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Didier Sornette
Critical Phenomena in Natural Sciences
Chaos, Fractals, Selforganization and Disorder: Concepts and Tools
2. Auflage, 528 Seiten, Paperback
Springer-Verlag GmbH & Co. KG | ISBN: 3540308822
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PrefaceSince its first edition, the ideas discussed in this book have expanded significantly as a result of very active research in the general domain of complex systems. I have also seen with pleasure different communities in the geo-, medical and social sciences becoming more aware of the usefulness of the concepts and techniques presented here. In this second edition, I have first corrected, made more precise and expanded a large number of points. I have also added a significant amount of nov...
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Springer: SPRINGER SERIES COMPLEXITY IN SYNERGETICS Sornette Critical Phenomena in Natural Sciences 2nd Edition Concepts, methods and techniques of statistical physics in the study of correlated, as well as uncorrected, phenomena are being applied ever increasingly in the natural sciences, biology and economics in an attempt to understand and model the large variability and risks of phenomena. This is the first textbook written by a well-known expert that provides a modern up-t... [weiter lesen] |
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Contents
1.Useful Notions of Probability Theory 1
1.1 What Is Probability?1
1.1.1 First Intuitive Notions 1
1.1.2 Objective Versus Subjective Probability 2
1.2 Bayesian View Point 6
1.2.1 Introduction 6
1.2.2 Bayes' Theorem 7
1.2.3 Bayesian Explanation for Change of Belief 9
1.2.4 Bayesian Probability and the Dutch Book 10
1.2.5 Probability Density Function 12
1.2.6 Measures of Central Tendency 13
1.2.7 Measure of Variations from Central Tendency 14
1.2.8 Moments and Characteristic Function 15
1.2.9 Cumulants 16
1.2.10 Maximum of Random Variables and Extreme Value Theory 18
1.8.1 Maximum Value Among TV Random Variables 19
1.8.2 Stable Extreme Value Distributions 23
1.8.3 First Heuristic Derivation of the Stable Gumbel Distribution 25
1.8.4 Second Heuristic Derivation of the Stable Gumbel Distribution 26
1.8.5 Practical Use and Expression of the Coefficients of the Gumbel Distribution 28
1.8.6 The Gnedenko-Pickands-Balkema-de Haan Theorem and the pdf of Peaks-Over-Thresho...
2.Sums of Random Variables, Random Walks and the Central Limit Theorem 33
2.1 The Random Walk Problem 33
2.1.1 Average Drift 34
2.1.2 Diffusion Law 35
2.1.3 Brownian Motion as Solution of a Stochastic ODE .35
2.1.4 Fractal Structure 37
2.1.5 Self-Affinity 39
2.2 Master and Diffusion (Fokker-Planck) Equations 41
2.2.1 Simple Formulation 41
2.2.2 General Fokker-Planck Equation 43
2.2.3 Ito Versus Stratonovich 44
2.2.4 Extracting Model Equations from Experimental Data 47
2.3 The Central Limit Theorem 48
2.3.1 Convolution 48
2.3.2 Statement 50
2.3.3 Conditions 50
2.3.4 Collective Phenomenon 51
2.3.5 Renormalization Group Derivation 52
2.3.6 Recursion Relation and Perturbative Analysis 55
3.Large Deviations 59
3.1 Cumulant Expansion 59
3.2 Large Deviation Theorem 60
3.2.1 Quantification of the Deviation from the Central Limit Theorem 61
3.2.2 Heuristic Derivation of the Large Deviation Theorem (3.9)61
3.2.3 Example: the Binomial Law 63
3.2.4 Non-identically Distributed Random Variables 64
3.3 Large Deviations with Constraints and the Boltzmann Formalism 66
3.3.1 Frequencies Conditioned by Large Deviations 66
3.3.2 Partition Function Formalism 68
3.3.3 Large Deviations in the Dice Game 70
3.3.4 Model Construction from Large Deviations 73
3.3.5 Large Deviations in the Gutenberg-Richter Law and the Gamma Law 76
3.4 Extreme Deviations 78
3.4.1 The "Democratic" Result 78
3.4.2 Application to the Multiplication of Random Variables: a Mechanism for Stretche...
3.4.3 Application to Turbulence and to Fragmentation 83
3.5 Large Deviations in the Sum of Variables with Power Law Distributions 87
3.5.1 General Case with Exponent μ > 287
3.5.2 Borderline Case with Exponent μ = 290
4.Power Law Distributions 93
4.1 Stable Laws: Gaussian and Lévy Laws 93
4.1.1 Definition 93
4.1.2 The Gaussian Probability Density Function 93
4.1.3 The Log-Normal Law 94
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Index
AAbelian sandpile model, 398
advection of passive scalars, 90
aftershocks, 278
aging, 356
anomalous diffusion, 112, 239
anti-ferromagnetic, 442
approximants, 288
ARCH: auto-regressive conditional heteroskedasticity, 381
Arrhenius activation law, 355, 443
asthenosphere, 411
autoregressive process, 232
avalanches, 386, 387, 401, 429
average, 13
BBak-Sneppen model, 307, 417, 427
Barkhausen noise, 386
Bayesian, 7, 70, 163
Bethe lattice, 314
bifurcation, 255
Binomial law, 63
Boltzmann formalism, 66, 358
Boltzmann function, 200, 211, 245, 378
