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Self-Organized Criticality

Book Description

Self-organized criticality (SOC) is based upon the idea that complex behavior can develop spontaneously in certain multi-body systems whose dynamics vary abruptly. This book is a clear and concise introduction to the field of self-organized criticality, and contains an overview of the main research results. The author begins with an examination of what is meant by SOC, and the systems in which it can occur. He then presents and analyzes computer models to describe a number of systems, and he explains the different mathematical formalisms developed to understand SOC. The final chapter assesses the impact of this field of study, and highlights some key areas of new research. The author assumes no previous knowledge of the field, and the book contains several exercises. It will be ideal as a textbook for graduate students taking physics, engineering, or mathematical biology courses in nonlinear science or complexity.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Dedication
  6. Contents
  7. Preface
  8. Acknowledgment
  9. 1. Introduction
  10. 2. Characterization of the SOC State
    1. 2.1 Response Distributions
    2. 2.2 Temporal Fluctuations
    3. 2.3 Power Spectrum and Distribution of Lifetimes
    4. 2.4 Spatial Correlation Functions
  11. 3. Systems Exhibiting SOC
    1. 3.1 Introduction
    2. 3.2 Sandpiles
    3. 3.3 Ricepiles
    4. 3.4 Superconducting Avalanches
    5. 3.5 Droplet Formation
    6. 3.6 Earthquakes
    7. 3.7 Evolution
  12. 4. Computer Models
    1. 4.1 Introduction
    2. 4.2 Sandpiles: Conservative Model
      1. 4.2.1 One-Dimensional Sandpile
      2. 4.2.2 Dimensions Larger than 1
      3. 4.2.3 Critical Response
      4. 4.2.4 Numerical Results: Distribution Functions
      5. 4.2.5 Power Spectrum
    3. 4.3 Earthquake Models: Nonconservative Models
      1. 4.3.1 Criticality of the OFC Model
      2. 4.3.2 Nearest Neighbor OFC Model
      3. 4.3.3 Random Neighbor OFC Model
      4. 4.3.4 Distributions and Fluctuations in the Nearest Neighbor OFC Model
      5. 4.3.5 The Effect of Disorder on the OFC Model
      6. 4.3.6 Physical Relevance of the OFC Model
    4. 4.4 Lattice Gas
      1. 4.4.1 Definition of the Lattice Gas Model
      2. 4.4.2 Properties of Lattice Gas
      3. 4.4.3 The Lesson of Lattice Gas
      4. 4.4.4 Physical Relevance of the Lattice Gas Model
    5. 4.5 Forest Fires
      1. 4.5.1 Definition of a Critical Forest Fire Model
      2. 4.5.2 Simulation Results for the Forest Fire Model
      3. 4.5.3 Physical Relevance of the Forest Fire Model
    6. 4.6 Extremum Dynamics
      1. 4.6.1 The Model of interface Growth in a Random Medium
      2. 4.6.2 The Evolution Model
  13. 5. The Search for a Formalism
    1. 5.1 Introduction
    2. 5.2 Mean Field Theory
      1. 5.2.1 Sandpile Models
      2. 5.2.2 Earthquake Models
      3. 5.2.3 Diffusive Description of Lattice Gas
      4. 5.2.4 Forest Fire Model
      5. 5.2.5 Model of Biological Evolution
    3. 5.3 Exact Solution of the Abelian Sandpile
      1. 5.3.1 The Δ Matrix and the Probability Measure on the Configuration Space
      2. 5.3.2 Correlation Functions
    4. 5.4 Langevin Equations
      1. 5.4.1 Conservative Models
      2. 5.4.2 Nonconservative Models
    5. 5.5 Dynamically Driven Renormalization Group Calculations
      1. 5.5.1 Renormalization Transformation
      2. 5.5.2 Exponents
      3. 5.5.3 Nonconservative Models
      4. 5.5.4 Forest Fire Models
  14. 6. Is It SOC or Not?
    1. 6.1 Where Is SOC to be Found?
    2. 6.2 What Is Tuning?
  15. Appendices
    1. A: Code for the BTW Sandpile
    2. B: Code for the Lattice Gas
    3. C: Code for the Bak−Sneppen Evolution Model
    4. D: Power Spectra and the Correlation Function
    5. E: Statistical Weights in the DDRG
  16. References
  17. Index