This book focuses on nonextensive statistical mechanics, a current generalization of Boltzmann-Gibbs (BG) statistical mechanics.
Conceived nearly 150 years ago by Maxwell, Boltzmann and Gibbs, the BG theory, one of the greatest monuments of contemporary physics, exhibits many impressive successes in physics, chemistry, mathematics, and computational sciences. Presently, several thousands of publications by scientists around the world have been dedicated to its nonextensive generalization. A variety of applications have emerged in complex systems and its mathematical grounding is by now well advanced.
Since the first edition release thirteen years ago, there has been a vast amount of new results in the field, all of which have been incorporated in this comprehensive second edition. Heavily revised and updated with new sections and figures, the second edition remains the go-to text on the subject.
A pedagogical introduction to the BG theory concepts and their generalizations – nonlinear dynamics, extensivity of the nonadditive entropy, global correlations, generalization of the standard CLT’s, complex networks, among others – is presented in this book, as well as a selection of paradigmatic applications in various sciences together with diversified experimental verifications of some of its predictions. Introduction to Nonextensive Statistical Mechanics is suitable for students and researchers with an interest in complex systems and statistical physics.
Author(s): Constantino Tsallis
Edition: 2
Publisher: Springer
Year: 2023
Language: English
Pages: 574
City: Cham
Preface
Contents
Part I Basics or How the Theory Works
1 Historical Background and Physical Motivations
1.1 An Overall Perspective
1.2 Introduction
1.3 Background and Indications in the Literature
1.4 Symmetry, Energy, and Entropy
1.5 A Few Words on the Foundations of Statistical Mechanics
2 Learning with Boltzmann-Gibbs Statistical Mechanics
2.1 Boltzmann–Gibbs Entropy
2.1.1 Entropic Forms
2.1.2 Properties
2.2 Kullback–Leibler Relative Entropy
2.3 Constraints and Entropy Optimization
2.3.1 Imposing the Mean Value of the Variable
2.3.2 Imposing the Mean Value of the Squared Variable
2.3.3 Imposing the Mean Values of Both the Variable and Its Square
2.3.4 Others
2.4 Boltzmann–Gibbs Statistical Mechanics and Thermodynamics
2.4.1 Isolated System—Microcanonical Ensemble
2.4.2 In the Presence of a Thermostat—Canonical Ensemble
2.4.3 Others
3 Generalizing What We Learnt: Nonextensive Statistical Mechanics
3.1 Playing with Differential Equations—A Metaphor
3.2 Nonadditive Entropy Sq
3.2.1 Definition
3.2.2 Properties
3.3 Correlations, Occupancy of Phase Space, and Extensivity of Sq
3.3.1 The Thermodynamical Limit
3.3.2 Einstein Likelihood Principle
3.3.3 The q-Product
3.3.4 The q-Sum and Related Issues
3.3.5 Extensivity of Sq—Effective Number of States
3.3.6 Extensivity of Sq—Binary Systems
3.3.7 Extensivity of Sq—Physical Realizations
3.3.8 An Epistemological Analogy
3.4 q-Generalization of the Kullback–Leibler Relative Entropy
3.5 Constraints and Entropy Optimization
3.5.1 Imposing the Mean Value of the Variable
3.5.2 Imposing the Mean Value of the Squared Variable
3.5.3 Imposing the Mean Values of Both the Variable and Its Square
3.5.4 Others
3.6 Nonextensive Statistical Mechanics and Thermodynamics
3.7 About the Escort Distribution and the q-Expectation Values
3.8 About Universal Constants in Physics
3.9 Comparing Various Entropic Forms
Part II Foundations or Why the Theory Works
4 Probabilistic and Stochastic Dynamical Foundations of Nonextensive Statistical Mechanics
4.1 Introduction
4.2 Diffusion
4.2.1 Normal Diffusion
4.2.2 Anomalous Diffusion
4.3 Stable Solutions of Fokker–Planck-Like Equations
4.4 Connection Between Entropic Functionals, Fokker–Planck Equations and H-theorems
4.5 Many-Body Interacting Systems with Overdamping
4.5.1 Models
4.5.2 Thermodynamics, Including Clausius Relation, First and Second Principles, Equation of States, and Carnot Cycle
4.5.3 Zeroth Principle of Thermodynamics
4.5.4 An Extended Model with Repulsive Power-Law Short-Ranged Interactions
4.5.5 Model with Direction-Depending Drag
4.5.6 Plasma in Spherical Capacitor with Overdamping
