Microscopic dynamics of the evacuation phenomena
MMicroscopic dynamics of the evacuation phenomena
F.E. Cornes a , G.A. Frank c , C.O. Dorso a,b a Departamento de F´ısica, Facultad de Ciencias Exactas y Naturales, Universidad deBuenos Aires, Pabell´on I, Ciudad Universitaria, 1428 Buenos Aires, Argentina. b Instituto de F´ısica de Buenos Aires, Pabell´on I, Ciudad Universitaria, 1428 BuenosAires, Argentina. c Unidad de Investigaci´on y Desarrollo de las Ingenier´ıas, Universidad Tecnol´ogicaNacional, Facultad Regional Buenos Aires, Av. Medrano 951, 1179 Buenos Aires,Argentina.
Abstract
We studied the room evacuation problem within the context of the SocialForce Model. We focused on a system of 225 pedestrians escaping from aroom in different anxiety levels, and analyzed the clogging delays as therelevant magnitude responsible for the evacuation performance. We linkedthe delays with the clusterization phenomenon along the faster is slower andthe faster is faster regimes. We will show that the faster is faster regimeis characterized by the presence of a giant cluster structure (composed bymore than 15 pedestrians), although no long lasting delays appear withinthis regime. For this system, we found that the relevant structures in the faster is slower regime are those blocking clusters that are somehow attachedto the two walls defining the exit. At very low desired velocities, very smallstructures become relevant (composed by less than 5 pedestrians), but atintermediate velocities ( v d (cid:39) Keywords:
Emergency evacuation, Social force model, Blocking clusters, Cloggingdelays
PACS:
1. Introduction
The problem of emergency evacuations has become relevant in the lastdecades due to the increasing number and size of mass events (religious
Preprint submitted to Elsevier July 15, 2020 a r X i v : . [ phy s i c s . s o c - ph ] J u l vents, music festivals, etc). The evacuation through narrow pathways ordoorways appears as one of the most problematic scenarios in the litera-ture. Many experiments and numerical simulations have been carried outfor a better understanding of the crowd behavior in a variety of situations.The door width [1–9], the crowd composition [6, 10], or even the positionof the exit [2, 11] raise as relevant magnitudes affecting the evacuation per-formance. The environmental and contextual realism, the pressure level, theoccupancy density, the sample size, etc. are some inherent limitations of theirexperiments. Thus, the comparison between the results of the controlled ex-periments and the emergency escape behavior is quite controversial [1].Ethical requirements in experimental setting impedes reproduce a realemergency situation. In this sense, the form that experimental researcherssimulates a competitive scenario differs from one to another experiment.Threfore, many controlled experiments on the pedestrians’ egress time showdissimilar results. For instance, Refs. [4, 12, 13] report that under a simulatedcompetitive scenario (say, allowing the contact between the pedestrians), themore the pedestrian’s anxiety to reach the exit, the greater the evacuationtime. But other experimental reports provide evidence of decreasing evacua-tion times for increasing escaping desires [1, 6]. The former correspond to a“faster-is-slower” behavior, while the latter correspond to a “faster-is-faster”behavior. Whether the dominant behavior is “faster-is-slower” or “faster-is-faster” seems to be a matter of the pushing aggressiveness, in spite of thedesire to reach the exit [1]. Thus, it has been proposed recently that theterm “faster-is-slower” should be replaced by the more precise one: “aggres-sive egress-is-slower”.The Social Force Model (SFM) introduced by Helbing [14] was the firstmodel to accomplish a “faster-is-slower” effect for moderate to high anxietylevels within the crowd. This occurs whenever friction dominates the pedes-trian dynamic [15–17]. The “faster-is-faster” phenomenon was also reportedwithin this model, but for very high pressures among the individuals [17].The time-lapse between consecutive evacuees and the egress curve are oneof the most common measures for experimental and computational research[6–8, 11–13, 16, 17]. The egress curve shows that the leaving flow may oc-cur either regularly or intermittently, depending on the pedestrians’ anxietyto reach the exit. Clogging can be observed as “arching structures” just2efore any narrowing of the leaving pathway [14, 16–18]. Clogging has alsobeen related to the pressure inside the bulk on this system [14, 16, 17, 19–24].Our aim is to analyze the relation between the delays of consecutive leav-ing pedestrians and the topological structures among the crowd during the faster is slower and faster is faster regimes. The work is organized as follows.In Section 2 we introduce the dynamics equations for evacuating pedestrians,in the context of the Social Force Model (SFM). We also define the meaningof spatial clusters. Section 3 details the simulation procedures used to study-ing the room evacuation of a crowd under increasing anxiety levels. Section 4displays the result of our investigation, while the conclusions are summarizein Section 5.
