Critical phenomena in natural sciences : chaos, fractals, selforganization and disorder : concepts and tools

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Critical phenomena in natural sciences : chaos, fractals, selforganization and disorder : concepts and tools / Didier Sornette

版:: 2nd ed
資料種別:: 図書
出版情報:: Berlin : Springer, c2006
形態:: xxii, 528 p. ; 24 cm
シリーズ名:: Springer series in synergetics <BA00004912>
Springer complexity <BA69338085>
著者名:: Sornette, Didier <DA14299887>
ISBN:: 9783540308829 [3540308822] (: pbk)
書誌ID:: BA76152291

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ISSN:

01727389

注記:

Includes bibliographical references (p. [477]-523) and index
"2nd printing of the hardcover edition with ISBN 3540407545"--T.p. verso

主題:

Critical phenomena (Physics)

言語:

英語

目次情報:

Useful Notions of Probability Theory / 1：

What Is Probability? / 1.1：

First Intuitive Notions / 1.1.1：

Objective Versus Subjective Probability / 1.1.2：

Bayesian View Point / 1.2：

Introduction / 1.2.1：

Bayes' Theorem / 1.2.2：

Bayesian Explanation for Change of Belief / 1.2.3：

Bayesian Probability and the Dutch Book / 1.2.4：

Probability Density Function / 1.3：

Measures of Central Tendency / 1.4：

Measure of Variations from Central Tendency / 1.5：

Moments and Characteristic Function / 1.6：

Cumulants / 1.7：

Maximum of Random Variables and Extreme Value Theory / 1.8：

Maximum Value Among N Random Variables / 1.8.1：

Stable Extreme Value Distributions / 1.8.2：

First Heuristic Derivation of the Stable Gumbel Distribution / 1.8.3：

Second Heuristic Derivation of the Stable Gumbel Distribution / 1.8.4：

Practical Use and Expression of the Coefficients of the Gumbel Distribution / 1.8.5：

The Gnedenko-Pickands-Balkema-de Haan Theorem and the pdfofPeaks-Over-Threshold / 1.8.6：

Sums of Random Variables, Random Walks and the Central Limit Theorem / 2：

TheRandomWalkProblem / 2.1：

AverageDrift / 2.1.1：

Diffusion Law / 2.1.2：

Brownian Motion as Solution of a Stochastic ODE / 2.1.3：

FractalStructure / 2.1.4：

Self-Affinity / 2.1.5：

Master and Diffusion (Fokker-Planck) Equations / 2.2：

Simple Formulation / 2.2.1：

GeneralFokker-PlanckEquation / 2.2.2：

ItoVersusStratonovich / 2.2.3：

Extracting Model Equations from Experimental Data / 2.2.4：

TheCentralLimit Theorem / 2.3：

Convolution / 2.3.1：

Statement / 2.3.2：

Conditions / 2.3.3：

CollectivePhenomenon / 2.3.4：

Renormalization Group Derivation / 2.3.5：

Recursion Relation and Perturbative Analysis / 2.3.6：

Large Deviations / 3：

CumulantExpansion / 3.1：

LargeDeviationTheorem / 3.2：

Quantification of the Deviation from the Central Limit Theorem / 3.2.1：

Heuristic Derivation of the Large Deviation Theorem(3.9) / 3.2.2：

Example: the Binomial Law / 3.2.3：

Non-identically Distributed Random Variables / 3.2.4：

Large Deviations with Constraints and the Boltzmann Formalism / 3.3：

Frequencies Conditioned byLargeDeviations / 3.3.1：

PartitionFunctionFormalism / 3.3.2：

LargeDeviationsintheDiceGame / 3.3.3：

Model Construction from Large Deviations / 3.3.4：

Large Deviations in the Gutenberg-Richter Law and the Gamma Law / 3.3.5：

Extreme Deviations / 3.4：

The "Democratic" Result / 3.4.1：

Application to the Multiplication of Random Variables: a Mechanism for Stretched Exponentials / 3.4.2：

