Critical Phenomena in Natural Sciences

Chaos, Fractals, Selforganization and Disorder: Concepts and Tools

By: Didier Sornette

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Paperback | 13 March 2006 | Edition Number 2

At a Glance

Paperback
528 Pages

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Revised

Dimensions(cm)
23.5 x 15.5 x 2.54

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$137.38

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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. This is the first textbook written by a well-known expert that provides a modern up-to-date introduction for workers outside statistical physics. 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. Although this book evolved out of a course for graduate students, it will be of great interest to researchers and engineers, as well as to post-docs in geophysics and meteorology.

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Non-Fiction Earth Sciences, Geography, Environment, Planning Earth Sciences Geology & The Lithosphere Mathematics Optimisation Game Theory Science Physics Materials & States of Matter Statistical Physics

Useful Notions of Probability Theory	p. 1
What Is Probability?	p. 1
First Intuitive Notions	p. 1
Objective Versus Subjective Probability	p. 2
Bayesian View Point	p. 6
Introduction	p. 6
Bayes' Theorem	p. 7
Bayesian Explanation for Change of Belief	p. 9
Bayesian Probability and the Dutch Book	p. 10
Probability Density Function	p. 12
Measures of Central Tendency	p. 13
Measure of Variations from Central Tendency	p. 14
Moments and Characteristic Function	p. 15
Cumulants	p. 16
Maximum of Random Variables and Extreme Value Theory	p. 18
Maximum Value Among N Random Variables	p. 19
Stable Extreme Value Distributions	p. 23
First Heuristic Derivation of the Stable Gumbel Distribution	p. 25
Second Heuristic Derivation of the Stable Gumbel Distribution	p. 26
Practical Use and Expression of the Coefficients of the Gumbel Distribution	p. 28
The Gnedenko-Pickands-Balkema-de Haan Theorem and the pdfofPeaks-Over-Threshold	p. 29
Sums of Random Variables, Random Walks and the Central Limit Theorem	p. 33
TheRandomWalkProblem	p. 33
AverageDrift	p. 34
Diffusion Law	p. 35
Brownian Motion as Solution of a Stochastic ODE	p. 35
FractalStructure	p. 37
Self-Affinity	p. 39
Master and Diffusion (Fokker-Planck) Equations	p. 41
Simple Formulation	p. 41
GeneralFokker-PlanckEquation	p. 43
ItoVersusStratonovich	p. 44
Extracting Model Equations from Experimental Data	p. 47
TheCentralLimit Theorem	p. 48
Convolution	p. 48
Statement	p. 50
Conditions	p. 50
CollectivePhenomenon	p. 51
Renormalization Group Derivation	p. 52
Recursion Relation and Perturbative Analysis	p. 55
Large Deviations	p. 59
CumulantExpansion	p. 59
LargeDeviationTheorem	p. 60
Quantification of the Deviation from the Central Limit Theorem	p. 61
Heuristic Derivation of the Large Deviation Theorem(3.9)	p. 61
Example: the Binomial Law	p. 63
Non-identically Distributed Random Variables	p. 64
Large Deviations with Constraints and the Boltzmann Formalism	p. 66
Frequencies Conditioned byLargeDeviations	p. 66
PartitionFunctionFormalism	p. 68
LargeDeviationsintheDiceGame	p. 70
Model Construction from Large Deviations	p. 73
Large Deviations in the Gutenberg-Richter Law and the Gamma Law	p. 76
Extreme Deviations	p. 78
The "Democratic" Result	p. 78
Application to the Multiplication of Random Variables: a Mechanism for Stretched Exponentials	p. 80
Application to Turbulence and to Fragmentation	p. 83
Large Deviations in the Sum of Variables with Power Law Distributions	p. 87
General Case with Exponent ¿ > 2	p. 87
Borderline Case with Exponent ¿ = 2	p. 90
Power Law Distributions	p. 93
Stable Laws: Gaussian and Lévy Laws	p. 93
Definition	p. 93
The Gaussian Probability Density Function	p. 93
TheLog-NormalLaw	p. 94
The Lévy Laws	p. 96
Truncated Lévy Laws101
PowerLaws	p. 104
How Does One Tame "Wild" Distributions?	p. 105
Multifractal Approach	p. 110
Anomalous Diffusion of Contaminants in the Earth's Crust and the Atmosphere	p. 112
General Intuitive Derivation	p. 113
More Detailed Model of Tracer Diffusion in the Crust	p. 113
Anomalous Diffusion in a Fluid	p. 115
Intuitive Calculation Tools for Power Law Distributions	p. 116
