## Critical Phenomena in Natural Sciences: Chaos, Fractals, Selforganization and Disorder: Concepts and ToolsConcepts, 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 LÃ©vy 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 |

477 | |

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applied approximation asymptotic avalanche average Bak-Sneppen model behavior central limit theorem characterized cluster condition configuration consider control parameter convergence correlation corresponds critical exponents critical point cumulative decay defined density derivative described deviation diffusion directed percolation discussed in Chap disorder dynamics earthquakes elements energy equation estimate exponent expression factor fault field finite fixed point fluctuations fractal dimension function Gaussian law given global infinite interactions Ising model larger lattice leading Levy law log-normal log-periodic mechanism method multifractal noise observed obtained order parameter Pareto distribution particle percolation model phase physical plates power law distribution probability problem properties quenched disorder random variables random walk regime renormalization group result rupture sample sandpile scale invariance self-affine self-organized criticality self-similar solution spins stable statistical stochastic stress stretched exponential structure tail temperature theory threshold tion transform transition typical variance velocity Weibull Weibull distribution zero

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Page iv - Institute of Geophysics and Planetary Physics and Department of Earth and Space Sciences University of California Los Angeles...