Application to Turbulence and to Fragmentation / 3.4.3：

Large Deviations in the Sum of Variables with Power Law Distributions / 3.5：

General Case with Exponent ? > 2 / 3.5.1：

Borderline Case with Exponent ? = 2 / 3.5.2：

Power Law Distributions / 4：

Stable Laws: Gaussian and LÃ©vy Laws / 4.1：

Definition / 4.1.1：

The Gaussian Probability Density Function / 4.1.2：

TheLog-NormalLaw / 4.1.3：

The LÃ©vy Laws / 4.1.4：

Truncated LÃ©vy Laws101 / 4.1.5：

PowerLaws / 4.2：

How Does One Tame "Wild" Distributions? / 4.2.1：

Multifractal Approach / 4.2.2：

Anomalous Diffusion of Contaminants in the Earth's Crust and the Atmosphere / 4.3：

General Intuitive Derivation / 4.3.1：

More Detailed Model of Tracer Diffusion in the Crust / 4.3.2：

Anomalous Diffusion in a Fluid / 4.3.3：

Intuitive Calculation Tools for Power Law Distributions / 4.4：

Fox Function, Mittag-Leffler Function and LÃ©vy Distributions / 4.5：

Fractals and Multifractals / 5：

Fractals / 5.1：

A First Canonical Example: the Triadic Cantor Set / 5.1.1：

How Long Is the Coast of Britain? / 5.1.3：

The Hausdorff Dimension / 5.1.4：

ExamplesofNaturalFractals / 5.1.5：

Multifractals / 5.2：

Correction Method for Finite Size Effects and Irregular Geometries / 5.2.1：

Origin of Multifractality and Some Exact Results / 5.2.3：

Generalization of Multifractality: Infinitely Divisible Cascades / 5.2.4：

ScaleInvariance / 5.3：

Relation with Dimensional Analysis / 5.3.1：

TheMultifractalRandomWalk / 5.4：

A First Step: the Fractional Brownian Motion / 5.4.1：

Definition and Properties of the Multifractal Random Walk / 5.4.2：

Complex Fractal Dimensions and Discrete Scale Invariance / 5.5：

Definition of Discrete Scale Invariance / 5.5.1：

Log-Periodicity and Complex Exponents / 5.5.2：

Importance and Usefulness of Discrete Scale Invariance / 5.5.3：

Scenarii Leading to Discrete Scale Invariance / 5.5.4：

Rank-Ordering Statistics and Heavy Tails / 6：

Probability Distributions / 6.1：

Definition of Rank Ordering Statistics / 6.2：

NormalandLog-NormalDistributions / 6.3：

TheExponentialDistribution / 6.4：

PowerLawDistributions / 6.5：

MaximumLikelihoodEstimation / 6.5.1：

QuantilesofLargeEvents / 6.5.2：

Power Laws with a Global Constraint: "Fractal Plate Tectonics" / 6.5.3：

The Gamma Law / 6.6：

The Stretched Exponential Distribution / 6.7：

Maximum Likelihood and Other Estimators ofStretchedExponentialDistributions / 6.8：

Two-Parameter Stretched Exponential Distribution / 6.8.1：

Three-Parameter Weibull Distribution / 6.8.3：

GeneralizedWeibullDistributions / 6.8.4：

Statistical Mechanics: Probabilistic Point of View and the Concept of "Temperature" / 7：