branching, 314, 363, 407
breaking of ergodicity, 441, 450
Breiman's theorem, 375
Brownian motion, 36, 378, 411
Burgers/adhesion model, 351
Burning method, 401
Burridge-Knopof-T model, 370, 405
Ccanonical ensemble, 205
Cantor set, 124
cascade, 235
catastrophe theory, 257, 353
Cauchy distribution, 97
cavity approach, 444
central charge, 146
central limit theorem, 48, 302, 320, 330, 356, 452, 470
chaos, 47
Chaoticity, 444
characteristic function, 16, 98
characteristic scale, 161
Charge-Density-Wave, 415
cloud, 139
clusters, 298, 368
Coast of Britain, 125
collective phenomena, 242
complex dimensions, 159
complex exponents, 157, 276
complex fractal dimension, 156
conditional probability, 7
conformai field theory, 145
conformai transformation, 145
contact processes, 305
continuous-time random walk, 121
control function, 290
control parameter, 407
convolution, 49, 57
correlation, 93, 223, 276
correlation function, 213, 223, 246
correlation length, 247
Coulomb solid friction law, 344, 413
cracks, 293, 313
Cram é r, 78
Cram é r function, 61
craters, 134
crisis, 255
critical exponent, 245, 275
critical phenomena, 245, 259, 294, 341
critical point, 268, 273, 330, 368, 448
cumulants, 16, 49, 54, 59
Ddamage, 313, 335
decimation, 53, 300
decision theory, 10
density of states, 201, 445
dependence, 229
depinning, 417
detailed balance, 214, 412
deviation theorem, 60
dice game, 70
Dieterich friction law, 343
diffusion, 41
Diffusion Limited Aggregation, 139, 148, 278
diffusion-reactions, 199
dimensional analysis, 150
directed percolation, 294, 304, 419
discrete scale invariance, 156, 276
dissipation function, 218
DNA, 225
droplets, 250
Dutch-book argument, 10
Eearthquake, 1, 12, 18, 19, 215, 335, 339, 343, 344, 405, 411
effective medium theory, 296, 313, 441
Ehrenfest classification, 236, 248
entropy, 68, 72, 199, 207, 448
epicenters, 133
epidemics, 306, 381
error function, 59
exceedance, 21
exponential distribution, 167
extended self-similarity, 146
extreme deviations, 78, 362
extreme value theory, 18, 319, 331
Ffaults, 129
feedback, 407
Fermi's theory of cosmic rays, 355
Fermi, Pasta and Ulam, 209
fiber bundle models, 318
Fick's law, 43
financial crash, 255
first order transition, 448
first-order phase transition, 248
first-order transition, 262, 263, 321
fixed point, 93, 273, 300, 330
Flinn-Engdahl regionalization, 76
floods, 107
fluctuation-dissipation, 213, 252
Fokker-Planck equation, 41, 43, 259, 309, 376 forest fires, 306, 391, 402
Fox function, 118
Fréchet distribution, 23 fractal, 282
fractal dimension, 37, 282 fractal growth phenomena, 331 fractals, 239, 336, 470
fractional Brownian motion, 153
fractional derivative, 236
Fractional diffusion equation, 239
fractional integral, 236
fractional noise, 153
fracture, 313
fragmentation, 83, 84, 362, 381, 453, 454
Fredholm integral equation, 471 free energy, 70 frustration, 442
GGamma law, 76, 103, 179, 315, 361
gap equation, 422, 429
Gauss distribution, 14, 21, 43, 87, 202, 246, 302, 356, 386
Gaussian law, 93
generic scale invariance, 409
Gibbs-Duhem relation, 208
glass transition, 356
global warming, 9
Gnedenko-Pickands-Balkema-de Haan theorem, 29
Goldstone modes, 406
grand canonical ensemble, 205
gravity altimetry, 473
Green function, 43, 410, 466, 468
Gumbel distribution, 23
Gutenberg-Richter, 19, 68, 76, 102, 104, 106, 164, 179, 215, 339, 412
Gutenberg-Richter, 2, 160
HHarvard catalog, 77
Hausdorff dimension, 127
Hermite polynomials, 57
hierarchical network, 269, 303, 323, 326, 341, 429
Hill estimator, 168
Holtsmark's gravitational force distribution, 91, 457
homogeneisation theory, 441
Hopf bifurcation, 261
hurricane, 9
Hurst, 153
Hurst effect, 153
Hurst exponent, 434
hyperbolic dynamical system, 435
hysteresis, 386
Ii.i.d., 33
imitation, 243
Infinitely divisible cascades, 146
infinitely divisible distributions, 65
instanton, 266
interacting particles, 243
Internet, 381
Ising model, 243, 341, 349, 368, 386
Iterated Function Systems, 435
Ito interpretation, 44
JJaynes analysis, 74
Jeffreys theory, 70
KKesten multiplicative process, 232, 374
Koch curve, 127
Kramers' problem, 266, 355
Kramers-Moyal expansion, 42
Kullback distance, 67
LLevy law, 96, 240, 365, 458, 459, 462
Levy walk, 110
Lagrange multipliers, 72, 205, 390
lambda point, 416
Landau-Ginzburg theory, 250, 264, 414
Langevin equation, 34, 41, 213, 252, 258, 309, 411
Laplace transform, 116, 364
large deviations, 73, 219
Ledrappier-Young, 436
Lee-Yang phenomenon, 438
Legendre transform, 80
likelihood, 9
Linear Fractional Stable Motion, 434
lithosphere, 411
localization, 332
log-normal, 94, 166, 380
log-periodicity, 152, 157, 276, 304, 341, 368, 455
Mmacrostate, 202
magnetization, 246
Markovian system, 47
Master equation, 41, 376
maximum value, 19
maximum-likelihood, 164, 168, 172, 179, 181
Maxwell construction rule, 249
mean, 13, 94
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