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Part I Characterization of Complex Systems |
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1 Primer on Complex Systems |
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3 | (38) |
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3 | (9) |
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1.1.1 What Is a Complex System? |
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3 | (2) |
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1.1.2 Examples of Complex Systems |
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5 | (2) |
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1.1.3 Complex Systems in Plasma Science |
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7 | (3) |
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10 | (2) |
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1.2 Key Concepts in the Study of Complex Systems |
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12 | (13) |
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1.2.1 Defining Characteristics |
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13 | (1) |
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14 | (7) |
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1.2.3 Main Emergent Features |
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21 | (4) |
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1.3 Self-organized Criticality |
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25 | (9) |
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1.3.1 The Running Sandpile |
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26 | (2) |
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1.3.2 Criticality in the SOC State |
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28 | (2) |
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1.3.3 Memory in the SOC State |
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30 | (2) |
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1.3.4 Transport in the SOC State |
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32 | (2) |
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1.4 Overview of the First Part of This Book |
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34 | (1) |
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Appendix 1 Fixed Points of a Dynamical System |
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35 | (1) |
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36 | (1) |
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37 | (4) |
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41 | (62) |
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41 | (1) |
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2.2 The Probability Density Function |
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42 | (9) |
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42 | (1) |
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2.2.2 Cumulative Distribution Function |
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43 | (1) |
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44 | (1) |
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2.2.4 Characteristic Function |
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45 | (2) |
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47 | (1) |
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47 | (1) |
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48 | (3) |
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2.3 Significance of Specific Pdfs |
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51 | (19) |
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2.3.1 Gaussian and Levy Pdfs: Additive Processes |
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52 | (5) |
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2.3.2 Log-Normal and Log-Stable Pdfs: Multiplicative Processes |
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57 | (4) |
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2.3.3 Weibull, Gumbel and Frechet Pdfs: Extreme Value Pdfs |
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61 | (3) |
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2.3.4 Exponential and Related Pdfs: Poisson Processes |
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64 | (6) |
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2.4 Techniques for the Practical Estimation of Pdfs |
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70 | (11) |
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2.4.1 Constant Bin Size Method |
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71 | (2) |
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2.4.2 Constant Bin Content Method |
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73 | (5) |
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2.4.3 Survival/Cumulative Distribution Function Method |
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78 | (3) |
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2.5 Techniques to Compare Experimentally Obtained Pdfs with Analytical Forms |
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81 | (9) |
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2.5.1 Maximum Likelihood Estimators |
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81 | (4) |
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2.5.2 Pearson's Goodness-of-Fit Test |
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85 | (4) |
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2.5.3 Minimum Chi-Square Parameter Estimation |
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89 | (1) |
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2.6 Case Study: The Running Sandpile |
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90 | (4) |
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94 | (1) |
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Appendix 1 The Fourier Transform |
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94 | (3) |
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Appendix 2 Numerical Generation of Series with Prescribed Statistics |
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97 | (2) |
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99 | (2) |
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101 | (2) |
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103 | (74) |
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103 | (3) |
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3.2 Scale-Invariance in Space |
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106 | (14) |
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107 | (4) |
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111 | (9) |
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3.3 Scale-Invariance in Time |
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120 | (25) |
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3.3.1 Self-Similar Time Random Processes |
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121 | (1) |
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3.3.2 Propagator of a Random Process |
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122 | (8) |
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3.3.3 Fractional Brownian Motion |
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130 | (3) |
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3.3.4 Fractional Levy Motion |
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133 | (2) |
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3.3.5 Stationarity and Self-Similarity |
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135 | (3) |
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3.3.6 Self-Similar Processes with Stationary Increments |
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138 | (2) |
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3.3.7 Multifractal Time Random Processes |
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140 | (5) |
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3.4 Techniques for the Practical Determination of Scale-Invariance |
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145 | (17) |
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3.4.1 Analysis of Non-stationary Processes |
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146 | (4) |
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3.4.2 Analysis of Stationary Processes |
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150 | (6) |
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3.4.3 Multifractal Analysis |
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156 | (6) |
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3.5 Case Study: The Running Sandpile |
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162 | (5) |
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167 | (1) |
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Appendix 1 Numerical Generation of Fractional Noises |
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167 | (2) |
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Appendix 2 Detrended Fluctuation Analysis |
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169 | (2) |
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Appendix 3 Multifractal Analysis via Wavelets |
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171 | (1) |
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172 | (1) |
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173 | (4) |
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177 | (44) |
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177 | (1) |
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4.2 Memory and Correlation |
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178 | (17) |
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4.2.1 The Autocorrelation Function |
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178 | (10) |
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188 | (5) |
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4.2.3 The Autodifference Function |
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193 | (2) |
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4.3 Memory in Self-Similar Time Random Processes |
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195 | (3) |
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4.3.1 Fractional Brownian Motion |
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195 | (1) |
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4.3.2 Fractional Levy Motion |
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196 | (2) |
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4.4 Techniques for Detecting Memory in Stationary Time Series |
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198 | (15) |
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4.4.1 Methods Based on the Autocorrelation Function |
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199 | (1) |
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4.4.2 Methods Based on the Power Spectrum |
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200 | (2) |
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4.4.3 Methods Based on the Autodifference |
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202 | (1) |
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4.4.4 Hurst's Rescaled Range (R/S) Method |
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203 | (9) |
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4.4.5 Waiting Time Statistics |
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212 | (1) |
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4.5 Case Study: The Running Sandpile |
