Critical Phenomena in Natural Sciences

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ISBN: 9783540407546 | 3540407545
Cover: Hardcover
Copyright: 2/1/2004

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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 L??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.

1. Useful Notions of Probability Theory

(32)

1.1 What Is Probability?

(5)

1.1.1 First Intuitive Notions

(1)

1.1.2 Objective Versus Subjective Probability

(4)

1.2 Bayesian View Point

(6)

1.2.1 Introduction

(1)

1.2.2 Bayes' Theorem

(2)

1.2.3 Bayesian Explanation for Change of Belief

(1)

1.2.4 Bayesian Probability and the Dutch Book

(2)

1.3 Probability Density Function

(1)

1.4 Measures of Central Tendency

(1)

1.5 Measure of Variations from Central Tendency

(1)

1.6 Moments and Characteristic Function

(1)

1.7 Cumulants

(2)

1.8 Maximum of Random Variables and Extreme Value Theory

(15)

1.8.1 Maximum Value Among N Random Variables

(4)

1.8.2 Stable Extreme Value Distributions

(2)

1.8.3 First Heuristic Derivation of the Stable Gumbel Distribution

(1)

1.8.4 Second Heuristic Derivation of the Stable Gumbel Distribution

(2)

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

(1)

1.8.6 The Gnedenko-Pickands-Balkema-de Haan Theorem and the pdf of Peaks-Over-Threshold

(4)

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

(26)

2.1 The Random Walk Problem

(8)

2.1.1 Average Drift

(1)

2.1.2 Diffusion Law

(1)

2.1.3 Brownian Motion as Solution of a Stochastic ODE

(2)

2.1.4 Fractal Structure

(2)

2.1.5 Self-Affinity

(2)

2.2 Master and Diffusion (Fokker-Planck) Equations

(7)

2.2.1 Simple Formulation

(2)

2.2.2 General Fokker-Planck Equation

(1)

2.2.3 Ito Versus Stratonovich

(3)

2.2.4 Extracting Model Equations from Experimental Data

(1)

2.3 The Central Limit Theorem

(11)

2.3.1 Convolution

(2)

2.3.2 Statement

(1)

2.3.3 Conditions

(1)

2.3.4 Collective Phenomenon

(1)

2.3.5 Renormalization Group Derivation

(3)

2.3.6 Recursion Relation and Perturbative Analysis

(4)

3. Large Deviations

(34)

3.1 Cumulant Expansion

(1)

3.2 Large Deviation Theorem

(6)

3.2.1 Quantification of the Deviation from the Central Limit Theorem

(1)

3.2.2 Heuristic Derivation of the Large Deviation Theorem (3.9)

(2)

3.2.3 Example: the Binomial Law

(1)

3.2.4 Non-identically Distributed Random Variables

(2)

3.3 Large Deviations with Constraints and the Boltzmann Formalism

(12)

3.3.1 Frequencies Conditioned by Large Deviations

(2)

3.3.2 Partition Function Formalism

(2)

3.3.3 Large Deviations in the Dice Game

(3)

3.3.4 Model Construction from Large Deviations

(3)

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

(2)

3.4 Extreme Deviations

(9)

3.4.1 The "Democratic" Result

(2)

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

(3)

3.4.3 Application to Turbulence and to Fragmentation

(4)

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

(6)

3.5.1 General Case with Exponent µ>2

(3)

3.5.2 Borderline Case with Exponent µ=2

(3)

4. Power Law Distributions

(30)

4.1 Stable Laws: Gaussian and Lévy Laws

(11)

4.1.1 Definition

(1)

4.1.2 The Gaussian Probability Density Function

(1)

4.1.3 The Log-Normal Law

(2)

4.1.4 The Levy Laws

(5)

4.1.5 Truncated Levy Laws

101

(3)

4.2 Power Laws

104

(8)

4.2.1 How Does One Tame "Wild" Distributions?

105

(5)

4.2.2 Multifractal Approach

110

(2)

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

112

(4)

4.3.1 General Intuitive Derivation

113

(1)

4.3.2 More Detailed Model of Tracer Diffusion in the Crust

113

(2)

4.3.3 Anomalous Diffusion in a Fluid

115

(1)

4.4 Intuitive Calculation Tools for Power Law Distributions

116

(2)

4.5 Fox Function, Mittag-Leffler Function and Lévy Distributions

118

(5)

5. Fractals and Multifractals

123

(40)

5.1 Fractals

123

(18)

5.1.1 Introduction

123

(1)

5.1.2 A First Canonical Example: the Triadic Cantor Set

124

(1)

5.1.3 How Long Is the Coast of Britain?

125

(2)

5.1.4 The Hausdorff Dimension

127

(1)

5.1.5 Examples of Natural Fractals

127

(14)

5.2 Multifractals

141

(7)

5.2.1 Definition

141

(2)

5.2.2 Correction Method for Finite Size Effects and Irregular Geometries

143

(2)

5.2.3 Origin of Multifractality and Some Exact Results

145

(1)

5.2.4 Generalization of Multifractality: Infinitely Divisible Cascades

146

(2)

5.3 Scale Invariance

148

(5)