2. Theoretical background
Our research was carried out in the context of the “social force model”(SFM) proposed by Helbing and co-workers [14]. This model states thathuman motion is caused by the desire of people to reach a certain destinationat a certain velocity, as well as other environmental factors. It is a generalizedforce model, including socio-psychological forces, as well as “physical” forceslike friction. These forces enter the Newton equation as follows. m i d v ( i ) dt = f ( i ) d + N (cid:88) j =1 f ( ij ) s + N (cid:88) j =1 f ( ij ) g (1)where the i, j subscripts correspond to any two pedestrians in the crowd. v ( i ) ( t ) means the current velocity of the pedestrian ( i ), while f d and f s cor-respond to the “desired force” and the “social force”, respectively. f g is thegranular force.The “desired force” f d describes the pedestrians own desire to move atthe desired velocity v d . But, due to environmental factors ( i.e. obstacles,visibility), he (she) actually moves at the current velocity v ( i ) ( t ). Thus, he(she) will accelerate (or decelerate) to reach the desired velocity v d that willmake him (her) feel more comfortable. Thus, in the Social Force Model, the3esired force reads [14]. f ( i ) d ( t ) = m i v ( i ) d e ( i ) d ( t ) − v ( i ) ( t ) τ (2)where m i is the mass of the pedestrian i and τ represents the relaxation timeneeded to reach his (her) desired velocity. e d is the unit vector pointing tothe target position. For simplicity, we assume that v d remains constant dur-ing an evacuation process, but e d changes according to the current positionof the pedestrian. Detailed values for m i and τ can be found in Refs. [14, 25].The “social force” f ( ij ) s represents the psychological tendency of any twopedestrians, say i and j , to stay away from each other ( private sphere preser-vation). It is represented by a repulsive interaction force f ( ij ) s = A i e ( R ij − r ij ) /B i n ij (3)where ( ij ) represents any pedestrian-pedestrian pair, or pedestrian-wall pair. A i and B i are two fixed parameters (see Ref. [16]). The distance R ij = R i + R j is the sum of the pedestrians radius, while r ij is the distance between thecenter of mass of the pedestrians i and j . n ij stands for the unit vector in the (cid:126)ji direction. For the case of repulsive feelings with the walls, d ij correspondsto the shortest distance between the pedestrian and the wall [14, 26].Any two pedestrians touch each other if their mutual distance r ij issmaller than R ij . Also, any pedestrian touches a wall if his (her) distance r ij to the wall j is smaller than R i . In these cases, an additional force isincluded in the model, called the granular force f g . It is composed by twoforces: a sliding friction and a body force. The expression for this force is f ( ij ) g = f sliding + f body = κ t g ( R ij − r ij ) (∆ v ( ij ) · ˆ t ij ) ˆ t ij + κ n g ( R ij − r ij ) ˆ n ij (4)where κ t and κ n are fixed parameters. The function g ( R ij − r ij ) is equalto its argument if R ij > r ij ( i.e. if pedestrians are in contact) and zero forany other case. ∆ v ( ij ) · ˆ t ij represents the relative tangential velocities of thesliding bodies (or between the individual and the walls).4otice that the sliding friction occurs in the tangential direction whilethe body force occurs in the normal direction. Both are assumed to be linearwith respect to the net distance between contacting pedestrians. The coef-ficients k t (for the sliding friction) and k n (for the body force) are supposedto be related to the areas of contact and the clothes material, among others. In this section we define magnitudes which we have found useful in orderto explore the system properties. We will focus our attention in two quantities • In Section 2.2.1 we will define and characterize the concept of clusteringstructures. • In Section 2.2.2 we will introduce the meaning of a clogging delay andwe will explain different kinds of it. • In Section 2.2.3 we will relate the clustering structures with cloggingdelays.
Human clustering arises when pedestrians get in contact between eachother. These morphological structures are responsible for the time delaysduring the evacuation process [15–17]. Thus, for future analysis a precisedefinition of this kind of structures is needed.
Clusters
Many previous works showed that clusters of pedestrians play a funda-mental role during the evacuation process [15–17]. In this sense, researchersdemonstrated that there exist a relation between them and the clogging up ofpeople. Clusters of pedestrians can be defined as the set of individuals thatfor any member of the group (say, i ) there exists at least another memberbelonging to the same group ( j ) in contact with the former. Thus, we definea spatial cluster ( C g ) following the mathematical formula given in Ref. [27]. C g : p i (cid:15) C g ⇔ ∃ p j (cid:15) C g /r ij < ( R i + R j ) (5)where ( p i ) indicate the ith pedestrian and R i is his (her) radius (shoulderwidth). This means that C g is a set of pedestrians that interact not only5ith the social force, but also with physical forces ( i.e. friction force andbody force).Fig. 1a shows four spatial clusters. We will label a spatial cluster com-posed by n pedestrians as a n - spatial cluster . Isolated pedestrians ( i.e. with-out contact with their surrounding individuals) corresponds to a 1- spatialcluster (green circles in Fig. 1a). (a) (b) (c)Figure 1: (a,b,c) Schematic representation of different spatial structures clusters. Eachcolor represent a different spatial cluster. Isolated pedestrians are represented by greencircles (1-spatial cluster). (a) Three 2-spatial cluster are represented by black, orange andred circles. One 3-spatial cluster is represented by blue circles. (b) One 6-spatial clusteris represented by red circles. This one corresponds to a blocking cluster. (c) Multiplespatial clusters founded through the simulations. 3-spatial cluster, 2-spatial cluster and14-spatial cluster are represented by blue, black and red circles, respectively. Also, the14-spatial cluster includes a blocking cluster (labeled as bc). The blue boxes represent thewalls. Blocking clusters
The simulations show that some spatial clusters are able to block thedoor. We will call blocking cluster to these kind of spatial clusters. In thissense, a “blocking cluster” is defined as the minimum subset of clusterizedparticles ( i.e. spatial cluster) closest to the door whose first and last compo-nent particles are in contact with the walls at both sides of the door [16].6ig. 1b shows a typical blocking cluster composed by 6 pedestrians (redcircles). As a special type of a spatial cluster, these can be composed bydifferent number of pedestrians. Our simulations show that they can includefrom 5 to a maximum of 15 pedestrians for a door width of 0.92 m. Ref. [15]shows that this depends on door width.A blocking cluster may coexist simultaneously with other spatial clusters.In this sense, we can observe in Fig. 1c three spatial clusters: 2, 3 and 14-spatial cluster are present in it. Notice that the blocking cluster belongs toa greater spatial cluster of 14 pedestrians. As we will see in Section 4.3, thisoccurs more frequently for high desired velocities.
