part i background for lattice gas automata · lattice gas automata the lattice gasautomaton is an...

The invention of a totally dis-crete model for natural phenom-ena was made by Ulam and vonNeumann in the early fifties and

was developed to the extent possible atthe time. A few years earlier von Neu-mann had designed the architecture forthe first serial digital computers contain-ing stored programs and capable of mak-ing internal decisions. These machinesare built of electronic logic devices thatunderstand only packets of binary bits.Hierarchies of stored translators arrangethem into virtual devices that can do or-dinary or radix arithmetic at high speed.By transcribing continuum equations intodiscrete form, using finite difference tech-niques and their variants, serial digitalcomputers can solve complex mathemat-ical systems such as partial differentialequations. Since most physical systems

Los Alamos Science Special Issue 1987

PART IBACKGROUND FOR

LATTICE GAS AUTOMATA

The lattice gas automaton is an approachto computing fluid dynamics that is stillin its infancy. In this three-purr articleone of the inventors of the model presentsits theoretical foundations and its promiseas a general approach to solving partialdifferential equations and to parallel com-puting. Readers less theoretically inclinedmight begin by reading “Calculations Us-ing Lattice Gas Techniques” at the endof Part II. This sidebar offers a summaryof the model’s advantages and limitationsand a graphic display of two- and three-dimensional lattice gas simulations.

with large numbers of degrees of freedom

memories have become the standard wayto simulate such phenomena.

As the architecture of serial machinesdeveloped, it became clear to both Ulamand von Neumann that such machineswere not the most natural or powerfulway to solve many problems. They wereespecially influenced by biological exam-ples. Biological systems appear to per-form computational tasks using methodsthat avoid both arithmetical operationsand discrete approximations to continu-ous systems.

Though motivated by the complex in-formation processing of biological sys-tems, Ulam and von Neumann did notstudy how such systems actually solvetasks. Biological processes have beenoperating in hostile environments for a

can be described by such equations, se- long time, finding the most efficient andrial digital machines equipped with large often devious way to do something, a

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way that is also resistant to disturbanceby noise. The crucial principles of theiroperation are hidden by the evolutionaryprocess. Instead, von Neumann chosethe task of simulating on a computer theleast complex discrete system capable ofself-reproduction. It was Ulam who sug-gested an abstract setting for this problemand many other totally discrete models,namely. the idea of cellular spaces. Thereasoning went roughly like this.

The question is simple: Find a mini-mal logic structure and devise a dynam-ics for it that is powerful enough to simu-late complex systems. Break this up intoa series of sharper and more elementarypictures. We begin by setting up a collec-tion of very simple finite-state machineswith, for simplicity, binary values. Con-nect them so that given a state for eachof them, the next state of each machinedepends only on its immediate environ-ment. In other words, the state of anymachine will depend only on the statesof machines in some small neighborhoodaround it. This builds in the constraintthat we only want to consider local dy-namics.

We will need rules to define how statescombine in a neighborhood to uniquelyfix the state of every machine, but thesecan be quite simple. The natural spaceon which to put all this is a lattice, withelementary, few-bit, finite-state machinesplaced at the vertices. The rules for up-dating this array of small machines canbe done concurrently in one clock step,that is, in parallel.

One can imagine such an abstract ma-chine in operation by thinking of a fishnetmade of wires. The fishnet has some reg-ular connection geometry, and there arelights at the nodes of the net. Each lightcan be on or off. Draw a disk aroundeach node of the fishnet, and let it have al-node radius. On a square net there arefour lights on the edge of each disk, ona triangular net six lights (Fig. 1). Thenext state of the light at the center of thedisk depends on the state of the lights on

CELLULAR SPACES

Fig. 1. Two examples of fishnets made of wires

with lights at the nodes. The lights are either

on or off. In each example a disk with a radius

of 1 node is drawn around one of the lights.

The next state of the light at the center de-

pends on the states of the lights on the edge of

the disk and on nothing else. Thus these are

examples of nearest-neighbor-connected cel-

lular spaces.

the edge of the disk and on nothing else.Imagine all the disks in the fishnet ask-ing their neighbors for their state at thesame time and switching states accord-ing to a definite rule. At the next tick ofan abstract clock, the pattern of lights onthe fishnet would in general look differ-ent. This is what Ulam and von Neumanncalled a nearest-neighbor-connected cel-lular space. It is the simplest case of aparallel computing space. You can alsosee that it can be imaged directly in hard-ware, so it is also the architecture for aphysical parallel computing machine.

We have not shown that such a devicecan compute. At worst, it is an elaboratelight display. Whether or not such acellular space can compute depends on

the definition of computation. The shortanswer is that special cases of fishnetsare provably universal computers in thestandard Turing machine sense; that is,they can simulate the architecture of anyother sequential machine.

But there are other interpretations ofcomputation that lie closer to the idea ofsimulation. For any given mathematicalsituation, we want to find the minimumcellular space that can do a simulation ofit: At what degree of complexity doesrepeated iteration of the space, on whichare coded both data and a solution algo-rithm, possess the power to come close tothe solution of a complex problem? Thisdepends on the complexity or degrees offreedom present in the problem.

An extreme case of complexity is phys-ical systems with many degrees of free-dom. These systems are ordinarily de-scribed by field theories in a continuumfor which the equations of motion arehighly nonlinear partial differential equa-tions. Fluid dynamics is an example, andwe will use it as a theoretical paradigmfor many “large” physical systems. Be-cause of the high degree of nonlinearity,analytic solutions to the field equationsfor such systems are known only in spe-cial cases. The standard way to studysuch models is either to perform experi-ments or simulate them on computers ofthe usual digital type.

Suppose a cellular space existed thatevolved to a solution of a fluid systemwith given boundary conditions. sup-pose also that we ask for the simplestpossible such space that captured at leastthe qualitative and topological aspects ofa solution. Later, one can worry aboutspaces that agree quantitatively with or-dinary simulations. The problem is three-fold: Find the least complex set of rulesfor updating the space; the simplest ge-ometry for a neighborhood; and a methodof analysis for the collective modes andtime evolution of such a system.

At first sight, modeling the dynamics oflarge systems by cellular spaces seems far

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too difficult to attempt. The general prob-lem of a so-called “inverse compiler”-given a partial differential system, findthe rules and interconnection geometrythat give a solution—would probably useup a non-polynomial function of comput-ing resources and so be impractical if notimpossible. Nevertheless cellular spaceshave been actively studied in recent years.Their modern name is cellular automata,and specific instances of them have sim-ulated interesting nonlinear systems. Butuntil recently there was no example of acellular automaton that simulated a largephysical system, even in a rough, quali-tative way.

Knowing that special cases of cellu-lar automata are capable of arbitrarilycomplex behavior is encouraging, but notvery useful to a physicist. The impor-tant phenomenon in large physical sys-tems is not arbitrarily complex behav-ior, but the collective motion that de-velops as the system evolves, typicallywith a characteristic size of many ele-mentary length scales. The problem is tosimulate such phenomena and, by usingsimulations, to try to understand the ori-gins of collective behavior from as manypoints of view as possible. Fluid dy-namics is filled with examples of collec-tive behavior—shocks, instabilities, vor-tices, vortex streets, vortex sheets, tur-bulence, to list a few. Any determin-istic cellular-automaton model that at-tempts to describe non-equilibrium fluiddynamics must contain in it an itera-tive mechanism for developing collec-tive motion. Knowing this and usingsome very basic physics, we will con-struct a cellular automaton with the ap-propriate geometry and updating rules forfluid behavior. It will also be the sim-plest such model. The methods we useto do this are very conservative from theviewpoint of recent work on cellular au-tomata, but rather drastic compared tothe approaches of standard mathematicalphysics. Presently there is a large gapbetween these two viewpoints. The sim-

ulation of fluid dynamics by cellular au-tomata shows that there are other comple-mentary and powerful ways to model phe-nomena that would normally be the exclu-sive domain of partial differential equa-tions.