Fox Function, Mittag-Leffler Function and Lévy Distributions	p. 118
Fractals and Multifractals	p. 123
Fractals	p. 123
Introduction	p. 123
A First Canonical Example: the Triadic Cantor Set	p. 124
How Long Is the Coast of Britain?	p. 125
The Hausdorff Dimension	p. 127
ExamplesofNaturalFractals	p. 127
Multifractals	p. 141
Definition	p. 141
Correction Method for Finite Size Effects and Irregular Geometries	p. 143
Origin of Multifractality and Some Exact Results	p. 145
Generalization of Multifractality: Infinitely Divisible Cascades	p. 146
ScaleInvariance	p. 148
Definition	p. 148
Relation with Dimensional Analysis	p. 150
TheMultifractalRandomWalk	p. 153
A First Step: the Fractional Brownian Motion	p. 153
Definition and Properties of the Multifractal Random Walk	p. 154
Complex Fractal Dimensions and Discrete Scale Invariance	p. 156
Definition of Discrete Scale Invariance	p. 156
Log-Periodicity and Complex Exponents	p. 157
Importance and Usefulness of Discrete Scale Invariance	p. 159
Scenarii Leading to Discrete Scale Invariance	p. 160
Rank-Ordering Statistics and Heavy Tails	p. 163
Probability Distributions	p. 163
Definition of Rank Ordering Statistics	p. 164
NormalandLog-NormalDistributions	p. 166
TheExponentialDistribution	p. 167
PowerLawDistributions	p. 170
MaximumLikelihoodEstimation	p. 170
QuantilesofLargeEvents	p. 173
Power Laws with a Global Constraint: "Fractal Plate Tectonics"	p. 174
The Gamma Law	p. 179
The Stretched Exponential Distribution	p. 180
Maximum Likelihood and Other Estimators ofStretchedExponentialDistributions	p. 181
Introduction	p. 182
Two-Parameter Stretched Exponential Distribution	p. 185
Three-Parameter Weibull Distribution	p. 194
GeneralizedWeibullDistributions	p. 196
Statistical Mechanics: Probabilistic Point of View and the Concept of "Temperature"	p. 199
Statistical Derivation of the Concept of Temperature	p. 200
Statistical Thermodynamics	p. 202
Statistical Mechanics as Probability Theory with Constraints	p. 203
GeneralFormulation	p. 203
First Law of Thermodynamics	p. 206
ThermodynamicPotentials	p. 207
Does the Concept of Temperature Apply to Non-thermal Systems?	p. 208
Formulation of the Problem	p. 208
AGeneralModelingStrategy	p. 210
DiscriminatingTests	p. 211
Stationary Distribution with External Noise	p. 213
Effective Temperature Generated byChaoticDynamics	p. 214
Principle of Least Action for Out-Of-Equilibrium Systems	p. 218
Superstatistics	p. 219
Long-Range Correlations	p. 223
Criterion for the Relevance of Correlations	p. 223
StatisticalInterpretation	p. 226
An Application: Super-Diffusion in a Layered Fluid with Random Velocities	p. 228
AdvancedResultsonCorrelations	p. 229
CorrelationandDependence	p. 229
Statistical Time Reversal Symmetry	p. 231
Fractional Derivation and Long-Time Correlations	p. 236
Phase Transitions: Critical Phenomena and First-Order Transitions	p. 241
Definition	p. 241
SpinModelsat TheirCriticalPoints	p. 242
Definition of the Spin Model	p. 242
CriticalBehavior	p. 245
Long-Range Correlations of Spin Models at their Critical Points	p. 246
First-OrderVersusCriticalTransitions	p. 248
Definition and Basic Properties	p. 248
Dynamical Landau-Ginzburg Formulation	p. 250
The Scaling Hypothesis: Dynamical Length Scales for Ordering	p. 253
Transitions, Bifurcations and Precursors	p. 255
"Supercritical" Bifurcation	p. 256
Critical PrecursoryFluctuations	p. 258
"Subcritical" Bifurcation	p. 262
Scaling and Precursors Near Spinodals	p. 264
SelectionofanAttractorintheAbsence of a Potential	p. 265
The Renormalization Group	p. 267
General Framework	p. 267
An Explicit Example: Spins on a Hierarchical Network	p. 269
Renormalization Group Calculation	p. 269
Fixed Points, Stable Phases and Critical Points	p. 273
Singularities and Critical Exponents	p. 275
Complex Exponents and Log-Periodic Correctionsto Scaling	p. 276
"Weierstrass-Type Functions" from Discrete Renormalization Group Equations	p. 279
Criticality and the Renormalization Group on Euclidean Systems	p. 283
A Novel Application to the Construction of Functional Approximants	p. 287
GeneralConcepts	p. 287
Self-Similar Approximants	p. 288
Towards a Hierarchical View of the World	p. 291
The Percolation Model	p. 293
Percolationas a Model ofCracking	p. 293
Effective Medium Theory and Percolation	p. 296
Renormalization Group Approach to Percolation and Generalizations	p. 298
