Simulation of pedestrian crowds in normal and evacuation situations

Full text

Simulation of Pedestrian Crowds in
Normal and Evacuation Situations
Dirk Helbing1, Ill´es J. Farkas2, P´eter Moln´ar3, and Tam´as Vicsek2
1Institute for Economics and Traffic, Faculty of Traffic Sciences “Friedrich List”,
Dresden University of Technology, D-01062 Dresden, Germany
2Department of Biological Physics, E¨otv¨os University,
P´azm´any P´eter S´et´any 1A, H-1117 Budapest, Hungary
3Center for Theoretical Studies of Physical Systems, Clark Atlanta University,
James P. Brawley Drive, Atlanta, Georgia 30314, USA
Starting with a short review of the available literature in the field of
pedestrian and evacuation research, an overview is given over the ob-
served collective phenomena in pedestrian crowds. This includes lane for-
mation in corridors and oscillations at bottlenecks in normal situations,
while different kinds of blocked states are produced in panic situations.
By means of molecular-dynamic-like microsimulations based on a general-
ized force model of interactive pedestrian dynamics, the spatio-temporal
patterns in pedestrian crowds are successfully reproduced and interpreted
as self-organized phenomena. In contrast to previous socio-psychological
approaches, this allows a physical understanding of the observations. De-
spite the significantly different phenomena occuring in normal and panic
situations, the main effects can be described by a unified model containing
only well interpretable and plausible terms. The transition between the “ra-
tional” normal behavior and the apparently “irrational” panic behavior is
controlled by a single parameter, the “nervousness”, which influences fluc-
tuation strengths, desired speeds, and the tendency of herding. Thereby,
it causes paradoxial effects like “freezing by heating”, “faster is slower”,
and the ignorance of available exits. Nevertheless, there are measures to
improve pedestrian flows, both in normal and panic situations. For exam-
ple, the suitable placement of columns can help, although they reduce the
accessible space.
1 Introduction
The various collective phenomena observed in pedestrian crowds have recently at-
tracted the interest of a rapidly increasing number of scientists. In this review, we
will always distinguish the dynamics of pedestrians in normal and panic situations.
Since both problems are characterized by different characteristic phenomena, they
have often been investigated by different scientific communities. However, as we will
show in the following, they can be treated in a consistent way by one and the same
pedestrian model.

2 Dirk Helbing, Ill´es J. Farkas, P´eter Moln´ar, and Tam´as Vicsek
1.1 Research in normal pedestrian behavior
Pedestrian crowds have been empirically studied for more than four decades now [1–
8]. The evaluation methods applied were based on direct observation, photographs,
and time-lapse films. Apart from behavioral investigations [9,10], the main goal of
these studies was to develop a level-of-service concept [11–13], design elements of
pedestrian facilities [14–19], or planning guidelines [20–24]. The latter have usually
the form of regression relations, which are, however, not very well suited for the
prediction of pedestrian flows in pedestrian zones and buildings with an exceptional
architecture, or in extreme conditions such as evacuation. Therefore, a number of
simulation models have been proposed, e.g. queueing models [25–27], transition ma-
trix models [28], and stochastic models [29,30], which are partly related to each other.
In addition, there are models for the route choice behavior of pedestrians [31–34].
None of these concepts adequately takes into account the self-organization effects
occuring in pedestrian crowds. These may, however, lead to unexpected obstructions
due to mutual disturbances of pedestrian flows. More promising with regard to this
is the approach by Henderson. He conjectured that pedestrian crowds behave sim-
ilar to gases or fluids ([35–38], see also [39,40]). This could be partly confirmed
(see Sec. 2.3). However, a realistic gas-kinetic or fluid-dynamic theory for pedes-
trians must contain corrections due to their particular interactions (i.e. avoidance
and deceleration maneuvers) which, of course, do not obey momentum and energy
conservation. Although such a theory can be actually formulated [34,41,42], for prac-
tical applications a direct simulation of individual pedestrian motion is favourable,
since this is more flexible. As a consequence, current research focusses on the mi-
crosimulation of pedestrian crowds, which also allows us to consider incoordination
by excluded volume effects related to the discrete, “granular” structure of pedestrian
flows. In this connection, a behavioral force model of individual pedestrian dynamics
has been developed [18,43–53] (see Sec. 3). A discrete and simple forerunner of this
model was proposed by Gipps and Marksj¨o (1985). We also like to mention recent
cellular automata of pedestrian dynamics [54–63], and AI-based models [64–67].
1.2 Evacuation and panic research
Computer models for emergency and evacuation situations have been developed as
well [62,68–76]. Most research into panics, however, has been of empirical nature
(see, e.g. Refs. [77–80]), carried out by social psychologists and others.
With some exceptions, panics are observed in cases of scarce or dwindling re-
sources [81,77], which are either required for survival or anxiously desired. They are
usually distinguished into escape panics (“stampedes”, bank or stock market panics)
and acquisitive panics (“crazes”, speculative manias) [82,83], but in some cases this
classification is questionable [84].
It is often stated that panicking people are obsessed by short-term personal
interests uncontrolled by social and cultural constraints [77,82]. This is possibly
a result of the reduced attention in situations of fear [77], which also causes that
options like side exits are mostly ignored [78]. It is, however, frequently attributed to
social contagion [77,79,81–89], i.e., a transition from individual to mass psychology,
in which individuals transfer control over their actions to others [83], leading to

Simulation of Pedestrian Crowds in Normal and Evacuation Situations 3
conformity [90]. This “herding behavior” is in some sense irrational, as it often leads
to bad overall results like dangerous overcrowding and slower escape [83,84,78]. In
this way, herding behavior can increase the fatalities or, more generally, the damage
in the crisis faced.
The various socio-psychological theories for this contagion assume hypnotic ef-
fects, rapport, mutual excitation of a primordial instinct, circular reactions, social
facilitation (see the summary by Brown [88]), or the emergence of normative sup-
port for selfish behavior [89]. Brown [88] and Coleman [83] add another explanation
related to the prisoner’s dilemma [91,92] or common goods dilemma [93], showing
that it is reasonable to make one’s subsequent actions contingent upon those of
others, but the socially favourable behavior of walking orderly is unstable, which
normally gives rise to rushing by everyone. These thoughtful considerations are well
compatible with many aspects discussed above and with the classical experiments by
Mintz [81], which showed that jamming in escape situations depends on the reward
structure (“payoff matrix”).
Nevertheless and despite of the frequent reports in the media and many published
investigations of crowd disasters (see Table 1), a quantitative understanding of the
observed phenomena in panic stampedes has been lacking. In this study, we will add
another aspect to the explanation of panics by simulating a computer model for the
crowd dynamics of pedestrians.
2 Observations
2.1 Normal situations
We have investigated pedestrian motion for several years and evaluated a number of
video films. Despite the sometimes more or less “chaotic” appearance of individual
pedestrian behavior, one can find regularities, some of which become best visible in
time-lapse films like the ones produced by Arns [94]. While describing these, we also
summarize results of other pedestrian studies and observations [18,19,45,95]:
1. Pedestrians feel a strong aversion to taking detours or moving opposite to the
desired walking direction, even if the direct way is crowded. However, there is
also some evidence that pedestrians normally choose the fastest route to their
next destination, but not the shortest one [96]. In general, pedestrians take into
account detours as well as the comfort of walking, thereby minimizing the effort
to reach their destination [97]. Their ways can be approximated by polygons.
2. Pedestrians prefer to walk with an individual desired speed, which corresponds to
the most comfortable (i.e. least energy-consuming) walking speed (see Ref. [8]) as
long as it is not necessary to go faster in order to reach the destination in time.
The desired speeds within pedestrian crowds are Gaussian distributed with a
mean value of approximately 1.34 m/s and a standard deviation of about 0.26 m/s
[35]. However, the average speed depends on the situation [21], sex and age, the
time of the day, the purpose of the trip, the surrounding, etc. [8].
3. Pedestrians keep a certain distance to other pedestrians and borders (of streets,
walls, and obstacles; see [22,24]). This distance is smaller the more a pedestrian
is in a hurry, and it decreases with growing pedestrian density.

4 Dirk Helbing, Ill´es J. Farkas, P´eter Moln´ar, and Tam´as Vicsek
Table 1: Incomplete list of major crowd disasters after J. F. Dickie in Ref. [98],
http://ourworld.compuserve.com/homepages/G Keith Still/disaster.htm,
http://SportsIllustrated.CNN.com/soccer/world/news/2000/07/09/stadium
disasters ap/, and other internet sources (from [99]). The number of injured people was
usually a multiple of the fatalities.
Date Place Venue Deaths Injured Reason
1863 Santiago, Chile Church 2000
1881 Vienna, Austria Theatre 570
1883 Sunderland, UK Theatre 182
1902 Ibrox, UK Stadium 26 517 Collapse of West Stand
1903 Chicago, USA Theatre 602
1943 London, UK Subway Sta-
tion
173 Stampede while air raid
1946 Bolton, UK Stadium 33 400 Collapse of a wall
1955 Santiago, Chile Stadium 6 Fans trying to force their way into
the stadium
1961 Rio de Janeiro,
Brazil
Circus 250
1964 Lima, Peru Stadium 318 500 Goal disallowed
1967 Kayseri, Turkey Stadium 40
1968 Buenos Aires,
Argentina
Stadium 75 150 Fans fleeing from fire
1970 St. Laurent-du-
Pont, France
Dance Hall 146
1971 Ibrox, UK Stadium 66 140 Collapse of barriers
1971 Salvador, Brazil Stadium 4 1500 Fight and wild rush
1974 Cairo, Egypt Stadium 48 Crowds break barriers
1976 Port-au-Prince,
Haiti
Stadium 2 Firecracker
1979 Nigeria Stadium 24 27 Light failure
1979 Cincinatti, USA Stadium 11 Fans trying to force their way into
the stadium
1981 Piraeus, Greece Stadium 24 Rush of leaving fans
1981 Sheffield, UK Stadium 38 Crowd surge
1982 Cali, Columbia Stadium 24 250 Provocation by drunken fans
1982 Moscow, USSR Stadium 340 Re-entering fans after last minute
goal
1985 Bradford, UK Stadium 56 Fire in wooden terrace section
1985 Mexico City,
Mexico
Stadium 10 29 Fans trying to force their way into
the stadium
1985 Brussels, Bel-
gium
Stadium 38 >400 Riots break out
1987 Tripoli, Libya Stadium 2 16 Collapse of a wall
1988 Katmandu,
Nepal
Stadium 93 >100 Stampede due to hailstorm
1989 Hillsborough,
Sheffield, UK
Stadium 96 Fans trying to force their way into
the stadium
1990 Mecca, Saudi
Arabia
Pedestrian
Tunnel
1425 Overcrowding
1991 Orkney, South
Africa
Stadium >40 Fans trying to escape fighting

