Renormalization methods: Critical Phenomena, Chaos, Fractal Structures

RENORMALIZATION METHODS: CRITICAL PHENOMENA, CHAOS, FRACTAL STRUCTURES......Page 1
Half-title......Page 2
Title Page......Page 3
Copyright Page......Page 4
Contents......Page 6
Foreword, by Pierre Collet......Page 12
Preface......Page 14
1.1 Introduction......Page 16
1.2.1 The critical liquid–gas transition of a pure substance......Page 17
1.2.2 Other examples of critical phenomena......Page 23
1.2.3 Checkerboards and balloons: two classes of phenomena......Page 24
1.2.4 Universality of the critical properties......Page 26
1.3.1 Systems, structure rules and genericity......Page 27
1.3.2 Characteristic scales......Page 29
1.3.3 Thermodynamic limit and asymptotic regime......Page 31
1.3.4 Scaling laws and exponents......Page 32
1.3.5 Control parameter......Page 33
1.3.6 A remark on the term \"renormalization\"......Page 34
1.4.1 Why?......Page 35
1.4.2 How?......Page 38
1.4.3 When?......Page 39
1.4.5 A brief history......Page 40
Remarks and bibliographical notes......Page 43
2.1.1 The model and its importance......Page 46
2.1.2 Analysis via renormalization......Page 48
2.1.3 Results and extensions......Page 51
2.2.1 Discrete dynamical systems......Page 54
2.2.2 The pitchfork bifurcation......Page 56
2.2.3 The period-doubling scenario......Page 58
2.2.4 Renorrnalization analysis......Page 61
2.3 A comparative table......Page 66
Remarks and bibliographical notes......Page 68
3.1.1 Goals and typical approaches......Page 70
3.1.2 Transfer of limits......Page 74
3.1.3 Fixed points and critical manifolds......Page 75
3.1.4 Renormalization of a one-parameter family......Page 78
3.2.1 The probabilistic approach to a critical phenomenon......Page 82
3.2.2 Fluctuations and correlations......Page 86
3.2.3 Mean-field theories and the law of large numbers......Page 89
3.2.4 Critical phenomena and finite scaling laws......Page 92
3.3.1 Simulation of critical phenomena......Page 94
3.3.3 Numerical resolution of the renormalization equations......Page 95
3.3.4 Renormalization and the Monte Carlo method......Page 96
3.4.1 Group law, Lie groups and infinitesimal generators......Page 100
3.4.2 Transformation groups and associated symmetries......Page 103
3.4.3 Representations and symmetry groups......Page 105
Remarks and bibliographical notes......Page 107
4.1.1 Magnetic media and ferromagnetic transition......Page 110
4.1.2 Langevin theory and Curie\'s law......Page 113
4.1.3 Mean-field theory and the Curie–Weiss law......Page 114
4.2.1 Goals of a theory of critical phenomena......Page 117
4.2.2 Ferromagnetic and liquid–gas transitions......Page 119
4.2.3 Limits of the methods preceding renormalization......Page 120
4.3.1 Methods in real space......Page 122
4.3.2 Fields and path integrals......Page 125
4.3.3 Renormalization in conjugate space......Page 127
4.3.4 The main universality classes......Page 128
4.4.1 Qualitative aspects......Page 131
4.4.2 Renormalization methods......Page 133
Remarks and bibliographical notes......Page 137
4A.1 The Gaussian model and perturbative expansions......Page 138
4A.2 Analytic expression of the graphs......Page 140
4A.3 Approximations and results......Page 142
Remarks and bibliographical notes......Page 143
4B.1 Paramagnetism, ferromagnetism and spin glasses......Page 144
4B.2 Frustrated systems......Page 145
4B.3 Transition to the spin–glass phase......Page 148
4B.4 Renormalization methods......Page 149
Remarks and bibliographical notes......Page 150
5.1.1 Asymptotic dynamics......Page 152
5.1.2 Discretization methods......Page 154
5.1.3 Chaotic properties......Page 155
5.1.4 Scenarios towards chaos and adapted renormalization methods......Page 160
5.1.5 Transition towards chaos and critical phase transitions......Page 163
5.1.6 Power spectrum and correlation time......Page 166
5.1.7 Dynamical systems with noise......Page 170
5.2.1 Self-similarity of the bifurcation scheme......Page 172