The Example of Fluid Dynamics

Fluid dynamics is an especially goodlarge system for a cellular automaton for-mulation because there are two rich andcomplementary ways to picture fluid mo-tion. The kinetic picture (many simpleatomic elements colliding rapidly withsimple interactions) coincides with our in-tuitive picture of dynamics on a cellularspace. Later we will exploit this analogyto construct a discrete model.

The other and older way of approach-ing flow phenomena is through the partialdifferential equations that describe col-lective motions in dissipative fluids-theNavier-Stokes equations. These can bederived without any reference to an un-derlying atomic picture. The derivationrelies on the idea of the continuum; itis simpler to grasp than the kinetic pic-ture and mathematically cleaner. Becausethe continuum argument leads to the cor-rect functional form of the Navier-Stokesequations, we spend some time describ-ing why it works. The continuum viewof fluids will be called “coming downfrom above,” and the microphysical view“coming up from below” (Fig. 2). Inthe intersection of these two very differ-ent descriptions, we can trap the essentialelements of a cellular-automaton modelthat leads to the Navier-Stokes equations.Through this review we wish to show thatcellular automaton models are a naturaland evolutionary idea and not an inven-tion come upon by accident.

Coming down from Above—The Continuum Description

The notion of a smooth flow of somequantity arises naturally from a contin-

uum description. A flow has physicalconservation laws built-in, at least con-servation of mass and momentum. Witha few additional remarks one can includeconservation of energy. The basic strat-egy for deriving the Euler and Navier-Stokes equations of fluid dynamics is toimbed these conservation laws into state-ments about special cases of the gen-eralized Stokes theorem. We use theusual Gauss and Stokes theorems, de-pending on dimension, and apply themto small surfaces and volumes that arestill large enough to ignore an underly-ing microworld. The equations of fluiddynamics are derived with no referenceto a ball-bearing picture of an underly-ing atomic world, but only with a serenereliance on the idea of a smooth flowin a continuum with some of Newton’slaws added to connect to the observedworld. As a model (for it is not a the-ory), the Navier-Stokes equations are agood example of how concepts derivedfrom the intuition of daily experience canbe remarkably successful in building ef-fective phenomenological models of verycomplex phenomena. It is useful to gothrough the continuum derivation of theEuler and Navier-Stokes equations pre-sented in “The Continuum Argument” forseveral reasons: First, the reasoning isshort and clear; second, the concepts in-troduced such as the momentum flux ten-sor, will appear pervasively when we passto discrete theories of fluids; third, welearn how few ingredients are really nec-essary to build a fluid model and so markout that which is essential—the role ofconservation laws.

It is clear from its derivation that theEuler equation describing inviscid flowsis essentially a geometrical equation. Theextension to the full Navier-Stokes equa-tions, for flows with dissipation, containsonly a minimal reference to an underlyingfluid microphysics, through the stress-rateof strain relation in the momentum stresstensor. So we see that continuum reason-ing alone leads to nonlinear partial differ-

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ential equations for large-scale physicalobservable that are a phenomenologicaldescription of fluid flow. This descriptionis experimentally quite accurate but the-oretically incomplete. The coupling con-stants that determine the strength of thenonlinear terms-that is, the transport co-efficients such as viscosity—have a directphysical interpretation in a microworldpicture. In the continuum approach how-ever, these must be measured and put inas data from the outside world. If we donot use some microscopic model for thefluid, the transport coefficients cannot bederived from first principles.

Solution Techniques—The Creationof a Microworld. The Navier-Stokesequations are highly nonlinear; this isprototypical of field-theoretical descrip-tions of large physical systems. The non-linearity allows analytic solutions onlyfor special cases and, in general, forcesone to solve the system by approximationtechniques. Invariably these are someform of perturbation methods in what-ever small parameters can be devised.Since there is no systematic way of apply-ing perturbation theory to highly nonlin-ear partial differential systems, the anal-ysis of the Navier-Stokes equations hasbeen, and still remains, a patchwork ofingenious techniques that are designed tocover special parameter regimes and lim-ited geometries.

After an approximation method is cho-sen, the next step toward a solution is todiscretize the approximate equations in aform suitable for use on a digital com-puter. This discretization is equivalent tointroducing an artificial microworld. Itsparticular form is fixed by mathematicalconsiderations of elegance and efficiencyapplied to simple arithmetic operationsand the particular architecture of avail-able machines. So, even if we adopt theview that the molecular kinetics of a fluidis unimportant for describing the generalfeatures of many fluid phenomena, we arenevertheless forced to describe the sys-

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Fig. 2. Both the continuum view of fluids tions (dashed lines). The text emphasizes how

and the atomic picture lead to the Navier- cellular-automaton models embody the essen-

Stokes equations but not without approxima- tials of both points of view.

tem by a microworld with a particular Coming up from Below—microkinetics. The idea of a partial dif- The Kinetic Theory Descriptionferential equation as a physical model istied directly to finding an analytic solu- Kinetic theory models a fluid by us-tion and is not particularly suited to ma- ing an atomic picture and imposing New-chine computation. In a sense, the geo- tonian mechanics on the motions of themetrically motivated continuum picture is atoms. Atomic interactions are controlledonly a clever and convenient way of en- by potentials, and the number of atomiccoding conservation laws into spaces with elements is assumed to be very large.which we are comfortable. This attempt at fluid realism has an imme-

Continued on page 181

Los AIamos Science Special Issue 1987

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THECONTINUUMARGUMENT

Let V(X, t) be a vector-valued field referred to a fixed origin in space, which weidentify with the velocity of a “macroscopic” fluid cell. The cell is not smallenough to notice a particle structure for the fluid, but it is small enough to be

treated as a mathematical pointTo derive the properties of

generalized Stokes theorem:

and still agree with physics.a flow defined by the vector field, one now invokes the

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by this description but keep to a level ofprecision consistent with a general under-standing of the basic ideas.

The distribution function f is basicallya weighting function that is used to definethe mean values of physical observable.The relation

(2)

where d/dt is a total derivative. In anisolated system with no external fields,we can expand the total derivative as

Liouville statement now is modified tobecome the transport equation:

(4)

Similarly, particle gain into the phase

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differential equation:

Solutions to the Boltzmann Trans-port Equation. Even though the Boltz-mann equation is intractable in general,by using entropy arguments (Boltzmann’sH theorem), the following can be statedabout possible functional forms for f, theone-particle distribution function. If thesystem is uniform in space, any form forf will relax monotonically to the globalMaxwell-Boltzmann form:

in which the macroscopic variables p,v, and T (density, macrovelocity, andtemperature) are independent of position,or global. In the non-equilibrium case,with a soft space dependence, any distri-bution function will relax monotonically

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in velocity space to a local Maxwell- izes cells whose length L in any directionBoltzmann form. This means that p, v,and T will depend on space as well astime. These local distribution functionsare solutions to the Boltzmann transportequation. For the non-uniform case, onegets a picture of the full solution as an en-semble of local Maxwell-Boltzmann dis-tributions covering the description spaceof the fluid, with some gluing conditionsproviding the consistency of the patching.

The second integral gives the momentumtensor equation:

In order to derive the Euler equationfor ideal gases with the usual form for


Discrete Fluids

THE HILBERT CONTRACTION

it is constructive. It explicitly displays arecursive closed tower of constraint rela-tions on the moments off that come di-rectly from the Boltzmann equation. Theproof also shows that such a contracteddescription is unique—a very powerfulresult.

It must be pointed out that Hilbert’sconstruction is on the time-evolved solu-tion to the Boltzmann transport equation,not on the equation itself, which still re-quires a complete specification of f. Itamounts to a hard mathematical statementon an effective field-theory descriptionfor times much greater than elementarycollision times, but with space gradientsstill smooth enough to entertain a seriousgradient perturbation expansion. As such,it says nothing about the turbulent regime,for example, where all these assumptionsfail.

In standard physics texts one can readall kinds of plausibility arguments as towhy this contraction process should ex-ist, but they lack force, for, by arguingtightly, one can make the conclusion gothe other way. This is why the Hilbertcontraction is important. It is really apowerful and mathematically unexpectedresult about a highly nonlinear integro-differential equation of very special form.Beyond Hilbert’s theorem and within theBoltzmann transport picture, we can saynothing more about the contraction of de-scriptions.