Simulation of Pedestrian Crowds in Normal and Evacuation Situations 5
Date Place Venue Deaths Injured Reason
1991 New York, USA Stadium 9 Overcrowding at concert
1992 Rio de Janeiro,
Brazil
Stadium 50 Part of the fence giving way
1992 Bastia, Corsica Stadium 17 1900
1994 Mecca, Saudi Arabia 270 “Stoning the devil” ritual
1995 New Delhi, India Tent ≈400 >100 Indian fire
1996 Lusaka, Zambia Stadium 9 78 Stampede after Zambia’s victory
over Sudan
1996 Tembisa, South
Africa
Railway Sta-
tion
15 >20 Electric cattle prods used by secu-
rity guards
1996 Guatemala City,
Guatemala
Stadium 80 180 Fans trying to force their way into
the stadium
1997 Las Vegas, USA Hotel 1 50 Gunshot
1997 D¨usseldorf, Germany Stadium 1 >300 Overcrowding at concert
1998 Dhaka, Bangladesh Multi-Storey
Building
1 15 Fire stampede
1998 Mecca, Saudi Arabia 107 “Stoning the devil” ritual
1998 Harare, Zimbabwe Stadium 4 10 Spectators scrambled for seats
1998 Manila, Phillipines Presidential
Action Cen-
ter
2 Large crowd waiting for jobs and
housing
1998 Chervonohrad,
Ukraine
Cinema 4 Stampede due to in- and outcom-
ing children
1998 Lima, Peru Disco 9 7 Tear gas
1999 Minsk, Belarus Subway Sta-
tion
51 150 Heavy rain at rock concert
1999 Kerala, India Hindu
Shrine
>50 Collapse of parts of the shrine
1999 Benin, Nigeria Religious
Place
14 Stampede at a Christian revivalist
rally
1999 Innsbruck, Austria Stadium 5 25 Fans re-entering the stadium?
2000 Kaloroa, Bangladesh Examination
Place
5 Stampede to enter an examination
hall
2000 Mecca, Saudi Arabia Holy Place 2 4 Pilgrim overcrowding
2000 Durban, South Africa Disco 13 44 Tear gas
2000 Chiaquelane, Mozam-
bique
Chiaquelane
Camp
5 10 Aid chaos
2000 Lisbon, Portugal Nightclub 7 65 Poisonous gas bombs
2000 Seville, Spain 30 Good Friday procession
2000 Monrovia, Liberia Stadium 3 Fans trying to force their way into
the stadium
2000 Lahore, Pakistan Circus 8 3 Guards used batons
2000 Addis Abeba,
Ethiopia
Memorial
Place
14 Children trying to cover from a
rainstorm
2000 Roskilde, Denmark Stadium 8 25 Failure of loud speakers
2000 Harare, Zimbabwe Stadium 12 Tear gas
2000 S˜ao Janu´ario, Brazil Stadium 200 Oversold stadium
Resting individuals (waiting on a railway platform for a train, sitting in a din-
ing hall, or lying at a beach) are uniformly distributed over the available area if
there are no acquaintances among the individuals. Pedestrian density increases
(i.e. interpersonal distances lessen) around particularly attractive places. It de-

6 Dirk Helbing, Ill´es J. Farkas, P´eter Moln´ar, and Tam´as Vicsek
creases with growing velocity variance, e.g., on a dance floor [41,18]. Individuals
knowing each other may form groups which are entities behaving similar to single
pedestrians. Group sizes are Poisson distributed [100–102].
2.2 Panic situations
Panic stampede is one of the most tragic collective behaviors [79,81–83,85–89], as
it often leads to the death of people who are either crushed or trampled down
by others. While this behavior is comprehensible in life-threatening situations like
fires in crowded buildings [77,78], it is hard to understand in cases of a rush for
good seats at a pop concert [84] or without any obvious reasons. Unfortunately, the
frequency of such disasters is increasing [84] (see Table 1), as growing population
densities combined with easier transportation lead to greater mass events like pop
concerts, sporting events, and demonstrations. Nevertheless, systematic studies of
panics [81,103] are rare [77,82,84], and there is a scarcity of quantitative theories
capable of predicting the dynamics of human crowds [62,68,69,72,73,76]. In spite of
this, the following features appear to be typical [52,53]:
1. In situations of escape panics, individuals are getting nervous, i.e. they tend to
develop blind actionism.
2. People try to move considerably faster than normal [21].
3. Individuals start pushing, and interactions among people become physical in
nature.
4. Moving and, in particular, passing of a bottleneck frequently becomes incoordi-
nated [81].
5. At exits, jams are building up [81]. Sometimes, arching and clogging are observed
[21], see Fig. 1.
6. The physical interactions in jammed crowds add up and can cause dangerous
pressures up to 4,500 Newtons per meter [78,98], which can bend steel barriers
or tear down brick walls.
7. Escape is slowed down by fallen or injured people turning into “obstacles”.
8. People tend to show herding behavior, i.e., to do what other people do [77,86].
9. Alternative exits are often overlooked or not efficiently used in escape situations
[77,78].
The following quotations give a more personal impression of the conditions during
escape panics:
1. “They just kept pushin’ forward and they would just walk right on top of you,
just trample over ya like you were a piece of the ground.” (After the panic at
“The Who Concert Stampede” in Cincinatti.)
2. “People were climbin’ over people ta get in ... an’ at one point I almost started
hittin’ ’em, because I could not believe the animal, animalistic ways of the people,
you know, nobody cared.” (After the panic at “The Who Concert Stampede”.)
3. “Smaller people began passing out. I attempted to lift one girl up and above to
be passed back ... After several tries I was unsuccessful and near exhaustion.”
(After the panic at “The Who Concert Stampede”.)

Simulation of Pedestrian Crowds in Normal and Evacuation Situations 7
Figure 1: Panicking football fans trying to escape the football stadium in Sheffield. Be-
cause of a clogging effect, it is difficult to pass the open door.
4. I “couldn’t see the floor because of the thickness of the smoke.” (After the “Hilton
Hotel Fire” in Las Vegas.)
5. “The club had two exits, but the young people had access to only one, said Narend
Singh, provincial minister for agriculture and environmental affairs. However, the
club’s owner, Rajan Naidoo, said the club had four exits, and that all were open.
‘I think the children panicked and headed for the main entrance where they
initially came in,’ he said.” (After the “Durban Disco Stampede”.)
2.3 Analogies with gases, fluids, and granular media
When the density is low, pedestrians can move freely, and crowd dynamics can be
compared with the behavior of gases. At medium and high densities, the motion
of pedestrian crowds shows some striking analogies with the motion of fluids and
granular flows:
1. Footprints of pedestrians in snow look similar to streamlines of fluids [34].
2. At borderlines between opposite directions of walking one can observe “viscous
fingering” [104,105].
3. The emergence of pedestrian streams through standing crowds [94,18,19,45] ap-
pears analogous to the formation of river beds [106–108] (see Fig. 2).
4. Similar to segregation or stratification phenomena in granular media [109,110],
pedestrians spontaneously organize in lanes of uniform walking direction, if the
pedestrian density is high enough [2,18,19,43,45] (see Fig. 3).
5. At bottlenecks (e.g. corridors, staircases, or doors), the passing direction of pedes-
trians oscillates [46,47]. This may be compared to the “saline oscillator” [111] or
the granular “ticking hour glass” [112,113].

8 Dirk Helbing, Ill´es J. Farkas, P´eter Moln´ar, and Tam´as Vicsek
Figure 2: The long-term photograph of a standing crowd in front of a cinema taken by
Thomas Arns shows that crossing pedestrians form a river-like stream (from [18,19,45]).
Figure 3: At sufficiently high densities, pedestrians form lanes of uniform walking direc-
tion (from [18,19,45]).

Simulation of Pedestrian Crowds in Normal and Evacuation Situations 9
6. One can find the propagation of shock waves in dense pedestrian crowds pushing
forward (see also [114]).
7. The arching and clogging in panicking crowds [53] is similar to the outflow of
rough granular media through small openings [115,116].
In summary, one could say that fluid-dynamic analogies work well in normal situa-
tions, while granular aspects become important in panic situations.
3 Generalized force model of pedestrian dynamics
3.1 The social force concept
Human behavior often seems to be “chaotic”, irregular, and unpredictable. So, why
and under what conditions can we model it by means of forces? First of all, we need to
be confronted with a phenomenon of motion in some (quasi-)continuous space, which
may be also an abstract behavioral space or opinion scale [34,117–119]. Moreover, it
is favourable to have a system where the fluctuations due to unknown influences are
not large compared to the systematic, deterministic part of motion. This is usually
the case in pedestrian and vehicle traffic, where people are confronted with standard
situations and react automatically rather than taking complicated decisions between
various possible alternatives. For example, an experienced driver would not have to
think about the detailled actions to be taken when turning, accelerating, or changing
lanes.
This automatic behavior can be interpreted as the result of a learning process
based on trial and error [19], which can be simulated with evolutionary algorithms
[120–123]. For example, pedestrians have a preferred side of walking [2,4,8], since
an asymmetrical avoidance behavior turns out to be profitable [54]. The related
formation of a behavioral convention can be described by means of evolutionary
game theory [34,43,119,124–127].
Another requirement is the vectorial additivity of the separate force terms re-
flecting different environmental influences. This is probably an approximation, but
there is some experimental evidence for it. Based on quantitative measurements for
animals and test persons subject to separately or simultaneously applied stimuli of
different nature and strength, one could show that the behavior in conflict situations
can be described by a superposition of forces [128–130]. This fits well into a concept
by Lewin [131], according to which behavioral changes are guided by so-called so-
cial fields or social forces, which has been put into mathematical terms by Helbing
[34,43,47,117–119]. In some cases, social or behavioral forces, which determine the
amount and direction of systematic behavioral changes, can be expressed as gradients
of dynamically varying potentials, which reflect the social or behavioral fields result-
ing from the interactions of individuals. The behavioral force concept was applied to
opinion formation [34,117–119] and migration [34,118,119], but it was particularly
successful in the description of pedestrian and vehicle traffic [43,46,47,132,133].
For reliable simulations of pedestrian crowds we do not need to know whether a
certain pedestrian, say, turns to the right at the next intersection. It is sufficient to
have a good estimate what percentage of pedestrians turns to the right. This can be
either empirically measured or calculated by means of route choice models like the