Statistical Derivation of the Concept of Temperature / 7.1：

Statistical Thermodynamics / 7.2：

Statistical Mechanics as Probability Theory with Constraints / 7.3：

GeneralFormulation / 7.3.1：

First Law of Thermodynamics / 7.3.2：

ThermodynamicPotentials / 7.3.3：

Does the Concept of Temperature Apply to Non-thermal Systems? / 7.4：

Formulation of the Problem / 7.4.1：

AGeneralModelingStrategy / 7.4.2：

DiscriminatingTests / 7.4.3：

Stationary Distribution with External Noise / 7.4.4：

Effective Temperature Generated byChaoticDynamics / 7.4.5：

Principle of Least Action for Out-Of-Equilibrium Systems / 7.4.6：

Superstatistics / 7.4.7：

Long-Range Correlations / 8：

Criterion for the Relevance of Correlations / 8.1：

StatisticalInterpretation / 8.2：

An Application: Super-Diffusion in a Layered Fluid with Random Velocities / 8.3：

AdvancedResultsonCorrelations / 8.4：

CorrelationandDependence / 8.4.1：

Statistical Time Reversal Symmetry / 8.4.2：

Fractional Derivation and Long-Time Correlations / 8.4.3：

Phase Transitions: Critical Phenomena and First-Order Transitions / 9：

SpinModelsat TheirCriticalPoints / 9.1：

Definition of the Spin Model / 9.2.1：

CriticalBehavior / 9.2.2：

Long-Range Correlations of Spin Models at their Critical Points / 9.2.3：

First-OrderVersusCriticalTransitions / 9.3：

Definition and Basic Properties / 9.3.1：

Dynamical Landau-Ginzburg Formulation / 9.3.2：

The Scaling Hypothesis: Dynamical Length Scales for Ordering / 9.3.3：

Transitions, Bifurcations and Precursors / 10：

"Supercritical" Bifurcation / 10.1：

Critical PrecursoryFluctuations / 10.2：

"Subcritical" Bifurcation / 10.3：

Scaling and Precursors Near Spinodals / 10.4：

SelectionofanAttractorintheAbsence of a Potential / 10.5：

The Renormalization Group / 11：

General Framework / 11.1：

An Explicit Example: Spins on a Hierarchical Network / 11.2：

Renormalization Group Calculation / 11.2.1：

Fixed Points, Stable Phases and Critical Points / 11.2.2：

Singularities and Critical Exponents / 11.2.3：

Complex Exponents and Log-Periodic Correctionsto Scaling / 11.2.4：

"Weierstrass-Type Functions" from Discrete Renormalization Group Equations / 11.2.5：

Criticality and the Renormalization Group on Euclidean Systems / 11.3：

A Novel Application to the Construction of Functional Approximants / 11.4：

GeneralConcepts / 11.4.1：

Self-Similar Approximants / 11.4.2：

Towards a Hierarchical View of the World / 11.5：

The Percolation Model / 12：

Percolationas a Model ofCracking / 12.1：

Effective Medium Theory and Percolation / 12.2：

Renormalization Group Approach to Percolation and Generalizations / 12.3：

Cell-to-Site Transformation / 12.3.1：

A Word of Caution on Real Space Renormalization Group Techniques / 12.3.2：

The Percolation Model on the Hierarchical Diamond Lattice / 12.3.3：

Directed Percolation / 12.4：

Definitions / 12.4.1：

UniversalityClass / 12.4.2：

Field Theory: Stochastic Partial Differential Equation with Multiplicative Noise / 12.4.3：

Self-Organized Formulation of Directed Percolation and Scaling Laws / 12.4.4：

Rupture Models / 13：

TheBranchingModel / 13.1：

Mean Field Version or Branching on the Bethe Lattice / 13.1.1：

A Branching-Aggregation Model Automatically Functioning at Its Critical Point / 13.1.2：

Generalization of Critical Branching Models / 13.1.3：

Fiber Bundle Models and the Effects of Stress Redistribution / 13.2：

One-Dimensional System of Fibers Associated in Series / 13.2.1：

Democratic Fiber Bundle Model (Daniels, 1945) / 13.2.2：

Hierarchical Model / 13.3：

The Simplest Hierarchical Model of Rupture / 13.3.1：

Quasi-Static Hierarchical Fiber Rupture Model / 13.3.2：

Hierarchical Fiber Rupture Model with Time-Dependence / 13.3.3：

Quasi-Static Models in Euclidean Spaces / 13.4：

A Dynamical Model of Rupture Without Elasto-Dynamics: the "Thermal Fuse Model" / 13.5：