Clogging delays are a quite relevant quantity during the evacuation pro-cess. They are defined as the period of time between the egress of twoconsecutive pedestrians [15, 16, 25]. Animations shows that there are twogenerating mechanisms for a delay.
Frictional clogging delays
As stated in the last Section, blocking clusters are those spatial structuresthat block the exit. Refs. [15–17] show that they are responsible of worseningthe evacuation performance. Thus, blocking clusters are one of the mecha-nism of production of a delay. In Fig. 2 we illustrate how this mechanismoperates. Notice that the clogging delay starts when individual “1” left theroom. At the same time ( i.e. , t = 0 s), we can observe that individual “2”belongs to a blocking cluster, delaying his (her) outgoing. Later, at t = 0 . fric-tional clogging delay as the clogging delays produce by a blocking cluster.We remark that the frictional clogging delay consists of two contribu-tions. One corresponds to the time that spends the individual jammed inthe blocking cluster. And the other corresponds to the time lapse between7he breakup of the blocking structure and the exit time of the pedestrians(belonging to this blocking structure). This time corresponds to the transittime (0 . Figure 2: Schematic representation of the generating mechanism of a frictional cloggingdelay. The black line corresponds to the direction of the desired velocity v d . A blockingcluster (red circles) impedes ind. 2 from leaving the room. It breaks at t = 0 . t = 1 s he (she) escapes from the room. See text for more details. Social clogging delays
Animations show that there exists another generating mechanism for adelay. On one side, this can occur without the presence of a blocking cluster,as we illustrate in Fig. 3a. As in Fig. 2, the delay of 1 s corresponds to theoutgoing process between individuals 1 and 2. But, in this case, he (she) isdelayed due to the presence of individual 1. Notice that this occurs due tothe social force.We stress the fact that this situations occur for low pedestrian density.This may be the case when the anxiety level is small and therefore, individu-als do not each other. Thus, we can observe an unstable equilibrium betweenthe desired force and the social force near the door.On other side, we have the situation in which the social force delayed theoutgoing of people can be reached through the breaking of a blocking cluster.We illustrate this mechanism in Fig. 3b. Again, a blocking cluster impedesmany pedestrians to leave the room, particularly individuals labeled as 2 ans3. But, unlike Fig. 2, it releases two pedestrians after it breaks, like a burst ,at t = 0 . t = 1 s corresponds to the same situation showed in8ig. 3a at t = 0 s. But, in this case the clogging delay of 0 . i.e. , ∆ t = 1 s). Finally, notice that the cloggingdelay of 1 s between individuals 1 and 2 corresponds to a frictional cloggingdelay ( i.e. , produced by a blocking cluster).Summarizing, we conclude that the clogging delay of 1 s and 0 . social clogging delay as those clogging delaysproduced by the social interaction. (a) (b)Figure 3: (a,b) Schematic representation of the production mechanism of a social cloggingdelay. Black lines correspond to the direction of the desired velocity v d . The red continueline corresponds to the social force f s exerted by individual “1” over “2”. Green (red)circles indicates that this pedestrian is (is not) in contact with his (her) surroundingindividuals. (a) At t = 0 s, pedestrians cannot leave the room due to the presence ofindividual 1. When he (she) escapes, pedestrian labeled as 2 leaves the room at 1 s. (b) Ablocking cluster (red circles) impedes individual 2 and 3 (among others) to leave the room.It breaks at t = 0 . t = 1 s, individual 2 escapes from the room. Finally,pedestrian 3 escapes at t = 1 . i.e. ∆ t = 0 . In order to quantify the relationship between blocking cluster and cloggingdelay we define the arch-clogging correlation coefficient as follows [16] c ac = 1 N N (cid:88) cd =1 f ( t bc , t cd , t cd ) , (6)9 able 1: Computation of the arch-clogging correlation coefficient c ac (Eq. 6) in the caseof Figs. 2 and 3. Fig. Pairwise (i,j) t cd (s) t bc (s) t cd (s) f ( i,j ) ∆ t (s) clogging type2 (1,2) 0 0.5 1 1 1 frictional3a (1,2) 0 - 1 0 1 social3b (1,2) 0 0.5 1 1 1 frictional3b (2,3) 1 - 1.3 0 0.3 socialwhere N is the total number of clogging delays during the reported timeinterval, t bc corresponds to the time stamp for the blocking cluster breakingand t cd ( t cd ) corresponds to the time stamp for the delay beginning (ending).The function f is equal to 1 if t cd ≤ t bc ≤ t cd , and zero otherwise. This meansthat f is equal to 1 if the outgoing individual belonged to a blocking clusterduring the interval ( t cd , t cd ). Table 1 resumes the diagrams shown in Figs. 2and 3.This correlation c ac represents the fraction of frictional clogging delays (or social clogging delays ) with respect to all the clogging delays appearingduring the (stationary) evacuation process. Any value close to one indicatesthat most of the clogging delays belong to blocking clusters. On the contrary,if c ac is close to zero, most of them belong to social force interactions.
3. Numerical simulations
The simulations were performed on a 20 m ×
20 m square room with 225pedestrians inside. The occupancy density was set to 0 . , as sug-gested by healthy indoor environmental regulations [28]. The room had asingle exit on one side, placed in the middle of it in order to avoid cornereffects. The door width was L = 0 .
