Fractal structures are found everywhere in nature, and as a consequence anomalous diffusion has far reaching implications in a host of phenomena. This book describes diffusion and transport in disordered media such as fractals, porous rocks and random resistor networks. Part I contains material of general interest to statistical physics: fractals, percolation theory, regular random walks and diffusion, continuous time random walks and Levy walks, and flights. Part II covers anomalous diffusion in fractals and disordered media, while Part III serves as an introduction to the kinetics of diffusion-limited reactions. Part IV discusses the problem of diffusion-limited coalescence in one dimension. This book will be of particular interest to researchers requiring a clear introduction to the field. It will also be a valuable source to graduate students studying in areas of physics, chemistry, and engineering.
Author(s): Daniel ben-Avraham, Shlomo Havlin
Publisher: CUP
Year: 2000
Language: English
Pages: 331
Cover......Page 1
About......Page 2
Diffusion and Reactions in Fractals and Disordered Systems......Page 4
Goto 4 /FitH 555521617200......Page 5
Contents......Page 8
Preface......Page 14
Part one: Basic concepts......Page 16
1.1 Deterministic fractals......Page 18
1.2 Properties of fractals......Page 21
1.3 Random fractals......Page 22
1.4 Self-affine fractals......Page 24
1.5 Exercises......Page 26
1.7 Further reading......Page 27
2.1 The percolation transition......Page 28
2.2 The fractal dimension of percolation......Page 33
2.3 Structural properties......Page 36
2.4 Percolation on the Cayley tree and scaling......Page 40
2.5 Exercises......Page 43
2.6 Open challenges......Page 45
2.7 Further reading......Page 46
3.1 The simple random walk......Page 48
3.2 Probability densities and the method of characteristic functions......Page 50
3.3 The continuum limit: diffusion......Page 52
3.4 Einstein's relation for diffusion and conductivity......Page 54
3.5 Continuous-time random walks......Page 56
3.6 Exercises......Page 58
3.7 Open challenges......Page 59
3.8 Further reading......Page 60
4.1 Random walks as fractal objects......Page 61
4.2 Anomalous continuous-time random walks......Page 62
4.3 Levy flights and Levy walks......Page 63
4.4 Long-range correlated walks......Page 65
4.5 One-dimensional walks and landscapes......Page 68
4.7 Open challenges......Page 70
4.8 Further reading......Page 71
Part two: Anomalous diffusion......Page 72
5.1 Anomalous diffusion......Page 74
5.2 The first-passage time......Page 76
5.3 Conductivity and the Einstein relation......Page 78
5.4 The density of states: fractons and the spectral dimension......Page 80
5.5 Probability densities......Page 82
5.6 Exercises......Page 85
5.7 Open challenges......Page 86
5.8 Further reading......Page 87
6.1 The analogy with diffusion in fractals......Page 89
6.2 Two ensembles......Page 90
6.3 Scaling analysis......Page 92
6.4 The Alexander-Orbach conjecture......Page 94
6.5 Fractons......Page 97
6.6 The chemical distance metric......Page 98
6.7 Diffusion probability densities......Page 102
6.8 Conductivity and multifractals......Page 104
6.10 Dynamical exponents in continuum percolation......Page 107
6.11 Exercises......Page 109
6.12 Open challenges......Page 110
6.13 Further reading......Page 111
7.1 Loopless fractals......Page 113
7.2 The relation between transport and structural exponents......Page 116
7.3 Diffusion in lattice animals......Page 118
7.4 Diffusion in DLAs......Page 119
7.5 Diffusion in combs with infinitely long teeth......Page 121
7.6 Diffusion in combs with varying teeth lengths......Page 123
7.7 Exercises......Page 125
7.8 Open challenges......Page 127
7.9 Further reading......Page 128
8.1 Types of disorder......Page 129
8.2 The power-law distribution of transition rates......Page 132
8.3 The power-law distribution of potential barriers and wells......Page 133