10 Dirk Helbing, Ill´es J. Farkas, P´eter Moln´ar, and Tam´as Vicsek
one by Borgers and Timmermans [31,32]. In some sense, the uncertainty about the
individual behaviors is averaged out at the macroscopic level of description, as in
fluid dynamics. Nevertheless, instead of a fluid-dynamic model, we will use the more
flexible microscopic simulation approach based on the generalized force concept.
According to this, the temporal change of the location xi(t) of pedestrian iobeys
the equation of motion
dxi(t)
dt =vi(t).(1)
Moreover, if fi(t) denotes the sum of forces influencing pedestrian i,miis the mass of
pedestrian i, and ξi(t) are individual fluctuations reflecting unsystematic behavioral
variations, the velocity changes are given by the acceleration equation
mi
dvi
dt =fi(t) + ξi(t).(2)
Particular advantages of this approach are that we can take into account
–the flexible usage of space by pedestrians, requiring a (quasi-)continous treatment
of motion, and
–excluded volume effects due to granular properties of panicking pedestrian
crowds.
It turns out that these points are essential to reproduce the above mentioned phe-
nomena in a natural way.
3.2 Social force model for normal pedestrian dynamics
We will now describe the different motivations of and influences on a pedestrian iby
separate force terms. First of all, the desire to adapt the actual velocity vi(t) to the
desired speed v0
iand direction e0
i(t) within a certain “relaxation time” τiis reflected
by the acceleration term [v0
i(t)e0
i(t)−vi(t)]/τi. Herein, the contribution v0
i(t)e0
i(t)/τi
can be interpreted as driving term, while −vi(t)/τihas the meaning of a friction
term with friction coefficient 1/τi.
Next, the tendency of pedestrians to keep a certain distance to other pedestrians
(“territorial effect”) may be described by repulsive social forces
fsoc
ij (t) = Aiexp[(rij −dij )/Bi]nij µλi+ (1 −λi)1 + cos(ϕij )
2¶.(3)
Herein, Aidenotes the interaction strength and Bithe range of the repulsive inter-
actions, which are culture-dependent and individual parameters. dij (t) = kxi(t)−
xj(t)kis the distance between the centers of mass of pedestrians iand j,rij = (ri+rj)
the sum of their radii riand rj, and
nij (t) = ¡n1
ij (t), n2
ij (t)¢=xi(t)−xj(t)
dij (t)(4)
the normalized vector pointing from pedestrian jto i. Finally, with the choice
λi<1, we can reflect the anisotropic character of pedestrian interaction. In other

Simulation of Pedestrian Crowds in Normal and Evacuation Situations 11
words, with the parameter λiwe can model that the situation in front of a pedes-
trian has a larger impact on his or her behavior than things happening behind.
The angle ϕij (t) denotes the angle between the direction ei(t) = vi(t)/kvi(t)kof
motion and the direction −nij (t) of the object exerting the repulsive force, i.e.
cos ϕij (t) = −nij (t)·ei(t). One may, of course, take into account other details such
as a velocity-dependence of the forces and non-circular shaped pedestrian bodies,
but this does not have qualitative effects on the dynamical phenomena resulting in
the simulations. In fact, most observed self-organization phenomena are quite insen-
sitive to the specification of the interaction forces, as different studies have shown
[47,49,51–53].
In addition, we may also take into account time-dependent attractive interactions
towards window displays, sights, or special attractions kby social forces fatt
ik (t)
of the type (3). However, compared with repulsive interactions, the corresponding
interaction range Bik is usually larger and the strength parameter Aik(t) typically
smaller, negative, and time-dependent. Additionally, the joining behavior [134] of
families, friends, or tourist groups can be reflected by forces of the type fatt
ij (t) =
−Cij nij (t), which guarantee that acquainted individuals join again, after they have
accidentally been separated by other pedestrians.
In summary, the force model of pedestrian motion in normal situations corre-
sponds to Eqs. (1) and (2) with
fi(t) = v0
i(t)e0
i(t)−vi(t)
τi
+X
j(6=i)
[fsoc
ij (t) + fatt
ij (t)] + X
b
fib(t) + X
k
fatt
ik (t).(5)
In the following, we will use a simplified version of this model by dropping attraction
effects and assuming λi= 0, so that the interaction forces become isotropic and
conform with Newton’s 3rd law.
3.3 Force model for panicking pedestrians
Additional, physical interaction forces fph
ij come into play when pedestrians get so
close to each other that they have physical contact (rij ≥dij). In this case, which
is mainly relevant to panic situations, we assume also a “body force” k(rij −dij )nij
counteracting body compression and a “sliding friction force” κ(rij −dij )∆vt
ji tij im-
peding relative tangential motion. Inspired by the formulas for granular interactions
[115,116], we assume
fph
ij (t) = kΘ(rij −dij )nij +κΘ(rij −dij )∆vt
ji tij ,(6)
where the function Θ(z) is equal to its argument z, if z≥0, otherwise 0. More-
over, tij = (−n2
ij , n1
ij ) means the tangential direction and ∆vt
ji = (vj−vi)·tij the
tangential velocity difference, while kand κrepresent large constants.
Strictly speaking, friction effects already set in before pedestrians touch each
other, because of the psychological tendency not to pass other individuals with a
high relative velocity, when the distance is small. This is, however, not important
for the effects we are going to reproduce later on.
The interactions with the boundaries of walls and other obstacles are treated
analogously to pedestrian interactions, i.e., if dib(t) means the distance to boundary

12 Dirk Helbing, Ill´es J. Farkas, P´eter Moln´ar, and Tam´as Vicsek
b,nib(t) denotes the direction perpendicular to it, and tib (t) the direction tangential
to it, the corresponding interaction force with the boundary reads
fib ={Aiexp[(ri−dib)/Bi] + kΘ(ri−dib )}nib −κΘ(ri−dib )(vi·tib)tib .(7)
Finally, fire fronts are reflected by repulsive social forces similar those describing
walls, but they are much stronger. The physical interactions, however, are qualita-
tively different, as people reached by the fire front become injured and immobile
(vi= 0).
4 Simulation results
The generalized force model of pedestrian dynamics has been simulated on a com-
puter for a large number of interacting pedestrians confronted with different sit-
uations. In spite of its simplifications, it describes a lot of observed phenomena
quite realistically. Especially, it allows us to explain various self-organized spatio-
temporal patterns that are not externally planned, prescribed, or organized, e.g. by
traffic signs, laws, or behavioral conventions. Instead, the spatio-temporal patterns
discussed below emerge due to the non-linear interactions of pedestrians even with-
out assuming strategical considerations or communication of pedestrians. Many of
these collective patterns of motion are symmetry-breaking phenomena, although the
model was formulated completely symmetric with respect to the right-hand and the
left-hand side [18,19,45,48].
4.1 Self-organized pedestrian dynamics in normal situations
Lane formation. Our microsimulations reproduce the empirically observed for-
mation of lanes consisting of pedestrians with the same desired walking direction
[18,19,44,46–51] (see Fig. 4). For open boundary conditions, these lanes are dynam-
ically varying. Their number depends on the width of the street [18,47], on pedes-
trian density, and on the noise level. Interestingly, one finds a noise-induced ordering
[51,135]: Compared to small noise amplitudes, medium ones result in a more pro-
nounced segregation (i.e., a smaller number of lanes), while large noise amplitudes
lead to a “freezing by heating” effect (see Fig. 7).
Figure 4: Formation of lanes in initially disordered pedestrian crowds with opposite walk-
ing directions (after [52,136]; cf. also [46,47,95,99,136]). White disks represent pedestri-
ans moving from left to right, black ones move the other way round. Lane formation
does not require the periodic boundary conditions applied above, see the Java applet
http://www.helbing.org/Pedestrians/Corridor.html.

Simulation of Pedestrian Crowds in Normal and Evacuation Situations 13
The conventional interpretation of lane formation assumes that pedestrians tend
to walk on the side which is prescribed in vehicular traffic. However, the above
model can explain lane formation even without assuming a preference for any side
[51,52]. The most relevant point is the higher relative velocity of pedestrians walking
in opposite directions. Pedestrians moving against the stream or in areas of mixed
directions of motion will have frequent and strong interactions. In each interaction,
the encountering pedestrians move a little aside in order to pass each other. This
sidewards movement tends to separate oppositely moving pedestrians, which leads
to segregation. The resulting collective pattern of motion minimizes the frequency
and strength of avoidance maneuvers, if fluctuations are weak. Assuming identical
desired velocities v0
i=v0, the most stable configuration corresponds to a state with
a minimization of the average interaction strength
−1
NX
i6=j
τfij ·e0
i≈1
NX
i
(v0−vi·e0
i) = v0(1 −E).(8)
It is related with a maximum efficiency
E=1
NX
i
vi·ei
v0
(9)
of motion corresponding to optimal self-organization [51], where the efficiency E
with 0 ≤E≤1 describes the average fraction of the desired speed v0with which
pedestrians actually approach their destinations (N=Pi1 is the respective number
of pedestrians i). As a consequence, lane formation “globally” maximizes the average
velocity into the respectively desired direction of motion, although the model does
not even assume that pedestrians would try to optimize their behavior locally. This
is a consequence of the symmetrical interactions among pedestrians with opposite
walking directions. One can even show that a large class of driven many-particle
systems, if they self-organize at all, tend to globally optimize their state [51].
Finally, note that lane formation is hard to describe by cellular automata. How-
ever, Burstedde et al. have recently found a way to reproduce this collective phe-
nomenon by introducing an additional floor field inspired by trail formation models
[97,137,138,45], which mimics individual intelligence. In the limit of vanishing diffu-
sion and fast decay of the floor field, this cellular automaton is similar to Helbing and
Bolay’s implementation of an efficient, discretized version of the social force model
[54], where the interaction effects of boundaries and pedestrians are represented by
a global potential. Some interactive Java applets based on this model are available
at www.helbing.org.
Oscillations at bottlenecks. In simulations of bottlenecks like doors, we observe
oscillatory changes of the passing direction, if people do not panic [18,19,44,46–
50] (see Fig. 5). Once a pedestrian is able to pass the narrowing, pedestrians with
the same walking direction can easily follow. Hence, the number and “pressure” of
waiting and pushing pedestrians becomes less than on the other side of the narrow-
ing where, consequently, the chance to occupy the passage grows. This leads to a
deadlock situation which is followed by a change in the passing direction.

