by defining a metric on a circle [4].

  2. 2.

    We remark that in [59–61] one finds several slightly different definitions of pseudochaos. Here we refer to the first one stated in [59].

  3. 3.

    This should not be confused with topological chaos as defined in [7].

  4. 4.

    In detail, the definition of a generalized diffusion coefficient is a bit more subtle [26].

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Acknowledgment

Each of the four sections reflects the collaboration with colleagues, without whom the work presented here would not have been possible. The second section benefitted very much from discussions with R. Zweimüler, whom the author thanks very much for a lot of mathematical insight into aspects of infinite ergodic theory. Particularly, the author is indebted to his former postdoc P. Howard, who did brilliant work on calculating generalized chaos quantities for the Pomeau–Manneville map. Regarding the third section, credit goes to his former Ph.D student N.Korabel for joint work that formed part of his Ph.D thesis. A.V. Chechkin significantly contributed to the same section as well as performed major research on the topic covered by the fourth one. The author is deeply indebted to him for his long-term collaboration on anomalous stochastic processes. P. Dieterich was the driving force behind the project reviewed in the fifth section. The author thanks him for much insight into the biophysical aspects of biological cell migration. Finally, he wishes to thank the editors of this book for their patience with this book chapter.

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  1. School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London, E1 4NS, UK

    Rainer Klages

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  1. , Centre de Physique Théorique, Aix-Marseille Université, Luminy Case 907, Marseille Cedex 09, 13288, France

    Xavier Leoncini

  2. Scientifique, Institut de Recherche sur le Phénomènes, Centre National de la Recherche, rue F. Joliot-Curie 49, Marseille Cedex 13, 13384, France

    Marc Leonetti

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Klages, R. (2013). Weak Chaos, Infinite Ergodic Theory, and Anomalous Dynamics. In: Leoncini, X., Leonetti, M. (eds) From Hamiltonian Chaos to Complex Systems. Nonlinear Systems and Complexity, vol 5. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6962-9_1

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