Cellular Automata Modeling of Physical Systems

Contents......Page 5
Preface......Page 9
1.1.1 Self-reproducing systems......Page 11
1.1.2 Simple dynamical systems......Page 13
1.1.3 A synthetic universe......Page 14
1.1.4 Modeling physical systems......Page 15
1.1.5 Beyond the cellular automata dynamics: lattice Boltzmann methods and multiparticle models......Page 17
1.2 A simple cellular automaton: the parity rule......Page 18
1.3.1 Cellular automata......Page 22
1.3.2 Neighborhood......Page 24
1.3.3 Boundary conditions......Page 25
1.3.4 Some remarks......Page 26
1.4 Problems......Page 28
2.1.1 Cellular automata as simple dynamical systems......Page 31
2.1.2 Cellular automata as spatially extended systems......Page 34
2.1.3 Several levels of reality......Page 36
2.1.4 A fictitious microscopic world......Page 37
2.2 Modeling of simple systems: a sampler of rules......Page 38
2.2.2 Probabilistic cellular automata rules......Page 39
2.2.3 The Q2R rule......Page 43
2.2.4 The annealing rule......Page 47
2.2.5 The HPP rule......Page 48
2.2.6 The sand pile rule......Page 52
2.2.7 The ant rule......Page 56
2.2.8 The road traffic rule......Page 61
2.2.9 The solid body motion rule......Page 66
2.3 Problems......Page 71
3.1 The one-dimensional diffusion automaton......Page 76
3.1.1 A random walk automaton......Page 77
3.1.2 The macroscopic limit......Page 78
3.1.3 The Chapman-Enskog expansion......Page 81
3.1.4 Spurious invariants......Page 84
3.2.1 The collision rule......Page 85
3.2.2 The microdynamics......Page 88
3.2.3 From microdynamics to macrodynamics......Page 90
3.2.4 The collision matrix and semi-detailed balance......Page 115
3.2.5 The FHP-III model......Page 117
3.2.6 Examples of fluid flows......Page 120
3.3.1 Multispeed models......Page 122
3.3.2 Thermo-hydrodynamical equations......Page 125
3.3.3 Thermal FHP lattice gases......Page 126
3.4 The staggered invariants......Page 127
3.5.1 Introduction......Page 132
3.5.2 A simple two-dimensional lattice Boltzmann fluid......Page 135
3.5.3 Lattice Boltzmann flows......Page 144
3.6 Problems......Page 145
4.1 Introduction......Page 148
4.2 The diffusion model......Page 149
4.2.1 Microdynamics of the diffusion process......Page 150
4.2.2 The mean square displacement and the Green-Kubo formula......Page 157
4.2.3 The three-dimensional case......Page 159
4.3 Finite systems......Page 160
4.3.1 The stationary source-sink problem......Page 161
4.3.2 Telegraphist equation......Page 163
4.3.3 The discrete Boltzmann equation in 2D......Page 167
4.3.4 Semi-infinite strip......Page 169
4.4.1 Study of the diffusion front......Page 173
4.4.2 Diffusion-limited aggregation......Page 176
4.4.3 Diffusion-limited surface adsorption......Page 181
4.5 Problems......Page 185
5.1 Introduction......Page 188
5.2 A model for excitable media......Page 189
5.3 Lattice gas microdynamics......Page 191
5.3.1 From microdynamics to rate equations......Page 194
5.4 Anomalous kinetics......Page 197
5.4.1 The homogeneous A + B -> \\emptyset process......Page 198
5.4.2 Cellular automata or lattice Boltzmann modeling......Page 200
5.4.3 Simulation results......Page 201
5.5 Reaction front in the A + B -> \\emptyset process......Page 203
5.5.1 The scaling solution......Page 205
5.6.1 What are Liesegang patterns......Page 208
5.6.2 The lattice gas automata model......Page 211
5.6.3 Cellular automata bands and rings......Page 212
5.6.4 The lattice Boltzmann model......Page 216
5.6.5 Lattice Boltzmann rings and spirals......Page 218
5.7 Multiparticle models......Page 219
5.7.1 Multiparticle diffusion model......Page 220
5.7.2 Numerical implementation......Page 222
5.7.3 The reaction algorithm......Page 224
5.7.4 Rate equation approximation......Page 226
5.7.5 Turing patterns......Page 227
5.8 From cellular automata to field theory......Page 229
5.9 Problems......Page 238
6.1 Introduction......Page 242
6.2.1 The A model......Page 244
6.2.2 The contact process model (CPM)......Page 251
6.3.1 The Ziff model......Page 252
6.3.2 More complicated models......Page 258
6.4 Critical behavior......Page 259
6.4.1 Localization of the critical point......Page 260
6.4.2 Critical exponents and universality classes......Page 262
6.5 Problems......Page 264
7.1.1 One-dimensional waves......Page 266
7.1.2 Two-dimensional waves......Page 268
7.1.3 The lattice BGK formulation of the wave model......Page 277
7.1.4 An application to wave propagation in urban environments......Page 288
7.2.1 Multiphase flows......Page 290
7.2.2 The problem of wetting......Page 292
7.2.3 An FHP model with surface tension......Page 294
7.2.4 Mapping of the hexagonal lattice on a square......Page 297
7.2.5 Simulations of wetting phenomena......Page 299
7.2.6 Another force rule......Page 302
7.2.7 An Ising cellular automata fluid......Page 303
7.3 Multiparticle fluids......Page 308
7.3.1 The multiparticle collision rule......Page 309
7.3.2 Multiparticle fluid simulations......Page 312
7.4.1 The wind model......Page 315
7.4.2 The snow model......Page 318
7.4.3 Simulations of snow transport......Page 322
References......Page 323
Glossary......Page 337
Index......Page 347