Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World Book

Part I Basics or How the Theory Works

1 Historical Background and Physical Motivations 3

1.1 Introduction 3

1.2 Background and Indications in the Literature 6

1.3 Symmetry, Energy, and Entropy 12

1.4 A Few Words on the Foundations of Statistical Mechanics 13

2 Learning with Boltzmann-Gibbs Statistical Mechanics 19

2.1 Boltzmann-Gibbs Entropy 19

2.1.1 Entropic Forms 19

2.1.2 Properties 21

2.2 Kullback-Leibler Relative Entropy 28

2.3 Constraints and Entropy Optimization 30

2.3.1 Imposing the Mean Value of the Variable 30

2.3.2 Imposing the Mean Value of the Squared Variable 31

2.3.3 Imposing the Mean Values of both the Variable and Its Square 32

2.3.4 Others 33

2.4 Boltzmann-Gibbs Statistical Mechanics and Thermodynamics 33

2.4.1 Isolated System - Microcanonical Ensemble 35

2.4.2 In the Presence of a Thermostat - Canonical Ensemble 35

2.4.3 Others 36

3 Generalizing What We Learnt: Nonextensive Statistical Mechanics 37

3.1 Playing with Differential Equations - A Metaphor 37

3.2 Nonadditive Entropy Sq 41

3.2.1 Definition 41

3.2.2 Properties 43

3.3 Correlations, Occupancy of Phase-Space, and Extensivity of Sq 54

3.3.1 A Remark on the Thermodynamical Limit 54

3.3.2 The q-Product 61

3.3.3 The q-Sum 64

3.3.4 Extensivity of Sq - Effective Number of States 66

3.3.5 Extensivity of Sq - Binary Systems 69

3.3.6 Extensivity of Sq - Physical Realizations 77

3.4 q-Generalization of the Kullback-Leibler Relative Entropy 84

3.5 Constraints and Entropy Optimization 88

3.5.1 Imposing the Mean Value of the Variable 88

3.5.2 Imposing the Mean Value of the Squared Variable 89

3.5.3 Others 90

3.6 Nonextensive Statistical Mechanics andThermodynamics 90

3.7 About the Escort Distribution and the q-Expectation Values 98

3.8 About Universal Constants in Physics 102

3.9 Various Other Entropic Forms 105

Part II Foundations or Why the Theory Works

4 Stochastic Dynamical Foundations of Nonextensive Statistical Mechanics 109

4.1 Introduction 109

4.2 Normal Diffusion 110

4.3 Lévy Anomalous Diffusion 111

4.4 Correlated Anomalous Diffusion 111

4.4.1 Further Generalizing the Fokker-Planck Equation 117

4.5 Stable Solutions of Fokker-Planck-Like Equations 117

4.6 Probabilistic Models with Correlations - Numerical and Analytical Approaches 119

4.6.1 The MTG Model and Its Numerical Approach 120

4.6.2 The TMNT Model and Its Numerical Approach 125

4.6.3 Analytical Approach of the MTG and TMNT Models 129

4.6.4 The RSTI Model and Its Analytical Approach 132

4.6.5 The RST2 Model and Its Numerical Approach 133

4.7 Central Limit Theorems 135

4.8 Generalizing the Langevin Equation 144

4.9 Time-Dependent Ginzburg-Landau d-Dimensional O(n) Ferromagnet with n = d 149

5 Deterministic Dynamical Foundations of Nonextensive Statistical Mechanics 151

5.1 Low-Dimensional Dissipative Maps 151

5.1.1 One-Dimensional Dissipative Maps 151

5.1.2 Two-Dimensional Dissipative Maps 164

5.2 Low-Dimensional Conserative Maps 165

5.2.1 Strongly Chaotic Two-Dimensional Conservative Maps 166

5.2.2 Strongly Chaotic Four-Dimensional Conservative Maps 172

5.2.3 Weakly Chaotic Two-Dimensional Conservative Maps 173

5.3 High-Dimensional Conservative Maps 179

5.4 Many-Body Long-Range-Interacting Hamiltonian Systems 182

5.4.1 Metastability, Nonergodicity, and Distribution of Velocities 186

5.4.2 Lyapunov Spectrum 186

5.4.3 Aging and Anomalous Diffusion 188

5.4.4 Connection with Glassy Systems 190

5.5 The q-Triplet 191

5.6 Connection with Critical Phenomena 195

5.7 A Conjecture on the Time and Size Dependences of Entropy 196

6 Generalizing Nonextensive Statistical Mechanics 209

6.1 Crossover Statistics 209

6.2 Further Generalizing 211

6.2.1 Spectral Statistics 212

6.2.2 Beck-Cohen Superstatistics 216

Part III Applications or What for the Theory Works

7 Thermodynamical and Nonthermodynamical Applications 221

7.1 Physics 222

7.1.1 Cold Atoms in Optical Lattices 222

7.1.2 High-Energy Physics 223

7.1.3 Turbulence 227

7.1.4 Fingering 233

7.1.5 Granular Matter 233

7.1.6 Condensed Matter Physics 235

7.1.7 Plasma 237

7.1.8 Astrophysics 241

7.1.9 Geophysics 244

7.1.10 Quantum Chaos 254

7.1.11 Quantum Entanglement 255

7.1.12 Random Matrices 255

7.2 Chemistry 258

7.2.1 Generalized Arrhenius Law and Anomalous Diffusion 258

7.2.2 Lattice Lotka-Volterra Model for Chemical Reactions and Growth 259

7.2.3 Re-Association in Folded Proteins 263

7.2.4 Ground State Energy of the Chemical Elements (Mendeleev's Table) and of Doped Fullerenes 265

7.3 Economics 266

7.4 Computer Sciences 269

7.4.1 Optimization Algorithms 269

7.4.2 Analysis of Time Series and Signals 275

7.4.3 Analysis of Images 279

7.4.4 PING Internet Experiment 280

7.5 Biosciences 281

7.6 Cellular Automata 282

7.7 Self-Organized Criticality 283

7.8 Scale-Free Networks 283

7.8.1 The Natal Model 290

7.8.2 Albert-Barabasi Model 291

7.8.3 Non-Growing Model 294

7.8.4 Lennard-Jones Cluster 295

7.9 Linguistics 295

7.10 Other Sciences 295

Part IV Last (But Not Least)

8 Final Comments and Perspectives 305

8.1 Falsifiable Predictions and Conjectures, and Their Verifications 305

8.2 Frequently Asked Questions 308

8.3 Open Questions 326

Appendix A Useful Mathematical Formulae 329

Appendix B Escort Distributions and q-Expectation Values 335

B.1 First Example 335

B.2 Second Example 339

B.3 Remarks 339

Bibliography 343

Index 381