The construction of towers of momentconstraints, coupled to a perturbation ex-pansion that Hilbert developed for hisproof of contraction, was used in a some-what different form by Chapman and En-skog. Their main purpose was to devisea perturbation expansion with side con-straints in such a way as to pick off thevalues of the coupling constants-whichare called transport coefficients in stan-dard terminology-for increasingly moresophisticated forms of macrodynamicalequations.

One makes the usual kinetic assump-

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THE HILBERT CONTRACTION (continued)

which turns out to be explicitly a spatialgradient expansion:

which is of the form

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Boltzmann equation. The zero-order re-lation gives the Euler equations and thesecond-order relation gives the Navier-Stokes equations. However, Hilbert’smethod is an asymptotic functional ex-pansion, so that the higher order termstake one away from ordinary fluids ratherthan closer to them. Nevertheless, solv-ing explicitly for the terms in the func-tional expansion provides a way of eval-uating transport coefficients such as vis-cosity. (See the "Hilbert Contraction” formore discussion.)

Summary of the Kinetic Theory Pic-ture. Our review of the kinetic theorydescription of fluids introduced a num-ber of important concepts: the idea oflocal thermal equilibrium; the character-ization of an equilibrium state by a fewmacroscopic observable; the Boltzmanntransport equation for systems of manyidentical objects (with ordinary statistics)in Collision; and the fact that a solutionto the Boltzmann transport equation isan ensemble of equilibrium states. In“The Hilbert Contraction” we introducedthe linear approximation to the Boltz-mann equation with which one can de-rive the Navier-Stokes equations for sys-tems not too far (in an appropriate sense)from equilibrium in terms of these samemacroscopic observable (density, pres-sure. temperature, etc.). We then outlineda method for calculating the couplingconstants in the Navier-Stokes system—that is, the strengths of the nonlinearterms-as a function of any particular mi-crodynamics.

This review was intended to give a fla-vor for the chain of reasoning involved.We will use this chain again in the totally discrete lattice world. However, justas important as understanding the kinetictheory viewpoint is keeping in mind itslimitations. In particular, notice that per-turbation theory was the main tool usedfor going from the exact Boltzrnann trans-port equation to the Navier-Stokes equa-tions. We did not discover more pow-


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erful techniques for finding solutions tothe Navier-Stokes equations than we hadbefore. To go from the Boltzmann tothe Navier-Stokes description, we mademany smoothness assumptions in variousprobabilistic disguises; in other words,we recreated an approximation to the con-tinuum. It is true one could compute (atleast for relatively simple systems) thetransport coefficients, but in a sense thesecoefficients are a property of microkinet-ics, not macrodynamics.

We are at a point where we can asksome questions about the emergence ofmacrodynamics from microscopic phys-ics. It is clear by now that microscopicconservation laws, those of mass, mo-mentum, and energy are crucial in fix-ing the form of large-scale dynamics.These are in a sense sacred. But onecan question the importance of the de-scription of individual collisions. Howdetailed must micromechanics be to gen-erate the qualitative behavior predictedby the Navier-Stokes equations’? Canit be done with simple collisions andvery few classes of them? There existsa whole collection of equations whosefunctional form is very nearly that ofthe Navier-Stokes equations. What mi-croworlds generate these? Do we haveto be exactly at the Navier-Stokes equa-tions to generate the qualitative behaviorand numerical values that we derive fromthe Navier-Stokes equations or from realfluid experiments? Is it possible to de-sign a collection of synthetic microworldsthat could be considered local-interactionboard games, all having Navier-Stokesmacrodynamics? In other words, doesthe detailed microphysics of fluids getwashed out of the macrodynamical pic-ture under very rapid iteration of the de-terministic system’? If the microgame issimple enough to update it deterministi-cally on a parallel machine, is the densityof states required to see everything wesee in ordinary Navier-Stokes simulationsmuch smaller than the density of atoms inreal physical fluids? If so, these synthetic

LOS Alamos Science Special Issue 1987

Fig. 3. Three ingredients are needed for the scale separation between microkinetics andemergence of macrodynamics: local thermo- collective motion.dynamic equilibrium, conservation laws, and

microworlds become a potentially power-ful analytic tool.

Our approach in building a cellularspace is to move away from the ideaof a fluid state and focus instead on theidea of the macrodynamics of a many-element system. In abstract terms, wewant to devise the simplest determinis-tic local game made of a collection offew-bit, finite-state machines that has the

Navier-Stokes equations as its macrody -namical description. From our brief lookat kinetic transport theory, we can ab-stract the essential features of such agame (Fig. 3). The many-element sys-tem must be capable of supporting anotion of local thermodynamic equilib-rium and must also include local micro-scopic conservation laws. The state ofa real fluid can be imagined as a col-

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consistency conditions among their pa-rameters, which become constrained vari-ables. These consistency conditions arethe macrodynamical equations necessaryto put a consistent equilibrium functiondescription onto the many-element sys-tem. In physical fluids they are theNavier-Stokes equations. This is the gen-eral setup that will guide us in creating alattice model.

Evolution of DiscreteFluid Models

Continuous Network Models. The Na-vier-Stokes equations, however derived,are analytically intractable, except in afew special cases for especially clean ge-ometries. Fortunately, one can avoidthem altogether for many problems, suchas shocks in certain geometries. Thestrategy is to rephrase the problem in avery simple phase space and solve theBoltzmann transport equation directly. Ifa single type of particle is constrainedto move continuously only along a reg-ular grid, the Boltzmann equation is sotightly constrained that it has simple ana-lytic solutions. In the early 1960s Broad-well and others applied this simplifiedmethod of analysis to the dynamics ofshock problems. Their numerical resultsagreed closely with much more elabo-rate computer modeling from the Navier-Stokes equations. However, there was noreal insight into why such a calculationin such a simplified microworld shouldgive such accurate answers. The accu-racy of the limited phase-space approachwas considered an anomaly.


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LOS Alamos Science Special Issue 1987

PART II THE SIMPLEHEXAGONAL MODELTheory and Simulation

The Minimal Totally Discrete Model of Navier-Stokesin Two Dimensions

We can now list the ingredients we need to build the simplest cellular-space worldwith a dynamics that reproduces the collective behavior predicted by the compressibleand incompressible Navier-Stokes equations:

1.

2.

3.

4.

5.

A population of identical particles, each with unit mass and moving with the sameaverage speed c.A totally discrete phase space (discrete values of x, y and discrete particle-velocitydirections) and discrete time t. Discrete time means that the particles hop fromsite to site.A lattice on which the particles reside only at the vertices. In the simplest casethe lattice is regular and has a hexagonal neighborhood to guarantee an isotropicmomentum flux tensor. We use a triangular lattice for convenience.A minimum set of collision rules that define symmetric binary and triple collisionssuch that momentum and particle number are conserved (Fig. 4).An exclusion principle so that at each vertex no two particles can have identicalvelocities. This limits the maximum number of particles at a vertex to six, each onehaving a velocity that points in one of the six directions defined by the hexagonalneighborhood.

The only way to make this hexagonal lattice gas simpler is to lower the rotationsymmetry of the lattice, remove collision rules, or break a conservation law. In atwo-dimensional universe with boundaries, any such modification will not give Navier-Stokes dynamics. Left as it is, the model will. Adding attributes to the model, such

187

as different types of particles, different speeds, enlarged neighborhoods, or weightedcollision rules, will give Navier-Stokes behavior with different equations of state anddifferent adjustable parameters such as the Reynolds number (see the discussion in PartIII). The hexagonal model defined by the five ingredients listed above is the simplestmodel that gives Navier-Stokes behavior in a sharply defined parameter regime.

