Towards the fundamentals of car following theory

arXiv:cond-mat/0212382v2[cond-mat.soft]15Apr2003TowardsthefundamentalsofcarfollowingtheoryIhorLubashevsky,1,2PeterWagner,2andReinhardMahnke31TheoryDepartment,GeneralPhysicsInstitute,RussianAcademyofSciences,VavilovStr.38,Moscow,119991Russia2InstituteofTransportResearch,GermanAerospaceCenter(DLR),Rutherfordstrasse2,12489Berlin,Germany.3FachbereichPhysik,Universit¨atRostock,D–18051Rostock,Germany(Dated:February1,2008)Theproblemofacarfollowingaleadcardrivenwithconstantvelocityisconsidered.Toderivethegoverningequationsforthefollowingcardynamicsacostfunctionalisconstructed.Thisfunctionalrankstheoutcomesofdifferentdrivingstrategies,whichappliestofairlygeneralpropertiesofthedriverbehavior.Assumingrationaldriverbehavior,theexistenceoftheNashequilibriumisproved.Rationaldrivingisdefinedbysupposingthatadrivercorrectscontinuouslythecarmotiontofollowtheoptimalpathminimizingthecostfunctional.Thecorrespondingcar-followingdynamicsisdescribedquitegenerallybyaboundaryvalueproblembasedontheobtainedextremalequations.Linearizationoftheseequationsaroundthestationarystateresultsinageneralizationofthewidelyusedoptimalvelocitymodel.Undercertainconditions(the“densetraffic”limit)therationalcardynamicscomprisestwostages,fastandslow.Duringthefaststageadrivereliminatesthevelocitydifferencebetweenthecars,thesubsequentslowstageoptimizestheheadway.Inthe“densetraffic”limitaneffectiveHamiltoniandescriptionisconstructed.Thisallowsamoredetailednonlinearanalysis.Finally,thedifferencesbetweenrationalandboundedrationaldriverbehaviorarediscussed.Thelatter,inparticular,justifiessomebasicassumptionsusedrecentlybytheauthorstoconstructacar-followingmodellyingbeyondtheframeworksofrationality.PACSnumbers:89.40.-a,45.70.Vn,02.50.LeKeywords:carfollowingmodel,motivatedbehavior,cost(utility)function,rationality,optimaldrivingconditions,HamiltoniandescriptionofcardynamicsI.CAR-FOLLOWINGTHEORIESANDBASICPROPERTIESOFDRIVERBEHAVIORRecently,thetheoreticalandempiricalfoundationsofthephysicsoftrafficflow(forareviewseeRefs.[1,2])hascomeintothefocusofthephysicalcommunity.Themotionofindividualcarshasmanypeculiarities,sinceitiscontrolledbymotivateddriverbehavior,togetherwithsomephysicalboundaries.Nevertheless,onmacro-scopicscalesthevehicleensemblesdisplaysphenomenalikephaseformationandphasetransitionswidelymetinphysicalsystems(see,e.g.,Refs.[1,2,3]).So,thecoop-erativebehaviorofcarstreatedasactiveparticlesseemstobeofamoregeneralnaturethanthemechanicallawsandconstructingaconsistenttheoryoftrafficflow“fromscratch”isuptonowachallengingproblem.Todescribeindividualcardynamicsagreatvarietyofmicroscopicmodelshavebeenproposed.Thesemodelsdifferinthedetailsoftheinteractionbetweencarsandthetimeupdaterule,rangingfromdifferentialequationstocellularautomata[1,2].Therehasbeenabigdealofworkonthemacroscopicbehavioremergingfromthemi-croscopicdynamicswhenexploringthebehaviorofsys-temsofinteractingcars.However,thereisstillalotofcontroversyinboththemacroscopicbehaviorwhencom-paredtoreality[4],andinthemicroscopicfoundationsoftheindividualcardynamicsitself[5].Onecurrentlyadoptedapproachtospecifythemicro-scopicgoverningequationsoftheindividualcarmotionistheso-calledsocialforcemodel.MoredetailscanbefoundinRefs.[6,7,8];hereonlythebasicideasaretouchedon.Ateachmomenttoftime,adriverαchangesthespeedvαofhercardependingontheroadconditionsandthearrangementoftheneighboringcars:dvαdt=gα(vα)+Xα′6=αgαα′(xα,vα|xα′,vα′).(1.1)Thetermgα(vα)describesthemotionofcarαontheemptyroad,whereasthetermgαα′(xα,vα|xα′,vα′)al-lowsfortheinteractionofcarαwithcarα′(α′6=α).Theinteractionisduetothenecessityfordriverαtokeepacertainsafeheadwaydistancebetweenthecars.AllthemodelsmentionedaboveusevariousAns¨atzeforthelastterm.Themostinterestingspecialcase,whichcoversthemajorityofalltrafficflowsituations,isthatofsingle-lanetraffic.Here,allcarscanbeorderedaccordingtotheirpositionontheroadinthecarmotiondirectionxαxα+1,hereα=1,...,N.Mostmodelstakeintoaccountsolelynearestneighboringcarsαandα+1,i.e.,gαα′6=0forα′=α+1and,maybe,α′=α−1only.However,morecomplicatedmodelsexistthatcanbede-scribedasmodelswithanticipation[9,10,11,12,13]ortheso-calledintelligentdrivermodel[14,15].Theearliest“follow-the-leader”models[16,17]relatetheaccelerationaαofcarαtothevelocitydifference(vα−vα+1)only,i.e.,dvαdt=−1τv(vα−vα+1),(1.2)whereτvisthecharacteristictimescaleoftheveloc-ityrelaxation.Insubsequentgeneralizationsofthismodelτvbecameafunctionofthecarmotionstate,in2particular,ofthecurrentvelocityvαandtheheadwayhα=xα+1−xα−ℓ(forareviewseeRefs.[5,18]).Here,ℓisthecarlength.InRefs.[19,20]anotherapproachcalledtheoptimalvelocitymodelisproposedthatde-scribestheindividualcarmotionbydvαdt=−1τv[vα−ϑopt(hα)],(1.3)whereϑopt(h)isthesteady-statevelocity(theoptimalvelocity)chosenbydriversasfunctionoftheheadwayhbetweenthecars.Itshouldbenotedthatthisapproachisrelatedtomuchearliersafetydistancemodels[21,22,23].Concerningthefundamentalsofapproximationssuchasaα=a(vα,vα+1,xα,xα+1),itisnotedthatthereareactuallytwostimuliaffectingthedriverbehavior.Oneofthemisthenecessitytomoveatthemeanspeedoftrafficflow,inthegive