14 Dirk Helbing, Ill´es J. Farkas, P´eter Moln´ar, and Tam´as Vicsek
Figure 5: Oscillations of the passing direction at a bottleneck (after [95,99,136];
cf. also [46,47]). Dynamic simulations are available at http://www.helbing.org/
Pedestrians/Door.html.
Dynamics at intersections. At intersections one is confronted with various al-
ternating collective patterns of motion which are very short-lived and unstable. For
example, phases during which the intersection is crossed in “vertical” or “horizon-
tal” direction alternate with phases of temporary roundabout traffic (see Fig. 6)
[46,44,18,49,50,48,19]. This self-organized round-about traffic is similar to the emer-
gent rotation found for self-driven particles [139]. It is connected with small detours
but decreases the frequency of necessary deceleration, stopping, and avoidance ma-
neuvers considerably, so that pedestrian motion becomes more efficient on average.
Figure 6: Self-organized, short-lived roundabout traffic in intersecting pedestrian streams
(from [18,19,44,48,95]; see also [46,49,50]).
4.2 Collective phenomena in panic situations
In panic situations (e.g. in some cases of emergency evacuation) the following char-
acteristic features of pedestrian behavior are often observed:

Simulation of Pedestrian Crowds in Normal and Evacuation Situations 15
1. People are getting nervous, resulting in a higher level of fluctuations.
2. They are trying to escape from the source of panic, which can be reflected by a
significantly higher desired velocity.
3. Individuals in complex situations, who do not know what is the right thing to do,
orient at the actions of their neighbours, i.e. they tend to do what other people
do. We will describe this by an additional herding interaction, but attractive
interactions have probably a similar effect.
We will now discuss the fundamental collective effects which fluctuations, increased
desired velocities, and herding behavior can have. In contrast to other approaches,
we do not assume or imply that individuals in panic or emergency situations would
behave relentless and asocial, although they sometimes do.
“Freezing by heating”. The effect of getting nervous has been investigated in
Ref. [52]. Let us assume the individual level of fluctuations is given by
ηi= (1 −ni)η0+niηmax ,(10)
where niwith 0 ≤ni≤1 measures the nervousness of pedestrian i. The parame-
ter η0means the normal and ηmax the maximum fluctuation strength. It turns out
that, at sufficiently high pedestrian densities, lanes are destroyed by increasing the
fluctuation strength (which is analogous to the temperature). However, instead of
the expected transition from the “fluid” lane state to a disordered, “gaseous” state,
a solid state is formed. It is characterized by a blocked situation with a regular
(i.e. “crystallized” or “frozen”) structure so that we call this paradoxial transition
“freezing by heating” (see Fig. 7). Notably enough, the blocked state has a higher
degree of order, although the internal energy is increased and the resulting state is
metastable with respect to structural perturbations such as the exchange of oppo-
sitely moving particles [52]. Therefore, “freezing by heating” is just opposite to what
one would expect for equilibrium systems, and different from fluctuation-driven or-
dering phenomena in some granular systems [140–142], where fluctuations lead from
a disordered metastable to an ordered stable state [135].
Figure 7: Noise-induced formation of a crystallized, “frozen” state in a periodic corridor
used by oppositely moving pedestrians (after [52,95,99,136]).
The precondition for the unusual freezing-by-heating transition are the driving
term v0
ie0
i/τiand the dissipative friction −vi/τi, while the sliding friction force is not
required. Inhomogeneities in the channel diameter or other impurities which tem-
porarily slow down pedestrians can further this transition at the respective places.

16 Dirk Helbing, Ill´es J. Farkas, P´eter Moln´ar, and Tam´as Vicsek
Finally note that a transition from fluid to blocked pedestrian counter flows is also
observed, when a critical density is exceeded [52,59].
Transition to incoordination due to clogging. The simulated outflow from a
room is well-coordinated and regular, if the desired velocities v0
i=v0are normal.
However, for desired velocities above 1.5 m/s, i.e. for people in a rush, we find an
irregular succession of arch-like blockings of the exit and avalanche-like bunches
of leaving pedestrians, when the arches break (see Fig. 8a, b). This phenomenon
is compatible with the empirical observations mentioned above and comparable to
intermittent clogging found in granular flows through funnels or hoppers [115,116]
(although this has been attributed to static friction between particles without remote
interactions, and the transition to clogging has been observed for small enough
openings rather than for a variation of the driving force).
50
55
60
65
70
75
0 1 2 3 4
Leaving Times T (s)
Desired Velocity v0 (m/s)
0
50
100
150
200
250
300
350
400
0 1 2 3 4 5 6 7 8 9 10
Desired Velocity v0 (m/s)
Leaving Time for 200 People (s)
Number of Injured People 0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0 1 2 3 4 5 6 7 8 9 10
Pedestrian Flow J (s-1m-1),
Divided by Desired Velocity v0
Desired Velocity v0 (m/s)
a b
c d
Figure 8: Simulation of pedestrians moving with identical desired velocity v0
i=v0towards
the 1 m wide exit of a room of size 15 m ×15 m (from [53], see also [95,99,136]). aSnapshot
of the scenario. Dynamic simulations are available at http://angel.elte.hu/˜panic/.
bIllustration of leaving times of pedestrians for various desired velocities v0. Irregular
outflow due to clogging is observed for high desired velocities (v0≥1.5 m/s, see dark
plusses). cUnder conditions of normal walking, the time for 200 pedestrians to leave the
room decreases with growing v0. Desired velocities higher than 1.5 m/s reduce the efficiency
of leaving, which becomes particularly clear, when the outflow Jis divided by the desired
velocity (see d). This is due to pushing, which causes additional friction effects. Moreover,
above a desired velocity of about v0= 5 m/s (– –), people are injured and become non-
moving obstacles for others, if the sum of the magnitudes of the radial forces acting on
them divided by their circumference exceeds a pressure of 1600 N/m [98].

Simulation of Pedestrian Crowds in Normal and Evacuation Situations 17
“Faster-is-slower effect” due to impatience. Since clogging is connected with
delays, trying to move faster (i.e., increasing v0
i) can cause a smaller average speed
of leaving, if the friction parameter κis large enough (see Fig. 8c, d). This “faster-
is-slower effect” is particularly tragic in the presence of fires, where fleeing people
sometimes reduce their own chances of survival. The related fatalities can be esti-
mated by the number of pedestrians reached by the fire front (see Fig. 9).
Fire
0
20
40
60
80
100
0 0.05 0.1 0.15 0.2 0.25
Number of Injured People
Propagation Velocity V of FireFront (m/s)
Pedestrian Flow J, Divided
by Desired Velocity v0
a b
Figure 9: Simulation of N= 200 individuals fleeing from a linear fire front, which prop-
agates from the left to the right wall with velocity V, starting at time t= 5 s [99] (for
a Java simulation applet, see http://angel.elte.hu/˜panic/). aSnapshot of the sce-
nario for a 15 m ×15 m large room with one door of width 1 m. The fire is indicated by
dark grey color. Pedestrians reached by the fire front are injured and symbolized by black
disks, while the white ones are still active. The socio-psychological effect of the fire front is
assumed 10 times stronger than that of a normal wall (AF= 10Ai). bNumber of injured
persons (casualities) as a function of the propagation velocity Vof the fire front, averaged
over 10 simulation runs. Up to a critical propagation velocity Vcrit (here: about 0.1 m/s),
nobody is injured. However, for higher velocities, we find a fast increase of the number of
casualties with increasing V. The transition is continuous.
Since our friction term has, on average, no deceleration effect in the crowd, if
the walls are sufficiently remote, the arching underlying the clogging effect requires
acombination of several effects:
1. slowing down due to a bottleneck such as a door and
2. strong inter-personal friction, which becomes dominant when pedestrians get too
close to each other. It is noteworthy that the faster-is-slower effect also occurs
when the sliding friction force changes continuously with the distance rather than
being “switched on” at a certain distance rjas in the model above.
The danger of clogging can be minimized by avoiding bottlenecks in the construc-
tion of stadia and public buildings. Notice, however, that jamming can also occur
at widenings of escape routes! This surprising result is illustrated in Fig. 10. It orig-
inates from disturbances due to pedestrians, who try to overtake each other and
expand in the wide area because of their repulsive interactions. They squeeze into
the main stream again at the end of the widening, which acts like a bottleneck and
leads to jamming. The corresponding drop of efficiency Eis more pronounced,

18 Dirk Helbing, Ill´es J. Farkas, P´eter Moln´ar, and Tam´as Vicsek
1. if the corridor is narrow,
2. if the pedestrians have different or high desired velocities, and
3. if the pedestrian density in the corridor is high.
φ
0.6
0.7
0.8
0.9
1
0 10 20 30 40 50
Relative Efficiency E of Escape
Angle φ (deg)
a b
Figure 10: Simulation of an escape route with a wider area (from [53,99], see also the Java
applets supplied at http://angel.elte.hu/˜panic/). aIllustration of the scenario with
v0
i=v0= 2 m/s. The corridor is 3 m wide and 15 m long, the length of the triangular
pieces in the middle being 2×3 m = 6 m. Pedestrians enter the simulation area on the
left-hand side with an inflow of J= 5.5 s−1m−1and flee towards the right-hand side. b
Efficiency of leaving as a function of the angle φcharacterizing the width of the central
zone, i.e., the difference from a linear corridor. The relative efficiency E=hvi·e0
ii/v0
measures the average velocity along the corridor compared to the desired velocity and lies
between 0 and 1 (—). While it is almost one (i.e., maximal) for a linear corridor (φ= 0),
the efficiency drops by about 20%, if the corridor contains a widening. The decrease of
efficiency Eis even more pronounced in the area of the widening where pedestrian flow is
most irregular (– –).
“Phantom panics”. Sometimes, panics have occured without any comprehensible
reasons such as a fire or another threatening event (e.g., in Moscow, 1982; Inns-
bruck, 1999). Due to the “faster-is-slower effect”, panics can be triggered by small
pedestrian counterflows [78], which cause delays to the crowd intending to leave.
Consequently, stopped pedestrians in the back, who do not see the reason for the
temporary slowdown, are getting impatient and pushy. In accordance with observa-
tions [43,18], one may describe this by increasing the desired velocity, for example,
by the formula
v0
i(t) = [1 −ni(t)]v0
i(0) + ni(t)vmax
i.(11)
Herein, vmax
iis the maximum desired velocity and v0
i(0) the initial one, corresponding
to the expected velocity of leaving. The time-dependent parameter
ni(t) = 1 −vi(t)
v0
i(0) (12)
reflects the nervousness, where vi(t) denotes the average speed into the desired
direction of motion. Altogether, long waiting times increase the desired velocity,