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213 | (5) |
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218 | (1) |
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218 | (1) |
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219 | (2) |
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5 Fundamentals of Fractional Transport |
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221 | (58) |
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221 | (2) |
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5.2 Diffusive Transport: Fundamentals |
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223 | (11) |
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5.2.1 The Continuous-Time Random Walk |
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223 | (7) |
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5.2.2 The Langevin Equation |
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230 | (4) |
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5.3 Scale Invariant Formulations of Transport |
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234 | (13) |
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5.3.1 Scale Invariant Continuous-Time Random Walks |
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234 | (7) |
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5.3.2 The Fractional Langevin Equation |
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241 | (2) |
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5.3.3 The Fractional Transport Equation |
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243 | (4) |
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5.4 Techniques for the Characterization of Fractional Transport |
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247 | (8) |
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247 | (5) |
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252 | (3) |
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5.5 Case Study: The Running Sandpile |
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255 | (6) |
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5.5.1 fTe for the Directed Running Sandpile |
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260 | (1) |
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261 | (3) |
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Appendix 1 The Laplace Transform |
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264 | (1) |
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Appendix 2 Riemann-Liouville Fractional Derivatives and Integrals |
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265 | (4) |
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Appendix 3 The Riesz-Feller Fractional Derivative |
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269 | (1) |
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Appendix 4 Discrete Approximations for Fractional Derivatives |
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270 | (3) |
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273 | (1) |
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274 | (5) |
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Part II Complex Dynamics in Magnetized Plasmas |
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6 Laboratory Fusion Plasmas: Dynamics of Near-Marginal Turbulent Radial Transport |
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279 | (34) |
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279 | (1) |
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6.2 Nuclear Fusion Processes |
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280 | (4) |
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6.3 Primer on Magnetic Confinement Fusion |
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284 | (7) |
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285 | (2) |
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287 | (1) |
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6.3.3 Main Transport Processes in Toroidal MCF Plasmas |
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288 | (3) |
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6.4 Are MCF Plasmas Complex Systems? |
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291 | (8) |
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6.4.1 Tokamak Transport Phenomenology |
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292 | (3) |
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6.4.2 Stellarator Confinement Phenomenology |
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295 | (1) |
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6.4.3 Self-organized Criticality and Toroidal MCF Plasmas |
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296 | (3) |
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6.5 Case Study: Analysis of Turbulent Fluctuations from the Edge of the W7-AS Stellarator |
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299 | (9) |
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300 | (3) |
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303 | (1) |
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304 | (2) |
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6.5.4 Multifractal Analysis |
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306 | (2) |
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308 | (1) |
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309 | (4) |
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7 Space Plasmas: Complex Dynamics of the Active Sun |
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313 | (26) |
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313 | (1) |
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7.2 Our Own Star: The Sun |
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313 | (6) |
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7.2.1 Structure of the Sun |
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314 | (3) |
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7.2.2 The Active Magnetic Sun |
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317 | (2) |
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7.3 Is Our Sun a Complex System? |
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319 | (5) |
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7.3.1 The Tachocline: A Case of Self-Organization |
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320 | (1) |
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7.3.2 Scale-Invariance of Solar Flare Data |
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321 | (1) |
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7.3.3 Lu-Hamilton SOC Flaring Model |
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322 | (2) |
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7.4 Case Study: Analysis of the SOHO-LASCO CME Database (1996--2016) |
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324 | (11) |
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7.4.1 Waiting-Time Statistics |
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326 | (1) |
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7.4.2 Linear Speed Analysis |
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326 | (3) |
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7.4.3 Ejected Mass Analysis |
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329 | (4) |
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7.4.4 Ejected Energy Analysis |
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333 | (2) |
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335 | (1) |
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335 | (4) |
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8 Planetary Plasmas: Complex Dynamics in the Magnetosphere of the Earth |
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339 | (42) |
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339 | (1) |
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8.2 The Magnetosphere of the Earth |
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340 | (12) |
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8.2.1 The Geomagnetic Field |
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341 | (1) |
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8.2.2 Structure of the Magnetosphere of the Earth |
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342 | (5) |
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8.2.3 Dynamics of the Magnetosphere of the Earth |
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347 | (5) |
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8.3 Is the Magnetosphere of the Earth a Complex System? |
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352 | (4) |
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8.3.1 Chang's SOC Substorming Model |
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353 | (3) |
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8.3.2 Evidence of Critical Dynamics in the Magnetotail |
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356 | (1) |
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8.4 Case Study: Magnetospheric and Solar Wind Indices |
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356 | (22) |
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8.4.1 Analysis of the Dst Index (1957--2008) |
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357 | (6) |
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8.4.2 Analysis of the AE Index (1990--2008) |
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363 | (6) |
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8.4.3 Analysis of the Scalar Bin the Solar Wind (1963--2017) |
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369 | (5) |
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8.4.4 Analysis of the Proton Density in the Solar Wind (1963--2017) |
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374 | (4) |
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378 | (1) |
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378 | (3) |
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9 Laboratory Plasmas: Dynamics of Transport Across Sheared Flows |
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381 | (20) |
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381 | (1) |
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382 | (6) |
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9.2.1 Differential Rotation and Magnetic Fields |
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383 | (1) |
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9.2.2 Turbulence Suppression by Sheared Flows |
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384 | (2) |
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9.2.3 Zonal Flows in Tokamaks |
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386 | (2) |
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9.3 Non-diffusive Transport Across Sheared Flows |
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388 | (3) |
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9.4 Case Study: Transport Across Self-Consistent Zonal Flows in Ion-Temperature-Gradient (ITG) Tokamak Turbulence |
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391 | (8) |
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9.4.1 Radial Propagator Analysis |
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395 | (2) |
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9.4.2 Other Considerations |
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397 | (2) |
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399 | (1) |
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399 | (2) |
Index |
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401 | |