4.6 Generalizing the Langevin Equation
4.7 Other Complex Dissipative Systems
4.7.1 Coherent Noise Model
4.7.2 Ehrenfest Dog-Flea Model
4.7.3 Random Walk Avalanches
4.7.4 Kuramoto Model
4.8 Probabilistic Models with Strong Correlations—Numerical and Analytical Approaches
4.8.1 The MTG Model and Its Numerical Approach
4.8.2 The TMNT Model and Its Numerical Approach
4.8.3 Analytical Approach of the MTG and TMNT Models
4.8.4 The MFMT Models and Their Numerical Approach
4.8.5 The RST1 Model and Its Analytical Approach
4.8.6 The RST2 Model and Its Numerical Approach
4.8.7 The HTT Model and Its Analytical Approach
4.8.8 Comparative Considerations
4.9 Central Limit Theorems
4.10 Large Deviation Theory
4.10.1 Purely Probabilistic Models
4.10.2 Dynamical Models
4.11 Time-Dependent Ginzburg–Landau d-Dimensional O(n) Ferromagnet with n=d
5 Deterministic Dynamical Foundations of Nonextensive Statistical Mechanics
5.1 Dissipative Maps
5.1.1 One-Dimensional Dissipative Maps
5.1.2 Two-Dimensional Dissipative Maps
5.1.3 High-Dimensional Dissipative Maps
5.2 Conservative Maps
5.2.1 Strongly Chaotic Two-Dimensional Conservative Maps
5.2.2 Weakly Chaotic Two-Dimensional Conservative Maps
5.2.3 Strongly Chaotic Four-Dimensional Conservative Maps
5.2.4 High-Dimensional Conservative Maps
5.3 Many-Body Long-Range-Interacting Hamiltonian Systems
5.3.1 Planar Rotators (Inertial XY-Like Ferromagnetic Model)
5.3.2 Three-Dimensional Rotators (Inertial Heisenberg-Like Ferromagnetic Model)
5.3.3 Fermi-Pasta-Ulam-Like Model for Coupled Oscillators
5.4 The q-Triplet and More
5.5 Connection with Critical Phenomena
5.6 A Conjecture on the Time and Size Dependences of Entropy
6 Generalizing Nonextensive Statistical Mechanics
6.1 Crossover Statistics
6.2 Further Generalizing
6.2.1 Spectral Statistics
6.2.2 Beck-Cohen Superstatistics
Part III Applications or What for the Theory Works
7 Thermodynamical and Nonthermodynamical Applications
7.1 Physics
7.1.1 Cold Atoms in Optical Lattices
7.1.2 Trapped Ions
7.1.3 Granular Matter
7.1.4 High-Energy Physics
7.1.5 Turbulence
7.1.6 Fingering
7.1.7 Condensed Matter Physics
7.1.8 Plasma
7.1.9 Astrophysics and Astronomy
7.1.10 Geophysics
7.1.11 Quantum Chaos
7.1.12 Quantum Information
7.1.13 Random Matrices
7.1.14 Nonlinear Quantum Mechanics
7.2 Chemistry
7.2.1 Generalized Arrhenius Law and Anomalous Diffusion
7.2.2 Lattice Lotka–Volterra Model for Chemical Reactions and Growth
7.2.3 Re-association in Folded Proteins
7.2.4 Ground State Energy of the Chemical Elements (Mendeleev's Table) and of Doped Fullerenes
7.2.5 Compton Profiles
7.3 Economics
7.3.1 Inter-Occurrence Times in Economics, Geophysics, Genome Structure, Civil Engineering, and Turbulence
7.4 Computer Sciences
7.4.1 Optimization Algorithms
7.4.2 Analysis of Time Series and Signals
7.4.3 Analysis of Images
7.4.4 Ping Internet Experiment
7.5 Biosciences
7.6 Cellular Automata
7.7 Self-Organized Criticality
7.8 Asymptotically Scale-Free Networks
7.8.1 The Natal Model
7.8.2 Albert–Barabási Model
7.8.3 Non-growing Model
7.8.4 Connection Between Asymptotically Scale-Invariant Networks with Weighted Links and q-Statistics
7.8.5 Lennard-Jones Cluster
7.9 Linguistics
7.10 Other Sciences
Part IV Last (But Not Least)
8 Final Comments and Perspectives
8.1 Falsifiable Predictions and Conjectures, and Their Verification
8.1.1 The Scaling Relation γ=23-q
8.1.2 q-Gaussian Distributions of Velocities
8.1.3 Generalized Central Limit Theorem Leading to Stable q-Gaussian Distributions
8.1.4 Existence of q, λq and Kq, and the Identity Kq=λq
8.1.5 Scaling with N* for Long-Range-Interacting Systems
8.1.6 Vanishing Lyapunov Spectrum for Classical Long-Range-Interacting Many-Body Hamiltonian Systems
8.1.7 Nonuniform Convergence for Long-Range Hamiltonians Associated with the (N,t) to(infty,infty) Limits
8.1.8 q-Duplets, q-Triplets and q-Nplets
8.1.9 Degree Distributions of the q-Exponential Type for Scale-Free Networks
8.2 Frequently Asked Questions
8.3 Open Questions
Appendix A Useful Mathematical Formulae
Appendix B Escort Distributions and q-Expectation Values
B.1 First Example
B.2 Second Example
B.3 Remarks
Appendix References
Index
