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arXiv:cond-mat/9509075v113Sep1995ParticlehoppingmodelsandtrafficflowtheoryKaiNagelTSA-DO/SA,MSM997,LosAlamosNationalLaboratory,NM87545,U.S.A.,kai@lanl.govandSantaFeInstitute,1399HydeParkRd.,SantaFe,NM87501,U.S.A.,kai@santafe.eduThisversion:February1,2008Thispapershowshowparticlehoppingmodelsfitintothecontextoftrafficflowtheory.Connectionsbetweenfluid-dynamicaltrafficflowmodels,whichderivefromtheNavier-Stokes-equations,andparticlehoppingmodelsareshown.Insomecases,theseconnectionsareexactandhavelongbeenestablished,buthaveneverbeenviewedinthecontextoftraffictheory.Inothercases,criticalbehavioroftrafficjamclusterscanbecomparedtoinstabilitiesinthepartialdifferentialequations.Finally,itisshownhowthisleadstoaconsistentpictureoftrafficjamdynamics.I.INTRODUCTIONTrafficjamshavealwaysbeenannoying.Atleastintheindustrializedcountries,thestandardreactionhasbeentoexpandthetransportationinfrastructuretomatchdemand.Inthisphaseoffastgrowth,relativelyroughplanningtoolsweresufficient.However,inthelastyearsmostindustrializedsocietiesstartedtoseethelimitsofsuchgrowth.Indenselypopulatedareas,thereisonlylimitedspaceavailableforextensionsofthetransportationsystem;andwefaceincreasingpollutionandgrowingaccidentfrequenciesasthedownsidesofmobility.Inconsequence,planningisnowturningtoafine-tuningoftheexistingsystems,withoutmajorextensionsoffacilities.ThisisforexamplereflectedintheUnitedStatesbytheCleanAirActandbytheISTEA(IntermodalSurfaceTransportationandEfficiencyAct)legislation.Theformersetsstandardsofairqualityforurbanareas,whereasthelatterforcesplanningauthoritiestoevaluatelandusepolicies,intermodalconnectivity,andenhancedtransitservicewhenplanningtransportation.Inconsequence,planningandpredictiontoolswithamuchhigherreliabilitythaninthepastarenecessary.Duetothehighcomplexityoftheproblems,analyticalapproachesareinfeasible.Currentapproachesaresimulation-based(e.g.[1–4]),whichisdrivenbynecessity,butlargelyenhancedbythewidespreadavailabilityofcomputingpowernowadays.Yet,alsoforcomputersoneneedsgoodsimplifiedmodelsofthephenomenaofinterest:Justcodingaperfectrepresentationofrealityintothecomputerisnotpossiblebecauseoflimitsofknowledge,limitsofhumanresourcesforcodingallthesedetails,andbecauseoflimitsofcomputationalresources.Practicalsimulationhastoobservetrade-offsbetweenresolution,fidelity,andscale[5].Resolutionreferstothesmallestentities(objects,particles,processes)resolvedinasimulation,whereasfidelitymeansthedegreeofrealisminmodelingeachoftheseentities,andscalemeansthe(spatial,temporal,...)sizeoftheproblem.Itisempiricallywellknown,forexamplefromfluiddynamics,thattoacertainextentalowfidelityhighresolutionmodel(latticegasautomata[6,7])candoaswellasahighfidelitylowresolutionmodel(discretizationoftheNavier-Stokes-equations),orinshort:Resolutioncanreplacefidelity.Currentstate-of-the-arttrafficmodelinghasafixedunitof(minimal)resolution,andthatistheindividualtraveler.Sinceoneisaimingforratherlargescales(forexampletheLosAngelesareaconsistsofapprox.10millionpotentialtravelers),itisratherobviousthatonehastosacrificefidelitytoachievereasonablecomputingtimes.1Oneimportantpartoftransportationmodelingisroadtraffic.ForexampleinGermany,roadtrafficcurrentlycontributesmorethan81%ofallpassengerand52.7%ofallfreighttransportation[8].Anddespitewidespreadefforts,theshareofroadtransportationisstillincreasing.Forthatreason,itmakessensetostartwithroadtrafficwhendealingwithtransportationsystems.Puttingtheseargumentstogether,onethingwhichisneededforlargescaletransportationsimulationsisaminimalrepresentationofroadtraffic.Particlehoppingmodelsclearlyarecandidatesforthis,andevenifnot,buildingaminimaltheoryofroadtrafficiscertainlytherightstartingpoint.Thispapershowshowparticlehoppingmodelsfitintothecontextoftrafficflowtheory.Itstartsoutwithahistoricaloverviewoftrafficflowtheory(SectionII),followedbyasystematicreviewoffluid-dynamicalmodelsfortrafficflow(SectionIII)startingfromtheNavier-Stokes-equations.SectionIVdefinesdifferentparticlehoppingmodelswhichareofinterestinthecontextoftrafficflow.SectionVthenshowsthedifferentconnectionsbetweenthefluid-dynamicaltrafficflowmodelsandparticlehoppingmodels.Insomecases,theseconnectionsareexactandhavelongbeenestablished,buthaveneverbeenviewedinthecontextoftraffictheory.Inothercases,criticalbehavioroftrafficjamclusterscanbecomparedtoinstabilitiesinthepartialdifferentialequations.Finally,itisshownhowthisleadstoaconsistentpictureoftrafficjamdynamics(SectionVI).Adiscussionoftheconsequencesfortrafficsimulations(SectionVII)servesassummaryanddiscussion,andacollectionofopenquestions(SectionVIII)concludethepaper.II.HISTORICALOVERVIEWVehiculartraffichasbeenawidelyandthoroughlyresearchedareainthe1950sand60s.Forareviewoftraffictheory,see,forexample,oneof[9–11].Vehiculartraffictheorycanbebroadlyseparatedintotwobranches:Trafficflowtheory,andcar-followingtheory.Trafficflowtheoryisconcernedwithfindingrelationsbetweenthethreefundamentalvariablesoftrafficflow,whicharevelocityv,densityρ,andcurrentorthroughputorflowj.Onlytwoofthesevariablesareindependentsincetheyarerelatedthroughj=ρv.Thefirsttaskoftrafficflowtheoryhistoricallywastosearchfortime-indepen