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Received22Jaccepted21July2000.
1.Mould,J.R.etal.TheHubbleSpaceTelescopekeyprojectontheextragalacticdistancescale.XXVIII.
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velocitydata.Astron.Astrophys.108,25±39(1994).
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analysisofCepheidvariables.Publ.Astron.Soc.Pacif.98,881±888(1986).
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visualsurfacebrightnesstechnique.Astrophys.J.342,467±475(1989).
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Acknowledgements
WethankR.Akeson,T.Armstrong,A.Bouchez,M.Colavita,T.Nordgren,M.NunezandD.Sasselovforvaluablecomments.PartoftheworkdescribedinthispaperwasperformedattheJetPropulsionLaboratoryundercontractwiththeNationalAeronauticsandSpaceAdministration.ThisresearchhasmadeuseoftheSimbaddatabase,operatedatCentrede
ÂesastronomiquesdeStrasbourg,Strasbourg,France.B.F.L.gratefullyacknowledgesDonne
thesupportofNASAthroughtheMichelsonfellowshipprogramme.B.F.L.acknowledgesthesupportofNASAandtheNSF.
CorrespondenceandrequestsformaterialsshouldbeaddressedtoB.F.L.(e-mail:b¯@astro.caltech.edu).
.................................................................
Simulatingdynamicalfeaturesofescapepanic
ÂsFarkas³&TamaÂsVicsek*³DirkHelbing*²,Ille
ÂromsaÂgu.2,*CollegiumBudapest±InstituteforAdvancedStudy,Szentha
H-1014Budapest,Hungary
²InstituteforEconomicsandTraf®c,DresdenUniversityofTechnology,D-01062Dresden,Germany
³DepartmentofBiologicalPhysics,EoÂzmaÂnyPeÂterSeÂtaÂny1A,ÈtvoÈsUniversity,PaH-1117Budapest,Hungary
..............................................................................................................................................
neersare®ndingwaystoalleviatethescaleofsuchdisasters,theirfrequencyseemstobeincreasingwiththenumberandsizeofmassevents2,5.Butsystematicstudiesofpanicbehaviour6±9andquan-titativetheoriescapableofpredictingsuchcrowddynamics5,10±12arerare.Hereweuseamodelofpedestrianbehaviourtoinvestigatethemechanismsof(andpreconditionsfor)panicandjammingbyuncoordinatedmotionincrowds.Oursimula-tionssuggestpracticalwaystopreventdangerouscrowdpres-sures.Moreover,we®ndanoptimalstrategyforescapefromasmoke-®lledroom,involvingamixtureofindividualisticbehaviourandcollective`herding'instinct.
Uptonow,panicasaparticularformofcollectivebehaviouroccurringinsituationsofscarceordwindlingresources1,6hasbeenmainlystudiedfromtheperspectiveofsocialpsychology7±9.Panick-ingindividualstendtoshowmaladaptiveandrelentlessmassbehaviourlikejammingandlife-threateningovercrowding1±4,8,whichhasoftenbeenattributedtosocialcontagion1,4,8(seeref.9foranoverviewoftheories).Theobservedjammingisaresultofuncoordinatedmotion(`incoordination')anddependsontherewardstructure6.
Wehavestudiedrelatedsocio-psychologicalliterature6±9,reportsinthemediaandavailablevideomaterials(seehttp://angel.elte.hu/,panic/),empiricalinvestigations1±3,andengineeringhandbooks13,14.Thecharacteristicfeaturesofescapepanicscanbesummarizedasfollows:(1)Peoplemoveortrytomoveconsider-ablyfasterthannormal13.(2)Individualsstartpushing,andinter-actionsamongpeoplebecomephysicalinnature.(3)Movingand,inparticular,passingofabottleneckbecomesuncoordinated6.(4)Atexits,archingandcloggingareobserved13.(5)Jamsbuildup7.(6)Thephysicalinteractionsinthejammedcrowdaddupandcausedangerouspressuresupto4,450Nm-1(refs2,5)whichcanbendsteelbarriersorpushdownbrickwalls.(7)Escapeisfurtherslowedbyfallenorinjuredpeopleactingas`obstacles'.(8)Peopleshowatendencytowardsmassbehaviour,thatis,todowhatotherpeopledo1,8.(9)Alternativeexitsareoftenoverlookedornotef®cientlyusedinescapesituations1,2.
Theseobservationshaveencouragedustomodelthecollectivephenomenonofescapepanicintheframeworkofself-drivenmany-particlesystems.Ourcomputersimulationsofthecrowddynamicsofpedestriansarebasedonageneralizedforcemodel15,whichisparticularlysuitedtodescribingthefatalbuildupofpressureobservedduringpanics2±5.Weassumeamixtureofsocio-psychological16andphysicalforcesin¯uencingthebehaviourinacrowd:eachofNpedestriansiofmassmilikestomovewitha
certaindesiredspeedv0iinacertaindirectionei,andthereforetendstocorrespondinglyadapthisorheractualvelocityviwithacertaincharacteristictimeti.Simultaneously,heorshetriestokeepavelocity-dependentdistancefromotherpedestriansjandwallsW.Thiscanbemodelledby`interactionforces'fijandfiW,respectively.Inmathematicalterms,thechangeofvelocityintimetisthengivenbytheaccelerationequation
Oneofthemostdisastrousformsofcollectivehumanbehaviouris
thekindofcrowdstampedeinducedbypanic,oftenleadingtofatalitiesaspeoplearecrushedortrampled.Sometimesthisbehaviouristriggeredinlife-threateningsituationssuchas®resincrowdedbuildings1,2;atothertimes,stampedescanariseduringtherushforseats3,4orseeminglywithoutcause.Althoughengi-NATURE|VOL407|28SEPTEMBER2000|http://www.wendangwang.com
whilethechangeofpositionri(t)isgivenbythevelocityvi
dri=dt.WedescribethepsychologicaltendencyoftwopedestriansiandjtostayawayfromeachotherbyarepulsiveinteractionforceAiexp
nij,whereAiandBiarecon-stants.dij
jjri2rjjjdenotesthedistancebetweenthepedestrians'
centresofmass,andnij
=dijisthenormalizedvectorpointingfrompedestrianjtoi.Thepedestrianstoucheachotheriftheirdistancedijissmallerthanthesumrij
oftheirradiiriandrj.Inthiscase,weassumetwoadditionalforcesinspiredbygranularinteractions17,18,whichareessentialforunder-standingtheparticulareffectsinpanickingcrowds:a`bodyforce'k
nijcounteractingbodycompressionanda`slidingfrictionforce'k
Dvtjitijimpedingrelativetangentialmotion,ifpedes-487
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