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

Note: Supplemental materials are not guaranteed with Rental or Used book purchases.

ISBN: 9783540674627 | 3540674624
Cover: Hardcover
Copyright: 12/1/2000

Rent

(Recommended)

$60.93

Term

Due

Price

Semester

Dec 13

$64.14

Quarter

Aug 16

$60.93

Short Term

Jul 3

$57.72

*This item is part of an exclusive publisher rental program and requires an additional convenience fee. This fee will be reflected in the shopping cart.
Buy New

Usually Ships in 3-5 Business Days

$87.62
eBook

Available Instantly

Online: 180 Days

Downloadable: 180 Days

$65.08

A textbook written by experts detailing concepts, methods and techniques of statistical physics in the study of correlated and uncorrelated phenomena in the natural sciences. Provides tools that can be of use in immediate applications and may be of interest to professionals in geophysics and metrology. DLC: Critical phenomena (Physics).

Useful Notions of Probability Theory

(22)

What Is Probability?

(1)

Bayesian View Point

(5)

Introduction

(2)

Bayes' Theorem

(2)

Bayesian Explanation for Change of Belief

(1)

Probability Density Function

(1)

Measures of Central Tendency

(2)

Measure of Variations from Central Tendency

(1)

Moments and Characteristic Function

(2)

Cumulants

(1)

Maximum of Random Variables and Extreme Value Theory

(11)

Maximum Value Among N Random Variables

(3)

Stable Extreme Value Distributions

(2)

First Heuristic Derivation of the Stable Gumbel Distribution

(1)

Second Heuristic Derivation of the Stable Gumbel Distribution

(2)

Practical Use and Expression of the Coefficients of the Gumbel Distribution

(2)

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

(24)

The Random Walk Problem

(8)

Average Drift

(1)

Diffusion Law

(1)

Brownian Motion as Solution of a Stochastic ODE

(2)

Fractal Structure

(1)

Self-Affinity

(3)

Master and Diffusion (Fokker-Planck) Equations

(5)

Simple Formulation

(2)

General Fokker-Planck Equation

(1)

Ito Versus Stratonovich

(2)

The Central Limit Theorem

(11)

Convolution

(2)

Statement

(1)

Conditions

(1)

Collective Phenomenon

(1)

Renormalization Group Derivation

(3)

Recursion Relation and Perturbative Analysis

(4)

Large Deviations

(32)

Cumulant Expansion

(1)

Large Deviation Theorem

(6)

Quantification of the Deviation from the Central Limit Theorem

(1)

Heuristic Derivation of the Large Deviation Theorem (3.9)

(2)

Example: the Binomial Law

(1)

Non-identically Distributed Random Variables

(2)

Large Deviations with Constraints and the Boltzmann Formalism

(11)

Frequencies Conditioned by Large Deviations

(2)

Partition Function Formalism

(3)

Large Deviations in the Dice Game

(1)

Model Construction from Large Deviations

(3)

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

(2)

Extreme Deviations

(9)

The ``Democratic'' Result

(2)

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

(2)

Application to Turbulence and to Fragmentation

(5)

Large Deviations in the Sum of Variables with Power Law Distributions

(5)

General Case with Exponent μ > 2

(2)

Borderline Case with Exponent μ = 2

(3)

Power Law Distributions

(24)

Stable Laws: Gaussian and Levy Laws

(10)

Definition

(1)

The Gaussian Probability Density Function

(1)

The Log-Normal Law

(2)

The Levy Laws

(5)

Truncated Levy Laws

(2)

Power Laws

(7)

How Does One Tame ``Wild'' Distributions?

(3)

Multifractal Approach

(3)

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

(3)

General Intuitive Derivation

(1)

More Detailed Model of Tracer Diffusion in the Crust

(1)

Anomalous Diffusion in a Fluid

(1)

Calculation Tools for Power Law Distributions

(4)

Fractals and Multifractals

103

(34)

Fractals

103

(17)

Introduction

103

(1)

A First Canonical Example: the Triadic Cantor Set

104

(1)

How Long Is the Coast of Britain?

