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To Carol J. Kersten

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIII

1. Self-Organized Criticality Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The Concept of Self-Organized Criticality . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 SOC Laboratory Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 SOC in Human Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 SOC in Biophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5 SOC in Geophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.6 SOC in Magnetospheric Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.7 SOC in Planetary Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.8 SOC in Solar Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.9 SOC in Stellar Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.10 SOC in Galaxies and Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.12 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2. Numerical SOC Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.1 SOC Simulations of Laboratory Experiments . . . . . . . . . . . . . . . . . . . . . . . . 38

2.1.1 Coupled Pendulums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.1.2 The Bak-Tang-Wiesenfeld 1-D Sandpile Model . . . . . . . . . . . . . . . . 392.1.3 The Bak–Tang–Wiesenfeld 2-D Sandpile Model . . . . . . . . . . . . . . . 412.1.4 The Lattice-Gas Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.2 SOC Simulations of Human Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.2.1 Conway’s Game of Life Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.2.2 Traffic Jam Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.2.3 Financial Market Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.3 SOC Simulations in Biophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.3.1 The Punctuated Equilibrium (Bak–Sneppen Model) . . . . . . . . . . . . 51

2.4 SOC Simulations in Geophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.4.1 Slider-Block Spring Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.4.2 The Forest-Fire Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

VIII Contents

2.5 SOC Simulations in Magnetospheric Physics . . . . . . . . . . . . . . . . . . . . . . . . 572.5.1 SOC Model with Finite System Size . . . . . . . . . . . . . . . . . . . . . . . . . 572.5.2 Cellular Automaton Model with Discretized MHD . . . . . . . . . . . . . 58

2.6 SOC Simulations in Solar Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632.6.1 Isotropic Cellular Automaton Models . . . . . . . . . . . . . . . . . . . . . . . . 632.6.2 Anisotropic Cellular Automaton Models . . . . . . . . . . . . . . . . . . . . . . 672.6.3 Discretized MHD Cellular Automaton Models . . . . . . . . . . . . . . . . . 702.6.4 Divergence-Free Field Braiding Models . . . . . . . . . . . . . . . . . . . . . . 732.6.5 Branching Process Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

2.7 SOC Simulations in Astrophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772.7.1 Cellular Automaton Model of Accretion Disk Fluctuations . . . . . . 78

2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3. Analytical SOC Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.1 The Exponential-Growth Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.2 The Powerlaw-Growth Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893.3 The Logistic-Growth Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.4 Analytical Fit to Numerical SOC Simulations . . . . . . . . . . . . . . . . . . . . . . . . 983.5 Inertial Range, Lower and Upper Cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023.6 Continuum Limit of Cellular Automaton Model . . . . . . . . . . . . . . . . . . . . . . 1053.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4. Statistics of Random Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.1 Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.2 Gaussian Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.3 Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174.4 Exponential Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.5 Count Rate Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1224.6 White Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1224.7 1/f Power Spectra Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1264.8 Shot Noise or Flicker Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

4.8.1 Derivation of Schottky’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.8.2 Shot Noise Spectrum for Rectangular Pulses . . . . . . . . . . . . . . . . . . 1314.8.3 Shot Noise Spectrum for Exponential-Decay Pulses . . . . . . . . . . . . 1324.8.4 Shot Noise Spectrum and Distribution of Pulse Durations . . . . . . . 133

4.9 Log-Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1354.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1374.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5. Waiting-Time Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395.1 Waiting Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1405.2 Nonstationary Waiting-Time Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1425.3 Measurement of Waiting Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Contents IX

5.4 Waiting-Time Statistics in Geophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1495.5 Waiting-Time Statistics in Magnetospheric Physics . . . . . . . . . . . . . . . . . . . 1515.6 Waiting-Time Statistics in Solar Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

5.6.1 Solar Flare Hard X-Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1545.6.2 Solar Flare Soft X-Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1595.6.3 Coronal Mass Ejections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1625.6.4 Solar Radio Bursts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1635.6.5 Solar Wind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

5.7 Waiting-Time Statistics in Astrophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1655.7.1 Flare Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1655.7.2 Black Hole Accretion Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167


6. Event Detection Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1716.1 Test Data for Event Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1726.2 Threshold-Based Event Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1746.3 Highpass-Filtered Event Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1806.4 Peak-Based Event Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1826.5 Fourier-Filtered Event Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1826.6 Time Scale Statistics from Power Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . 1846.7 Wavelet-Based Time Scale Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1876.8 Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1916.9 Image-Based Event Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1936.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1986.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

7. Occurrence Frequency Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2017.1 Basics of Frequency Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 202

7.1.1 Differential Frequency Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 2027.1.2 Cumulative Frequency Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 2037.1.3 Rank-Order Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2067.1.4 Numerical Generation of Frequency Distributions . . . . . . . . . . . . . . 2087.1.5 Integrals of Powerlaw Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 2107.1.6 Powerlaw Scaling Laws and Correlations . . . . . . . . . . . . . . . . . . . . . 2117.1.7 Accuracy of Powerlaw Fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