8.4 Barriers and wells in strips (n x infty) and in d >= 2......Page 134
8.5 Barriers and wells in fractals......Page 136
8.6 Random transition rates in one dimension......Page 137
8.7 Exercises......Page 139
8.8 Open challenges......Page 140
8.9 Further reading......Page 141
9 Biased anomalous diffusion......Page 142
9.1 Delay in a tooth under bias......Page 143
9.2 Combs with exponential distributions of teeth lengths......Page 144
9.3 Combs with power-law distributions of teeth lengths......Page 146
9.4 Topological bias in percolation clusters......Page 147
9.5 Cartesian bias in percolation clusters......Page 148
9.6 Bias along the backbone......Page 150
9.7 Time-dependent bias......Page 151
9.8 Exercises......Page 153
9.9 Open challenges......Page 154
9.10 Further reading......Page 155
10.1 Tracer diffusion......Page 156
10.2 Tracer diffusion in fractals......Page 158
10.3 Self-avoiding walks......Page 159
10.4 Flory's theory......Page 161
10.5 SAWs in fractals......Page 163
10.6 Exercises......Page 166
10.7 Open challenges......Page 167
10.8 Further reading......Page 168
Part three: Diffusion-limited reactions......Page 170
11.1 The limiting behavior of reaction processes......Page 172
11.2 Classical rate equations......Page 174
11.3 Kinetic phase transitions......Page 176
11.4 Reaction-diffusion equations......Page 178
11.5 Exercises......Page 179
11.7 Further reading......Page 181
12.1 Smoluchowski's model and the trapping problem......Page 182
12.2 Long-time survival probabilities......Page 183
12.3 The distance to the nearest surviving particle......Page 186
12.5 Imperfect traps......Page 189
12.6 Exercises......Page 190
12.7 Open challenges......Page 191
12.8 Further reading......Page 192
13.1 One-species reactions: scaling and effective rate equations......Page 194
13.2 Two-species annihilation: segregation......Page 197
13.3 Discrete fluctuations......Page 200
13.4 Other models......Page 202
13.6 Open challenges......Page 204
13.7 Further reading......Page 205
14.1 The mean-field description......Page 207
14.2 The shape of the reaction front in the mean-field approach......Page 209
14.3 Studies of the front in one dimension......Page 210
14.4 Reaction rates in percolation......Page 211
14.5 A + Bs_{static} -> C with a localized source of A particles......Page 215
14.6 Exercises......Page 216
14.7 Open challenges......Page 217
14.8 Further reading......Page 218
Part four: Diffusion-limited coalescence: an exactly solvable model......Page 220
15.1 The one-species coalescence model......Page 222
15.2 The IPDF method......Page 223
15.3 The continuum limit......Page 226
15.4 Exact evolution equations......Page 227
15.5 The general solution......Page 228
15.6 Exercises......Page 230
15.8 Further reading......Page 231
16.1 Simple coalescence, A + A -> A......Page 232
16.2 Coalescence with input......Page 237
16.3 Rate equations......Page 238
16.4 Exercises......Page 242
16.6 Further reading......Page 243
17.1 The equilibrium steady state......Page 244
17.2 The approach to equilibrium: a dynamical phase transition......Page 246
17.3 Rate equations......Page 248
17.4 Finite-size effects......Page 249
17.5 Exercises......Page 251
17.7 Further reading......Page 252
18.1 Inhomogeneous initial conditions......Page 253
18.2 Fisher waves......Page 255
18.3 Multiple-point correlation functions......Page 258
18.4 Shielding......Page 260
18.6 Open challenges......Page 262
18.7 Further reading......Page 263
19.1 A model for finite coalescence rates......Page 264
19.2 The approximation method......Page 265
19.3 Kinetics crossover......Page 266
19.4 Finite-rate coalescence with input......Page 269
19.5 Exercises......Page 271
19.7 Further reading......Page 272
Appendix A - The fractal dimension......Page 273
Appendix B - The number of distinct sites visited by random walks......Page 275
Appendix C - Exact enumeration......Page 278
Appendix D - Long-range correlations......Page 281
References......Page 287
Index......Page 328
