Definition of the Subject
The computational science community has always been faced with thechallenge of bringing efficient numerical tools to solve problems ofincreasing difficulty. Nowadays, the investigation and understandingof the so‐called complex systems, and the simulation of all kinds ofphenomena originating from the interaction of many components are ofcentral importance in many area of science.
Cellular automata turn out to be a very fruitful approach to addressmany scientific problems by providing an efficient way to model andsimulate specific phenomena for which more traditional computationaltechniques are hardly applicable.
The goal of this article is to provide the reader with the foundationof this approach, as well as a selection of simple applications ofthe cellular automata approach to the modeling of physical systems.We invite the reader to consult the web sitehttp://cui.unige.ch/%7Echopard/CA/Animations/imgroot.htmlin orderto view short movies about several of the...
Abbreviations
 BGK models:

Lattice Boltzmann models where the collision termΩ is expressed as a deviation from a given local equilibriumdistribution \( { f^{(0)} } \), namely \( { \Omega=(f^{(0)}f)/\tau } \), where fis the unknown particle distribution and τ a relaxation time(which is a parameter of the model). BGK stands for Bhatnager, Grossand Krook who first considered such a collision term, but notspecifically in the context of lattice systems.
 CA:

Abbreviation for cellular automata or cellular automaton.
 Cell:

The elementary spatial component of a CA. The cell ischaracterized by a state whose value evolves in time according to theCA rule.
 Cellular automaton:

System composed of adjacent cells or sites(usually organized as a regular lattice) which evolves in discrete timesteps. Each cell is characterized by an internal state whose value belongsto a finite set. The updating of these states is made in parallel accordingto a local rule involving only a neighborhood of each cell.
 Conservation law:

A property of a physicalsystem in which some quantity (such as mass, momentum or energy) is locallyconserved during the time evolution.These conservation laws should be included in the microdynamicsof a CA model because they are essential ingredients governing themacroscopic behavior of any physical system.
 Collision:

The process by which the particles of a LGAchange their direction of motion.
 Continuity equation:

An equation of the form \( \partial_t\rho+\operatorname{div}\rho \mathbf{u} = 0 \) expressing the mass (or particle number) conservation law.The quantity ρ is the local density of particles and \( { \mathbf{u} } \) the localvelocity field.
 Critical phenomena:

The phenomena which occur in the vicinity ofa continuous phase transition, and are characterized by very long correlationlength.
 Diffusion:

A physical process described by the equation \( { \partial_t\rho=D\nabla^2\rho } \), where ρ is the density of a diffusing substance.Microscopically, diffusion can be viewed as a random motion of particles.
 DLA:

Abbreviation of Diffusion Limited Aggregation. Model ofa physical growth process in which diffusing particles stick on an existingcluster when they hit it. Initially, the cluster is reduced to a singleseed particle and grows as more and more particles arrive. A DLA cluster isa fractal object whose dimension is typically 1.72 if the experiment isconducted in a two‐dimensional space.
 Dynamical system:

A system of equations (differential equations or discretized equations) modeling the dynamical behavior of a physical system.
 Equilibrium states:

States characterizing a closed system ora system in thermal equilibrium with a heat bath.
 Ergodicity:

Property of a system or process for which thetime‐averages of the observables converge, in a probabilistic sense, totheir ensemble averages.
 Exclusion principle:

A restriction which is imposed on LGA or CAmodels to limit the number of particles per site and/or lattice directions.This ensures that the dynamics can be described with a cellular automatarule with a given maximum number of bits. The consequence of this exclusionprinciple is that the equilibrium distribution of the particle numbersfollows a Fermi–Diraclike distribution in LGA dynamics.
 FHP model:

Abbreviation for the Frisch, Hasslacher and Pomeaulattice gas model which was the first serious candidate to simulatetwo‐dimensional hydrodynamics on a hexagonal lattice.
 Fractal:

Mathematical object usually having a geometricalrepresentation and whose spatial dimension is not an integer. The relationbetween the size of the object and its “mass” does not obey that of usualgeometrical objects. A DLA cluster is an example of a fractal.
 Front:

The region where some physical process occurs. Usuallythe front includes the locations in space that are first affected by thephenomena. For instance, in a reaction process between two spatiallyseparated reactants, the front describes the region where the reactiontakes place.
 HPP model:

