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2013 | OriginalPaper | Buchkapitel

1. Weak Chaos, Infinite Ergodic Theory, and Anomalous Dynamics

verfasst von : Rainer Klages

Erschienen in: From Hamiltonian Chaos to Complex Systems

Verlag: Springer New York

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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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Nächstes Kapitel Directed Transport in a Stochastic Layer
Fußnoten
1
This condition could be eliminated by defining a metric on a circle [4].
 
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
This should not be confused with topological chaos as defined in [7].
 
4
In detail, the definition of a generalized diffusion coefficient is a bit more subtle [26].
 
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Metadaten
Titel
Weak Chaos, Infinite Ergodic Theory, and Anomalous Dynamics
verfasst von
Rainer Klages
Copyright-Jahr
2013
Verlag
Springer New York
Buch
From Hamiltonian Chaos to Complex Systems

Print ISBN: 978-1-4614-6961-2

Electronic ISBN: 978-1-4614-6962-9

Copyright-Jahr: 2013

DOI
https://doi.org/10.1007/978-1-4614-6962-9_1