96 m, enough to allow up to two pedes-trians to escape simultaneously (side by side).The pedestrians were modeled as soft spheres. They were initially placedin a regular square arrangement along the room with random velocities, re-sembling a Gaussian distribution with null mean value. The rms value forthe Gaussian distribution was close to 1 m/s. The desired velocity v d was10he same for all the individuals. At each time-step, however, the desireddirection e d was updated, in order to point to the exit.We used periodic boundary condition (re-entering mechanism) for theoutgoing pedestrians. That is, those individuals who were able to leave theroom were reinserted at the back end of the room and placed at the veryback of the bulk with velocity v = 0.1 m/s, in order to cause a minimal bulkperturbation. This mechanism was carried out in order to keep the crowdsize unchanged, and therefore, the pressure among pedestrians.According to the literature (see Ref [14]), the model parameters used were τ = 0 . A = 2000 N, B = 0 .
08 m and κ t = 2 . × kg m − s − . However,the pedestrian’s mass and radius were set according to the more realist valuesof 70 kg and 0 .
23 m, as in Ref. [29]. Also, the compression coefficient κ n wasset equal to 2.62 × N m − , according to the experimental value of thehuman torso stiffness [30].The simulations were performed using Lammps molecular dynamics sim-ulator with parallel computing capabilities [31]. The time integration algo-rithm followed the velocity Verlet scheme with a time step of 10 − s. Weimplemented special modules in C++ for upgrading the Lammps capabili-ties to attain the “social force model” simulations. We also checked over the
Lammps output with previous computations (see Refs. [15, 16]).Data recording was done at time intervals of 0.05 τ , that is, at intervalsas short as 10% of the pedestrian’s relaxation time. The simulating processlasted until 7000 pedestrians left the room.The explored anxiety levels ranged from relaxed situations ( v d = 1 m/s)to highly stressing ones ( v d = 10 m/s). This last value can hardly be reachedin real life situations. However, in Ref. [17], it was shown that similar pres-sure levels can be reached in a big crowd with a moderate anxiety level.Thus, this wide range of desired velocities allows us to study different pres-sure scenarios. We also assumed that the pedestrians were not able to falldue to the crowd pressure as in Ref. [30].11 . Results Our results run along four major streams as follows: • In Section 4.1 we revisit the context of the evacuation time. We furtherintroduce other novel concepts that will be used throughout the work. • In Section 4.2 we show the differences between the faster is slower and faster is faster regimes in terms of the clogging delays. • In Section 4.3 we focus on the clusterization phenomenon at either the faster is slower and faster is faster regimes. • In Section 4.4 we analyze the relationship between the size of the clus-tering structures and the corresponding clogging delays.
As a first step, we computed the evacuation time for a wide range ofdesired velocities v d . This is shown in Fig. 4. We stress the fact that the ex-plored range corresponds to the one analyzed by Sticco et. al. (see Ref. [17])but, considering periodic boundary conditions. This means that any pedes-trian who left the room is introduced on the opposite side of the bottleneck(see Section 3).It can be seen in Fig. 4 the faster is slower (blue circles) and the faster isfaster (white and yellow circles) phenomena as reported in Ref. [17]. Recallthat the faster is slower regime, introduced by Helbing et. al. [14] corre-sponds to the increase in the evacuation time as the pedestrian’s anxietylevel increases. Thus, the evacuation efficiency reduces within this region. Itwas shown in Ref. [15] that this is related to the presence of sliding frictionamong pedestrians (and the walls).Besides, it was reported in Ref. [17] that the evacuation time decreases forvery high desired velocities ( i.e. , high pressures). This behavior correspondsto the faster is faster regime. It was concluded (see Ref. [17]) that as thecrowd pushing force increases, the sliding friction at the “blocking cluster”12 v d ( m/s ) e v a c u a t i o n t i m e ( s ) faster is fasterfaster is slowerfaster is faster Figure 4: Evacuation time for the first 7000 evacuees as a function of the desired velocity v d . The simulated room was 20 m ×
20 m with a single door of 0 .