Abbreviation for the Hardy, de Pazzis and Pomeaumodel. The first two‐dimensional LGA aimed at modeling the behavior ofparticles colliding on a square lattice with mass and momentumconservation. The HPP model has several physical drawbacks that have beenovercome with the FHP model.
 Invariant:

A quantity which is conserved during the evolution ofa dynamical system. Some invariants are imposed by the physical laws (mass,momentum, energy) and others result from the model used to describephysical situations (spurious, staggered invariants). Collisionalinvariants are constant vectors in the space where the Chapman–Enskogexpansion is performed, associated to each quantity conserved by thecollision term.
 Ising model:

Hamiltonian model describing the ferromagneticparamagnetic transition. Each local classical spin variables \( { s_i=\pm 1 } \)interacts with its neighbors.
 Isotropy:

The property of continuous systems to beinvariant under any rotations of the spatial coordinate system. Physicalquantities defined on a lattice and obtained by an averaging procedure mayor may not be isotropic, in the continuous limit. It depends on the type oflattice and the nature of the quantity. Second‐order tensors areisotropic on a 2D square lattice but fourth‐order tensors need a hexagonallattice.
 Lattice:

The set of cells (or sites) making up the spatialarea covered by a CA.
 Lattice Boltzmann model:

A physical model defined on a latticewhere the variables associated to each site represent an averagenumber of particles or the probability of the presence of a particlewith a given velocity. Lattice Boltzmann models can be derived fromcellular automata dynamics by an averaging and factorizationprocedure, or be defined per se, independently of a specificrealization.
 Lattice gas:

A system defined on a lattice where particlesare present and follow given dynamics. Lattice gas automata (LGA)are a particular class of such a system where the dynamics areperformed in parallel over all the sites and can be decomposed intwo stages: (i) propagation: the particles jump toa nearest‐neighbor site, according to their direction of motionand (ii) collision: the particles entering the same site at thesame iteration interact so as to produce a new particledistribution. HPP and FHP are wellknown LGA.
 Lattice spacing:

The separation between two adjacent sites ofa regular lattice. Throughout this book, it is denoted by the symbol \( { \Delta_\mathrm{r} } \).
 LB:

Abbreviation for Lattice Boltzmann.
 LGA:

Abbreviation for Lattice Gas Automaton. See lattice gas modelfor a definition.
 Local equilibrium:

Situation in which a large system can bedecomposed into subsystems, very small on a macroscopic scale but large ona microscopic scale such that each sub‐system can be assumed to be in thermalequilibrium. The local equilibrium distribution is the function which makesthe collision term of a Boltzmann equation vanish.
 Lookup table:

A table in which all possible outcomes ofa cellular automata rule are pre‐computed. The use of a lookup table yieldsa fast implementation of a cellular automata dynamics since howevercomplicated a rule is, the evolution of any configuration of a site and itsneighbors is directly obtained through a memory access. The size of a lookuptable grows exponentially with the number of bits involved in the rule.
 Margolus neighborhood:

A neighborhoodmade of twobytwo blocks of cells, typically in a two‐dimensional squarelattice. Each cell is updated according to the values of theother cells in the same block. A different rule may possibly be assigneddependent on whetherthe cell is at the upper left, upper right, lower left or lower rightlocation. After each iteration, the lattice partition definingthe Margolus blocs is shifted one cell right and one cell down so that atevery other step, information can be exchanged across the lattice.Can be generalized to higher dimensions.
 Microdynamics:

The Boolean equation governing the time evolutionof a LGA model or a cellular automata system.
 Moore neighborhood:

A neighborhood composed of the central celland all eight nearest and next‐nearest neighbors in a two‐dimensionalsquare lattice. Can be generalized to higher dimensions.
 Multiparticle models:

Discrete dynamics modeling a physicalsystem in which an arbitrary number of particles is allowed ateach site. This is an extension of an LGA where no exclusionprinciple is imposed.
 Navier–Stokes equation:

The equation describing the velocityfield \( { \mathbf{u} } \) in a fluid flow. For an incompressible fluid (\( { \partial_t\rho=0 } \)), itreads
$$ \partial_t \mathbf{u} + (\mathbf{u}\cdot\nabla)\mathbf{u}={1\over\rho}\nabla P +\nu\nabla^2\mathbf{u} $$where ρ is the density and P the pressure. The Navier–Stokesequation expresses the local momentum conservation in the fluid and, asopposed to the Euler equation, includes the dissipative effects witha viscosity term \( { \nu\nabla^2\mathbf{u} } \). Together with the continuity equation,this is the fundamental equation of fluid dynamics.
 Neighborhood:

