Complex Physical, Biophysical and Econophysical Systems: Proceedings of the 22nd Canberra International Physics Summer School

Preface......Page 6
Contents......Page 10
Contents......Page 12
1.1. Introduction......Page 13
1.2. An Example of Complex Systems: Financial Markets......Page 16
1.3. Probabilities and Improbabilities......Page 17
1.3.1. Log-returns......Page 18
1.3.2. Leptokurtic distributions......Page 19
1.3.3. Distribution tails......Page 20
1.4.1. Tendency towards normal distribution......Page 21
1.4.2. Violation of central limit theorem......Page 22
1.4.4. Stable distributions......Page 23
1.5.1. Extreme uctuations......Page 24
1.5.3. Fat-tailed distributions......Page 25
1.5.4. Sum of power-law-tailed distributions......Page 27
1.6.2. Rank frequency plot......Page 28
1.7.1. Price uctuations as random walks......Page 29
1.7.3. High frequency data......Page 31
1.8.1. Super-diffusive processes......Page 32
1.8.2. Sub-diffusive processes......Page 33
1.8.4. Multi-scaling......Page 34
1.9. Complex Networked Systems......Page 35
1.9.2. Small and ultra-small worlds......Page 36
1.9.4. Dependency......Page 37
1.9.6. Significance......Page 39
1.9.7. Building the network......Page 40
1.9.8. Disentangling the network: minimum spanning tree......Page 41
1.9.9. Disentangling the network: planar maximally  filtered graph......Page 43
1.10. Conclusion......Page 44
References......Page 45
Contents......Page 48
2.1.1. Brownian motion and the Langevin equation......Page 49
2.1.2.1. Random walks and the binomial distribution......Page 52
2.1.2.2. Random walks and the normal distribution......Page 53
2.1.2.3. Random walks in the continuum approximation......Page 54
2.1.2.4. Central limit theorem......Page 56
2.1.3. Fick's law and the diffusion equation......Page 57
2.1.3.1. Generalized diffusion equations......Page 59
2.1.4. Master equations and the Fokker–Planck equation......Page 60
2.1.5. The Chapman–Kolmogorov equation and Markov processes......Page 61
2.1.5.1. Wiener process......Page 62
2.1.5.2. Poisson process......Page 63
2.2. Fractional diffusion......Page 64
2.2.1. Diffusion on fractals......Page 65
2.2.2. Fractional Brownian motion......Page 66
2.2.3.1. CTRW master equations......Page 68
2.2.3.2. Exponential waiting times and standard diffusion......Page 71
2.2.3.3. Power law waiting times and fractional subdiffusion......Page 73
2.2.3.4. Subordinated diffusion......Page 75
2.2.3.5. Levy flights and fractional superdiffusion......Page 76
2.2.4. Simulating random walks for fractional diffusion......Page 78
2.2.4.1. Generation of waiting-times......Page 79
2.2.4.2. Generation of jump-lengths......Page 80
2.2.4.4. Probability density functions......Page 81
2.2.5. Fractional Fokker–Planck equations......Page 82
2.2.6. Fractional Reaction-Diffusion equations......Page 85
2.2.8. Power laws and fractional diffusion......Page 87
2.3. Appendix: Introduction to fractional calculus......Page 89
2.3.1. Riemann–Liouville fractional integral......Page 90
2.3.2.1. Tautochrone problem......Page 91
2.3.4. Fourier and Laplace transforms and fractional Calculus......Page 93
2.3.5. Special functions for fractional calculus......Page 94
Acknowledgments......Page 97
References......Page 98
3.1. Introduction......Page 102
3.2. Complex Systems Models for Global Tokamak Phenomenology......Page 108
3.3. Complex Systems Modelling of Earth's Magnetosphere......Page 111
3.4. A Global Complex Systems Model for Aspects of Solar Coronal Plasma and Magnetic Fields......Page 114
3.5. Scaling Properties of Turbulent Fluctuations in Fusion Plasmas......Page 117
3.6. Large Scale Astrophysical Objects as Complex Systems......Page 119
3.7. Solar Wind Plasma as a Complex System......Page 123
3.8. Information Theory and Plasma Physics......Page 125
3.9. Conclusion......Page 127
References......Page 128
4.1. Scientific Inference......Page 132
4.2.1. Bayes's theorem......Page 133
4.2.2. Bayesian parameter estimation......Page 134
4.2.4. Is this coin fair?......Page 137
4.2.5. Markov chain Monte Carlo (MCMC)......Page 140
4.2.6. Relationship to maximum likelihood and least squares......Page 141
4.2.7. Classical hypothesis testing......Page 143
4.3.1. Background......Page 144
4.3.2. Flare statistics......Page 145
4.3.3. Event statistics method of prediction......Page 146
4.3.4. Whole-Sun prediction of GOES ares......Page 149
4.4. Summary......Page 151
References......Page 152
Contents......Page 154
5.1.2. Reinventing reductionism......Page 155
5.1.3. The modelling spectrum......Page 157
5.1.4. A dichotomy......Page 160
5.1.5. A big picture?......Page 162
5.2.1. Earth......Page 164
5.2.2. Questions......Page 165
5.2.3. Implications of complexity for earth system science......Page 166
5.2.4. Inverse problems......Page 169