105

(2)

The Hausdorff Dimension

107

(1)

Examples of Natural Fractals

107

(13)

Multifractals

120

(6)

Definition

120

(1)

Correction Method for Finite Size Effects and Irregular Geometries

121

(2)

Origin of Multifractality and Some Exact Results

123

(1)

Generalization of Multifractality: Infinitely Divisible Cascades

124

(2)

Scale Invariance

126

(5)

Definition

126

(2)

Relation with Dimensional Analysis

128

(3)

Complex Fractal Dimensions and Discrete Scale Invariance

131

(6)

Definition of Discrete Scale Invariance

131

(1)

Log-Periodicity and Complex Exponents

131

(2)

Importance and Usefulness of Discrete Scale Invariance

133

(1)

Scenarios Leading to Discrete Scale Invariance

134

(3)

Rank-Ordering Statistics and Heavy Tails

137

(26)

Probability Distributions

137

(1)

Definition of Rank-Ordering Statistics

138

(2)

Normal and Log-Normal Distributions

140

(1)

The Exponential Distribution

141

(2)

Power Law Distributions

143

(5)

The Gamma Law

148

(1)

The Stretched Exponential Distribution

148

(3)

Maximum Likelihood and Other Estimators of Stretched Exponential Distributions

151

(12)

Introduction

151

(2)

Two-Parameter Stretched Exponential Distribution

153

(6)

Three-Parameter Weibull Distribution

159

(1)

Generalized Weibull Distributions

160

(3)

Statistical Mechanics: Probabilistic Point of View and the Concept of ``Temperature''

163

(20)

Statistical Derivation of the Concept of Temperature

164

(2)

Statistical Thermodynamics

166

(1)

Statistical Mechanics as Probability Theory with Constraints

167

(6)

General Formulation

167

(3)

First Law of Thermodynamics

170

(1)

Thermodynamic Potentials

171

(2)

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

173

(10)

Formulation of the Problem

173

(1)

A General Modeling Strategy

174

(1)

Discriminating Tests

175

(2)

Stationary Distribution with External Noise

177

(1)

Effective Temperature Generated by Chaotic Dynamics

178

(2)

Principle of Least Action for Out-of-Equilibrium Systems

180

(3)

Long-Range Correlations

183

(16)

Criterion for the Relevance of Correlations

183

(3)

Statistical Interpretation

186

(2)

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

188

(1)

Advanced Results on Correlations

189

(10)

Correlation and Dependence

189

(1)

Statistical Time Reversal Symmetry

190

(5)

Fractional Derivation and Long-Time Correlations

195

(4)

Phase Transitions: Critical Phenomena and First-Order Transitions

199

(14)

Definition

199

(1)

Spin Models at Their Critical Points

200

(6)

Definition of the Spin Model

200

(3)

Critical Behavior

203

(1)

Long-Range Correlations of Spin Models at Their Critical Points

204

(2)

First-Order Versus Critical Transitions

206

(7)

Definition and Basic Properties

206

(2)

Dynamical Landau-Ginzburg Formulation

208

(3)

The Scaling Hypothesis: Dynamical Length Scales for Ordering

211

(2)

Transitions, Bifurcations and Precursors

213

(12)

``Supercritical'' Bifurcation

214

(2)

Critical Precursory Fluctuations

216

(3)

``Subcritical'' Bifurcation

219

(2)

Scaling and Precursors Near Spinodals

221

(1)

Selection of an Attractor in the Absence of a Potential

222

(3)

The Renormalization Group

225

(14)

General Framework

225

(2)

An Explicit Example: Spins on a Hierarchical Network

227

(9)

Renormalization Group Calculation

227

(3)

Fixed Points, Stable Phases and Critical Points

230

(2)

Singularities and Critical Exponents

232

(1)

Complex Exponents and Log-Periodic Corrections to Scaling

233

(3)

Criticality and the Renormalization Group on Euclidean Systems

236

(3)

The Percolation Model

239

(18)

Percolation as a Model of Cracking

239

(2)

Effective Medium Theory and Percolation

241

(3)

Renormalization Group Approach to Percolation and Generalizations

244

(6)

Cell-to-Site Transformation

245

(2)