5.2.2 Self-similarity of the critical attractor......Page 175
5.2.3 Renormalization analysis......Page 177
5.3.1 Description of the scenario......Page 179
5.3.2 Renormalization analysis......Page 181
5.3.3 The influence of noise......Page 182
5.4.1 Quasi-periodic attractors and strange attractors......Page 183
5.4.2 The theorem of Ruelle and Takens......Page 184
5.5.1 The Hamiltonian formalism......Page 186
5.5.2 The KAM theorem and Moser\'s \"twist\" theorem......Page 187
5.5.3 Renormalization and universal properties......Page 190
Remarks and bibliographical notes......Page 193
5A.1 Invariant measures......Page 196
5A.2 Ergodicity and the law of large numbers......Page 198
5A.3 Temporal correlations and the mixing property......Page 201
5A.4 The microcanonical hypothesis and ergodicity......Page 203
Remarks and bibliographical notes......Page 204
5B.1 External influences and fibered formalism......Page 206
5B.2 Renormalization in the presence of noise......Page 207
5B.3 Linear analysis and spectral decomposition......Page 209
5B.4 Critical exponents: results and conclusions......Page 211
Remarks and bibliographical notes......Page 212
5C.1 Circle mappings and rotation numbers......Page 214
5C.2 Continued fraction expansions and the golden ratio......Page 216
5C.3 Asymptotic stability and renormalization......Page 218
5C.4 Universal properties......Page 220
Remarks and bibliographical notes......Page 221
5D.1 Fully developed turbulence and deterministic chaos......Page 222
5D.2 Hydrodynamic equations and Richardson\'s cascade......Page 223
5D.3 Renormalization for large-scale dynamics......Page 225
Remarks and bibliographical notes......Page 232
6.1.1 The example of Brownian motion......Page 234
6.1.2 Random motion and diffusion laws......Page 238
6.1.3 Formalization: stochastic processes and random walks......Page 240
6.2.1 Renormalization of Brownian motion......Page 242
6.2.2 Asymptotic behavior and renormalization of the process......Page 244
6.2.3 Self-similar processes......Page 247
6.2.4 Renormalization of the probabilities of transition......Page 250
6.2.5 The example of diffusion in a disordered medium......Page 252
Remarks and bibliographical notes......Page 260
6A.1 Polymer physics......Page 262
6A.2 Polymer chains and random walks......Page 263
6A.3 Geometric renormalization methods......Page 269
6A.4 Polymers in statistical mechanics......Page 274
Remarks and bibliographical notes......Page 278
7.1.1 Critical aspects of fractal structures......Page 280
7.1.2 Real inhomogeneous fractals......Page 284
7.2.1 Local dimension and dimension spectrum......Page 286
7.2.2 Multifractal analysis......Page 288
7.2.3 Renormalization of a measure......Page 291
7.3 The wavelet transform......Page 292
7.3.1 Transformation formulas......Page 293
7.3.2 Local scale invariance and renormalization......Page 294
Remarks and bibliographical notes......Page 298
7A.1 Percolation models: clusters and percolation threshold......Page 300
7A.2 Static aspects......Page 305
7A.3 Renormalization methods......Page 309
7A.4 Dynamic aspects......Page 315
Remarks and bibliographical notes......Page 317
1.1 Measurable spaces and measures......Page 320
1.2 Random variables and stochastic convergence......Page 321
1.3 Stochastic processes and Markov chains......Page 323
Remarks and bibliographical notes......Page 325
2.1 Discrete dynamical systems......Page 326
2.2 Continuous dynamical systems......Page 328
2.3 Stable and unstable manifolds......Page 330
2.4 Ergodic theory......Page 332
Remarks and bibliographical notes......Page 334
3.1 The microcanonical ensemble......Page 336
3.2 The canonical ensemble......Page 337
Remarks and bibliographical notes......Page 339
4.1 Transformation formulas......Page 340
4.2 Basic properties......Page 341
Remarks and bibliographical notes......Page 342
Bibliography......Page 344
Index [missing]......Page 364
Back Cover......Page 365