At this point it is instructive to look at the complete table of allowed states forthe model (Fig. 5). The states and collision rules can be expressed by Boolean logic

Other Configurations Don’t ScatterPure Transport


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operations with the two allowed values taken as O and 1. From this organization schemewe see that the hexagonal lattice gas can be seen as a Boolean parallel computer. Infact, a large parallel machine can be constructed to implement part or all of the statetable locally with Boolean operations alone. Our simulations were done this way andprovide the first example of the programming of a cellular-automaton, or cellular-space,machine that evolves the dynamics of a many-degrees-of-freedom, nonlinear physicalsystem.

STATE TABLE FOR HEXAGONAL MODEL

Right Three Bits

(000) (001) (010) (01 1) (loo) (101) (1 10) (111)

(000)Scattering Rules for Simple

Hexagonal Model in 6-Bit Notation

Additional Rules for ExtendedHexagonal Model

Fig. 5. All possible states of the hexagonal lat- and the maximally occupied state shown in the are written beside the table. All other states do

tice gas are shown in the state table. Each lower right hand corner of the table is written not result in scattering. The extended hexag-

state can be expressed in 6-bit notation (a (111, 111). Collision states for the simplest onal model includes scattering rules for four-

combination of 3 right bits and 3 left bits). For hexagonal model are shown in red and shaded body states (shaded in gray). The extended

example, the empty state is written (000,000) in gray. The scattering rules for these states model lowers the viscosity of the lattice gas.


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3 2 Theoretical Analysis of the Discrete Lattice Gas

particle

can point in one of six possible directions. All

particles have the same speed c.

(8)


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To lowest order in h and k, we have

So far we have kept only the leading terms of the Taylor series expansion inthe scaling factors that relate to the discreteness of the lattice. It’s easy to show thatkeeping quadratic terms in this lattice-size expansion leaves the continuity equationinvariant but alters the momentum equation by introducing a free-streaming correctionto the measured viscosity. This rather elegant way of viewing this correction was firstdeveloped by D. Levermore. The correction comes from breaking the form of a Galileancovariant derivative and is a geometrical effect. Specifically, to second order in thelattice size expansion, the momentum equation does not decompose simply into factorsof these covariant derivatives but instead the expansion introduces a nonvanishingcovariant-breaking term:

This term is of the same order as those terms that contribute to the viscosity. Later wewill show how to use the Chapman-Enskog expansion to compute an explicit form forthe lattice-gas viscosity.


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The systematic expansion becomes

The Lattice Collision Operator and the Solution to the Lattice BoltzmannTransport Equation. We will write down the discrete form of the Boltzmann equation,especially noting the collision operator, for a number of reasons. First, writing theexplicit form of the collision kernel builds up an intuition of how the heart of the modelworks; second, we can show in a few lines that the Femi-Dirac distribution satisfies thelattice gas Boltzmann equation; third, knowing this, we can quickly compute the latticeform of the Euler equations; fourth, we can see that many properties of the lattice-gasmodel are independent of the types of collisions involved and come only from the formof the Fermi-Dirac distribution.

Collision operators for lattice gases with continuous speeds were derived by Broad-well, Harris, and other early workers on continuum lattice-gas systems. For totallydiscrete lattice gases with an exclusion principle, we must be careful to apply this prin-ciple correctly. It is similar to the case of quasi-particles in quantum Fermi liquids.

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(16)

where we have taken the particle speed as 1 (c = 1). The coefficient g(p) is

3 – pg ( p ) . —6 – p“

The lattice Euler equation (Eq. 12) thus becomes

(17)

In the usual Euler equation g(p) = 1. Here g(p) is the lattice correction to the convectiveterm due to the explicit lattice breaking of Galilean invariance. The equation of statefor Eq. 17 is


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and

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(18)

and

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(19)

We end this theoretical analysis by showing under what conditions we recover theseequations for lattice gases. One way is to freeze the density everywhere except in thepressure term of the momentum equation (Eq. 18). Then, in the low-velocity limit, wecan write the lattice Navier-Stokes equations as


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SIMULATED VELOCITY PROFILE

Fig. 7. The predicted velocity profile was ob-

tained in a low-velocity lattice gas simulation

of two- dimensional flow in a channel with vis-

cous boundaries (Kadanoff, McNamara, and

Zanetti 1987).

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0 10 20 30

Lattice Row along Channel Width

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SIMULATED AND THEORETICAL KINEMATIC VISCOSITIES

Fig. 8. Theoretical shear (solid line) and bulk (dashed line) reduced viscosities as a function ofreduced density compared with the results of hexagonal lattice gas simulation with rest particlesand all possible collisions (d’ Humieres and Lallemand 1987).

functional forms for various fluid dynamical laws. The question of quantitative accuracyof various known constants is harder to answer, and we will take it up in detail later.

The next broad class of flows studied are flows past objects. Here, we look fordistinctive qualitative behavior characteristic of a fluid or gas obeying Navier-Stokesdynamics. The geometries studied, through a wide range of Reynolds numbers, wereflows past flat plates placed normal to the flow, flows past plates inclined at variousangles to the flow, and flows past cylinders, 60-degree wedges, and typical airfoils. Theexpected scenario changes as a function of increasing Reynolds number: recirculatingflow behind obstacles should develop into vortices, growing couples of vortices shouldeventually break off to form von Karman streets with periodic oscillation of the vonKarman tails; finally, and as the Reynolds number increases, the periodic oscillationsshould become aperiodic, and the complex phenomena characteristic of turbulent flowshould appear. The lattice gas exhibits all these phenomena with no non-Navier-Stokesanomalies in the range of lattice-gas parameters that characterize near incompressibility.

The next topic is quantitative self-consistency. We used the Boltzmann transportapproximation for the discrete model to calculate viscosities for the simple hexagonalautomaton as well as models with additional scattering rules and rest particles. We thenchecked these analytic predictions against the viscosities deduced from two kinds of

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simulations. We ran plane-parallel Poiseuille flow in a channel, saw that it developed theexpected parabolic velocity profile (Fig. 7) and then deduced the viscosity characteristicof this type of flow. We also ran an initially flat velocity distribution and deduced aviscosity from the observed velocity decay. These two simulations agree with each otherto within a few percent and agree with the analytic predictions from the Boltzmanntransport calculation to within 10 percent. Viscosity was also measured by observing thedecay of sound waves of various frequencies (Fig. 8). The level of agreement betweensimulation and the computed Boltzmann viscosity is generic: we see a systematic errorof approximately 10 percent. Monte Carlo calculations of viscosities computed frommicroscopic correlation functions improve agreement with simulations to at least 3percent and indicate that the Boltzmann description is not as accurate an analytic toolfor the automaton as are microscopic correlation techniques. One would call this type ofviscosity disagreement a Boltzmann-induced error. Other consistency checks betweenthe automaton simulation and analytic predictions display the same level of agreement.

Detailed quantitative comparisons between conventional discretizations of theNavier-Stokes equations and lattice-gas simulations have yet to be done for severalreasons. The simple lattice-gas automaton has a Fermi-Dirac distribution rather thanthe standard Maxwell-Boltzmann distribution. This difference alone causes deviationsof 0(v2) in the macrovelocity from standard results. For the same reason and unlikestandard numerical spectral codes for fluid dynamics, the simple lattice-gas automa-ton has a velocity-dependent equation of state. A meaningful comparison between thetwo approaches requires adjusting the usual spectral codes to compute with a velocity-dependent equation of state. This rather considerable task has yet to be done. So far oursimulations can be compared only to traditional two-dimensional computer simulationsand analytic results derived from simple equations of state.

Some simple quantities such as the speed of sound and velocity profiles have beenmeasured in the automaton model. The speed of sound agrees with predicted values andfunctional forms for channel velocity profiles and D’Arcy’s law agree with calculationsby standard methods. The automaton reaches local equilibrium in a few time steps andreaches global equilibrium at the maximum information-transmission speed, namely, atthe speed of sound.

Simulations with the two-dimensional lattice-gas model hang together rather wellas a simulator of Navier-Stokes dynamics. The method is accurate enough to testtheoretical turbulent mechanisms at high Reynolds number and as a simulation tool forcomplex geometries, provided that velocity-dependent effects due to the Fermi natureof the automaton are correctly included. Automaton models can be designed to fitspecific phenomena, and work along these lines is in progress.