7.2 Frequency Distributions in Magnetospheric Physics . . . . . . . . . . . . . . . . . . . 2147.3 Frequency Distributions in Solar Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

7.3.1 Solar Flare Hard X-rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2177.3.2 Solar Flare Soft X-rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2247.3.3 Solar Flare Extreme Ultraviolet Emission . . . . . . . . . . . . . . . . . . . . . 2297.3.4 Solar Radio Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2337.3.5 Solar Energetic Particle (SEP) Events . . . . . . . . . . . . . . . . . . . . . . . . 237

7.4 Frequency Distributions in Astrophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2387.4.1 Stellar Flares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

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7.4.2 Pulsar Glitches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2427.4.3 Soft Gamma-Ray Repeaters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2447.4.4 Black Hole Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2457.4.5 Blazars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246


8. Fractal Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2498.1 1-D Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

8.1.1 The Cantor Set and Koch Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2508.1.2 Irregularity of Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2518.1.3 Variability of Solar Radio Emission . . . . . . . . . . . . . . . . . . . . . . . . . . 253

8.2 2-D Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2568.2.1 Hausdorff Dimension and Box-Counting Method . . . . . . . . . . . . . . 2578.2.2 Solar Photosphere and Chromosphere . . . . . . . . . . . . . . . . . . . . . . . . 2598.2.3 Solar Flares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

8.3 3-D Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2678.3.1 Cellular Automaton Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2688.3.2 Solar Flares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

8.4 Multifractal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2738.5 Spatial Power Spectrum Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2758.6 Statistics of Spatial Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

8.6.1 Solar Photosphere and Chromosphere . . . . . . . . . . . . . . . . . . . . . . . . 2778.6.2 Solar Flares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2798.6.3 Lunar Craters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2808.6.4 Asteroid Belt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2828.6.5 Saturn Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283


9. Physical SOC Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2879.1 A General (Physics-Free) Definition of SOC . . . . . . . . . . . . . . . . . . . . . . . . . 2889.2 Astrophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

9.2.1 Galaxy Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2899.2.2 Star Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2909.2.3 Blazars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2919.2.4 Neutron Star Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2939.2.5 Blackhole Objects and Accretion Disks . . . . . . . . . . . . . . . . . . . . . . . 2959.2.6 Cosmic Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

9.3 Solar and Stellar Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2989.3.1 Maxwell’s Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2989.3.2 The Solar Dynamo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2999.3.3 Magnetic Field Braiding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3019.3.4 Magnetic Reconnection in Solar/Stellar Flares . . . . . . . . . . . . . . . . . 3049.3.5 Thermal Energy of Flare Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

Contents XI

9.3.6 Nonthermal Energy of Flares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3089.3.7 Particle Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3119.3.8 Coherent Radio Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3139.3.9 Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

9.4 Magnetospheric Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3159.4.1 Coronal Mass Ejections and Magnetospheric Storms . . . . . . . . . . . . 3159.4.2 Heliospheric Field and Magnetospheric Substorms . . . . . . . . . . . . . 316


10. SOC-Like Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32110.1 Hierarchical SOC Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32210.2 Self-Organization without Criticality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32410.3 Brownian Motion and Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32610.4 MHD Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

10.4.1 Solar Corona . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32910.4.2 Solar Wind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33210.4.3 Magnetospheric Substorms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33410.4.4 Interstellar Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

10.5 Forced Criticality Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33710.5.1 Magnetospheric Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

10.6 Percolation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33810.6.1 Solar Active Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

10.7 Nonlinear Chaotic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34010.7.1 Astrophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34110.7.2 Solar Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342


Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347Appendix A: Physical Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349Physical Units Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349Latin Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349Greek Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

Image Credit: Public Websites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391