A Word of Caution on Real Space Renrmalization Group Techniques

247

(1)

The Percolation Model on the Hierarchical Diamond Lattice

248

(2)

Directed Percolation

250

(7)

Definitions

250

(2)

Universality Class

252

(2)

Field Theory: Stochastic Partial Differential Equation with Multiplicative Nose

254

(1)

Self-Organized Formulation of Directed Percolation and Scaling Laws

255

(2)

Rupture Models

257

(28)

The Branching Model

258

(4)

Mean Field Version or Branching on the Bethe Lattic

258

(2)

A Branching-Aggregation Model Automatically Functioning at Its Critical Point

260

(1)

Generalization of Critical Branching Models

261

(1)

Fiber Bundle Models and the Effects of Stress Redistribution

262

(5)

One-Dimensional System of Fibers Associated in Series

263

(1)

Democratic Fiber Bundle Model (Daniels, 1945)

264

(3)

Hierarchical Model

267

(7)

The Simplest Hierarchical Model of Rupture

267

(3)

Quasi-Static Hierarchical Fiber Rupture Model

270

(2)

Hierarchical Fiber Rupture Model with Time-Dependence

272

(2)

Quasi-Static Models in Euclidean Spaces

274

(5)

A Dynamical Model of Rupture Without Elasto-Dynamics: The ``Thermal Fuse Model''

279

(3)

Time-to-Failure and Rupture Criticality

282

(3)

Critical Time-to-Failure Analysis

282

(1)

Time-to-Failure Behavior in the Dieterich Friction Law

283

(2)

Mechanisms for Power Laws

285

(36)

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

286

(2)

Change of Variable

288

(7)

Power Law Change of Variable Close to the Origin

288

(6)

Combination of Exponentials

294

(1)

Maximization of the Generalized Tsallis Entropy

295

(3)

Superposition of Distributions

298

(2)

Power Law Distribution of Widths

298

(2)

Sum of Stretched Exponentials (Chap.3)

300

(1)

Random Walks: Distribution of Return Times to the Origin

300

(4)

Derivation

300

(2)

Applications

302

(2)

Sweeping of a Control Parameter Towards an Instability

304

(3)

Multiplicative Noise with Constraints

307

(6)

Definition of the Process

307

(1)

Random Walk Analogy

308

(2)

Exact Derivation, Generalization and Applications

310

(3)

Avalanches in Hysteretic Loops and First-Order Transitions with Randomness

313

(4)

Highly Optimized Tolerant (HOT) Systems

317

(4)

Self-Organized Criticality

321

(40)

What Is Self-Organized Criticality?

321

(3)

Introduction

321

(2)

Definition

323

(1)

Sandpile Models

324

(4)

Generalities

324

(1)

The Abelian Sandpile

324

(4)

Threshold Dynamics

328

(3)

Generalization

328

(1)

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

329

(2)

Scenarios for Self-Organized Criticality

331

(27)

Generalities

331

(1)

Nonlinear Feedback of the ``Order Parameter'' onto the ``Control Parameter''

332

(1)

Generic Scale Invariance

333

(5)

Mapping onto a Critical Point

338

(8)

Mapping to Contact Processes

346

(1)

Critical Desynchronization

347

(4)

Extremal Dynamics

351

(5)

Dynamical System Theory of Self-Organized Criticality

356

(2)

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

358

(3)

Introduction to the Physics of Random Systems

361

(14)

Generalities

361

(4)

The Random Energy Model

365

(4)

Non-self-averaging Properties

369

(6)

Definitions

369

(1)

Fragmentation Models

370

(5)

Randomness and Long-Range Laplacian Interactions

375

(18)

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

375

(5)

Holtsmark's Gravitational Force Distribution

375

(4)

Generalization to Other Fields (Electric, Elastic, Hydrodynamics)

379

(1)

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

380

(13)

Problem and Main Results

380

(1)

Calculation Methods

381

(7)

Applications

388

(5)

References

393

(38)

Index

431

Please wait while the item is added to your bag...

FREE SHIPPING

Amazon no longer offers textbook rentals. We do!

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

Summary

Table of Contents

Supplemental Materials