5.3.1 Definition

148

(2)

5.3.2 Relation with Dimensional Analysis

150

(3)

5.4 The Multifractal Random Walk

153

(3)

5.4.1 A First Step: the Fractional Brownian Motion

153

(1)

5.4.2 Definition and Properties of the Multifractal Random Walk

154

(2)

5.5 Complex Fractal Dimensions and Discrete Scale Invariance

156

(7)

5.5.1 Definition of Discrete Scale Invariance

156

(1)

5.5.2 Log-Periodicity and Complex Exponents

157

(2)

5.5.3 Importance and Usefulness of Discrete Scale Invariance

159

(1)

5.5.4 Scenarii Leading to Discrete Scale Invariance

160

(3)

6. Rank-Ordering Statistics and Heavy Tails

163

(36)

6.1 Probability Distributions

163

(1)

6.2 Definition of Rank Ordering Statistics

164

(2)

6.3 Normal and Log-Normal Distributions

166

(1)

6.4 The Exponential Distribution

167

(3)

6.5 Power Law Distributions

170

(9)

6.5.1 Maximum Likelihood Estimation

170

(3)

6.5.2 Quantiles of Large Events

173

(1)

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

174

(5)

6.6 The Gamma Law

179

(1)

6.7 The Stretched Exponential Distribution

180

(1)

6.8 Maximum Likelihood and Other Estimators of Stretched Exponential Distributions

181

(18)

6.8.1 Introduction

182

(3)

6.8.2 Two-Parameter Stretched Exponential Distribution

185

(9)

6.8.3 Three-Parameter Weibull Distribution

194

(2)

6.8.4 Generalized Weibull Distributions

196

(3)

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

199

(24)

7.1 Statistical Derivation of the Concept of Temperature

200

(2)

7.2 Statistical Thermodynamics

202

(1)

7.3 Statistical Mechanics as Probability Theory with Constraints

203

(5)

7.3.1 General Formulation

203

(3)

7.3.2 First Law of Thermodynamics

206

(1)

7.3.3 Thermodynamic Potentials

207

(1)

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

208

(15)

7.4.1 Formulation of the Problem

208

(2)

7.4.2 A General Modeling Strategy

210

(1)

7.4.3 Discriminating Tests

211

(2)

7.4.4 Stationary Distribution with External Noise

213

(1)

7.4.5 Effective Temperature Generated by Chaotic Dynamics

214

(4)

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

218

(1)

7.4.7 Superstatistics

219

(4)

8. Long-Range Correlations

223

(18)

8.1 Criterion for the Relevance of Correlations

223

(3)

8.2 Statistical Interpretation

226

(2)

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

228

(1)

8.4 Advanced Results on Correlations

229

(12)

8.4.1 Correlation and Dependence

229

(2)

8.4.2 Statistical Time Reversal Symmetry

231

(5)

8.4.3 Fractional Derivation and Long-Time Correlations

236

(5)

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

241

(14)

9.1 Definition

241

(1)

9.2 Spin Models at Their Critical Points

242

(6)

9.2.1 Definition of the Spin Model

242

(3)

9.2.2 Critical Behavior

245

(1)

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

246

(2)

9.3 First-Order Versus Critical Transitions

248

(8)

9.3.1 Definition and Basic Properties

248

(2)

9.3.2 Dynamical Landau-Ginzburg Formulation

250

(3)

9.3.3 The Scaling Hypothesis: Dynamical Length Scales for Ordering

253

(2)

10. Transitions, Bifurcations and Precursors

255

(12)

10.1 "Supercritical" Bifurcation

256

(2)

10.2 Critical Precursory Fluctuations

258

(4)

10.3 "Subcritical" Bifurcation

262

(2)

10.4 Scaling and Precursors Near Spinodals

264

(1)

10.5 Selection of an Attractor in the Absence of a Potential

265

(2)

11. The Renormalization Group

267

(26)

11.1 General Framework

267

(2)

11.2 An Explicit Example: Spins on a Hierarchical Network

269

(14)

11.2.1 Renormalization Group Calculation

269

(4)

11.2.2 Fixed Points, Stable Phases and Critical Points

273

(2)

11.2.3 Singularities and Critical Exponents

275

(1)

11.2.4 Complex Exponents and Log-Periodic Corrections to Scaling

276

(3)

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

279

(4)

11.3 Criticality and the Renormalization Group on Euclidean Systems

283

(4)

11.4 A Novel Application to the Construction of Functional Approximants

287

(4)

11.4.1 General Concepts

287

(1)

11.4.2 Self-Similar Approximants

288

(3)

11.5 Towards a Hierarchical View of the World

291

(2)

12. The Percolation Model

293

(20)

12.1 Percolation as a Model of Cracking

293

(3)

12.2 Effective Medium Theory and Percolation

296

(2)

12.3 Renormalization Group Approach to Percolation and Generalizations

298

(6)

12.3.1 Cell-to-Site Transformation

299

(2)

12.3.2 A Word of Caution on Real Space Renormalization Group Techniques

301

(2)

12.3.3 The Percolation Model on the Hierarchical Diamond Lattice

303

(1)

