Particle-hopping-Models-of-Vehicular-Traffic-Distr

arXiv:cond-mat/9706094v1[cond-mat.stat-mech]10Jun1997Particle-hoppingModelsofVehicularTraffic:DistributionsofDistanceHeadwaysandDistanceBetweenJamsDebashishChowdhury1,2,KingshukGhosh1,ArnabMajumdar1,ShishirSinha1andR.B.Stinchcombe31PhysicsDepartment,IndianInstituteofTechnologyKanpur208016,India2InstituteforTheoreticalPhysics,UniversityofCologne,D-50937K¨oln,Germany3DepartmentofTheoreticalPhysics,UniversityofOxfordOxfordOx13NP,EnglandAbstractWecalculatethedistributionofthedistanceheadways(i.e.,theinstan-taneousgapbetweensuccessivevehicles)aswellasthedistributionofin-stantaneousdistancebetweensuccessivejamsintheNagel-Schreckenberg(NS)modelofvehiculartraffic.Whenthemaximumallowedspeed,Vmax,ofthevehiclesislargerthanunity,overanintermediaterangeofdensitiesofvehicles,ourMonteCarlo(MC)dataforthedistanceheadwaydistri-butionexhibittwopeaks,whichindicatethecoexistenceof”free-flowing”trafficandtrafficjams.Ouranalyticalargumentsclearlyruleoutthepos-sibilityofoccurrenceofmorethanonepeakinthedistributionofdistanceheadwaysintheNSmodelwhenVmax=1aswellasintheasymmetricsimpleexclusionprocess.ModifyingandextendinganearlieranalyticalapproachfortheNSmodelwithVmax=1,andintroducinganoveltrans-fermatrixtechnique,wealsocalculatetheexactanalyticalexpressionforthedistributionofdistancebetweenthejamsinthismodel;thecor-respondingdistributionsforVmax1havebeencomputednumericallythroughMCsimulation.PACS.05.40.+j-Fluctuationphenomena,randomprocesses,andBrownianmotion.PACS.05.60.+w-Transportprocesses:theory.PACS.89.40.+k-Transportation.11.INTRODUCTION:Themotionofeachvehicleintrafficisinfluencedbyothersaroundit;thestatisticsanddynamicsoftrafficflow[1-5]cruciallydependonthese”interac-tions”betweenthevehicles.However,thedynamicsofvehiclesismorecompli-catedthanthatofinteractingparticles[6]becauseofthedependenceonhumanbehaviour;usually,thedriverhastheaimofreachinghisdestinationsafelyintheshortesttimeandcanalsolearnfromexperience.Nevertheless,signifi-cantprogresshasbeenmadeinmodellingtrafficbyextendingtheconceptsandtechniquesofstatisticalmechanics[4,5]bycapturingsomeofthecommonandessentialfeaturesofthedrivinghabitsoftheindividualdrivers.Thetheoreticalmodelsoftrafficcanbebroadlydividedintotwoclasses:(i)continuummodels,and(ii)car−followingmodels;theformeristheanalogueofthe”hydrodynamic”modelsoffluidswhilethelatteristheanalogueofthe”microscopic”modelsinstatisticalmechanics.Theolderversionofthecar-followingtheory[7]wasformulatedmathematicallyusingdifferentialequationsfordescribingthetimeevolutionofthepositionsandspeedsoftheindividualvehiclesintime,whichwastreatedasacontinuousrealvariable.Inthemodernversions,namely,theparticle−hoppingmodels,notonlythepositionandspeedofthevehiclesbuttimealsoisassumedtobediscrete.Someoftheseparticle-hoppingmodels[8]mayberegardedasstochasticcellularautomata[9].Thedistanceheadwayisoneofthemostimportantcharacteristicsofvehic-ulartraffic.Itisdefinedasthedistancefromaselectedpointontheleadvehicletothesamepointonthefollowingvehicle.Usually,thefrontedgesorbumpersareselected[3].Oneoftheaimsofthispaperistocalculatethedistributionsofthedistanceheadwaysinthesteadystate,usingaclassofparticle-hoppingmodels,whichareknowntocapturesomeoftheessentialfeaturesoftrafficflowonhighways.Wealsoreportanalyticalaswellasnumericalresultsonthedistributionofinstantaneousdistancebetweensuccessivevehicularjamsinthesteadystateofthesemodels.Wedefinethemodelsandthecharacteristicquantitiesofinterestinsection2.Wecomputethedistanceheadwaydistributioninsection3andthedistributionofdistancebetweenjamsinsection4.Wesummarizetheresultsandconclusioninsection5.2.THEMODELSANDCHARACTERISTICQUANTITIESOFINTEREST:Inallthe”microscopic”modelsunderourconsideration,alaneisrepresentedbyaone-dimensionallatticeofLsites(i.e.,achainofLequispacedpoints).Eachofthelatticesitescanbeeitheremptyoroccupiedbya”vehicle”.Thereisan”exclusionprinciple”,namely,thatnotwovehiclescanoccupythesamelatticesitesimultaneously.Theratioofthenumberofoccupiedlatticesitestothetotalnumberoflatticesitesisdefinedasthedensityofthevehiclesonthelane.Thedensityistime-dependentforafinitelatticewithopenboundarieswherevehiclesenterfromoneendandleavefromtheother.Inthispaperweshallconsideronly2idealizedsingle-lanehighwayswithperiodicboundaryconditions.Therefore,inthisgeometryofthehighway,thetime-independentdensitycofthevehiclesisN/LwhereN(≤L)isthetotalnumberofvehiclesinthetraffic.2.1.TheAsymmetricSimpleExclusionProcess:Oneofthesimplestdynamicalmodelsofinteractingparticlesystemsistheso-calledasymmetricsimpleexclusionprocess(ASEP)whichmayberegardedasacaricatureofvehiculartraffic.TherulesforupdatingthepositionsoftheparticlesintheASEPareasfollows:oneparticleispickedatrandomandmovedforwardbyonelatticesiteifthenewsiteisempty.Inthisrandomsequentialupdate,itistherandompickingthatintroducesstochasticity(noise)intothemodel.Althoughspeedsoftheparticlesdonotexplicitlyenterintotherules,theeffectivespeedofaparticleintheASEPcantakeonlytwovalues,namely,0andVmax=1.2.2.TheNagel-SchreckenbergModel:Wenowrecallthe”rules”describingthedynamicsofvehiclesintheNagel-Schrecken