Three-dimensional hydrodynamics is being simulated, both on serial and parallelmachines, and early results show that we can easily simulate flows with Reynoldsnumbers of a few thousand. How accurately this model reproduces known instabilitiesand flows remains to be seen, but there is every reason to believe agreement will begood since the ingredients to evolve to Navier-Stokes dynamics are all present. We endPart II of this article with a graphical display of two- and three-dimensional simulationsin “Calculations Using Lattice-Gas Techniques. ” My Los Alamos collaborators and I

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have accompanied this display with a summary of the known advantages and presentlimitations of lattice gas methods. (Part 111 begins on page 21 1.)


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over the last few years the tantaliz-ing prospect of being able to per-form hydrodynamic calculations

orders-of-magnitude faster than presentmethods allow has prompted considerableinterest in lattice gas techniques. A fewdozen published papers have presentedboth advantages and disadvantages, andseveral groups have studied the possibil-ities of building computers specially de-signed for lattice gas calculations. Yet thehydrodynamics community remains gen-erally skeptical toward this new approach.The question is often asked, “What cal-culations can be done with lattice gastechniques?” Enthusiasts respond that inprinciple the techniques are applicable toany calculation, adding cautiously that in-creased accuracy requires increased com-putational effort. Indeed, by adding moreparticle directions, more particles per site,more particle speeds, and more varietyin the interparticle scattering rules, latticegas methods can be tailored to achievebetter and better accuracy. So the realproblem is one of tradeoff: How muchaccuracy is gained by making lattice gasmethods more complex, and what is thecomputational price of those complica-tions? That problem has not yet been wellstudied. This paper and most of the re-search to date focus on the simplest latticegas models in the hope that knowledge ofthem will give some insight into the es-sential issues.

We begin by examining a few of thefeatures of the simple models. We thendisplay results of some calculations. Fi-nally, we conclude with a discussion oflimitations of the simple models.

Features of SimpleLattice Gas Methods

We will discuss in some depth thememory efficiency and the parallelism oflattice gas methods, but first “we will touchon their simplicity, stability, and ability to

model complicated boundaries.Computer codes’ for lattice gas meth-

ods are enormously simpler than thosefor other methods, Usually the essentialparts of the code are contained in only afew dozen lines of FORTRAN. And thosefew lines of code are much less com-plicated than the several hundred linesof code normally required for two- andthree-dimensional hydrodynamic calcula-tions.

There are many hydrodynamic prob-lems that cause most standard codes (suchas finite-difference codes, spectral codes,and particle-in-cell codes) to crash. Thatis, the code simply stops running becausethe algorithm becomes unstable. Stabilityis not a problem with the codes for latticegas methods. In addition, such methodsconserve energy and momentum exactly,with no roundoff errors.

Boundary conditions are quite easy toimplement for lattice gas methods, andthey do not require much computer time.One simply chooses the cells to whichboundary- conditions apply and updatesthose cells in a slightly different way.One of three boundary conditions is com-monly chosen: bounce-back, in whichthe directions of the reflected particlesare simply reversed; specular, in whichmirror-like reflection is simulated; or dif-fusive, in which the directions of the re-flected particles are chosen randomly.

We consider next the memory effi-ciency of the lattice gas method, Whenthe two-dimensional hydrodynamic lat-tice gas algorithm is programmed on acomputer with a word length of, say,64 bits (such as the Cray X-MP), twoimpressive efficiencies occur. The firstarises because every single bit of mem-ory is used equally effectively. Coined“bit democracy” by von Neumann, suchefficient use of memory should be con-trasted with that attainable in standardcalculations, where each number requiresa whole 64-bit word. The lattice gasis “bit democratic” because all that one


Discrete Fluids

fact that lattice gas operations are bit ori-ented rather than floating-point-numberoriented and therefore execute more natu-rally on a computer. Most computers cancarry out logic operations bit by bit. Forexample, the result of the logic operationAND on the 64-bit words A and B is anew 64-bit word in which the ith bit hasa value of 1 only if the ith bits of bothA and B have values of 1. Hence in oneclock cycle a logic operation can be per-formed on information for 64 cells. Sincea Cray X-MP/416 includes eight logicalfunction units, information for 8 times64, or 512, cells can be processed dur-ing each clock cycle, which lasts about10 nanoseconds. Thus information for51,200,000,000 cells can be processedeach second. The two-dimensional latticegas models used so far require from aboutthirty to one hundred logic operations toimplement the scattering rules and aboutanother dozen to move the particles to thenext cells. So the number of cells thatcan be updated each second by logic op-erations is near 500,000,000. Cells canalso be updated by table-lookup meth-ods. The authors have a table-lookupcode for three-dimensional hydrodynam-ics that processes about 30,000,000 cellsper second.

A final feature of the lattice gas methodis that the algorithm is inherently parallel.The rules for scattering particles withina cell depend only on the combinationof particle directions in that cell. Thescattering can be done by table lookup,in which one creates and uses a table ofscattering results-one for each possiblecell configuration. Or it can be done bylogic operations.

Using Lattice Gas MethodsTo Approximate Hydrodynamics

In August 1985 Frisch, Hasslacher, andPomeau demonstrated that one can ap-proximate solutions to the Navier-Stokesequations by using lattice gas methods,

D

c

E

but their demonstration applied only tolow-velocity incompressible flows nearequilibrium. No one knew whether moreinteresting flows could be approximated.Consequently, computer codes were writ-ten to determine the region of validity ofthe lattice gas method. Results of some ofthe first simulations done at Los Alamosand of some later simulations are shownin Figs. 1 through 6. (Most of the earlycalculations were done on a Celerity com-puter, and the displays were done on aSun workstation.) All the results indicatequalitatively correct fluid behavior.

Figure la demonstrates that a stabletrailing vortex pattern develops in a two-dimensional lattice gas flowing past aplate. Figure lb shows that without athree-particle scattering rule, which re-moves the spurious conservation of mo-mentum along each line of particles, novortex develops. (Scattering rules are de-scribed in Part II of the main text.)

Figure 2 shows that stable vortices de-velop in a lattice gas at the interface be-tween fluids moving in opposite direc-tions. The Kelvin-Helmholtz instabilityis known to initiate such vortices. Thefact that lattice gas methods could simu-late vortex evolution was reassuring andcaused several scientists to begin to studythe new method.

Figure 3 shows the complicated wakethat develops behind a V-shaped wedge ina uniform-velocity flow.

Figure 4 shows the periodic oscillationof a low-velocity wake behind a cylin-

der. With a Reynolds number of 76, theflow has a stable period of oscillation thatslowly grows to its asymptotic limit.

Figure 5 shows a flow with a higherReynolds number past an ellipse. Thewake here becomes chaotic and quite sen-sitive to details of the flow.

Figure 6 shows views of a three-dimensional flow around a square plate,which was one of the first results fromLos Alamos in three-dimensional latticegas hydrodynamic simulations.

Rivet and Frisch and other French sci-entists have developed a similar codethat measures the kinematic shear viscos-ity numerically; the results compare wellwith theoretical predictions (see Fig. 8 inthe main text).

The lattice gas calculations of a groupat the University of Chicago (Kadanoff,McNamara, and Zanetti) for two-dimen-sional flow through a channel (Fig. 7of the main text) agree with the knownparabolic velocity profile for low-velocitychannel flows.

The above calculations, and many oth-ers, have established some confidencethat qualitative features of hydrodynamicflows are simulated by lattice-gas meth-ods. Problems encountered in detailedcomparisons with other types of calcula-tions are discussed in the next section.

Limitations of SimpleLattice Gas Models

As we discussed earlier, lattice gasmethods can be made more accurate bymaking them more complicated—by, forexample, adding more velocity directionsand magnitudes. But the added complica-tions degrade the efficiency. We mentionin this section some of the difficulties (as-sociated with limited range of speed, ve-locity dependence of the equation of state,and noisy results) encountered in the sim-plest lattice-gas models.

