Physics and fractal structures

Foreword......Page 3
Preface......Page 8
1.1 Introduction......Page 10
1.2 The notion of dimension......Page 11
1.3 Metric properties: Hausdorff dimension, topological dimension......Page 13
1.3.2 The Hausdorff–Besicovitch dimension......Page 14
1.3.3 The Bouligand–Minkowski dimension......Page 15
1.3.4 The packing dimension......Page 17
1.4.1 Deterministic fractals......Page 19
1.4.2 Random fractals......Page 28
1.4.3 Scale invariance......Page 30
1.4.4 Ambiguities in practical measurements......Page 31
1.5.1 Spreading dimension, dimension of connectivity......Page 32
1.5.3 The lacunarity L......Page 34
1.6 Multifractal measures......Page 35
1.6.1 Binomial fractal measure......Page 36
1.6.2 Multinomial fractal measure......Page 39
1.6.3 Two scale Cantor sets......Page 45
1.6.4 Multifractal measure on a set of points......Page 46
2.1 Distribution of galaxies......Page 50
2.1.1 Distribution of clusters in the universe......Page 51
2.1.2 Olbers’ blazing sky paradox......Page 52
2.2 Mountain reliefs, clouds, fractures......Page 55
2.2.1 Brownian motion, its fractal dimension......Page 56
2.2.2 Scalar Brownian motion......Page 58
2.2.4 Fractional Brownian motion......Page 59
2.2.5 Self-affine fractals......Page 62
2.2.6 Mountainous reliefs......Page 67
2.2.7 Spectral density of a fractional Brownian motion,the spectral exponent ß......Page 68
2.2.8 Clouds......Page 71
2.2.9 Fractures......Page 72
2.3 Turbulence and chaos......Page 75
2.3.1 Fractal models of developed turbulence......Page 76
2.3.2 Deterministic chaos in dissipative systems......Page 82
3.1.1 A model: percolation......Page 98
3.1.2 Evaporated films......Page 114
3.2 Porous media......Page 116
3.2.1 Monophasic flow in poorly connected media......Page 117
3.2.2 Displacement of a fluid by another in a porous medium......Page 118
3.2.3 Quasistatic drainage......Page 120
3.3.1 Diffusion fronts of noninteracting particles......Page 127
3.3.2 The attractive interaction case......Page 134
3.4.1 Definition of aggregation......Page 139
3.4.2 Aerosols and colloids......Page 141
3.4.3 Macroscopic aggregation......Page 149
3.4.4 Layers deposited by sputtering......Page 150
3.4.5 Aggregation in a weak field......Page 151
3.5.1 Fractal properties of polymers......Page 155
3.5.2 Fractal properties of membranes......Page 162
4.1 The Eden model......Page 167
4.1.1 Growth of the Eden cluster: scaling laws......Page 169
4.1.2 The Williams and Bjerknes model......Page 172
4.1.3 Growing percolation clusters......Page 174
4.2.1 Description of the DLA model......Page 175
4.2.2 Extensions of the Witten and Sander model......Page 177
4.2.3 The harmonic measure and multifractality......Page 182
4.3.2 Deposition models......Page 184
4.3.3 Analytical approach to the growth of rough surfaces......Page 186
4.4.1 Diffusion–limited cluster–cluster aggregation......Page 187
4.4.2 Reaction–limited cluster–cluster aggregation......Page 189
4.4.3 Ballistic cluster–cluster aggregation and other models......Page 190
5.1.1 Spectral dimension......Page 193
5.1.2 Diffusion and random walks......Page 198
5.1.3 Distinct sites visited by diffusion......Page 201
5.1.4 Phonons and fractons in real systems......Page 202
5.2.1 Conduction through a fractal......Page 204
5.2.2 Conduction in disordered media......Page 207
5.2.3 Dielectric behavior of composite media......Page 217
5.2.4 Response of viscoelastic systems......Page 218
5.3 Exchanges at interfaces......Page 221
5.3.1 The diffusion-limited regime......Page 222
5.3.2 Response to a blocking electrode......Page 223
5.4 Reaction kinetics in fractal media......Page 225
COLOR PLATE LEGENDS......Page 228
Bibliography......Page 233
Index......Page 245