Cell-to-Site Transformation	p. 299
A Word of Caution on Real Space Renormalization Group Techniques	p. 301
The Percolation Model on the Hierarchical Diamond Lattice	p. 303
Directed Percolation	p. 304
Definitions	p. 304
UniversalityClass	p. 306
Field Theory: Stochastic Partial Differential Equation with Multiplicative Noise	p. 308
Self-Organized Formulation of Directed Percolation and Scaling Laws	p. 309
Rupture Models	p. 313
TheBranchingModel	p. 314
Mean Field Version or Branching on the Bethe Lattice	p. 314
A Branching-Aggregation Model Automatically Functioning at Its Critical Point	p. 316
Generalization of Critical Branching Models	p. 317
Fiber Bundle Models and the Effects of Stress Redistribution	p. 318
One-Dimensional System of Fibers Associated in Series	p. 318
Democratic Fiber Bundle Model (Daniels, 1945)	p. 320
Hierarchical Model	p. 323
The Simplest Hierarchical Model of Rupture	p. 323
Quasi-Static Hierarchical Fiber Rupture Model	p. 326
Hierarchical Fiber Rupture Model with Time-Dependence	p. 328
Quasi-Static Models in Euclidean Spaces	p. 330
A Dynamical Model of Rupture Without Elasto-Dynamics: the "Thermal Fuse Model"	p. 335
Time-to-Failure and Rupture Criticality	p. 339
Critical Time-to-Failure Analysis	p. 339
Time-to-Failure Behavior in the Dieterich Friction Law	p. 343
Mechanisms for Power Laws	p. 345
Temporal Copernican Principle and ¿ = 1 Universal Distribution of Residual Lifetimes	p. 346
Change of Variable	p. 348
Power Law Change of Variable Close to the Origin	p. 348
CombinationofExponentials	p. 354
Maximization of the Generalized Tsallis Entropy	p. 356
Superposition of Distributions	p. 359
Power Law Distribution ofWidths	p. 359
Sum of Stretched Exponentials (Chap. 3)	p. 362
Double Pareto Distribution by Superposition of Log-Normalpdf's	p. 362
Random Walks: Distribution of Return Times to the Origin	p. 363
Derivation	p. 364
Applications	p. 365
Sweeping of a Control Parameter Towards an Instability	p. 367
Growth with Preferential Attachment	p. 370
Multiplicative Noise with Constraints	p. 373
Definition of the Process	p. 373
The Kesten Multiplicative Stochastic Process	p. 374
Random Walk Analogy	p. 375
Exact Derivation, Generalization and Applications	p. 378
The "Coherent-Noise" Mechanism	p. 381
Avalanches in Hysteretic Loops and First-Order Transitions with Randomness	p. 386
"Highly Optimized Tolerant" (HOT) Systems	p. 389
Mechanism for the Power Law Distribution of Fire Sizes	p. 390
"Constrained Optimization with Limited Deviations" (COLD)	p. 393
HOT versus Percolation	p. 393
Self-Organized Criticality	p. 395
What Is Self-OrganizedCriticality?	p. 395
Introduction	p. 395
Definition	p. 397
SandpileModels	p. 398
Generalities	p. 398
TheAbelianSandpile	p. 398
Threshold Dynamics	p. 402
Generalization	p. 402
Illustration of Self-Organized Criticality Within the Earth's Crust	p. 404
Scenarios for Self-Organized Criticality	p. 406
Generalities	p. 406
Nonlinear Feedback of the "Order Parameter" onto the "Control Parameter"	p. 407
Generic Scale Invariance	p. 409
Mapping onto a Critical Point	p. 414
Mapping to Contact Processes	p. 422
Critical Desynchronization	p. 424
Extremal Dynamics	p. 427
Dynamical System Theory of Self-Organized Criti-cality	p. 435
Tests of Self-Organized Criticality in Complex Systems: the Example of the Earth'sCrust	p. 438
Introduction to the Physics of Random Systems	p. 441
Generalities	p. 441
The Random Energy Model	p. 445
Non-Self-Averaging Properties	p. 449
Definitions	p. 449
Fragmentation Models	p. 451
Randomness and Long-Range Laplacian Interactions	p. 457
Levy Distributions from Random Distributions of Sources with Long-Range Interactions	p. 457
Holtsmark's Gravitational Force Distribution	p. 457
Generalization to Other Fields (Electric, Elastic, Hydrodynamics)	p. 461
Long-Range Field Fluctuations Due to Irregular Arrays of Sources at Boundaries	p. 463
Problem and Main Results	p. 463
Calculation Methods	p. 464
Applications	p. 471
References	p. 477
Index	p. 525
Table of Contents provided by Publisher. All Rights Reserved.

Critical Phenomena in Natural Sciences

Chaos, Fractals, Selforganization and Disorder: Concepts and Tools

At a Glance

Paperback

Shipping

How to return your order

Defective items

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