92 m (say, the diameterof two pedestrians). The number of individuals inside the room was 225. This curve wascomputed from a single simulation process. Re-entering mechanism was allowed (see textfor more details). The simulation lasted until 7000 individuals left the room. Desiredvelocities of v d = 1.2, 2.5, 5.0 and 10.0 m/s are indicated in red color. seems not enough to prevent this kind of structure from (completely) stop-ping the crowd movement. This is the reason for the improvement of theevacuation time at the exit. We call the attention that we are not consider-ing the possibility of fallen pedestrians due to high pressures (see Ref. [30]for details). Further readings on the role of the sliding friction can be foundin Refs. [15–17].As can be noticed from Fig. 4, the faster is slower regime appears fordesired velocities between 1 and 5 m/s (approximately). This is somewhatshifted from the range reported in Ref. [17] (2 < v d < faster is faster effects. This discrepancyis a consequence of the periodic boundary condition. The bulk pressure re-mains (almost) constant during the stationary state for periodic boundarycondition. But, it decreases for non-periodic conditions, since pedestrians es-cape from the room. This phenomenon is similar the one observed in Ref. [17]when varying the number of pedestrians inside the room.We will focus on four specific desired velocities in the next section. These13orrespond to the minimum evacuation time ( v d = 1 . v d = 5 m/s) and two desired velocities with similar evacua-tion times but on different regimes. The desired velocities v d = 2 . v d = 10 m/s were chosen as representative of the faster is slower and fasteris faster phenomena, respectively. Discharge curves
We proceed to a first microscopic analysis of the evacuation processthrough the discharge curves (say, the number of individuals that escapefrom the room over time). Fig. 5 shows the discharge curves for four desiresvelocities (see caption for details) [15–17]. For each curve, the time betweenconsecutive evacuees ( i.e. the delay) is represented by a horizontal line.It can be seen in Fig. 5 that the evacuation time for the first 180 evacueescorresponds to the upper end of each discharge curve. It can be seen that thedesired velocities v d = 1 . v d = 2 . faster is slower and faster is faster regimes (within a sig-nificance of 5%). However, a tendency towards uniformity could be noticed(at least) for the faster is faster regime as the individual’s anxiety level ( v d )increases.In summary, the discharge curves show that the pedestrian flow is neveruniform (within a significance level), but delays tends to become more regu-lar for very high anxiety levels (or bulk pressures).14
20 40 60 80 time (s) nu m b e r o f e v a c u ee s v d = 1.2 m/s v d = 2.5 m/s v d = 5.0 m/s v d = 10.0 m/s Figure 5: Number of pedestrians that left the room versus time, for four different desiredvelocities v d (see legend). The showed time-window corresponds to a representative sampleof 180 pedestrians that left the room from a set of 7000 evacuees. The analyzed desiredvelocities are indicated by red circles in Fig. 4. The curves were computed from a singlesimulation process (re-entering mechanism was allowed) In this section we turn to study the mechanism by which different typesof clogging delays are generated. We analyze the clogging delays distribu-tions and the contribution of each type of delay to the overall evacuation time.
As mentioned in Section 2.2.2, there are two categories of clogging delays.The first one corresponds to those generated as a consequence of the socialforce among individuals (see Fig. 3). The second one corresponds to thosethat occur due to blocking clusters (see Fig. 2). The former corresponds toa social clogging delay , while the latter corresponds for a frictional cloggingdelay . Fig. 6 plots the coefficient c ac (see Eq. 6) as a function of the cloggingdelays longer than (or equal to) any threshold t c .According to Fig. 6, the probability of finding frictional delays increasesamong long delays. In other words, long lasting delays commonly belong toblocking clusters. Thus, we can (almost) exclusively classify any delay longer15 .
00 0 .
25 0 .
50 0 .
75 1 . t c (s) . . . . . . c a c v d = 1.5m/s v d = 2.5m/s v d = 5.0m/s v d = 7.5m/s v d = 10.0m/s Figure 6: Arch-clogging correlation coefficient c ac as a function of the duration of thedelay (equal or greater than t c ) for five different desired velocities (see legend for details).All the curves were computed from a single simulation process (re-entering mechanismwas allowed). We computed 6999 clogging delays for each desired velocity, according tothe evacuation of 7000 pedestrians. than 1 s as a frictional clogging delay . Notice that this criterion is fulfilledregardless of the value of the desired velocity v d (within the explored range).The situation for short clogging delays appears somewhat mixed. Not alldelays less than 1 s belong to a blocking cluster.Furthermore, the fraction of frictional delays increases for increasing de-sired velocities (see Fig. 3). Recall from Appendix B that the probability ofblocking clusters also increases for increasing values of v d . This is the reasonfor the increase of the fraction of frictional clogging delays when increasingthe desired velocity.A careful examination of the evacuation animations shows that social de-lays may also appear after the breaking process of a blocking cluster. Thiscorresponds to the burst released after the rupture, as exhibit in Fig. 3b.We observed in the animations (not shown) that this phenomenon occurscommonly as the desired velocity increases. Therefore, social delays shouldnot be considered as “opposed” to frictional delays, but complementary tothese. 16he major conclusion from this Section is that long delays (say, greaterthan 1 s) can be associated to blocking clusters. But, those delays of short du-ration may either be associated to social interactions or granular interactions.Our next step focuses on the relevance of the frictional and social cloggingdelays on the evacuation time. We computed the evacuation time consideringthese two categories of delays separately. Fig. 7 shows the results. v d ( m/s ) e v a c u a t i o n t i m e ( s ) frictional clogging delayssocial clogging delaysevacuation time Figure 7: Evacuation time for the first 7000 evacuees as a function of the desired veloc-ity v d . The black line corresponds to the total evacuation time. Red and green circlescorrespond to the evacuation time associated with frictional and social clogging delays,respectively. These were recorded from a single simulation process (re-entering mechanismwas allowed). The evacuation time considering (adding) only the frictional delays (andthus the blocking clusters) is qualitatively similar to the total evacuationtime. This corresponds to (almost) the entire explored interval ( v d > . v d < . i.e. social cloggingdelays) decreases monotonically as the desired velocity increases. As men-17ioned above, the individuals are very seldom in contact for v d < . v d < . faster is slower effect takes placefor the latter at higher v s ’s (as expected). We further computed the delays’s probability distribution, as shown inFig. 8. We chose the same desired velocities as in Section 4.1 for the purposeof comparison. The distribution corresponding to all the clogging delays andthe frictional clogging delays can be seen in red and blue bars, respectively.The latter corresponds to a subset of the former. Notice that distributionsare valid for the specific door width mentioned in the caption (see Ref. [16]for more details).The delays shorter than 3 s in Fig. 8 (approximately) are present in allthe plotted situations. But, clogging delays greater than 3 s occur only forthe intermediate situations v d = 2.5 and 5 m/s (see Figs. 8b and 8c). Bothsituations exhibit delays up to 7 s approximately. Recall that these levelsof v d correspond to the faster is slower regime (see Fig. 4). The faster isslower regime includes long delays, while the faster is faster regime standsfor delays no longer than 3 s (under the explored conditions). We concluded in Section 4.2.1 that almost all the delays longer than 1 scorrespond to the presence of a blocking cluster. Therefore, they correspondto frictional clogging delays . In Section 4.2.2 we further noticed that longdelays (greater than 3 s) appear during the faster is slower regime. We focuson three categories: short (∆ t ≤ < ∆ t ≤ t > • Short clogging delays (∆ t ≤ . socialclogging delays . Also, as can be seen this kind of delay does not allow18 ∆ t (s) − − − − p ( ∆ t ) all clogging delaysfrictional clogging (a) v d = 1.2 m/s (minimum t ev ) ∆ t (s) − − − − p ( ∆ t ) all clogging delaysfrictional clogging (b) v d = 2.5 m/s ∆ t (s) − − − − p ( ∆ t ) all clogging delaysfrictional clogging (c) v d = 5 m/s (maximum t ev ) ∆ t (s) − − − − p ( ∆ t ) all clogging delaysfrictional clogging (d) v d = 10 m/sFigure 8: Normalized distributions of time lapses ∆ t between egresses of consecutivepedestrians up to 7000 people evacuated, for the same values of v d analyzed in Fig. 5. Thebin size is 0 .