The limited range of flow velocitiesis inherent in a model that assumes a

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continued from page 203

But in the incompressible, low-velocitylimit the single-speed hexagonal latticegas follows the equation

Conclusion

In the last few years lattice gas methodshave been shown to simulate the quali-tative features of hydrodynamic flows intwo and three dimensions. Precise com-parisons with other methods of calcula-tion remain to be done, but it is believedthat the accuracy of the lattice gas method

Note added in proof: Recently KadanoffMcNamara, and Zanetti reported precisecomparisons between theoretical predic-tions and lattice gas simulations (Univer-sity of Chicago preprint, October 1987)They used a seven-bit hexagonal modelon a small automaton universe to simulate forced two-dimensional channel flowfor long times. Three tests were used toprobe the hydrodynamic and statisticalmechanical behavior of the model. Thetests determined (1) the profile of mo-mentum density in the channel, (2) theequation of state given by the statisticalmechanics of the system, and (3) the log-arithmic divergence in the viscosity (a fa-mous effect in two- dimensional hydrody-namics and a deep test of the accuracy 01the model in the strong nonlinear regime)

The results were impressive, First,to within the accuracy of the simula-tion, there is no discrepancy betweenthe parabolic velocity profile predicted bymacroscopic theory and the lattice gassimulation data. Second, the equation ofstate derived from theory fits the simula-tion data to better than 1 percent. Finally,the measured logarithmic divergence inthe viscosity as a function of channelwidth agrees with prediction. These re-sults are at least one order of magni-tude more accurate than any previouslyreported calculations.


Discrete Fluids

In the sidebar “Calculations Using Lat-tice Gas Techniques” we displayedthe results of generalizing the sim-

ple hexagonal model to three dimensions.Here, in the last part of the article, wewill discuss numerous ways to extend andadapt the simple model. In particular, weemphasize its role as a paradigm for par-allel computing.

Adjusting the Model To Fitthe Phenomenon

There are several reasons for alteringthe geometry and rule set of the funda-mental hexagonal model. To understandthe mathematical physics of lattice gases,we need to know the class of functionallyequivalent models, namely those modelswith different geometries and rules thatproduce the same dynamics in the sameparameter range.

To explore turbulent mechanisms influids, the Reynolds number must be sig-nificantly higher than for smooth flow,so models must be developed that in-crease the Reynolds number in some way.The most straightforward method, otherthan increasing the size of the simula-tion universe, is to lower the effectivemean free path in the gas. This lowersthe viscosity and the Reynolds numberrises in inverse proportion. Increasingthe Reynolds number is also importantfor practical applications. In “ReynoldsNumber and Lattice Gas Calculations”


PART III THE PROMISE OFLATTICE GAS METHODS

we discuss the computational storage andwork needed to simulate high-Reynolds-number flows with cellular automata.

To apply lattice gas methods to sys-tems such as plasmas, we need to developmodels that can support widely separatedtime scales appropriate to, for example,both photon and hydrodynamical modes.The original hexagonal model on a singlelattice cannot do so in any natural waybut must be modified to include severallattices or the equivalent (see below).

Within the class of fluids, problems in-volving gravity on the gas, multi-compo-nent fluids, gases of varying density, andgases that undergo generalized chemicalreactions require variations of the hexago-nal model. Once into the subject of appli-cations rather than fundamental statisticalmechanics, there is an endless industryin devising clever gases that can simulatethe dynamics of a problem effectively.

We outline some of the possible ex-tensions to the hexagonal gas, but do soonly to give an overview of this develop-ing field. Nothing fundamental changesby making the gas more complex. Thismodel is very much like a language. Wecan build compound sentences and para-graphs out of simple sentences, but itdoes not change the fundamental rules bywhich the language works.

The obvious alterations to the hexago-nal model are listed below. They com-prise almost a complete list of what canbe done in two dimensions, since a lattice

Discrete Fluids

els are in general equivalent to single-species models operating on separate lat-tices. Colored collision rules couple thelattices so that information can be trans-ferred between them at different timescales. Certain statistical-mechanical phe-nomena such as phase transitions can bedone this way.

By altering the rule domain and addinggas species with distinct speeds, it is pos-sible to add independent energy conserva-tion. This allows one to tune gas modelsto different equations of state. Again, wegain no fundamental insight into the de-velopment of large collective models bydoing so. but it is useful for applications.

In using these lattice gas variations toconstruct models of complex phenomena,we can proceed in two directions. Thefirst direction is to study whether or notcomplex systems with several types ofcoupled dynamics are described by skele-tal gases. Can complex chemical reac-tions in fluids and gases, for example,be simulated by adding collision rulesoperating on colored multi-speed latticegases? Complex chemisty is set up in thegas in outline form, as a gross scheme ofclosed sets of interaction rules. The sameidea might be used for plasmas. From atheoretical viewpoint one wants to studyhow much of the known dynamics of suchsystems is reproduced by a skeletal gas;consequently both qualitative and quanti-tative results are important.

Exploring Fundamental Questions.Models of complex gas or fluid systems,like other lattice gas descriptions, may ei-ther be a minimalist description of mi-crophysics or simply have no relation tomicrophysics other than a mechanism forcarrying known conservation laws and re-actions. We can always consider such gasmodels to be pure computers, where wefit the wiring, or architecture, to the prob-lem, in the same fashion that ordinary dis-cretization schemes have no relation tothe microphysics of the problem. How-ever for lattice gas models, or cellular-

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212

REYNOLDSNUMBER

andLattice Gas

Calculations

The only model-dependent couplingconstant in the Navier-Stokes equa-tion is the viscosity. Its main role

in lattice gas computations is its influenceon the Reynolds number, an importantscaling concept for flows. Given a systemwith a fixed intrinsic global length scale,such as the size of a pipe or box, andgiven a flow, then the Reynolds numbercan be thought of as the ratio of a typicalmacrodynamic time scale to a time scaleset by elementary molecular processes inthe kinetic model.

Reynolds numbers characterize the be-havior of flows in general, irrespectiveof whether the system is a fluid or agas. At high enough Reynolds num-bers turbulence begins, and turbulencequickly loses all memory of molecularstructure, becoming universal across liq-

uids and gases. For this reason andbecause many interesting physical andmathematical phenomena happen in tur-bulent regimes, it is important to be ableto reach these Reynolds numbers in real-istic simulations without incurring a largeamount of computational work or storage.

Some simple arguments based on di-mensional analysis and phenomenolog-ical theories of turbulence indicate, atfirst glance, that any cellular automatonmodel has a high cost in computer re-sources when simulating high-Reynolds-number flows. These arguments appearedin the first paper on the subject (Frisch,Hasslacher, and Pomeau 1987) and werelater elaborated on by other authors. Wewill go through the derivation of someof the more severe constraints on simu-lating high-Reynolds-number flows with


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continued from page 212

automaton models in general, there al-ways seems to be a deep relation betweenthe abstract computer embodying the gasalgorithm for a physical problem and themathematical physics of the system itself.

This duality property is an importantone, and it is not well understood. Oneof the main aims of lattice gas theory isto make the underlying mathematics ofdynamical evolution clearer by providinga new perspective on it. One would, forexample, like to know the class of all lat-tice gas systems that evolve to a dynam-ics that is, in an appropriate sense, nearbythe dynamics actually evolved by nature.Doing this will allow us to isolate whatis common to such systems and identifyuniversal mathematical mechanisms.

Engineering Design Applications. Thesecond direction of study is highly ap-plied. In most engineering-design sit-uations with complicated systems, onewould like to know first the general qual-itative dynamical behavior taking placein some rather involved geometry andthen some rough numerics. Given both,one can plot out the zoo of dynamicaldevelopment within a design problem.Usually, one does not know what kindsof phenomena can occur as a parameterin the system varies. Analytic methodsare either unavailable, hard to computeby traditional methods, or simply breakdown. Estimating phenomena by scal-ing or arguments depending on order-of-magnitude dimensional analysis is ofteninaccurate or yields insufficient informa-tion. As a result, a large amount of ex-pensive and scarce supercomputer time isused just to scan the parameter space ofa system.