Appendix B: Plasma Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348

Preface

How did this book come about? In 1985, Brian Dennis published a review on solar flaresand presented a stunning figure that showed a perfect powerlaw distribution in the oc-currence of solar flares that extended over almost 4 orders of magnitude, with a slope of−1.8, for which no explanation could be found. Just two years later in 1987, Per Bak,the father of self-organized criticality (SOC), published his landmark paper on the inter-pretation of the ubiquitous powerlaw distributions, observed also in sandpile avalanchesand earthquakes (the so-called Gutenberg–Richter law), by relating the scale-free behav-ior to the 1/ f -flicker noise. A few years later, Per Bak gave a colloquium at the NASAGoddard Space Flight Center (GSFC), where he met Brian Dennis and heard about solarflare statistics; but he admitted in his book How Nature Works that he did not really un-derstand how solar flares work. Intuitively, there was the notion that the intricate details ofthe underlying physical processes could not provide the answer to the fundamental under-standing of the observed powerlaws. In 1991, the two students Ed Lu and Russ Hamiltonat Stanford University wrote the first paper where self-organized criticality was applied tosolar flare statistics, which was interpreted and modeled with a cellular automaton model.This approach offered an explanation of the observed powerlaws in terms of statistics ofnext-neighbor interactions of complex dissipative systems in a critical state. This universalaspect fascinated me more and more and I gave a number of colloquia on self-organizedcriticality applied to solar flares at the ETH Zurich, NASA GSFC, and the University ofMaryland in 1991–1993. Since powerlaw distributions were also observed for stellar flares,pulsar glitches, lunar craters, and asteroid sizes, I speculated that these may all be dissipa-tive systems with self-organized criticality. During one of the seminars at the University ofMaryland I remember that Lucy McFadden, an expert in solar system small bodies, com-mented that this was the most fascinating model she had ever heard of and asked whether itapplied also to the powerlaw distributions of asteroids and Saturn rings. I did not know theanswer at this time but an answer is given in this book. A textbook that explains the funda-mental aspects of self-organized criticality in terms of the statistics of nonlinear events hasnever been written in astrophysics, which motivated me to undertake such an endeavor.One of the major aims of this book is to convey a deeper understanding of the statistics ofnonlinear processes that is common to solar flares, sandpile avalanches, and earthquakes,although the underlying physics is completely different.

XIV Preface

This textbook is intended to be an introduction to the relatively new subject of self-organized criticality (SOC), suitable for students and post-docs, as well as for researcherswho want to know all the relevant literature references. The main applications are astro-physical phenomena, although we include also a few other phenomena from geophysicsor social sciences that provided important basic models, later applied to astrophysical phe-nomena. In Chapter 1 we give an introductory broad overview of SOC phenomena ob-served in the entire universe, wherever publications with SOC interpretations were foundin the scientific literature. The theoretical modeling of SOC phenomena can be pursued in3 different approaches: by numerical (mostly cellular automaton) simulations (Chapter 2),by analytical modeling of statistical distributions (Chapter 3), or by physical modeling(Chapter 9). The temporal aspects of SOC statistics includes random statistics (Chapter 4),waiting-time statistics (Chapter 5), and event-detection methods (Chapter 6). Using thesebasic prerequisites, we can then model and understand the occurrence frequency distribu-tions of SOC events, which reveal the ubiquitous powerlaws that are the hallmark of SOC(Chapter 7). The spatial aspects of SOC events entail the geometry of fractal structures(Chapter 8). Finally, we arrive at a general physics-free definition of SOC phenomena(Section 9.1). Individual physical processes for astrophysical SOC phenomena are sum-marized in Table 9.1 and discussed case by case in the remainder of Chapter 9, qualita-tively for astrophysical observations, and somewhat more quantitatively for solar physicsapplications. Alternatives to SOC processes are discussed in Chapter 10, which may alsoexhibit powerlaw distributions but can be discriminated from pure SOC processes usingthe criteria of our physics-free SOC definition (Table 10.1).

Do we understand SOC completely now? Although we hope to have established adeeper understanding of SOC phenomena in this book, there are still a lot of open ques-tions that can only be answered by large statistics of observations and by more detailedmodeling. For instance, how does the statistics of next-neighbor interactions result in theexponential growth characteristics of SOC avalanches? What determines the powerlawslopes? How much is the powerlaw slope determined by mathematical statistics, and howmuch by physical scaling laws? The relatively new scientific discipline of self-organizedcriticality is a very interdisciplinary field and we hope that this book stimulates a cross-fertilization in the data analysis and development of methods among the disciplines ofastrophysics, geophysics, biophysics, and social sciences.

The author is most indebted to invaluable discussions with, comments from, and re-viewing by colleagues and friends, who are listed in alphabetical order: Eric Buchlin, AnneCristina Cadavid, Sandra Chapman, Paul Charbonneau, Norma Crosby, Pablo Dmitruk,Manuel Gudel, Henrik Jeldtoft Jensen, Debbie Leddon, Yuri Litvinenko, William Liu,Nadege Meunier, Laura Morales, Jeff Scargle, Virginia Trimble, Astrid Veronig, NicolasWatkins, and Mike Wheatland. The author wishes to acknowledge the efficient and mosthelpful support provided by Springer/Praxis, especially by the publishers Clive Horwood(Praxis) and Ramon Khanna (Springer), who encouraged and supported the publication ofthis book. Extensive usage of scientific literature was enabled by the NASA AstrophysicsData System (ADS), operated by the Smithsonian Astrophysical Observatory (SAO), aswell as by numerous Wikipedia and Google searches. Special thanks go also to my family,to my children Pascal Dominique and Alexander Julian, and particularly to my wife, CarolJ. Kersten, for their enthusiastic support of this project.

Palo Alto, California, July 2010 Markus J. Aschwanden

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