Simulation of Pedestrian Crowds in Normal and Evacuation Situations 19
which can produce inefficient outflow. This further increases the waiting times, and
so on, so that this tragic feedback can eventually trigger so high pressures that people
are crushed or falling and trampled. It is, therefore, imperative, to have sufficiently
wide exits and to prevent counterflows, when big crowds want to leave [53].
Ignorance of available exits. Finally, we investigate a situation in which pedes-
trians are trying to leave a smoky room, but first have to find one of the invisible
exits (see Fig. 11a). Each pedestrian imay either select an individual direction ei
or follow the average direction he0
j(t)iiof his neighbours jin a certain radius Ri
[143,144], or try a mixture of both. We assume that both options are weighted with
the nervousness ni:
e0
i(t) = N£(1 −ni)ei+nihe0
j(t)ii¤,(13)
where N(z) = z/kzkdenotes the normalization of a vector z. As a consequence,
we have individualistic behavior if niis low, but herding behavior if niis high. (In
a somewhat different szenario, one may simulate a crowd with a proportion (1 −n)
of individualists and a proportion nof herd followers.)
Our model suggests that neither individualistic nor herding behavior performs
well (see Fig. 11b). Pure individualistic behavior means that each pedestrian finds an
exit only accidentally, while pure herding behavior implies that the complete crowd
is eventually moving into the same and probably blocked direction, so that available
exits are not efficiently used, in agreement with observations. According to Figs. 11b
and c, we expect optimal chances of survival for a certain mixture of individualistic
and herding behavior, where individualism allows some people to detect the exits
and herding guarantees that successful solutions are imitated by small groups of
others. If pedestrians follow the walls instead of “reflecting” at them, we expect
that herd following causes jamming and inefficient use of doors as well (see Fig. 8),
while individualists moving in opposite directions obstruct each other.
5 Optimization of pedestrian flows
The emerging pedestrian flows decisively depend on the geometry of the boundaries.
They can be simulated on a computer already in the planning phase of pedestrian
facilities. Their configuration and shape can be systematically varied, e.g. by means
of evolutionary algorithms [121,123,54] (see Fig. 12), and evaluated on the basis of
particular mathematical performance measures [18,48]. Apart from the efficiency E
with 0 ≤E≤1 defined in formula (9), we can, for example, define the measure of
comfort C= (1 −D) via the discomfort
D=1
NX
i
(vi−vi)2
(vi)2=1
NX
iÃ1−vi2
(vi)2!.(14)
The latter is again between 0 and 1 and reflects the frequency and degree of sudden
velocity changes, i.e. the level of discontinuity of walking due to necessary avoidance
maneuvers. Hence, the optimal configuration regarding the pedestrian requirements
is the one with the highest values of efficiency and comfort.
During the optimization procedure, some or all of the following can be varied:

20 Dirk Helbing, Ill´es J. Farkas, P´eter Moln´ar, and Tam´as Vicsek
Smoke
56
58
60
62
64
66
68
70
72
74
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
People Escaping within 30s
Degree of Herding, Nervousness n
36
38
40
42
44
46
48
50
52
54
56
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Leaving Time for 80 People (s)
Degree of Herding, Nervousness n
5
10
15
20
25
30
35
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Difference in the Usage of Two Doors
Degree of Herding, Nervousness n
a b
dc
Figure 11: Simulation of N= 90 pedestrians trying to escape a smoky room of area
A= 15m ×15 m (black) through two smoke-hidden doors of 1.5 m width, which have to be
found with a mixture of individualistic and herding behavior (from [53,95,99]). Java applets
are available at http://angel.elte.hu/˜panic/.aSnapshot of the simulation with v0
i=
v0= 5 m/s. Initially, each pedestrian selects his or her desired walking direction randomly.
Afterwards, a pedestrian’s walking direction is influenced by the average direction of the
neighbours within a radius of, for example, Ri=R= 5 m. The strength of this herding
effect grows with increasing nervousness parameter ni=nand increasing value of h=
πR2ρ, where ρ=N/A denotes the pedestrian density. When reaching a boundary, the
direction of a pedestrian is reflected. If one of the exits is closer than 2 m, the room is left.
bNumber of people who manage to escape within 30 s as a function of the nervousness
parameter n.cIllustration of the time required by 80 individuals to leave the smoky room.
If the exits are relatively narrow and the degree nof herding is small or large, leaving takes
particularly long, so that only some of the people escape before being poisoned by smoke.
Our results suggest that the best escape strategy is a certain compromise between following
of others and an individualistic searching behavior. This fits well into experimental data
on the efficiency of group problem solving [145–147], according to which groups normally
perform better than individuals, but masses are inefficient in finding new solutions to
complex problems. dAbsolute difference |N1−N2|in the numbers N1and N2of persons
leaving through the left exit or the right exit as a function of the degree nof herding.
We find that pedestrians tend to jam up at one of the exits instead of equally using all
available exits, if the nervousness is large.
1. the location and form of planned buildings,
2. the arrangement of walkways, entrances, exits, staircases, elevators, escalators,
and corridors,
3. the shape of rooms, corridors, entrances, and exits,
4. the function and time schedule of room usage. (Recreation rooms or restaurants
are continuously frequented, rooms for conferences or special events are mainly

Simulation of Pedestrian Crowds in Normal and Evacuation Situations 21
Figure 12: Different phases in the evolutionary optimization of a bottleneck (from [54,19]).
visited and left at peak periods, exhibition rooms or rooms for festivities require
additional space for people standing around, and some areas are claimed by
queues or through traffic.)
The proposed optimization procedure can not only be applied to the design of new
pedestrian facilities but also to a reduction of existing bottlenecks by suitable mod-
ifications.
5.1 Normal situations
Here, we discuss four simple examples of how to improve some standard elements of
pedestrian facilities ([18]; see Fig. 13):
1. At high pedestrian densities, the lanes of uniform walking direction tend to dis-
turb each other: Impatient pedestrians try to use any gap for overtaking, which
often leads to subsequent obstructions of the opposite walking directions. The
lanes can be stabilized by series of trees or columns in the middle of the road
(see Fig. 13a) which, in walking direction, looks similar to a wall (see Fig. 14).
Also, it takes some detour to reach the other side of the permeable wall, which
makes it less attractive to use gaps occuring in the opposite pedestrian stream.
2. The flow at bottlenecks can be improved by a funnel-shaped construction (see
Fig. 13b) which, at the same time, allows one to save expensive space. Inter-
estingly, the optimal form resulting from an evolutionary optimization is convex
[54] (see Fig. 12).
3. A broader door does not necessarily lead to a proportional increase of pedestrian
flow through it. It may rather lead to more frequent changes of the walking
direction which are connected with temporary deadlock situations. Therefore,
two doors close to the walls are more efficient than one single door with double
width. By self-organization, each door is used by one walking direction [18,48–50],
which is related to lane formation (see Fig. 15).

22 Dirk Helbing, Ill´es J. Farkas, P´eter Moln´ar, and Tam´as Vicsek
mmmll
••
•
•
XXXX
X
XXXX
X
XXXX
X
¤¤¤¤¤
¤¤¤¤¤
¤¤¤¤¤
X
X
X
X
XX
X
X
X
XX
X
X
X
X
¤
¤
¤
¤
¤
¤
¤
¤
¤
¤
¤
¤
¤
¤
¤
•
¤¤¤º
•
X
X
Xy
•
¤
¤
¤²
•
XX
Xz
c
b
»»»»
»
XXXX
X
»»»»
»
XXXX
X
»»»»
»
XXXX
X
»»»»
»
XXXX
X
»»»»
»
XXXX
X
•
•
6
?
•
•
-
-
a
±°
²¯
±°
²¯
n±°
²¯
±°
²¯
n±°
²¯
±°
²¯
n±°
²¯
±°
²¯
n
•-
•
¾
Figure 13: Conventional (left) and improved (right) elements of pedestrians facilities:
aways, bbottlenecks, and cintersections (from [18,19]). The exclamation marks stand
for attraction effects (e.g. interesting posters above the street). Empty circles represent
columns or trees, while full circles with arrows symbolize pedestrians and their walking
directions.
4. Oscillatory changes of the walking direction and periods of standstill in between
also occur when different flows cross each other. The loss of efficiency caused
by this can be reduced by psychological guiding measures or railings initializing
roundabout traffic (see Fig. 13c). Roundabout traffic can already be induced and
stabilized by planting a tree in the middle of a crossing, because it surpresses
the phases of “vertical” or “horizontal” motion in the intersection area. In our
simulations this increased efficiency up to 13%.
The complex interaction between various flows can lead to completely unexpected
results due to the nonlinearity of dynamics. (A very impressive and surprising re-
sult of evolutionary form optimization is presented by Klockgether and Schwefel in
Ref. [120].) This means, planning of pedestrian facilities with conventional meth-
ods does not always guarantee the avoidance of big jams, serious obstructions, and
catastrophic blockages (especially in emergency situations). In contrast, a skilful

Simulation of Pedestrian Crowds in Normal and Evacuation Situations 23
Figure 14: These photographs of a pedestrian tunnel connecting two subways in Budapest
at De´ak t´er illustrates that a series of columns acts similar to a wall and stabilizes lanes
by preventing that their width exceeds half of the total width of the walkway.
Figure 15: If two alternative passageways are available, pedestrians with opposite walking
directions use different doors as a result of self-organization (after [18,19,45,48–50,95]).
flow optimization not only enhances efficiency but also saves space that can be used
for kiosks, benches, or other purposes [18].
5.2 Panic situations
Similar design strategies can be developed for panic situations, where signifi-
cantly improved outflows can be reached by columns placed asymmetrically in
front of the exits [53,99]. These can prevent the build up of fatal pressures in
exit areas and, thereby, also reduce injuries (see Fig. 16 and the Java applets
at http://angel.elte.hu/˜panic/). The asymmetrical placement helps to avoid
equilibria of forces (blockages).
Additionally, one can guide people into the directions of usable exits by means of
optical and acoustic stimuli, i.e. by suitable arrangements of light and sound sources.
6 Summary and outlook
We have developed a continuous pedestrian model based on plausible interactions,
which is, due to its simplicity, robust with respect to parameter variations. It was
pointed out that pedestrian dynamics shows various collective phenomena, which

