Contents

1 Introduction

1.1 Brief history

1.1.1 Self-reproducing systems

1.1.2 Simple dynamical systems

1.1.3 A synthetic universe

1.1.4 Modeling physical systems

1.1.5 Beyond the cellular automata dynamics: lattice Boltzmann methods and multiparticle models

1.2 A simple cellular automaton: the parity rule

1.3 Definitions

1.3.1 Cellular automata

1.3.2 Neighborhood

1.3.3 Boundary conditions

1.3.4 Some remarks

1.4 Problems

2 Cellular automata modeling

2.1 Why cellular automata are useful in physics

2.1.1 Cellular automata as simple dynamical systems

2.1.2 Cellular automata as spatially extended systems

2.1.3 Several levels of reality

2.1.4 A fictitious microscopic world

2.2 Modeling of simple systems: a sampler of rules

2.2.1 The rule 184 as a model for surface growth

2.2.2 Probabilistic cellular automata rules

2.2.3 The Q2R rule

2.2.4 The annealing rule

2.2.5 The HPP rule

2.2.6 The sand pile rule

2.2.7 The ant rule

2.2.8 The road traffic rule

2.2.9 The solid body motion rule

2.3 Problems

3 Statistical mechanics of lattice gas

3.1 The one-dimensional diffusion automaton

3.1.1 A random walk automaton

3.1.2 The macroscopic limit

3.1.3 The Chapman-Enskog expansion

3.1.4 Spurious invariants

3.2 The FHP model

3.2.1 The collision rule

3.2.2 The microdynamics

3.2.3 From microdynamics to macrodynamics

3.2.4 The collision matrix and semi-detailed balance

3.2.5 The FHP-III model

3.2.6 Examples of fluid flows

3.2.7 Three-dimensional lattice gas models

3.3 Thermal lattice gas automata

3.3.1 Multispeed models

3.3.2 Thermo-hydrodynamical equations

3.3.3 Thermal FHP lattice gases

3.4 The staggered invariants

3.5 Lattice Boltzmann models

3.5.1 Introduction

3.5.2 A simple two-dimensional lattice Boltzmann fluid

3.5.3 Lattice Boltzmann flows

3.6 Problems

4 Diffusion Phenomena

4.1 Introduction

4.2 The diffusion model

4.2.1 Microdynamics of the diffusion process

4.2.2 The mean square displacement and the Green-Kubo formula

4.2.3 The three-dimensional case

4.3 Finite systems

4.3.1 The stationary source-sink problem

4.3.2 Telegraphist equation

4.3.3 The discrete Boltzmann equation in 2D

4.3.4 Semi-infinite strip

4.4 Applications of the diffusion rule

4.4.1 Study of the diffusion front

4.4.2 Diffusion-limited aggregation

4.4.3 Diffusion-limited surface adsorption

4.5 Problems

5 Reaction-diffusion processes

5.1 Introduction

5.2 A model for excitable media

5.3 Lattice gas microdynamics

5.3.1 From microdynamics to rate equations

5.4 Anomalous kinetics

5.4.1 The homogeneous process

5.4.2 Cellular automata or lattice Boltzmann modeling

5.4.3 Simulation results

5.5 Reaction Front in the process

5.5.1 The scaling solution

5.6 Liesegang patterns

5.6.1 What are Liesegang patterns

5.6.2 The lattice gas automata model

5.6.3 Cellular automata bands and rings

5.6.4 The lattice Boltzmann model

5.6.5 Lattice Boltzmann rings and spirals

5.7 Multiparticle models

5.7.1 Multiparticle diffusion model

5.7.2 Numerical implementation

5.7.3 The reaction algorithm

5.7.4 Rate equation approximation

5.7.5 Turing patterns

5.8 From cellular automata to field theory

5.9 Problems

6 Nonequilibrium phase transitions

6.1 Introduction

6.2 Simple interacting particle systems

6.2.1 The A model

6.2.2 The contact process model (CPM)

6.3 Simple models of catalytic surfaces

6.3.1 The Ziff model

6.3.2 More complicated models

6.4 Critical behavior

6.4.1 Localization of the critical point

6.4.2 Critical exponents and universality classes

6.5 Problems

7 Other models and applications

7.1 Wave propagation

7.1.1 One-dimensional waves

7.1.2 Two-dimensional waves

7.1.3 The lattice BGK formulation of the wave model

7.1.4 An application to wave propagation in urban environments

7.2 Wetting, spreading and two-phase fluids

7.2.1 Multiphase flows

7.2.2 The problem of wetting

7.2.3 An FHP model with surface tension

7.2.4 Mapping of the hexagonal lattice on a square lattice

7.2.5 Simulations of wetting phenomena

7.2.6 Another force rule

7.2.7 An Ising cellular automata fluid

7.3 Multiparticle fluids

7.3.1 The multiparticle collision rule

7.3.2 Multiparticle fluid simulations

7.4 Modeling snow transport by wind

7.4.1 The wind model

7.4.2 The snow model

7.4.3 Simulations of snow transport

References
Glossary
Index