25 s. Clogging delays and delays originated due to a blocking cluster (namedfrictional clogging) are indicated by red and blue bars, respectively. Frictional cloggingcorresponds to a subset of the clogging delays (see text for details). Thus, red bars areplotted behind the blue bars. The distribution area corresponding to the clogging delaysis 1. Data was recorded after the first 10 s of the beginning of the simulation process, inorder to avoid non-stationary effects. The door width was 0 .
92 m (say, the diameter oftwo pedestrians). us to distinguish the faster is slower and faster is faster regimes (above1 . • Intermediate clogging delays (1 s < ∆ t ≤ v d > . v d > . faster is slower and faster is faster regimes. Also, the cumulative time due to both delaysexplains almost all of the evacuation time. • Long clogging delays (∆ t > faster isslower effect. Also, the lack of long clogging delays may be partiallyresponsible for the faster is faster phenomenon. v d ( m/s ) e v a c u a t i o n t i m e ( s ) ∆ t ≤ < ∆ t ≤ ∆ t > (a) Short delays v d ( m/s ) e v a c u a t i o n t i m e ( s ) ∆ t ≤ < ∆ t ≤ ∆ t > (b) Intermediate and long delaysFigure 9: The black line corresponds to the evacuation time for the first 7000 evacueesas a function of the desired velocity v d . Red, green and blue circles (and dashed lines)correspond to the different delay categories. The following conclusions can be outlined from this Section. The analysisof the delays show that the overall evacuation time is mainly due to frictionalclogging delays for v d > < ∆ t ≤ faster is slower and faster is faster regimes (see Fig. 9). In this section we turn to study the clustering structures. We analyze thedifferent kind of spatial clusters and we explore how they change with thedesired velocity. 20 .3.1. The morphology in the bulk of the crowd
Fig. 10 shows four snapshots of the bulk close to the exit. The shownsituations correspond to the same desired velocities as in Section 4.2. Redand white circles represent the blocking and spatial clusters, respectively. Westress that the individuals in red (blocking cluster) also belong to the spatialcluster (white color) (see Fig. 10). (a) v d = 1.2 m/s (b) v d = 2.5 m/s (c) v d = 5.0 m/s (d) v d = 10.0 m/sFigure 10: Snapshots of the bulk for four desired velocities. Red and white circles rep-resent the blocking cluster and the spatial clusters, respectively. Green circles correspondto individuals without contact with his (her) surrounding pedestrians and walls. The or-ange lines represent the walls on the right of the room. The simulated room was 20 m ×
20 m with a single door of 0 .
92 m (say, the diameter of two individuals). The number ofindividuals inside the room was 225. The selected desired velocities are those indicated inred in Fig. 4.
As can be seen in Fig. 10, the number of individuals in contact increaseswith increasing desired velocities. This is the expected picture for peoplepushing towards the exit. The harder they push (increasing values of v d ),the larger the clogging region.Notice that two small spatial clusters appear at v d = 1 . v d = 2 . v d ( m/s ) e v a c u a t i o n t i m e ( s ) . . . . . n b c / n s c n bc /n sc evacuation time Figure 11: The black line corresponds to the evacuation time for 7000 evacuees, whilethe red circles correspond to the normalized size of a blocking cluster ( n bc ) with respect tothe spatial cluster ( n sc ), both as a function of the desired velocity v d . The spatial clustercorresponds to those in contact with the blocking cluster. The size of the blocking clusterand the spatial cluster were recorded every 0 .