24 Dirk Helbing, Ill´es J. Farkas, P´eter Moln´ar, and Tam´as Vicsek
t = 43
N = 159
V0 = 5
Inj.: 4
a b
Figure 16: aIn panicking crowds, high pressures build up due to physical interactions.
This can injure people (black disks), who turn into obstacles for other pedestrians trying
to leave (see also the lower curve in Fig. 8c). bA column in front of the exit (large black
disk) can avoid injuries by taking up pressure from behind. It can also increase the outflow
by 50%. In large exit areas used by many hundret people, several randomly placed columns
are needed to subdivide the crowd and the pressure. An asymmetric configuration of the
columns is most efficient, as it avoids equilibria of forces which may temporarily stop the
outflow. (From [99].)
every simulation model should reproduce in order to be realistic. For example, in
normal situations one finds
1. lane formation and
2. oscillatory flows through bottlenecks.
These and other empirical findings can be well described by our microscopic sim-
ulations of pedestrian streams based on a generalized force model. According to
this model, the collective patterns of motion can be interpreted as self-organization
phenomena, arising from the non-linear interactions among pedestrians.
We underline that self-organized flow patterns can significantly change the ca-
pacities of pedestrian facilities. They often lead to undesireable obstructions, but
they can also be utilized to reach more efficient pedestrian flows with less space.
Applications to the optimization of pedestrian facilities are, therefore, quite natu-
ral.
The proposed force model is also suitable for drawing conclusions about the pos-
sible mechanisms beyond escape panic (regarding an increase of the desired velocity,
strong friction effects during physical interactions, and herding). After calibration
of the model parameters to available data on pedestrian flows, we managed to re-
produce many observed phenomena including
1. the breakdown of fluid lanes (“freezing by heating”),
2. the build up of fatal pressures,
3. clogging effects at bottlenecks,
4. jamming at widenings,
5. the “faster-is-slower effect”,
6. “phantom panics” triggered by counterflows and impatience, and
7. the ignorance of available exits due to herding.
The underlying behavior could be called “irrational”, as all of these effects decrease
the chances of survival compared to normal pedestrian behavior. We were also able to

Simulation of Pedestrian Crowds in Normal and Evacuation Situations 25
simulate situations of dwindling resources and estimate the casualties (see Figs. 8c
and 9). Therefore, the model could be used to test buildings for their suitability in
emergency situations. It accounts for the considerably different dynamics both in
normal and panic situations just by changing a single parameter ni=n. In this way,
we have proposed a consistent theoretical approach allowing a continuous switching
between seemingly incompatible kinds of human behavior (individualistic, “rational”
behavior vs. “irrational” panic behavior). Thereby, however, we do not want to imply
that individuals would always behave irrational in emergency situations. It has been
observed that, even in such situations individuals can behave highly self-controlled,
coordinated, rational, and social. Our study just investigates the fundamental col-
lective effects which fluctuations, increased desired velocities, and herding behavior
can have, independently of whether all criteria of panics are fulfilled or not.
We believe that the above model can serve as an example linking collective behav-
ior as a phenomenon of mass psychology (from the socio-psychological perspective)
to the view of an emergent collective pattern of motion (from the perspective of
physics). Our simulations suggest that the optimal behavior in escape situations is a
suitable mixture of individualistic and herding behavior. This conclusion is probably
transferable to many cases of problem solving in new and complex situations, where
standard solutions fail. It may explain why both, individualistic and herding behav-
iors are common in human societies. For example, herding behavior is also relevant
to fashion and stock market dynamics (see Refs. [99,148–153]). Apart from that,
the competition of moving particles for limited space is analogous to the situation
in various socio-economic systems, where individuals or other entities compete for
limited resources as well. Therefore, conclusions from the above findings for self-
driven many-particle systems reach far into the realm of the social, economic, and
psychological sciences.
Finally, we are calling for quantitative data and, as far as possible, experimen-
tal studies of panic situations to make this model even more realistic. For exam-
ple, one could include direction- and velocity-dependent interpersonal interactions,
specify the individual variation of parameters, integrate acoustic information ex-
change, implement more complex strategies and interactions (also three-dimensional
ones), or allow for switching of strategies. One should also complement the pro-
gram by detailled fire and smoke propagation modules and model hazards, toxi-
city and behavioral reactions, as evacuation software tools like EXODUS do (see
http://fseg.gre.ac.uk/exodus).
Acknowledgments. The authors are grateful to the Collegium Budapest—
Institute for Advanced Study for the warm hospitality and the excellent scientific
working conditions. D.H. thanks the German Research Foundation (DFG) for finan-
cial support by the Heisenberg scholarship He 2789/1-1. T.V. and I.F. are grateful
for partial support by OTKA and FKFP. Last but not least, Tilo Grigat has helped
with formatting this manuscript.

26 Dirk Helbing, Ill´es J. Farkas, P´eter Moln´ar, and Tam´as Vicsek
References
1. B. D. Hankin and R. A. Wright, Passenger flow in subways, Operational Research
Quarterly 9, 81–88 (1958).
2. D. Oeding, Verkehrsbelastung und Dimensionierung von Gehwegen und anderen An-
lagen des Fußg¨angerverkehrs, in Straßenbau und Straßenverkehrstechnik, Heft 22
(Bundesministerium f¨ur Verkehr, Abt. Straßenbau, Bonn, 1963).
3. L. A. Hoel, Pedestrian travel rates in central business districts, Traffic Engineering
38, 10–13 (1968).
4. S. J. Older, Movement of pedestrians on footways in shopping streets, Traffic Engi-
neering & Control 10, 160–163 (1968).
5. P. D. Navin and R. J. Wheeler, Pedestrian flow characteristics, Traffic Engineering
39, 31–36 (1969).
6. R. L. Carstens and S. L. Ring, Pedestrian capacities of shelter entrances, Traffic
Engineering 41, 38–43 (1970).
7. C. A. O’Flaherty and M. H. Parkinson, Movement on a city centre footway, Traffic
Engineering & Control 13, 434–438 (1972).
8. U. Weidmann, Transporttechnik der Fußg¨anger, in Schriftenreihe des Instituts
f¨ur Verkehrsplanung, Transporttechnik, Straßen- und Eisenbahnbau (Institut f¨ur
Verkehrsplanung, Transporttechnik, Straßen- und Eisenbahnbau, ETH Z¨urich, 1993).
9. M. R. Hill, Walking, Crossing Streets, and Choosing Pedestrian Routes (University
of Nebraska, Lincoln, 1984).
10. M. Batty, Predicting where we walk, Nature 388, 19–20 (1997).
11. J. J. Fruin, Designing for pedestrians: A level-of-service concept, in Highway Research
Record, Number 355: Pedestrians, pp. 1–15 (Highway Research Board, Washington,
D.C., 1971).
12. A. Polus , J. L. Schofer, and A. Ushpiz, Pedestrian flow and level of service, Journal
of Transportation Engineering 109, 46–56 (1983).
13. M. M¯ori and H. Tsukaguchi, A new method for evaluation of level of service in
pedestrian facilities, Transportation Research A 21, 223–234 (1987).
14. H. Schubert, Planungsmaßnahmen f¨ur den Fußg¨angerverkehr in den St¨adten, in
Straßenbau und Straßenverkehrstechnik, Heft 56 (Bundesministerium f¨ur Verkehr,
Abt. Straßenbau, Bonn, 1967).
15. D. Boeminghaus, Fußg¨angerbereiche + Gestaltungselemente (Kr¨amer, Stuttgart,
1982).
16. J. Pauls, The movement of people in buildings and design solutions for means of
egress, Fire Technology 20, 27–47 (1984).
17. W. H. Whyte, City. Rediscovering the Center (Doubleday, New York, 1988).
18. D. Helbing, Verkehrsdynamik (Springer, Berlin, 1997).
19. D. Helbing, P. Moln´ar, I. Farkas, and K. Bolay, Self-organizing pedestrian movement,
Environment and Planning B 28, xxx (2001).
20. H. Kirsch, Leistungsf¨ahigkeit und Dimensionierung von Fußg¨angerwegen, in Straßen-
bau und Straßenverkehrstechnik, Heft 33 (Bundesministerium f¨ur Verkehr, Abt.
Straßenbau, Bonn, 1964).
21. W. M. Predtetschenski and A. I. Milinski, Personenstr¨ome in Geb¨auden – Berech-
nungsmethoden f¨ur die Projektierung – (Rudolf M¨uller, K¨oln-Braunsfeld, 1971).
22. Transportation Research Board, Highway Capacity Manual, Special Report 209
(Transportation Research Board, Washington, D.C., 1985).
23. D. G. Davis and J. P. Braaksma, Adjusting for luggage-laden pedestrians in airport
terminals, Transportation Research A 22, 375–388 (1988).

Simulation of Pedestrian Crowds in Normal and Evacuation Situations 27
24. W. Brilon, M. Großmann, and H. Blanke, Verfahren f¨ur die Berechnung der Leis-
tungsf¨ahigkeit und Qualit¨at des Verkehrsablaufes auf Straßen, in Straßenbau und
Straßenverkehrstechnik, Heft 669 (Bundesministerium f¨ur Verkehr, Abt. Straßenbau,
Bonn, 1993).
25. S. J. Yuhaski Jr., J. M. Macgregor Smith, Modelling circulation systems in buildings
using state dependent queueing models, Queueing Systems 4, 319–338 (1989).
26. J. R. Roy, Queuing in spatially dispersed public facilities, presented at the IV. World
Congress of the Regional Science Association International (Palma de Mallorca,
1992).
27. G. G. Løv˚as, Modelling and simulation of pedestrian traffic flow, Transportation
Research B 28, 429–443 (1994).
28. D. Garbrecht, Describing pedestrian and car trips by transition matrices, Traffic
Quarterly 27, 89–109 (1973).
29. A. J. Mayne, Some further results in the theory of pedestrians and road traffic,
Biometrika 41, 375–389 (1954).
30. N. Ashford, M. O’Leary, and P. D. McGinity, Stochastic modelling of passenger and
baggage flows through an airport terminal, Traffic Engineering & Control 17, 207–210
(1976).
31. A. Borgers and H. Timmermans, A model of pedestrian route choice and demand for
retail facilities within inner-city shopping areas, Geographical Analysis 18, 115–128
(1986).
32. A. Borgers and H. Timmermans, City centre entry points, store location patterns and
pedestrian route choice behaviour: A microlevel simulation model, Socio-Economic
Planning Science 20, 25–31 (1986).
33. H. Timmermans, X. van der Hagen, and A. Borgers, Transportation systems, retail
environments and pedestrian trip chaining behaviour: Modelling issues and applica-
tions, Transportation Research B 26, 45–59 (1992).
34. D. Helbing, Stochastische Methoden, nichtlineare Dynamik und quantitative Modelle
sozialer Prozesse, Ph.D. thesis (University of Stuttgart, 1992, published by Shaker,
Aachen, 1993).
35. L. F. Henderson, The statistics of crowd fluids, Nature 229, 381–383 (1971).
36. L. F. Henderson, On the fluid mechanics of human crowd motion, Transportation
Research 8, 509–515 (1974).
37. L. F. Henderson and D. J. Lyons, Sexual differences in human crowd motion, Nature
240, 353-355 (1972).
38. L. F. Henderson and D. M. Jenkins, Response of pedestrians to traffic challenge,
Transportation Research 8, 71–74 (1973).
39. R. L. Hughes, The flow of large crowds of pedestrians, Math. Comp. Simul. 53, 367–
370 (2000).
40. R. L. Hughes, A continuum theory for the flow of pedestrians, Transportation Re-
search B, in print (2001).
41. D. Helbing, A fluid-dynamic model for the movement of pedestrians, Complex Sys-
tems 6, 391–415 (1992).
42. S. P. Hoogendoorn and P. H. L. Bovy, Gas-kinetic modelling and simulation of pedes-
trian flows, Transportation Research Records 1710, 28–36 (2000).
43. D. Helbing, A mathematical model for the behavior of pedestrians, Behavioral Science
36, 298–310 (1991).
44. D. Helbing, Traffic modeling by means of physical concepts, in Traffic and Granular
Flow, pp. 87–104, D. E. Wolf, M. Schreckenberg, and A. Bachem (Eds.) (World
Scientific, Singapore, 1996).
45. D. Helbing, Pedestrian dynamics and trail formation, pp. 21–36, in Traffic and Gran-
ular Flow ’97, M. Schreckenberg and D. E. Wolf (Eds.) (Springer, Singapore, 1998).