Lattice gas models can perform suchtasks efficiently, since they simulate atthe same speed whether the geometry andsystem are simple or complex. Compli-cated geometries and boundary conditionsfor massively parallel lattice gas simula-tors involve only altering collision rulesin a region. This is easily coded and

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can be done interactively with a little in-vestment in expert systems. There is noquestion that for complex design prob-lems, lattice gas methods can be madeinto powerful design tools.

Beyond Two Dimensions

In two dimensions there exists a single-speed skeletal model for fluid dynamicswith a regular lattice geometry. It re-lies on the existence of a complete tilingof the plane by a domain of sufficientlyhigh symmetry to guarantee the isotropyof macroscopic modes in the model. Inthree dimensions this is not the case, forthe minimum appropriate domain symme-try is icosahedral and such polyhedra donot tile three-space. If we are willingto introduce multiple-speed models, theremay exist a model with high enough ro-tational symmetry, as in the square modelwith nearest and next-to-nearest neighborinteraction in two dimensions, but it is noteasy to find and may not be efficient forsimulations.

A tactic for developing an enlarged-neighborhood, three-dimensional model,which still admits a regular lattice, is tonotice that the number of regular polyhe-dra as a function of dimension has a max-imum in four dimensions. Examinationof the face-centered four-dimensional hy-percube shows that a single-speed modelconnected to each of twenty-four near-est neighbors has exactly the right in-variance group to guarantee isotropy infour dimensions. So four-dimensionalsingle-speed models exist on a regulartiling. Three-dimensional, or regular, hy-drodynamics can be recovered by taking athin one-site slice of the four-dimensionalcase, where the edges of the slice areidentified. Projecting such a scheme intothree-dimensional space generates a two-speed model with nearest and next-to-nearest neighbor interactions of the sortguaranteed to produce three-dimensionalNavier-Stokes dynamics.

Such models are straightforward ex-

tensions of all the ideas present in thetwo-dimensional case and are being sim-ulated presently on large Cray-class ma-chines and the Connection Machine 2.Preliminary results show good agreementwith standard computations at least forlow Reynolds numbers. In particular,simulation of Taylor-Green vortices ata Reynolds number of about 100 on a(128)3 universe (a three-dimensional cubewith 128 cells in each direction) agreeswith spectral methods to within 1 per-cent, the error being limited by MonteCarlo noise. The ultimate comparison isagainst laboratory fluid-flow experiments.As displayed at the end of Pant H, three-dimensional flows around flat plates havealso been done.

A more intriguing strategy is to giveup the idea of a regular lattice. Phys-ical systems are much more like a lat-tice with nodes laid down at random. Atpresent, we don’t know how to analyzesuch lattices, but an approximation can begiven that is intermediate between regu-lar and random grids. Quasi-tilings aresets of objects that completely tile spacebut the grids they generate are not peri-odic. Locally, various types of rotationsymmetry can be designed into such lat-tices, and in three dimensions there ex-ists such a quasi-tiling that has icosahe-dral symmetry everywhere. The beautyof quasi-tilings is that they can all beobtained by simple slices through hyper-cubes in the appropriate dimension. Forthree dimensions the parent hypercube issix-dimensional.

The idea is to run an automaton modelcontaining the conservation laws with assimple a rule set as possible on the six-dimensional cube and then take an appro-priately oriented three-dimensional sliceout of the cube so arranged as to gen-erate the icosahedral quasi-tiling. Sincewe only examine averaged quantities, it isenough to do all the averaging in six di-mensions along the quasi-slice and imagethe results. By such a method we guar-antee exact isotropy everywhere in three


Discrete Fluids

dimensions and avoid computing directlyon the extremely complex lattices that thequasi-tiling generates. Ultimately, onewould like to compute on truly randomlattices, but for now there is no simpleway of doing that efficiently.

The simple four-dimensional model isa good example of the limits of presentsuper-computer power. It is just barelytolerable to run a (1000)3 universe at aReynolds number of order a few thousandon the largest existing Cray’s. It is farmore efficient to compute in large paral-lel arrays with rather inexpensive custommachines, either embedded in an existingparallel architecture or on one designedespecially for this class of problems.

Lattice Gases as ParallelComputers

Let us review the essential features of alattice gas. The first property is the totallydiscrete nature of the description: Thegas is updated in discrete time steps, lat-tice gas elements can only lie on discretespace points arranged in a space-fillingnetwork or grid, velocities can also haveonly discrete values and these are usuallyaligned with the grid directions, and thestate of each lattice gas site is describedby a small number of discrete bits insteadof continuous values.

The second crucial property is the localdeterministic rules for updating the arrayin space and time. The value of a sitedepends only on the values of a few lo-cal neighbors so there is no need for in-formation to propagate across the latticeuniverse in a single step. Therefore, thereis no requirement for a hardwired inter-connection of distant sites on the grid.

The third element is the Boolean natureof the updating rules. The evolution ofa lattice gas can be done by a series ofpurely Boolean operations, without evercomputing with radix arithmetic.

To a computer architect, we have justdescribed the properties of an ideal con-current, or parallel, computer. The iden-


tical nature of particles and the localityof the rules for updating make it naturalto update all sites in one operation—thisis what one means by concurrent or par-allel computation. Digital circuitry canperform Boolean operations naturally andquickly. Advancing the array in time is asequence of purely local Boolean opera-tions on a deterministic algorithm.

Most current parallel computer designswere built with non-local operations inmind. For this reason the basic architec-ture of present parallel machines is over-laid with a switching network that en-ables all sites to communicate in variousdegrees with all other sites. (The usualmodel of a switching network is a tele-phone exchange.) The complexity of ma-chine architecture grows rapidly with thenumber of sites, usually as n log n at bestwith some time tradeoff and as O (n2) atworst. In a large machine, the complex-ity of the switching network quickly be-comes greater than the complexity of thecomputing configuration.

In a purely local architecture switch-ing networks are unnecessary, so two-dimensional systems can be built in atwo-dimensional, or planar configuration,which is the configuration of existinglarge-scale integrated circuits. Such anarchitecture can be made physically com-pact by arranging the circuit boards in anaccordion configuration similar to a pieceof folded paper. Since the type of geome-try chosen is vital to the collective behav-ior of the lattice gas model and no uniquegeometry fits all of parameter space, itwould be a design mistake to hardwire aparticular model into a machine architec-ture. Machines with programmable ge-ometries could be designed in which theswitching overhead to change geometriesand rules would be minimal and the gainin flexibility large (Fig. 9).

In more than two dimensions a purelytwo-dimensional geometry is still effi-cient, using a combination of massiveparallel updating in a two-dimensionalplane and pipelining for the extra dimen-

sions. As technology improves, it is easyto imagine fully three-dimensional ma-chines, perhaps with optical pathways be-tween planes, that have a long mean timeto failure.

The basic hardware unit in conven-tional computers is the memory chip,since it has a large storage capacity (256K bytes or 1 M bytes presently) and isinexpensive, reliable, and available com-mercially in large quantities. In fact,most modem computers have a memory-bound architecture, with a small numberof controlling processors either doing lo-cal arithmetic and logical operations orusing fast hashing algorithms on largelook-up tables. An alternative is the lo-cal architecture described above for lat-tice gas simulators. In computer archi-tecture terms it becomes very attractiveto build compact, cheap, very fast simu-lators which are general over a large classof problems such as fluids. Such ma-chines have a potential processing capac-ity much larger than the general-purposearchitectures of present or foreseen vec-torial and pipelined supercomputers. Anumber of such machines are in the pro-cess of being designed and built, and itwill be quite interesting to see how theseexperiments in non-von Neumann archi-tectures (more appropriately called super-von Neumann) turn out.

At present, the most interesting ma-chine existing for lattice gas work is theConnection Machine with around 65,000elementary processors and several giga-bytes of main memory. This machinehas a far more complex architecture thanneeded for pure lattice-gas work, but itwas designed for quite a different pur-pose. Despite this, some excellent simu-lations have been done on it. The simu-lations at Los Alamos were done mainlyon Crays with SUN workstations serv-ing as code generators, controllers, andgraphical units. The next generation ofmachines will see specialized lattice gasmachines whether parallel, pipelined, orsome combination, running either against

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ARCHITECTURE OF THELATTICE GAS SIMULATOR

Fig. 9. The lattice gas code is a virtual machinein the sense that the way the code works isexactly the way to build a machine.

