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Deterministic transport: from normal to anomalous diffusion

By Nickolay Korabel

Abstract

The way in which macroscopic transport results from microscopic dynamics is one of the important questions in statistical physics. Dynamical systems theory play a key role in a resent advance in this direction. Offering relatively simple models which are easy to study, dynamical systems theory became a standard branch of modern nonequilibrium statistical physics. In the present work the deterministic diffusion generated by simple dynamical systems is considered. The deterministic nature of these systems is more clearly expressed through the dependencies of the transport quantities as functions of systems parameters. For fully hyperbolic dynamical systems these dependencies were found to be highly irregular and, in fact, fractal. The main focus in this work is on nonhyperbolic and on intermittent dynamical systems. First, the climbing sine map is considered which is a nonhyperbolic system with many physical applications. Then we treat anomalous dynamics generated by a paradigmatic subdiffusive map. In both cases these systems display deterministic transport which, under variation of control parameters, is fractal. For both systems we give an explanation of the observed phenomena. The third part of the thesis is devoted to the relation between chaotic and transport properties of dynamical systems. This question lies at the heart of dynamical systems theory. For closed hyperbolic dynamical systems the Pesin theorem links the sum of positive Lyapunov exponents to the Kolmogorov-Sinai entropy. For open hyperbolic systems the escape rate formula is valid. In this work we have formulated generalizations of these formulas for a class of intermittent dynamical systems where the chaotic properties are weaker

Topics: info:eu-repo/classification/ddc/530, ddc:530, Anomale Diffusion; Diffusion; Dynamisches System; Fraktal; Statistische Physik, Intermittanz, anomalen Diffusion, deterministische Diffusion, fraktale Structur, nichthyperbolische Systeme, anomalous diffusion, deterministic diffusion, fractal structures, intermittency, nonhyperbolic systems
Publisher: Technische Universit├Ąt Dresden
Year: 2004
OAI identifier: oai:qucosa:de:qucosa:24417
Provided by: Qucosa
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