28 Dirk Helbing, Ill´es J. Farkas, P´eter Moln´ar, and Tam´as Vicsek
46. D. Helbing, P. Moln´ar, and F. Schweitzer, Computer simulations of pedestrian dy-
namics and trail formation, in Evolution of Natural Structures, pp. 229–234 (Sonder-
forschungsbereich 230, Stuttgart, 1994).
47. D. Helbing and P. Moln´ar, Social force model for pedestrian dynamics, Physical Re-
view E 51, 4282–4286 (1995).
48. D. Helbing and P. Moln´ar, Self-organization phenomena in pedestrian crowds, in
Self-Organization of Complex Structures: From Individual to Collective Dynamics,
pp. 569–577, F. Schweitzer (Ed.) (Gordon and Breach, London, 1997).
49. P. Moln´ar, Modellierung und Simulation der Dynamik von Fußg¨angerstr¨omen,
(Shaker, Aachen, 1996).
50. P. Moln´ar, Microsimulation of pedestrian dynamics in Social Science Microsimula-
tion, J. Doran, N. Gilbert, U. Mueller, and K. Troitzsch (Eds.) (Springer, Berlin,
1996).
51. D. Helbing and T. Vicsek, Optimal self-organization, New Journal of Physics 1, 13.1–
13.17 (1999).
52. D. Helbing, I. Farkas, and T. Vicsek, Freezing by heating in a driven mesoscopic
system, Physical Review Letters 84, 1240–1243 (2000).
53. D. Helbing, I. Farkas, and T. Vicsek, Simulating dynamical features of escape panic,
Nature 407, 487–490 (2000).
54. K. Bolay, Nichtlineare Ph¨anomene in einem fluid-dynamischen Verkehrsmodell (Mas-
ter’s thesis, University of Stuttgart, 1998).
55. V. J. Blue and J. L. Adler, Emergent fundamental pedestrian flows from cellular
automata microsimulation, Transportation Research Records 1644, 29–36 (1998).
56. V. J. Blue and J. L. Adler, Using cellular automata microsimulation to model pedes-
trian movements, in Proceedings of the 14th International Symposium on Transporta-
tion and Traffic Theory, pp. 235–254, A. Ceder (Ed.) (Pergamon, New York, N. Y.,
1999).
57. M. Fukui and Y. Ishibashi, Self-organized phase transitions in cellular automaton
models for pedestrians, Journal of the Physical Society of Japan 68, 2861–2863 (1999).
58. M. Fukui and Y. Ishibashi, Jamming transition in cellular automaton models for
pedestrian on passageway, Journal of the Physical Society of Japan 68, 3738–3739
(1999).
59. M. Muramatsu, T. Irie, and T. Nagatani, Jamming transition in pedestrian counter
flow, Physica A 267, 487–498 (1999).
60. M. Muramatsu and T. Nagatani, Jamming transition in two-dimensional pedestrian
traffic, Physica A 275, 281–291 (2000).
61. M. Muramatsu and T. Nagatani, Jamming transition of pedestrian traffic at a cross-
ing with open boundaries, Physica A 286, 377–390 (2000).
62. H. Kl¨upfel, M. Meyer-K¨onig, J. Wahle, and M. Schreckenberg, Microscopic simulation
of evacuation processes on passenger ships, in Theory and Practical Issues on Cellular
Automata, S. Bandini and T. Worsch (Eds.) (Springer, London, 2000).
63. C. Burstedde, K. Klauck, A. Schadschneider, and J. Zittartz, Simulation of pedestrian
dynamics using a 2-dimensional cellular automaton, Physica A, in print (2001).
64. S. Gopal and T. R. Smith, NAVIGATOR: An AI-based model of human way-finding
in an urban environment, in Spatial Choices and Processes, pp. 169–200, M. M.
Fischer, P. Nijkamp, and Y. Y. Papageorgiou (Eds.) (North-Holland, Amsterdam,
1990).
65. C. W. Reynolds, Evolution of corridor following behavior in a noisy world, in From
Animals to Animats 3: Proceedings of the Third International Conference on Simu-
lation of Adaptive Behavior, pp. 402–410, D. Cliff, P. Husbands, J.-A. Meyer, and S.
Wilson (Eds.) (MIT Press, Cambridge, Massachusetts, 1994).

Simulation of Pedestrian Crowds in Normal and Evacuation Situations 29
66. C. W. Reynolds, Steering behaviors for autonomous characters, in Proceedings of the
1999 Game Developer’s Conference, pp. 763–782, A. Yu (Ed.) (Miller Freeman, San
Fransisco, 1999).
67. T. Schelhorn, D. O’Sullivan, M. Haklay, and M. Thurstain-Goodwin, STREETS:
An agent-based pedestrian model, presented at the conference Computers in Urban
Planning and Urban Management (Venice, 1999).
68. K. H. Drager, G. Løv˚as, J. Wiklund, H. Soma, D. Duong, A. Violas, and V. Lan`er`es,
EVACSIM—A comprehensive evacuation simulation tool, in the Proceedings of the
1992 Emergency Management and Engineering Conference, pp. 101–108 (Society for
Computer Simulation, Orlando, Florida, 1992).
69. M. Ebihara, A. Ohtsuki, and H. Iwaki, A model for simulating human behavior during
emergency evacuation based on classificatory reasoning and certainty value handling,
Microcomputers in Civil Engineering 7, 63–71 (1992).
70. N. Ketchell, S. Cole, D. M. Webber, C. A. Marriott, P. J. Stephens, I. R. Brearley,
J. Fraser, J. Doheny, and J. Smart, The EGRESS Code for human movement and
behaviour in emergency evacuations, in Engineering for Crowd Safety, pp. 361–370,
R. A. Smith and J. F. Dickie (Eds.) (Elsevier, Amsterdam, 1993).
71. S. Okazaki and S. Matsushita, A study of simulation model for pedestrian movement
with evacuation and queuing, in Engineering for Crowd Safety, pp. 271–280, R. A.
Smith and J. F. Dickie (Eds.) (Elsevier, Amsterdam, 1993).
72. G. K. Still, New computer system can predict human behaviour response to building
fires, Fire 84, 40–41 (1993).
73. G. K. Still, Crowd Dynamics (Ph.D. thesis, University of Warwick, 2000).
74. P. A. Thompson and E. W. Marchant, Modelling techniques for evacuation, in Engi-
neering for Crowd Safety, pp. 259–269, R. A. Smith and J. F. Dickie (Eds.) (Elsevier,
Amsterdam, 1993).
75. G. G. Løv˚as, On the importance of building evacuation system components, IEEE
Transactions on Engineering Management 45, 181–191 (1998).
76. H. W. Hamacher and S. A. Tjandra, Modelling Emergency Evacuation—An
Overview, to be published (2001).
77. J. P. Keating, The myth of panic, Fire Journal, 57–61+147 (May/1982).
78. D. Elliott and D. Smith, Football stadia disasters in the United Kingdom: Learning
from tragedy?, Industrial & Environmental Crisis Quarterly 7(3), 205–229 (1993).
79. B. D. Jacobs and P. ’t Hart, Disaster at Hillsborough Stadium: a comparative analysis,
in Hazard Management and Emergency Planning, Chap. 10, D. J. Parker and J. W.
Handmer (Eds.) (James & James Science, London, 1992).
80. D. Canter (Ed.), Fires and Human Behaviour, (David Fulton, London, 1990).
81. A. Mintz, Non-adaptive group behavior, The Journal of Abnormal and Normal Social
Psychology 46, 150–159 (1951).
82. D. L. Miller, Introduction to Collective Behavior, Fig. 3.3 and Chap. 9 (Wadsworth,
Belmont, CA, 1985).
83. J. S. Coleman, Foundations of Social Theory, Chaps. 9 and 33 (Belkamp, Cambridge,
MA, 1990).
84. N. R. Johnson, Panic at “The Who Concert Stampede”: An empirical assessment,
Social Problems 34(4), 362–373 (1987).
85. G. LeBon, The Crowd (Viking, New York, 1960 [1895]).
86. E. Quarantelli, The behavior of panic participants, Sociology and Social Research 41,
187–194 (1957).
87. N. J. Smelser, Theory of Collective Behavior, (The Free Press, New York, 1963).
88. R. Brown, Social Psychology (The Free Press, New York, 1965).
89. R. H. Turner and L. M. Killian, Collective Behavior (Prentice Hall, Englewood Cliffs,
NJ, 3rd ed., 1987).

