Lattice Gases and Cellular Automata

comp-gas/99050015May1999LatticeGasesandCellularAutomataBruceM.BoghosianCenterforComputationalScience,BostonUniversity,3CummingtonStreet,Boston,MA02215,U.S.A.Electronicmail:[e-mail]:brucebAbstractWereviewtheclassofcellularautomataknownaslatticegases,andtheirap-plicationstoproblemsinphysicsandmaterialsscience.Thepresentationisself-contained,andassumesverylittlepriorknowledgeofthesubject.Hydrodynamiclatticegasesareemphasized,andnon-lattice-gascellularautomata{eventhosewithphysicalapplications{arenottreatedatall.Webeginwithareviewoflatticegasesasthetermisunderstoodinequilibriumstatisticalphysics.Wethendiscussthevariousmethodsthathavebeenproposedtosimulatehydrodynamicswithalatticegas,leadinguptothe1986discoveryofalatticegasfortheisotropicNavier-Stokesequations.Finally,wediscussvariantsoflattice-gasmodelsthathavebeenusedforthesimulationofcomplexuids.Keywords:Cellularautomata,latticegases,hydrodynamics,discretekinetictheory,Isingmodel,complexuids,microemulsions1HistoricalBackground1.1TheIsingModelTheuseoflatticegasesforthestudyofequilibriumstatisticalmechanicsdatesbacktoa1920paperofLenz[1]inwhichheproposedtomodelaferromagnetbyaregularD-dimensionallatticeLoftwo-state\spins.Physically,thesemaybethoughtofasthemagnetizationvectorsofelementalmagneticdo-mains,andthemodelconstrainsthemtopointinoneoftwodirections,say\upand\down.Foratwo-dimensionallattice,thisisillustratedschemati-callyinFig.1.Mathematically,thestateofthesystemcanbedescribedbythecollectionofvariablesS(x),indexedbythelatticepointsx2L,andtakingPreprintsubmittedtoElsevierPreprint6May1999Fig.1.D=2Isingmodel:Ateachlatticepointthereisaspin,representedherebyanarrow,pointingeitherupordown.theirvaluesfromthesetf1;+1g;hereS(x)=+1meansthatthespinatsitexispointingup,andS(x)=1meansthatitispointingdown.IfwesupposethatthelatticehasatotalofNjLjsites,thenthetotalnumberofpossiblestatesofthesystemis2NTousethesespinsasamodelofferromagnetism,itwasnecessarytoassignanenergytoeachofthese2Nstates,insuchawayastomakeitenergeticallyfavorableforeachspintoalignwithanexternallyappliedmagneticeld,andforneighboringspinstoalignwitheachother.TherstofthesegoalsisachievedbyincludinganenergycontributionS(x)foreachspinpresent,andthesecondbyincludinganenergycontributionJS(x)S(y)foreachpairofneighboringsitesxandy.Thus,thefullenergyofthesystemisH(S)=XxS(x)J2XxXy2N(x)S(x)S(y);whereN(x)denotesthesetofsitesneighboringsitex,andthefactorof1=2infrontofthesecondtermpreventsdouble-countingofthepairsofspins.Tousethistostudytheequilibriumpropertiesofaferromagnet,itisnecessarytocomputethepartitionfunctionZ(K;h)limN!1XSexpH(S)kBT#;2whereTisthetemperature,KJ=(kBT),h=(kBT),thesumoverSincludesall2Npossiblestatesofthesystem,andwehavetakenthethermo-dynamiclimitbylettingthenumberofspinsgotoinnity.LenzposedtheproblemofcalculatingthisquantitytohisstudentIsing,whosolveditforaone-dimensionallatticeofspinsin1925[2].WhileIsing’sD=1solutioniselementary,Onsager’sD=2solutionforh=0requiredalmostanothertwentyyears[3]tocomplete,andissignicantlymorecomplicated.ThesolutionforthecriticalexponentsforD=2withh6=0isamuchmorerecentdevelopment,rstpublishedbyZamalodchikov[4]in1989.TheproblemforD=3isoutstanding,evenforh=0.1.2UniversalityandMaterialsScienceOnemightwonderwhysomucheorthasbeendevotedtotheIsingmodelwhenitisclearlyonlyacrudeidealizationofarealferromagnet.Certainly,nobodyexpectsthedetailedfunctionalformof,say,thedependenceoftheIsingmodel’smagnetizationM(K;h)=PNxS(x)exphH(x)kBTiPNxexphH(x)kBTi=@lnZ(K;h)@honthetemperatureTtobevalidforanyrealmaterial.Thereare,however,goodreasonstobelievethatcertainfeaturesofthisfunctionalformareuni-versal{thatis,model-independent.Thisisparticularlytruenearcriticality(intheD=2andD=3Isingmodels),wherethespin-spincorrelationlengthdiverges,anductuationsatalllengthscalesarepresent.Forexample,atzeroappliedeldandnearcriticality,themagnetizationvariesasM=8:0forTTcM0(TcTTc)forTTc;whereTcisthecriticaltemperature,M0isaproportionalityconstant,andisanexampleofwhatiscalledacriticalexponent.ThescaleinvarianceoftheuctuationsatthecriticalpointallowarenormalizationgrouptreatmentwhichindicatesthatthecriticalexponentshouldberatherinsensitivetotheparticularmodelHamiltonianused.Infact,criticalexponentsshoulddependononlythedimensionalityofthespaceandthesymmetriesoftheunderlyingHamiltonianfunction.Forexam-ple,theunmagnetizedIsing-modelHamiltonianisinvariantunderthesymme-3trygroupZ2{thatis,multiplicationinthesetf1;+1g{becausetheenergyisinvariantundertheinversionofallthespinsinthesystem.SystemswithZ2symmetryareexpectedtohave=1=8inD=2,and0:33inD=3.Arelatedlatticespinmodel,calledtheHeisenbergmodel,endowseachspinwithavectororientationinthreedimensionsandhasaninteractionHamiltonianthatdependsonlyondotproductsofthesevectorsatneighboringsites.SincetheseareinvariantunderthecontinuousgroupofSO(3)rotations,wemightexpectadierentcriticalexponentfor,andinfactthisisthecase:0:36fortheD=3Heisenbergmodel.Thus,universalityteachesusthatitispossibletolearnsome\realph