Nonextensive Statistical Mechanics and Its Applications (Lecture Notes in Physics, 560)

✍ Scribed by Sumiyoshi Abe (editor), Yuko Okamoto (editor)

• Publisher: Springer
• Year: 2001
• Tongue: English
• Leaves: 268
• Edition: 2001
• Category: Library

✦ Synopsis

Nonextensive statistical mechanics is now a rapidly growing field and a new stream in the research of the foundations of statistical mechanics. This generalization of the well-known Boltzmann--Gibbs theory enables the study of systems with long-range interactions, long-term memories or multi-fractal structures. This book consists of a set of self-contained lectures and includes additional contributions where some of the latest developments -- ranging from astro- to biophysics -- are covered. Addressing primarily graduate students and lecturers, this book will also be a useful reference for all researchers working in the field.

✦ Table of Contents

Chapter 1
1 Introduction
2 Formalism
2.1 Entropy
2.2 Canonical Ensemble
3 Theoretical Evidence and Connections
3.1 Lévy-Type Anomalous Diffusion
3.2 Correlated-Type Anomalous Diffusion
3.3 Charm Quark Diffusion in Quark –Gluon Plasma
3.4 Self-Gravitating Systems
3.5 Zipf –Mandelbrot Law
3.6 Theory of Financial Decisions: Risk Aversion
3.7 Physiology of Vision
4 Experimental Evidence and Connections
4.1 D =2 Turbulence in Pure-Electron Plasma
4.2 Solar Neutrino Problem
4.3 Peculiar Velocities in Sc Galaxies
4.4 Nonlinear Inverse Bremsstrahlung Absorption in Low Pressure Argon Plasma
4.5 Cosmic Microwave Background Radiation
4.6 Electron –Positron Collisions
4.7 Emulsion Chamber Observation of Cosmic Rays
4.8 Reassociation of Heme –Ligands in Folded Proteins
4.9 Diffusion of Hydra Vulgaris
4.10 Citations of Scientific Papers
4.11 Electroencephalographic Signals of Epilepsy
4.12 Cognitive Psychology
4.13 Fully Developed Turbulence and Financial Markets
5 Computational Evidence and Connections
5.1 Thermalization of a Hot Gas Penetrating in a Cold Gas
5.2 Long-Range Classical Hamiltonian Systems:Static Properties
5.3 Long-Range Tight-Binding Systems
5.4 Granular Systems
5.5 d =1 Dissipative Systems
5.6 Self-Organized Criticality
5.7 Long-Range Classical Hamiltonian Systems: Dynamic Properties
5.8 Symbolic Sequences
5.9 Optimization Techniques; Simulated Annealing
6 Final Remarks
Acknowledgements
Appendix: q -Exponential and q -Logarithm Functions
References
Chapter 2
1 General Remarks
A. Motivation for Generalization of Boltzmann –Gibbs Description
B. Motivation for Using Density Matrix Description
C. Maximum Entropy Principle of Jaynes Constraints Replace Ensembles
2 Theory of Entangled States and Its Implications: Jaynes–Cummings Model
3 Variational Principle
4 Time-Dependence: Unitary Dynamics
5 Time-Dependence: Nonunitary Dynamics
6 Concluding Remarks
Appendix: q-Kullback –Leibler Entropy as a Guide?
References
Chapter 3
1 Introduction
2 Jaynes Maximum Entropy Principle
3 General Thermostatistical Formalisms
3.1 MaxEnt Formalisms with Standard Linear Constraints
3.2 MaxEnt Formalisms with Generalized Nonlinear Constraints
3.3 Tsallis Entropy Plus Escort Mean Values
3.4 Invariance Under Uniform Shifts of the Hamiltonian Eigenenergies
3.5 Other Universal Properties of General Thermostatistical Formalisms
4 Time Dependent MaxEnt
5 Time-Dependent Tsallis MaxEnt Solutions of the Nonlinear Fokker–Planck Equation
5.1 The Nonlinear Fokker-Planck Equation
5.2 Generalized Maximum Entropy Approach
6 Tsallis Nonextensive Thermostatistics and the Vlasov–Poisson Equations
6.1 Long-Range Interactions and Nonextensivity
6.2 The Vlasov–Poisson Equations
6.3 MaxEnt Stationary Solutions to the Vlasov–Poisson Equations
6.4 Tsallis MaxEnt Solutions to the Vlasov–Poisson Equations
6.5 D-Dimensional Schuster Spheres
6.6 Tsallis MaxEnt Time-Dependent Solutions
7 Conclusions
Acknowledgements
References
Chapter 4
1 Background and Focus
2 Basic Properties of Tsallis Statistics
2.1 From the Thermal Density Distribution to Tsallis Statistics
2.2 Specific Defining Properties of the Tsallis Statistical Distributions
2.3 Generalized Partition Functions for Ideal Systems
2.4 Problems Arise for Many-Body Systems
2.5 Enforcing Separability Using Maxwell–Tsallis Statistics
2.6 Corrected Ensemble Averages
3 General Properties of Mass Action and Kinetics
3.1 Transition State Theory for Rates of Barrier Crossing
3.2 Master Equations and Relaxation to Equilibrium
4 Tsallis Statistics and Simulated Annealing
4.1 Essential Features and Algorithms
4.2 Temperature Scaling in Simulated Annealing
4.3 Further Applications of Enhanced Simulated Annealing
4.4 The Importance of Detailed Balance
5 Tsallis Statistics and Monte Carlo Methods
5.1 Monte Carlo Estimates of Tsallis Statistical Averages
5.2 Monte Carlo Estimates of Gibbs–Boltzmann Statistical Averages
5.3 Monte Carlo Algorithm for Spin Systems
5.4 Further Applications of Generalized Monte Carlo
6 Tsallis Statistics and Molecular Dynamics
6.1 Molecular Dynamics Estimates of Gibbs–Boltzmann Statistical Averages
6.2 Rate Constants for Maxwell–Tsallis Statistics
6.3 Further Applications of Generalized Molecular Dynamics
7 Optimizing the Monte Carlo or Molecular Dynamics Algorithm Using the Ergodic Measure
8 Tsallis Statistics and Feynman Path Integral Quantum Mechanics
8.1 Connection to Tsallis Statistics
8.2 Feynman –Tsallis Path Integrals for the Simulation of Quantum Many-Body Systems
9 Simulated Annealing Using Cauchy –Lorentz “Density Packet ”Dynamics
References
Chapter 5
Acknowledgements
References
Chapter 6
1 Introduction
2 Nonextensive Thermodynamics
3 Nonlinear von Neumann Equation
4 Dynamic Stability
5 Thermodynamic Stability
6 Proof of Theorem 1
7 Minima of F^(3)
8 Proof of Theorem 2
9 Conclusions
Acknowledgements
Appendix
References
Chapter 7
1 Generalized Acceptance Probabilities
2 Model and Simulations
3 Results
4 Summary
Acknowledgements
References
Chapter 8
1 Introduction
2 Methods
2.1 Energy Function of Protein Systems
2.2 Generalized-Ensemble Algorithm with Tsallis Statistics
3 Results
4 Conclusions
Acknowledgements
References