(a) The basic processor unit in a lattice gassimular has five units: (1) a memory unit thatstores the state at each node of the lattice grid;(2) a propagation unit that advances particlesfrom one node to the next; (3) a scattering unitthat checks the state at each node and imple-ments the scattering rules where appropriate;(4) an averaging unit that averages velocitiesover a preassigned region of the lattice uni-verse; and (5) an output and display unit.

(b) Processors are arranged in a parallel ar-ray. Each processor operates independentlyexcept at nodes on shared boundaries of thelattice gas universe.

(c) Processor units are overlaid by units thatcan alter the geometry of the lattice, the col-lision rules and boundary conditions, and thetype of averaging.

Connection Machine style architecturesor using them as analyzing engines forprocessing data generated in lattice gas“black boxes.” This will be a learn-ing experience for everyone involved inmassive simulation and provide hardwareengines that will have many interestingphysics and engineering applications.

Unfortunately, fast hardware alone isnot enough to provide a truly useful ex-ploration and design tool. A large amountof data is produced in a typical many de-gree of freedom system simulation. Inthree dimensions the problems of access-ing, processing, storing, and visualizingsuch quantities of data are unsolved andare really universal problems even forstandard supercomputer technology. Asthe systems we study become more com-

(c) Control Unit Modifying Processors

plex, all these problems will also. It will that information and introduce artificialtake innovative engineering and physicsapproaches to overcome them.

Conclusion

To any system naturally thought ofas classes of simple elements interactingthrough local rules there should corre-spond a lattice-gas universe that can sim-ulate it. From such skeletal gas models,one can gain a new perspective on theunderlying mathematical physics of phe-nomena. So far we have used only theexample of fluids and related systems thatnaturally support flows. The analysis ofthese systems used the principle of max-imum ignorance: Even though we knowthe system is deterministic, we disregard

probabilistic methods. The reason is thatthe analytic tools for treating problemsin this way are well developed, and al-though tedious to apply, they require nonew mathematical or physical insight.

A deep problem in mathematical phys-ics now comes up. The traditional meth-ods of analyzing large probabilistic sys-tems are asymptotic perturbation expan-sions in various disguises. These containno information on how fast large-scalecollective behavior should occur. Weknow from computer simulations that lo-cal equilibrium in lattice gases takes onlya few time steps, global equilibrium oc-curs as fast as sound propagation will al-low, and fully developed hydrodynamicphenomena, including complex instabil-


Discrete Fluids

ities, happen again as fast as a traverseof the geometry by a sound wave. Onemight say that the gas is precociouslyasymptotic and that this is basically dueto the deterministic property that conveysinformation at the greatest possible speed.

Methods of analyzing the transient andinvariant states of such complex multi-dimensional cellular spaces, using de-terminism as a central ingredient, arejust beginning to be explored. They arenon-perturbative. The problem seems asthough some of the methods of dynam-ical systems theory should apply to it,and there is always the tempting shadowof renormalization-group ideas waiting tobe applied with the right formalism. Sofar we have been just nibbling around theedges of the problem. It is an extraordi-narily difficult one, but breaking it wouldprovide new insight into the origin of ir-reversible processes in nature.

The second feature of lattice gas mod-els, for phenomena reducible to naturalskeletal worlds, is their efficiency com-pared to standard computational meth-ods. Both styles of computing reduceto inventing effective microworlds, butthe conventional one is dictated and con-strained by a limited vocabulary of differ-ence techniques, whereas the lattice gasmethod designs a virtual machine insidea real one, whose architectural structure isdirectly related to physics. It is not a pri-ori clear that elegance equals efficiency.In many cases, lattice gas methods willbe better at some kinds of problems, es-pecially ones involving highly complexsystems, and in others not. Its usefulnesswill depend on cleverness and the prob-lem at hand. At worst the two ways oflooking at the microphysics are comple-mentary and can be used in various mix-tures to create a beautiful and powerfulcomputational tool.

We close this article with a series ofconjectures. The image of the physicalworld as a skeletal lattice gas is essen-tially an abstract mathematical frameworkfor creating algorithms whose dynamics


spans the same solution spaces as manyphysically important nonlinear partial dif-ferential equations that have a micrody-namical underpinning. There is no intrin-sic reason why this point of view shouldnot extend to those rich nonlinear sys-tems which have no natural many-bodypicture. The classical Einstein-Hilbert ac-tion, phrased in the appropriate space, isno more complex than the Navier-Stokesequations. It should be possible to in-vent appropriate skeletal virtual comput-ers for various gauge field theories, be-ginning with the Maxwell equations andproceeding to non-Abelian gauge mod-els. Quantum mechanics can perhapsbe implemented by using a variation onthe stochastic quantization formulation ofNelson in an appropriate gas. When suchmodels are invented, the physical mean-ing of the skeletal worlds is open to in-terpretation. It may be they are only apowerful mathematical device, a kind ofvirtual Turing machine for solving suchproblems. But it may also be that theywill provide a new point of view on thephysical origin and behavior of quan-tum mechanics and fundamental field-theoretic descriptions of the world. ■

Further Reading

G. E. Uhlenbeck and G. W. Ford. 1963. Lecturesin Statistical Mechanics. Providence, Rhode Island:American Mathematical Society.

Paul C. Martin. 1968. Measurements and Corre-lation Functions. New York: Gordon and BreachScience Publishers.

Stewart Harris. 1971. An Introduction to the The-ory of the Boltzmann Equation. New York: HoIt,Rinehart. and Winston.

E. M. Lifshitz and L. P. Pitaevskii. 1981. PhysicalKinetics. Oxford: Pergamon Press.

Stephen Wolfram, editor. 1986. Theory and Ap-plications of Cellular Automata. Singapore: WorldScientific Publishing Company.

L. D. Landau and E. M. Lifshitz. 1987. Fluid Me-chanics, second edition. Oxford: Pergamon Press.

Complex Systems, volume 1, number 4. The entiretyof this journal issue consists of papers presentedat the Workshop on Modem Approaches to LargeNonlinear Systems, Santa Fe, New Mexico, October27-29, 1986.

U. Frisch, B. Hasslacher, and Y. Pomeau. 1986.Lattice-gas automata for the Navier-Stokes equa-tion. Physical Review Letters 56: 1505–1508.

Stephen Wolfram. 1986. Cellular automaton flu-ids 1: Basic theory. Journal of Statistical Physics45: 471-526.

Victor Yakhot and Stephen A. Orszag. 1986. Phys-ical Review Letters 57: 1722.

U. Frisch, D. d’Humieres, B. Hasslacher, P. Lalle-mand, Y. Pomeau, and J. P. Rivet. 1987. ComplexSystems 1: 649-707.

Leo P. Kadanoff, Guy R. McNamara, and GianluigiZanetti. 1987. A Poiseuille viscometer for latticegas automata. Complex Systems 1: 791-803.

Dominique d’Humieres and Pierre Lallemand. 1987.Numerical simulations of hydrodynamics with lat-tice gas automata in two dimensions. Complex Sys-tems 1: 599–632.

Brosl Hasslacher was born and grew up in Man-hattan. He earned his Ph.D. in theoretical physics(quantum field theory) under Daniel Friedman, theinventor of supergravity, at State University of NewYork, Stonybrook. Most of his physics career hasbeen spent at the Institute for Advanced Study inPrinceton and at Caltech, with stops at the Univer-sity of Illinois, Ecole Normale Superieure in Paris,and CERN. He is currently a staff physicist at theLaboratory. Says Brosl, “I spend essentially all mytime thinking about physics. Presently I’m workingon nonlinear field theories, chaotic systems, latticegases, the architectures of very fast parallel comput-ers, and the theory of computation and complexity.”

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