From Deterministic Chaos to Anomalous Diffusion

Klages, Rainer

doi:10.1002/9783527630967.ch6

Cited by 9 publications

(6 citation statements)

References 64 publications

(194 reference statements)

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“…In the case of free diffusion, we can easily compute the displacement distribution function analytically as in Equation 11. [25,26] When plotting the measured displacement probability in the free diffusion case, it is indistinguishable from the analytically derived function.…”

Section: Freely Diffusing Tracer Particles With Nearest-neighbor Intementioning

confidence: 82%

Simulational study of anomalous tracer diffusion in hydrogels

2011

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In this article, we analyze different factors that affect the diffusion behavior of small tracer particles (as they are used e.g.in fluorescence correlation spectroscopy (FCS)) in the polymer network of a hydrogel and perform simulations of various simplified models. We observe, that under certain circumstances the attraction of a tracer particle to the polymer network strands might cause subdiffusive behavior on intermediate time scales. In theory, this behavior could be employed to examine the network structure and swelling behavior of weakly crosslinked hydrogels with the help of FCS.Comment: 11 pages, 11 figure

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Section: Freely Diffusing Tracer Particles With Nearest-neighbor Intementioning

confidence: 82%

Simulational study of anomalous tracer diffusion in hydrogels

2011

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“…Ref. [Kla10] provides a more tutorial exposition of most of these ideas. By switching to the Pomeau-Manneville map we find that generalizations of these concepts are needed in order to describe the model's weakly chaotic dynamics.…”

Section: Chaos and Anomalous Dynamicsmentioning

confidence: 99%

“…The subsequent second section is to some extent based on the review Ref. [Kla10] by combining it with ideas from Refs. [Kla07,How09].…”

Section: Introductionmentioning

confidence: 99%

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Weak Chaos, Infinite Ergodic Theory, and Anomalous Dynamics

Klages

2013

Nonlinear Systems and Complexity

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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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“…For the unbiased Markovian coin-toss case, equation (12) would simply be the well-known Bernoulli Map [15], i.e. f (x) = x (so x t+1 = 2x t ) and c ∈ R + .…”

Section: Deriving An Underlying Mapmentioning

confidence: 99%

Self-similarity and non-Markovian behavior in traded stock volumes

2015

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Abstract. The volume traded daily for 17 stocks is followed over a period of about half a century. We look at the volume of stocks traded in a certain time interval (day, week, month) and analyze how long that traded volume keeps monotonically increasing or decreasing. On all three times scales we find that the sequence of traded volumes behaves neither like a sequence of independent and identically distributed variables, nor like a Markov sequence. A compressed exponential survival function with the same parameters at all timescales is firmly established. A day with an increase (decrease) of traded volume is most likely followed by a day with a decrease (increase) of traded volume. We show how the apparent self-similarity results because the small day-to-day anticorrelation carries over when larger time intervals are considered. The observed small anticorrelation can be explained as a consequence of market forces and trader reactions.

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From Deterministic Chaos to Anomalous Diffusion

Cited by 9 publications

References 64 publications

Simulational study of anomalous tracer diffusion in hydrogels

Simulational study of anomalous tracer diffusion in hydrogels

Weak Chaos, Infinite Ergodic Theory, and Anomalous Dynamics

Self-similarity and non-Markovian behavior in traded stock volumes

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