Time-to-Failure and Rupture Criticality / 13.6：

Critical Time-to-Failure Analysis / 13.6.1：

Time-to-Failure Behavior in the Dieterich Friction Law / 13.6.2：

Mechanisms for Power Laws / 14：

Temporal Copernican Principle and ? = 1 Universal Distribution of Residual Lifetimes / 14.1：

Change of Variable / 14.2：

Power Law Change of Variable Close to the Origin / 14.2.1：

CombinationofExponentials / 14.2.2：

Maximization of the Generalized Tsallis Entropy / 14.3：

Superposition of Distributions / 14.4：

Power Law Distribution ofWidths / 14.4.1：

Sum of Stretched Exponentials (Chap. 3) / 14.4.2：

Double Pareto Distribution by Superposition of Log-Normalpdf's / 14.4.3：

Random Walks: Distribution of Return Times to the Origin / 14.5：

Derivation / 14.5.1：

Applications / 14.5.2：

Sweeping of a Control Parameter Towards an Instability / 14.6：

Growth with Preferential Attachment / 14.7：

Multiplicative Noise with Constraints / 14.8：

Definition of the Process / 14.8.1：

The Kesten Multiplicative Stochastic Process / 14.8.2：

Random Walk Analogy / 14.8.3：

Exact Derivation, Generalization and Applications / 14.8.4：

The "Coherent-Noise" Mechanism / 14.9：

Avalanches in Hysteretic Loops and First-Order Transitions with Randomness / 14.10：

"Highly Optimized Tolerant" (HOT) Systems / 14.11：

Mechanism for the Power Law Distribution of Fire Sizes / 14.11.1：

"Constrained Optimization with Limited Deviations" (COLD) / 14.11.2：

HOT versus Percolation / 14.11.3：

Self-Organized Criticality / 15：

What Is Self-OrganizedCriticality? / 15.1：

SandpileModels / 15.1.1：

Generalities / 15.2.1：

TheAbelianSandpile / 15.2.2：

Threshold Dynamics / 15.3：

Generalization / 15.3.1：

Illustration of Self-Organized Criticality Within the Earth's Crust / 15.3.2：

Scenarios for Self-Organized Criticality / 15.4：

Nonlinear Feedback of the "Order Parameter" onto the "Control Parameter" / 15.4.1：

Generic Scale Invariance / 15.4.3：

Mapping onto a Critical Point / 15.4.4：

Mapping to Contact Processes / 15.4.5：

Critical Desynchronization / 15.4.6：

Extremal Dynamics / 15.4.7：

Dynamical System Theory of Self-Organized Criti-cality / 15.4.8：

Tests of Self-Organized Criticality in Complex Systems: the Example of the Earth'sCrust / 15.5：

Introduction to the Physics of Random Systems / 16：

The Random Energy Model / 16.1：

Non-Self-Averaging Properties / 16.3：

Fragmentation Models / 16.3.1：

Randomness and Long-Range Laplacian Interactions / 17：

Levy Distributions from Random Distributions of Sources with Long-Range Interactions / 17.1：

Holtsmark's Gravitational Force Distribution / 17.1.1：

Generalization to Other Fields (Electric, Elastic, Hydrodynamics) / 17.1.2：

Long-Range Field Fluctuations Due to Irregular Arrays of Sources at Boundaries / 17.2：

Problem and Main Results / 17.2.1：

Calculation Methods / 17.2.2：

References / 17.2.3：

Index

Useful Notions of Probability Theory / 1：

What Is Probability? / 1.1：

First Intuitive Notions / 1.1.1：

Objective Versus Subjective Probability / 1.1.2：

Bayesian View Point / 1.2：

Introduction / 1.2.1：

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