arXiv:cond-mat/0009448 v1 28 Sep 2000Simulating Dynamical Features of Escape PanicDirk Helbing∗,+, Ill´es Farkas,‡and Tam´as Vicsek∗,‡∗Collegium Budapest – Institute for Advanced Study, Szenth´aroms´ag u. 2,H-1014 Budapest, Hungary+Institute for Economics and Traffic, Dresden University of Technology,D-01062 Dresden, Germany‡Department of Biological Physics, E¨otv¨os Unive rsity,P´azm´any P´eter S´et´any 1A, H-1117 Budapest, [email protected]; [email protected]; [email protected] of the most disastrous forms of collective human b ehaviour isthe kind of crowd stampede induced by panic, often leading to fatal-ities as people are crushed or trampled. Sometimes this behaviouris triggered in life-threatening sit uations such as fires in crowdedbuildings;1,2at other times, stampedes can arise from the rush forseats3,4or seemingly without causes. Tragic examples within re-cent months include the panics in Harare, Zimbabwe, and at theRoskilde rock concert in Denmark. Although engineers are findingways to alleviate the scale of such disasters, their frequency seemsto be increasing with the number and size of mass events.2,5Yet,systematic studies of panic behaviour,6−9and quant itative theoriescapable of predicting such crowd dynamics,5,10−12are rare. Here weshow that simulations based on a model of pedestrian behaviour canprovide valuable insights into the mechanisms of and preconditionsfor panic and jamming by incoordination. Our results suggest prac-tical ways of minimising the harmful consequences of such events1Helbing/Farkas/Vicsek: Simulating Dynamical Features of Escape Panic 2and the existence of an optimal escape strategy, corresponding to asuitable mixture of individualistic and collect ive behaviour.Up to now, panics as a particular form of collective behaviour occuring insituations of scarce or dwindling resources1,6has been mainly studied fromthe perspective of social psychology.7−9Panicking individuals tend to showmaladaptive and r elentless mass behaviour like jamming a nd life-threateningovercrowding ,1−4,8which has of t en been attributed to social contagion1,4,8(see Brown9for an overview of theories). According to Mint z,6the observedjamming is a result of incoordination and depends on the reward structure.After having carefully studied the related socio-psychological literature,6−9reports in the media and available video materials (see http://angel.elte.hu/epanic/), empirical investigations,1,2,3and engineering handbooks,13,14wecan summarise the following characteristic features of escape panics: (i) Peo-ple move or try to move considerably faster than normal.13(ii) Individualsstart pushing, and interactions among people become physical in nature. (iii)Moving and, in particula r , passing of a bottleneck becomes incoordinated.6(iv) At exits, arching and clogging are observed.13(v) Jams are building up.7(vi) The physical interactions in the jammed crowd a dd up and cause danger-ous pressur es up to 4,450 Newtons per meter,2,5which can bend steel barriersor tear down brick walls. (vii) Escape is further slowed down by fallen or in-jured people turning into “obstacles”. (viii) People show a tendency of massbehaviour, i.e., to do what other people do.1,8(ix) Alternative exits are oftenoverlooked or not efficiently used in escape situations.1,2These observations have encouraged us to model the collective phenomenon ofescape panic in the spirit of self-driven many-particle systems. Our computerHelbing/Farkas/Vicsek: Simulating Dynamical Features of Escape Panic 3simulations of the crowd dynamics of pedestrians are based on a generalisedforce model,15which is particularly suited to describe the fatal build up of pres-sure observed during panics.2−4,5We assume a mixture of socio-psychological16and physical forces influencing the b ehaviour in a crowd: Each of N pedestri-ans i of mass milikes to move with a certain desired speed v0iinto a certaindirection e0i, and therefore tends to correspondingly adapt his or her actualvelocity viwith a certain characteristic time τi. Simultaneously, he o r shetries to keep a velocity-dependent distance to other pedestrians j and wallsW . This can be modelled by “interaction forces” fijand fiW, respectively.In mathematical terms, the change o f velocity in time t is then given by theacceleration equationmidvidt= miv0i(t)e0i(t) − vi(t)τi+Xj(6=i)fij+XWfiW, (1)while the change of position ri(t) is given by the velocity vi(t) = dri/dt.We describe the psychological tendency of two pedestrians i and j to stayaway from each other by a repulsive interaction force Aiexp[(rij−dij)/Bi] nij,where Aiand Biare constants. dij= k ri− rjk denotes the distance betweenthe pedestrians’ centers of mass, and nij= (n1ij, n2ij) = (ri− rj)/dijis thenormalised vector pointing from pedestrian j to i. If their distance dijissmaller than the sum rij= (ri+ rj) o f their radii riand rj, the pedestrianstouch each other. In this case, we assume two additional forces inspired bygranular interactions,17,18which are essential for understanding the particulareffects in panicking crowds: a “body force” k(rij− dij) nijcounteracting bodycompression and a “sliding f ric tion force” κ(rij−dij) ∆vtjitijimpeding relativetangential motion, if pedestrian i comes close to j. Herein, tij= (−n2ij, n1ij)means t he tangential direction and ∆vtji= (vj−vi)·tijthe tangential velocityHelbing/Farkas/Vicsek: Simulating Dynamical Features of Escape Panic 4difference, while k and κ represent large constants. In summary, we havefij= {Aiexp[(rij− dij)/Bi] + kg(rij− dij)} nij+ κg(rij− dij)∆vtjitij, (2)where the function g(x) is zero, if the pedestrians do not touch each other(dij> rij), otherwise equal to the argument x.The interaction with the walls is treated analogously, i.e., if diWmeans thedistance to wall W , niWdenotes the direction perpendicular to it, and tiWthe direction tangential to it, the corresponding interaction force with the wallreadsfiW= { Aiexp[(ri− diW)/Bi] + kg(ri− diW)} niW− κg(ri− diW)(vi· tiW) tiW.(3)Probably due to the fact that escape panics are unexpected and danger ousevents, which also excludes real-life experiments, we could not find suitabledata on escape panics to test our model quantitatively. This scarcity of datacalls for reliable models. We have, therefore, specified the parameters as fol-lows: With a mass of mi= 80 kg, we represent an average soccer f