The set of all cells necessary to computea cellular automaton rule. A neighborhood is usually composed of severaladjacent cells organized in a simple geometrical structure. Moore, vonNeumann and Margolus neighborhoods are typical examples.
 Occupation numbers:

Boolean quantities indicating the presenceor absence of a particle in a given physical state.
 Open system:

A system communicating with the environment byexchange of energy or matter.
 Parallel:

Refers to an action which is performed simultaneouslyat several places. A parallel updating rule corresponds to the updating ofall cells at the same time as if resulting from the computations of severalindependent processors.
 Partitioning:

A technique consisting of dividing space inadjacent domains (through a partition) so that the evolution ofeach block is uniquely determined by the states of the elementswithin the block.
 Phase transition:

Change of state obtained when varyinga control parameter such as the one occurring in the boiling or freezing ofa liquid, or in the change between ferromagnetic and paramagnetic states ofa magnetic solid.
 Propagation:

This is the process by which the particles ofa LGA are moved to a nearest neighbor, according to the direction oftheir velocity vector \( { \mathbf{v_i} } \). In one time step Δ_{ t } the particletravel from cell \( { \mathbf{r} } \) to cell \( { \mathbf{r}+\mathbf{v_i}\Delta_t } \) where \( { \mathbf{r}+\mathbf{v_i}\Delta_t } \) is thenearest neighbor in lattice direction i.
 Random walk:

A series of uncorrelated steps of length unitydescribing a random path with zero average displacement but characteristicsize proportional to the square root of the number of steps.
 Reaction‐diffusion systems:

Systems made of one or severalspecies of particles which diffuse and react among themselves to producesome new species.
 Scaling hypothesis:

A hypothesis concerning the analyticalproperties of the thermodynamic potentials and the correlation functions ina problem invariant under a change of scale.
 Scaling law:

Relations among the critical exponents describing the power law behaviors of physical quantities in systems invariant under a change of scale.
 Self‐organized criticality:

Concept aimed at describing a classof dynamical systems which naturally drive themselves to a state whereinteresting physics occurs at all scales.
 Site:

Same as a cell, but preferred terminology in LGA and LBmodels.
 Spatially extended systems:

Physical systems involving manyspatial degrees of freedom and which, usually, have rich dynamicsand show complex behaviors. Coupled map lattices and cellularautomata provides a way to model spatially extended systems.
 Spin:

Internal degree of freedom associated to particles inorder to describe their magnetic state. A widely used case is the one ofclassical Ising spins. To each particle, one associates an “arrow” whichis allowed to take only two different orientations, up or down.
 Time step:

Interval of time separating two consecutiveiterations in the evolution of a discrete time process, like a CAor a LB model. Throughout this work the time step is denoted bythe symbol Δ_{ t }.
 Universality:

The phenomenon whereby many microscopicallydifferent systems exhibit a critical behavior with quantitatively identicalproperties such as the critical exponents.
 Updating:

operation consisting of assigning a new value toa set of variables, for instance those describing the states ofa cellular automata system. The updating can be done in paralleland synchronously as is the case in CA dynamics or sequentially,one variable after another, as is usually the case forMonte–Carlo dynamics. Parallel, asynchronous updating is lesscommon but can be envisaged too. Sequential and parallel updatingschemes may yield different results since the interdependenciesbetween variables are treated differently.
 Viscosity:

A property of a fluid indicating how much momentum“diffuses” through the fluid in a inhomogeneous flow pattern.Equivalently, it describes the stress occurring between two fluid layersmoving with different velocities. A high viscosity means that the resultingdrag force is important and low viscosity means that this force is weak.Kinematic viscosity is usually denoted by ν and dynamic viscosity isdenoted by \( { \eta=\nu\rho } \) where ρ is the fluid density.
 von Neumann neighborhood:

On a two‐dimensional squarelattice, the neighborhood including a central cell and its nearestneighbors north, south, east and west.
 Ziff model:

A simple model describing adsorption–dissociation–desorption on a catalytic surface.This model is based upon some of the known steps of the reaction A–B _{2} on a catalyst surface (for example CO–O_{2}).
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Chopard, B. (2009). Cellular Automata Modeling of Physical Systems. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/9780387304403_57
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