05 s. The acquisition was done only when ablocking cluster existed. The error bars corresponds to ± σ (one standard deviation). Thesewere computed from a single simulation process (re-entering mechanism was allowed). A sharp decay in the blocking-to-spatial ratio can be noticed in Fig. 11.We immediately identify two regimes according to Fig. 11. For v d < v d > .3.2. Size of spatial clusters Recall that as the desired velocity increases, more people gets into con-tact with each other, and therefore, the size of the spatial cluster increases forincreasing anxiety levels of the individuals. We classified the spatial clustersinto three categories (according to their size n ) as follows: small (1 < n ≤ < n <
15) and big ( n ≥ v d ( m/s ) e v a c u a t i o n t i m e ( s ) . . . . . . n b i n / n c l u s t e r s < n ≤ < n ≤ (a) small and medium spatial clusters v d ( m/s ) e v a c u a t i o n t i m e ( s ) . . . . . . n b i n / n c l u s t e r s n ≥ (b) big spatial clustersFigure 12: (a,b) The black line corresponds to the evacuation time for the first 7000evacuees, while the red, green and blue circles correspond to the normalized number ofspatial clusters ( n bin ) for three different size categories (see legend for details), as a functionof the desired velocity v d . The normalization was done with respect to the total numberof spatial clusters ( n clusters ). The size of each spatial cluster was obtained every 0 .
05 s.Data was acquired from a single simulation process (re-entering mechanism was allowed). • Small spatial clusters (1 < n ≤ v d < v d > v d > • Medium spatial clusters (5 < n < faster is slower effect and holds for a narrow range ofdesired velocities (say, between 1 < v d < • Big spatial clusters ( n ≥ v d =2 m/s, approximately. At this threshold, the pressure is high enoughto force the pedestrians contact each other. A single big cluster wouldbe expected to very high desired velocities. Recall from Section 4.2.3 that we classified the clogging delays into threecategories: short (∆ t ≤ < ∆ t ≤ t > < n ≤ < n <
15) and big( n ≥ i.e. a subset of the spatial cluster). Thus, many spatial clusters can be reportedduring a frictional clogging delay . The number of spatial clusters (for eachcategory) taking place between the beginning of a frictional clogging delay and the rupture of the corresponding blocking cluster is shown in Fig. 13.As a first insight, it becomes clear that some kind of correlation betweenthe size of the spatial cluster and the associated frictional clogging delay ex-ists. This correlation is different for different values of the desired velocity24 h o r t i n t e r m e d i a t e l o n g duration smallmediumbig s i z e
30 33 116 8 30 0 0 (a) v d = 1 . s h o r t i n t e r m e d i a t e l o n g duration smallmediumbig s i z e
338 200 12160 112 00 0 0 (b) v d = 1 . s h o r t i n t e r m e d i a t e l o n g duration smallmediumbig s i z e
378 296 56498 414 120 0 0 (c) v d = 1 . s h o r t i n t e r m e d i a t e l o n g duration smallmediumbig s i z e
172 112 24936 1144 21222 32 6 (d) v d = 1 . s h o r t i n t e r m e d i a t e l o n g duration smallmediumbig s i z e
10 10 2764 880 142388 696 124 (e) v d = 2 . s h o r t i n t e r m e d i a t e l o n g duration smallmediumbig s i z e (f) v d = 2 . s h o r t i n t e r m e d i a t e l o n g duration smallmediumbig s i z e (g) v d = 5 . s h o r t i n t e r m e d i a t e l o n g duration smallmediumbig s i z e (h) v d = 10 . frictional clogging delay and the sizeof the spatial cluster that produces it. In each tile the x-axis represents each of the threeclogging delay categories and the y-axis represents each of the three size of the spatialcluster categories. The number inside each cell corresponds to the amount of frictionalclogging delays that was generated by a specific spatial cluster (see text for details). (see Fig. 13). This is in agreement with Figs. 9 and 12, since both mag-nitudes depend on the desired velocity. Furthermore, we can recognize twovery different situations by reading the number of reported events inside eachcell (see caption for details). On the upper row of Fig. 13 (plots (a) to (d))only small and medium spatial clusters are relevant. But, on the lower row(plots (e) to (h)), the medium or big spatial clusters are the relevant ones. Acloser examination, though, shows that a single size is the only relevant onein all the plots except for plots (c) and (e) ( v d = 1 . v d may be sum-marized as follows:1. The patterns (in violet) at v d = 1 . . v d = 1 . v d = 1 . v d < . . < v d < v d > v d = 1 . faster is slower and faster is faster regimes. From the point of view of the spatial clusters, small and mediumspatial clusters ( n <
15) dominate the scene for desired velocities below3 m/s. These (almost always) match the blocking cluster definition. But, for v d > n ≥
15) becomes the relevant struc-ture during the evacuation process. We may associate the overall delays (for v d > . . < v d < v d = 1 . → medium stage, whilethe latter corresponds to a medium → big stage. All these regimes attainsmall/intermediate clogging delays.