30 Dirk Helbing, Ill´es J. Farkas, P´eter Moln´ar, and Tam´as Vicsek
90. J. L. Bryan, Convergence clusters, Fire Journal, 27–30+86–90 (Nov./1985).
91. R. Axelrod and W. D. Hamilton, The evolution of cooperation, Science 211, 1390–
1396 (1981).
92. R. Axelrod and D. Dion, The further evolution of cooperation, Science 242, 1385–
1390 (1988).
93. N. S. Glance and B. A. Huberman, The dynamics of social dilemmas, Scientific
American 270, 76–81 (1994).
94. T. Arns, Video films of pedestrian crowds, (Wannenstr. 22, 70199 Stuttgart, 1993).
95. D. Helbing, Traffic and related self-driven many-particle systems, e-print cond-
mat/0012229, to appear in Reviews of Modern Physics (2001).
96. J. Ganem, A behavioral demonstration of Fermat’s principle, The Physics Teacher
36, 76–78 (1998).
97. D. Helbing, J. Keltsch, and P. Moln´ar, Modelling the evolution of human trail systems,
Nature 388, 47–50 (1997).
98. R. A. Smith and J. F. Dickie (Eds.), Engineering for Crowd Safety (Elsevier, Ams-
terdam, 1993).
99. D. Helbing, I. J. Farkas, and T. Vicsek, Crowd disasters and simulation of panic
situations, in Science of Disaster: Climate Disruptions, Heart Attacks and Market
Crashes, A. Bunde, J. Kropp, and H. J. Schellnhuber (Eds.) (Springer, Berlin, 2001).
100. J. S. Coleman, Introduction to Mathematical Sociology, pp. 361–375 (The Free Press
of Glencoe, New York, 1964).
101. J. S. Coleman and J. James, The equilibrium size distribution of freely-forming
groups, Sociometry 24, 36–45 (1961).
102. L. A. Goodman, Mathematical methods for the study of systems of groups, American
Journal of Sociology 70, 170–192 (1964).
103. H. H. Kelley, J. C. Condry Jr., A. E. Dahlke, and A. H. Hill, Collective behavior
in a simulated panic situation, Journal of Experimental Social Psychology 1, 20–54
(1965).
104. L. P. Kadanoff, Simulating hydrodynamics: A pedestrian model, Journal of Statistical
Physics 39, 267–283 (1985).
105. H. E. Stanley and N. Ostrowsky (Eds.), On Growth and Form (Martinus Nijhoff,
Boston, 1986).
106. H.-H. Stølum, River meandering as a self-organization process, Nature 271, 1710–
1713 (1996).
107. I. Rodr´ıguez-Iturbe and A. Rinaldo, Fractal River Basins: Chance and Self-
Organization (Cambridge University, Cambridge, England, 1997).
108. G. Caldarelli, Cellular models for river networks, e-print cond-mat/0011086 (2000).
109. S. B. Santra, S. Schwarzer, and H. Herrmann, Fluid-induced particle segregation in
sheared granular assemblies, Physical Review E 54, 5066–5072 (1996).
110. H. A. Makse, S. Havlin, P. R. King, and H. E. Stanley, Spontaneous stratification in
granular mixtures, Nature 386, 379–382 (1997).
111. K. Yoshikawa, N. Oyama, M. Shoji, and S. Nakata, Use of a saline oscillator as a
simple nonlinear dynamical system: Rhythms, bifurcation, and entrainment, Ameri-
can Journal of Physics 59, 137–141 (1991).
112. X.-L. Wu, K. J. M˚aløy, A. Hansen, M. Ammi, and D. Bideau, Why hour glasses tick,
Physical Review Letters 71, 1363–1366 (1993).
113. T. L. Pennec, K. J.M˚aløy, A. Hansen, M. Ammi, D. Bideau D, and X.-L. Wu, Ticking
hour glasses: Experimental analysis of intermittent flow, Physical Review E 53, 2257–
2264 (1996).
114. M. R. Virkler and S. Elayadath, Pedestrian density characteristics and shockwaves,
in Proceedings of the Second International Symposium on Highway Capacity, Vol. 2,
pp. 671–684, R. Ak¸celik (Ed.) (Transportation Research Board, Washington, D.C.,
1994).

Simulation of Pedestrian Crowds in Normal and Evacuation Situations 31
115. G. H. Ristow and H. J. Herrmann, Density patterns in two-dimensional hoppers,
Physical Review E 50, R5–R8 (1994).
116. D. E. Wolf and P. Grassberger (Eds.), Friction, Arching, Contact Dynamics (World
Scientific, Singapore, 1997).
117. D. Helbing, Boltzmann-like and Boltzmann-Fokker-Planck equations as a foundation
of behavioral models, Physica A 196, 546–573 (1993).
118. D. Helbing, A mathematical model for the behavior of individuals in a social field,
Journal of Mathematical Sociology 19, 189–219 (1994).
119. D. Helbing, Quantitative Sociodynamics. Stochastic Methods and Models of Social
Interaction Processes (Kluwer Academic, Dordrecht, 1995).
120. J. Klockgether and H.-P. Schwefel, Two-phase nozzle and hollow core jet experiments,
in Proceedings of the Eleventh Symposium on Engineering Aspects of Magnetohy-
drodynamics, pp. 141–148, D. G. Elliott (Ed.) (California Institute of Technology,
Pasadena, CA, 1970).
121. I. Rechenberg, Evolutionsstrategie: Optimierung technischer Systeme nach Prinzipien
der biologischen Evolution (Frommann-Holzboog, Stuttgart, 1973).
122. H.-P. Schwefel, Numerische Optimierung von Computer-Modellen mittels Evolutions-
strategien (Birkh¨auser, Stuttgart, 1977).
123. T. Baeck, Evolutionary Algorithms in Theory and Practice (Oxford University Press,
New York, 1996).
124. D. Helbing, Physikalische Modellierung des dynamischen Verhaltens von Fußg¨angern
(Master’s thesis, Georg-August University G¨ottingen, 1990).
125. D. Helbing, A mathematical model for behavioral changes by pair interactions, in
Economic Evolution and Demographic Change. Formal Models in Social Sciences,
pp. 330–348, G. Haag, U. Mueller, and K. G. Troitzsch (Eds.) (Springer, Berlin,
1992).
126. D. Helbing, Stochastic and Boltzmann-like models for behavioral changes, and their
relation to game theory, Physica A 193, 241–258 (1993).
127. D. Helbing, A stochastic behavioral model and a ‘microscopic’ foundation of evolu-
tionary game theory, Theory and Decision 40, 149–179 (1996).
128. N. E. Miller, Experimental studies of conflict, in Personality and the behavior disor-
ders, Vol. 1, J. McV. Hunt (Ed.) (Ronald, New York, 1944).
129. N. E. Miller, Liberalization of basic S-R-concepts: Extension to conflict behavior,
motivation, and social learning in Psychology: A Study of Science, Vol. 2, S. Koch
(Ed.) (McGraw Hill, New York, 1959).
130. W. H. Herkner, Ein erweitertes Modell des Appetenz-Aversions-Konflikts, Z. Klin.
Psychol. 4, 50–60 (1975).
131. K. Lewin, Field Theory in Social Science, (Harper & Brothers, New York, 1951).
132. D. Helbing and B. Tilch, Generalized force model of traffic dynamics, Physical Review
E58, 133–138 (1998).
133. M. Treiber, A. Hennecke, and D. Helbing, Congested traffic states in empirical ob-
servations and microscopic simulations, Physical Review E 62, 1805–1824 (2000).
134. A. K. Dewdney, Diverse personalities search for social equilibrium at a computer
party, Scientific American 257(9), 104–107 (1987).
135. D. Helbing and T. Platkowski, Self-organization in space and induced by fluctuations,
International Journal of Chaos Theory and Applications 5, 25–39 (2000).
136. D. Helbing, Die wundervolle Welt aktiver Vielteilchensysteme, Physikalische Bl¨atter
57, 27–33 (2001).
137. F. Schweitzer, K. Lao, and F. Family, Active random walkers simulate trunk trail
formation by ants, BioSystems 41, 153–166 (1997).
138. D. Helbing, F. Schweitzer, J. Keltsch, and P. Moln´ar, Active walker model for the
formation of human and animal trail systems, Physical Review E 56, 2527–2539
(1997).

32 Dirk Helbing, Ill´es J. Farkas, P´eter Moln´ar, and Tam´as Vicsek
139. Y. L. Duparcmeur, H. Herrmann, and J. P. Troadec, Spontaneous formation of vortex
in a system of self motorised particles, Journal de Physique I France 5, 1119–1128
(1995).
140. J. Gallas, H. J. Herrmann, and S. SokoÃlowski, Convection cells in vibrating granular
media, Physical Review Letters 69, 1371–1374 (1992).
141. P. B. Umbanhowar, F. Melo, and H. L. Swinney, Localized excitations in a vertically
vibrated granular layer, Nature 382, 793–796 (1996).
142. A. Rosato, K. J. Strandburg, F. Prinz, and R. H. Swendsen, Why the Brazil nuts are
on top: Size segregation of particulate matter by shaking, Physical Review Letters
58, 1038–1041 (1987).
143. T. Vicsek, A. Czir´ok, E. Ben-Jacob, I. Cohen, and O. Shochet, Novel type of phase
transition in a system of self-driven particles, Physical Review Letters 75, 1226–1229
(1995).
144. A. Czir´ok, M. Vicsek, and T. Vicsek, Collective motion of organisms in three dimen-
sions, Physica A 264, 299–304 (1999).
145. N. H. Anderson, Group performance in an anagram task, Journal of Social Psychology
55, 67–75 (1961).
146. H. H. Kelley and J. W. Thibaut, Group problem solving, in The Handbook of Social
Psychology, Vol. 4, G. Lindzey and E. Aronson (Eds.) (Addison-Wesley, Reading,
MA, 1969).
147. P. R. Laughlin, N. L. Kerr, J. H. Davis, H. M. Halff, and K. A. Marciniak, Group
size, member ability, and social decision schemes on an intellective task, Journal of
Personality and Social Psychology 31, 522–535 (1975).
148. L. Mann, T. Nagel, and P. Dowling, A study of economic panic: The “run” on the
Hindmarsh building society, Sociometry 39(3), 223–235 (1976).
149. M. Youssefmir, B. A. Huberman, and T. Hogg, Bubbles and market crashes, Comp.
Econ. 12, 97–114 (1998).
150. J. D. Farmer, Market force, ecology, and evolution, e-print adap-org/9812005, sub-
mitted to the Journal of Economic Behavior and Organization (1998).
151. T. Lux and M. Marchesi, Scaling and criticality in a stochastic multi-agent model of
a financial market, Nature 397, 498–500 (1999).
152. J.-P. Bouchaud, A. Matacz, and M. Potters, The leverage effect in financial markets:
Retarded volatility and market panic, e-print cond-mat/0101120 (2001).
153. D. Sornette and J. V. Andersen, Quantifying Herding During Speculative Financial
Bubbles, e-print cond-mat/0104341 (2001).