5.2.5. Data assimilation in meteorology......Page 171
5.3.1. The carbon cycle......Page 172
5.3.2. Estimating carbon uxes......Page 173
5.3.3. Deducing emission targets......Page 175
5.3.4. Future directions......Page 176
5.4.1. Carbon cycle process inversions......Page 178
5.4.2. Feedbacks......Page 182
5.4.3. Analysing feedbacks......Page 183
5.4.4. Laplace transform analysis......Page 184
5.5. Gaia......Page 186
5.6. Concluding Remarks......Page 188
Acknowledgments......Page 189
References......Page 191
Contents......Page 198
6.1.1. What is chaotic advection?......Page 199
6.1.2. Examples of chaotic advection in practice......Page 200
6.1.2.1. Small scale......Page 201
6.1.2.2. Medium scale......Page 204
6.1.2.3. Large scale......Page 209
6.1.3. Objectives and major points......Page 211
6.2. Illustrating Basic Concepts of Transport in Planar Flows......Page 213
6.2.1. Motion, kinematics, continuity......Page 214
6.2.2. Steady ow local topologies......Page 216
6.2.3. Chaotic tangles......Page 217
6.2.3.1. Homoclinic connection......Page 218
6.2.3.2. Heteroclinic......Page 221
6.2.4. Where are applications?......Page 222
6.3.1. Poincar  sections......Page 224
6.3.2. Periodic points......Page 225
6.3.3. Dye advection and stretching......Page 227
6.4. Periodically Reoriented Flows: Mixing, Scalar Transport, and Parametric Variation......Page 229
6.4.1. Scalar transport......Page 230
6.4.2. Parametric variation......Page 232
6.5.1. Reection and rotation in the plane......Page 235
6.5.2. Time-reversal symmetry......Page 237
6.5.3. Origin of tongues......Page 240
6.6. Summary......Page 245
References......Page 247
7. Approaches to Modelling the Dynamical Activity of Brain Function Based on the Electroencephalogram D.T.J. Liley and F. Frascoli......Page 252
7.1. Introduction......Page 253
7.1.1. Chapter outline......Page 258
7.2. Overview of Macroscopic Neural Modelling Approaches......Page 259
7.2.1. The Wilson-Cowan neural field equations......Page 260
7.2.2. The lateral-inhibition type neural field theory of Amari......Page 262
7.2.3.1. Freeman's neural mass action and the K-set hierarchy......Page 264
7.2.3.2. Nunez's global model of electrorhythmogenesis......Page 266
7.2.3.3. Lopes da Silva's local model of the alpha rhythm......Page 267
7.2.3.5. Liley's coupled macrocolumnar continuum theory......Page 268
7.3. Linearisation and Parameter Space Search......Page 270
7.4. Characteristics of the Model Dynamics in the Liley Equations: Multistability, Chaos, Bifurcations and Emergence......Page 274
7.5. Conclusion......Page 286
Appendix A. Continuation Software and Selected Problems......Page 287
References......Page 289
8. Jaynes' Maximum Entropy Principle, Riemannian Metrics and Generalised Least Action Bound R.K. Niven and B. Andresen......Page 294
8.1. Introduction......Page 295
8.2.1. Theoretical principles......Page 296
8.2.2. The generalised free energy concept......Page 299
8.3.1. Generalised Riemannian metrics and arc lengths......Page 300
8.3.2. Generalised action concepts and least action bound......Page 303
8.4. Applications......Page 309
8.4.1. Equilibrium thermodynamic systems......Page 310
8.4.2. Flow systems......Page 315
8.5. Conclusion......Page 320
Appendix 1: Riemannian Geometric Considerations......Page 321
References......Page 323
Contents......Page 330
9.1.1. Organisms as complex systems......Page 331
9.2.1. Parts: genes and their products......Page 333
9.2.2. The central dogma of molecular biology......Page 334
9.2.3. The text, and its reader......Page 336
9.2.4. The real world: genetic variation......Page 337
9.3.1. Genome sequencing and its inuence......Page 338
9.3.3. Post-genomic biology as an information science?......Page 339
9.3.4. Confirmation or exploration?......Page 341
9.4.1. The eukaryotic nucleus and gene regulation......Page 342
9.4.2. Local control......Page 343
9.4.3. Extended control......Page 344
9.4.4. Large–scale control......Page 347
9.5. Methods for Identifying Regulatory Control......Page 348
9.6. A Simple Approach to Coordinated Control of Gene Expression......Page 352
9.7. Summary and Outlook......Page 356
References......Page 357
Contents......Page 362
10.1.1. Introduction......Page 363
10.1.2. Introduction to NetLogo......Page 364
10.1.3. NetLogo interface......Page 365
10.1.4. NetLogo programming......Page 367
10.1.5. The art of agent-based modelling......Page 372
10.2.1. Introduction to Pajek with reference to other network analysis tools......Page 373
10.2.2. Importing network data......Page 375
10.2.3. Visualising networks......Page 378
10.2.6. Generating a comparison network......Page 380
10.2.7. Common network measures......Page 381
References......Page 384
Author Index......Page 388
Subject Index......Page 390