12.4 Directed Percolation

304

(10)

12.4.1 Definitions

304

(2)

12.4.2 Universality Class

306

(2)

12.4.3 Field Theory: Stochastic Partial Differential Equation with Multiplicative Noise

308

(1)

12.4.4 Self-Organized Formulation of Directed Percolation and Scaling Laws

309

(4)

13. Rupture Models

313

(32)

13.1 The Branching Model

314

(4)

13.1.1 Mean Field Version or Branching on the Bethe Lattice

314

(2)

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

316

(1)

13.1.3 Generalization of Critical Branching Models

317

(1)

13.2 Fiber Bundle Models and the Effects of Stress Redistribution

318

(5)

13.2.1 One-Dimensional System of Fibers Associated in Series

318

(2)

13.2.2 Democratic Fiber Bundle Model (Daniels, 1945)

320

(3)

13.3 Hierarchical Model

323

(7)

13.3.1 The Simplest Hierarchical Model of Rupture

323

(3)

13.3.2 Quasi-Static Hierarchical Fiber Rupture Model

326

(2)

13.3.3 Hierarchical Fiber Rupture Model with Time-Dependence

328

(2)

13.4 Quasi-Static Models in Euclidean Spaces

330

(5)

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

335

(4)

13.6 Time-to-Failure and Rupture Criticality

339

(47)

13.6.1 Critical Time-to-Failure Analysis

339

(4)

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

343

(2)

14. Mechanisms for Power Laws

345

(50)

14.1 Temporal Copernican Principle and µ=1 Universal Distribution of Residual Lifetimes

346

(2)

14.2 Change of Variable

348

(8)

14.2.1 Power Law Change of Variable Close to the Origin

348

(6)

14.2.2 Combination of Exponentials

354

(2)

14.3 Maximization of the Generalized Tsallis Entropy

356

(3)

14.4 Superposition of Distributions

359

(4)

14.4.1 Power Law Distribution of Widths

359

(3)

14.4.2 Sum of Stretched Exponentials (Chap. 3)

362

(1)

14.4.3 Double Pareto Distribution by Superposition of Log-Normal pdf's

362

(1)

14.5 Random Walks: Distribution of Return Times to the Origin

363

(4)

14.5.1 Derivation

364

(1)

14.5.2 Applications

365

(2)

14.6 Sweeping of a Control Parameter Towards an Instability

367

(3)

14.7 Growth with Preferential Attachment

370

(3)

14.8 Multiplicative Noise with Constraints

373

(8)

14.8.1 Definition of the Process

373

(1)

14.8.2 The Kesten Multiplicative Stochastic Process

374

(1)

14.8.3 Random Walk Analogy

375

(3)

14.8.4 Exact Derivation, Generalization and Applications

378

(3)

14.9 The "Coherent-Noise" Mechanism

381

(5)

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

386

(3)

14.11 "Highly Optimized Tolerant" (HOT) Systems

389

(6)

14.11.1 Mechanism for the Power Law Distribution of Fire Sizes

390

(3)

14.11.2 "Constrained Optimization with Limited Deviations" (COLD)

393

(1)

14.11.3 HOT versus Percolation

393

(2)

15. Self-Organized Criticality

395

(46)

15.1 What Is Self-Organized Criticality?

395

(3)

15.1.1 Introduction

395

(2)

15.1.2 Definition

397

(1)

15.2 Sandpile Models

398

(4)

15.2.1 Generalities

398

(1)

15.2.2 The Abelian Sandpile

398

(4)

15.3 Threshold Dynamics

402

(4)

15.3.1 Generalization

402

(2)

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

404

(2)

15.4 Scenarios for Self-Organized Criticality

406

(32)

15.4.1 Generalities

406

(1)

15.4.2 Nonlinear Feedback of the "Order Parameter" onto the "Control Parameter"

407

(2)

15.4.3 Generic Scale Invariance

409

(5)

15.4.4 Mapping onto a Critical Point

414

(8)

15.4.5 Mapping to Contact Processes

422

(2)

15.4.6 Critical Desynchronization

424

(3)

15.4.7 Extremal Dynamics

427

(8)

15.4.8 Dynamical System Theory of Self-Organized Criticality

435

(3)

15.5 Tests of Self-Organized Criticality in Complex Systems: the Example of the Earth's Crust

438

(3)

16. Introduction to the Physics of Random Systems

441

(16)

16.1 Generalities

441

(4)

16.2 The Random Energy Model

445

(4)

16.3 Non-Self-Averaging Properties

449

(8)

16.3.1 Definitions

449

(2)

16.3.2 Fragmentation Models

451

(6)

17. Randomness and Long-Range Laplacian Interactions

457

(20)

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

457

(6)

17.1.1 Holtsmark's Gravitational Force Distribution

457

(4)

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

461

(2)

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

463

(14)

17.2.1 Problem and Main Results

463

(1)

17.2.2 Calculation Methods

464

(7)

17.2.3 Applications

471

(6)

References

477

(48)

Index

525

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Critical Phenomena in Natural Sciences

Summary

Table of Contents

Supplemental Materials