Critical Phenomena in Natural Sciences: Chaos, Fractals, Selforganization and Disorder: Concepts and Tools

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Springer Science & Business Media, 2004 - Science - 528 pages
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Concepts, methods and techniques of statistical physics in the study of correlated, as well as uncorrelated, phenomena are being applied ever increasingly in the natural sciences, biology and economics in an attempt to understand and model the large variability and risks of phenomena.The emphasis of the book is on a clear understanding of concepts and methods, while it also provides the tools that can be of immediate use in applications. The second edition is a significant expansion over the first one which meanwhile has become a standard reference in complex system research and teaching: Probability concepts are presented more in-depth and the sections on Lévy laws and the mechanisms for power laws have been greatly enlarged. Much material has been added to the chapter on renormalisation group ideas. Further improvements can be found in the applications to earthquake or rupture models.
  

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Contents

Useful Notions of Probability Theory
1
Large Deviations 59
58
Power Law Distributions
93
Fractals and Multifractals 123
122
and Discrete Scale Invariance
156
RankOrdering Statistics and Heavy Tails 163
162
Probabilistic Point of View
198
LongRange Correlations
223
The Renormalization Group
267
The Percolation Model
293
Rupture Models
313
Mechanisms for Power Laws
345
SelfOrganized Criticality
395
Introduction to the Physics of Random Systems
441
Randomness and LongRange Laplacian Interactions
457
References
477

Critical Phenomena
241
Transitions Bifurcations and Precursors
255

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Useful Notions of Probability Theory.- Sums of Random Variables, Random Walks and the Central Limit Theorem.- Large Deviations.- Power Law Distributions.- Fractals and Multifractals.- Rank-Ordering Statistics and Heavy Tails.- Statistical Mechanics: Probabilistic Point of View and the Concept of "Temperature."- Long-Range Correlations.- Phase Transitions: Critical Phenomena and First-Order Transitions.- Transitions, Bifurcations and Precursors.- The Renormalization Group.- The Percolation Model.- Rupture Models.- Mechanisms for Power Laws.- Self-Organized Criticality.- Introduction to the Physics of Random Systems.- Randomness and Long-Range Laplacian Interactions.

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