5. Conclusions
Our research focused on the microscopic analysis of the evacuation processof self-driven particles confined in a square room with a single exit door.26e simulated 225 individuals escaping through a door with a width of twotimes a pedestrian width. The simulations were done in the context of theSocial Force Model. We were mainly interested in properly understandingthe faster-is-slower and faster-is-faster phenomena, i.e. the relation betweenthe flow of pedestrians and the formation of structures (clusters) which mightimpede the motion of the walkers.We have found that as the desire velocity increases, the system evolvesfrom a condition in which the flow decreases with increasing v d to another inwhich the flow increases with increasing v d . The cause of such a change canbe traced to the characteristics of the above mentioned blocking structures.We have shown that the faster-is-slower and the faster-is-faster are quitedifferent phenomena from the standpoints of the “frictional” clogging de-lays and the underlying clusterization structure. The faster-is-slower occurswhenever the blocking clusters dominate the scene, i.e. there is a structurewhich anchors to the walls of the exit and is able to momentarily obstructthe exit, accomplishing moderate to long lasting clogging delays (say, above1 s). However, as the v d increases, the pressure increases as well and then,the blocking cluster becomes a giant cluster, involving (almost) all the crowd.The giant cluster is not anchored to the walls but resembles a very viscousfluid. Keep in mind that in the Social Force Model the agents are not rep-resented by hard spheres but by soft spheres and as such the resistance op-posed by the agents can be overcome by a force big enough. The harder thepedestrians push, the weaker the slowing-down, achieving the faster-is-faster phenomenon. This phenomenon has not been reported on other granularsystems, to our knowledge.For the set of parameters of the Social Force Model used in this workthe analysis of the correlation between the clogging delays and the spatialstructures showed that for v d > t ≤ < ∆ t ≤ faster-is-faster scenario, since the burst of leaving pedestrianstends to be increasingly regular. 27 by-product of our investigation is that we can distinguish between twocategories of clogging delays, according to their production mechanism. Thesocial clogging delays corresponds to those that are generated as a conse-quence of the social force among individuals. Instead, the frictional cloggingdelays are a consequence of the granular force among individual. However,social and frictional clogging delays are complementary delays whenever ablocking cluster breaks down, releasing a burst of individuals. We shouldcall the attention on this point when analyzing the overall delay of an evac-uation. Acknowledgments
This work was supported by the National Scientific and Technical Re-search Council (spanish: Consejo Nacional de Investigaciones Cient´ıficas yT´ecnicas - CONICET, Argentina) and grant Programaci´on Cient´ıfica 2018(UBACYT) Number 20020170100628BA. G. Frank thanks Universidad Tec-nol´ogica Nacional (UTN) for partial support through Grant PID NumberSIUTNBA0006595.
Appendix A. Statistical analysis of the discharge curves
This analysis was carried out by means of the Kolmogorov-Smirnov test.This non-parametric statistical test is a goodness of fit test, which allowscomparing the distribution of an empirical sample with another, or, to whatis expected to be obtained theoretically (null hypothesis) [32].The following statistic is considered D n = max | S n ( x ) − F o ( x ) | (A.1)where S n ( x ) and F o ( x ) represent the cumulative distributions of the sampleddata and the theoretical one, respectively. As can be seen, this test quantifiesthe maximum (absolute) difference between both distributions.In our case, the discharge curves are compared against a uniform flow ofindividuals. The latter corresponds to delays of the same duration. This isuseful for analyzing the relative significance of the delays during the evacu-ation process. 28o carry out the Kolmogorov-Smirnov test, 40 discharge curves were ana-lyzed, each of which correspond to the evacuation process of 180 individuals.The 40 selected processes were chosen separately in time, in order to be un-correlated. The results are presented in Fig. A.14. v d ( m/s ) e v a c u a t i o n t i m e ( s ) . . . . . D n medianevacuation time α = 0.05 Figure A.14: The black line corresponds to the evacuation time for the first 7000 evacueesas a function of the desired velocity v d . Red circles correspond to the Kolmogorov-Smirnovtest (see text for details). The blue horizontal dashed line corresponds to a significancelevel of 5%. 40 discharge curves were analyzed for each desired velocity Each dischargecurve corresponds to the evacuation process of 180 individuals. These were uncorrelatedsub-intervals from a single simulation process (re-entering mechanism was allowed). As can be seen, the behavior of the median coincides qualitatively withthat corresponding to the evacuation time. However, both curves are hori-zontally shifted from each other. The maximum of the curve correspondingto the median is approximately at v d = 2 m/s, unlike what happens with theevacuation time ( v d = 5 m/s). It should be noted that the former occurs inthe moment of the inversion of the concavity of the curve associated with theevacuation time.We can identify two regimes from Fig. A.14, depending on the desiredvelocity. For v d < v d > D n as the desireto escape from the room increases. Thus, the flow becomes more regular.It is worth mentioning that this behavior starts during the faster is slower effect (and continues during the faster is faster effect). In addition, it is pos-sible to notice a linear behavior of the median in this range of desired velocity.Finally, in order to quantify the discrepancy between both distributions,a significance level of 5% is indicated in Fig. A.14 (dashed line). As can beseen, the null hypothesis is rejected for approximately the entire exploredrange of desired velocity. On the other hand, it should be noted that thisis not possible before the minimum evacuation time ( i.e. v d < . Appendix B. Blocking cluster probability
We analyze here the probability of occurrence of a blocking cluster (seeFig. B.15). That is, the percentage of time that the system is in the presenceof a blocking cluster. Recall that in Fig. 6 it was observed that, for any givendelay value, as the desired velocity increases, so does the probability thatthis delay was generated by a blocking cluster.As can be seen, the higher the desired velocity, the greater the probabil-ity of attaining a blocking cluster. Moreover, it has an asymptotic behaviorabove v d = 5 m/s, approximately. This means that above this desired veloc-ity, the presence of blocking clusters is permanent over time. However, wecan note that despite the still presence of a blocking cluster in front of thedoor, evacuation time decreases. The explanation for this phenomenon areexplained in Section 4.3.The above results are in agreement with those reported in Refs. [15, 16], inspite of the different boundary conditions, ranges of desired velocity and/orthe value of the elastic constant k n [29]. But, unlike the results reportedin the literature where this behavior occurs in the presence of the faster isslower effect, in this case we observe that it is also satisfied during the faster v d ( m/s ) e v a c u a t i o n t i m e ( s ) . . . . . b l o c k i n g c l u s t e r p r o bab ili t y probabilityevacuation time Figure B.15: The black line and red circles correspond to the evacuation time for the first7000 evacuees and the probability of a blocking cluster as a function of the desired veloc-ity v d , respectively. 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