A study on pattern formation in crowd dynamics via ... · Kohta SUZUNO Graduate School of Advanced Mathematical Sciences, Meiji University, Japan January 2015 (ver. 150204) 1. Abstract

明治大学大学院先端数理科学研究科２０１４年度博士学位請求論文

A study on pattern formation in crowddynamics via mathematical modeling

（群集ダイナミクスにおけるパターン形成の数理モデリングによる研究)

学位請求者　現象数理学専攻鈴野　浩大

Dissertation

A study on pattern formation in crowd dynamics via mathematicalmodeling

Kohta SUZUNOGraduate School of Advanced Mathematical Sciences, Meiji University,

JapanJanuary 2015

(ver. 150204)

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Abstract

In this study the investigation of self-organized pattern formation in crowddynamics via mathematical modeling is presented. Crowd systems are ex-amples of non-equilibrium particle flow and show interesting spontaneouscollective motion. It is important to clarify the mechanism, to investigatethe universality, and to establish mathematical methods to analyze discreteflow. Based on these backgrounds, the main focus of the study is placedon the clarification of the mechanisms of self-organization phenomena in ac-tive particle systems that correspond to crowd motion. Through modeling,analysis and simulation, some of the spontaneous pattern dynamics will beexplained from mathematical and physical view. In this work, I also wouldlike to show that mathematical modeling is a useful tool for the investigationof pattern formation in active particle flow in which the conventional fluidapproximation does not hold. Many previous studies have been based oncase studies using numerical simulations and dedicate to finding new self-organized phenomena. On the other hand, the clarification of the underlyingmechanism draws less attention of crowd scientists. One of the reasons of thisdrawback is the absence of appropriate description methods to investigate thecollective motion of spontaneous order in dissipative particle systems. HereI would like to stress that mathematical modeling could be a useful alter-native method in crowd studies. The combination of simulations, modelingand analysis offers us the revelation of the fundamental mechanism of crowdmotion and new insights into pattern dynamics in discrete flow, as shown inthis study.

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This study is composed of (i) the historical survey of collective motion ofself-driven particles, (ii) the technical review of crowd pattern with particlesimulations, (iii) modeling self-organized phenomena, (iv) analyzing modelsto clarify the mechanism of phenomena, and (v) give a comparison of themodeling results with the simulation results.

Chapter 1 is the introduction of crowd dynamics. The purpose of thestudy is to give mathematical description for some self-organized phenomenain crowd dynamics and provide possible mechanisms. Crowd dynamics is adiscipline that treats spontaneous order in the flow of non-equilibrium par-ticles mathematically. The flow of human is the example of such kind. Oneof the importance of the crowd studies is that a crowd shows spontaneousspatio-temporal collective patterns. Interestingly, some patterns in crowdmotion show the phase transition, for instance, between fluid-like and solid-like configuration, depending on the density, noise and driving forces. Someof them are specific to crowd motion, whereas some others are universal sincesimilar patterns are observed in other systems. The comparison of the differ-ences and the similarities will give us new insights into the collective motionof crowd flow. Another importance of studying crowd motion is that crowddynamics provide a new mathematical and physical issue: how should we de-scribe and understand discrete particle flow? The numerical many-particlecalculations do not necessarily bring us deeper understanding of complexphenomena. Thus, we need to seek alternative methods to describe funda-mental properties of particle flow. The study of crowd dynamics would givea new example of collective motion and offers an opportunity of developingnew mathematical description of pattern dynamics. Chapter 1 also includesa historical review. Crowd dynamics has a long history and consist of mul-tiple viewpoints based on massive amount of contributions. The review isbased on the standpoint of self-organization in crowd systems. At the end ofthis chapter, the critical discussion on crowd dynamics is given. Note thatmodeling crowds via physics-based concepts only holds when we consider (i)

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panic situations, (ii) crowds have definite destinations but the ways to reachthere are less conscious, and (iii) the size of crowds is tremendously large.In such situations, the difference of each people diminishes and the averagedbehavior could be treated mathematically.

Chapter 2 is the review of spontaneous phenomena observed in crowd dy-namics. It is well known that crowd motion is accompanied by self-organizedorder. Here I would like to show the examples, fundamental features, anddetailed historical reviews for some phenomena in crowd dynamics. The dis-cussion provided here is mainly based on the social force model. Phenomenadiscussed here are as follows: (i) lane formation in pedestrian counterflow,(ii) the freezing-by-heating transition, (iii) self-excited oscillatory flow at bot-tlenecks, (iv) the faster-is-slower effect in bottleneck particle flow.

Chapter 3 gives a mathematical model of oscillatory crowd flow. Here wefocus on the oscillatory counterflow of pedestrians passing through a bottle-neck. This “crowd flow oscillator” has been observed both numerically andexperimentally and known as an example of self-organized order in particleflow. In this chapter, a model is proposed to describe the oscillatory flow andinvestigate its mechanism. Through an analysis of the model, it is shown thatthe oscillator is understood as a van der Pol-type self-excited oscillation.

In chapter 4, the investigation of the mechanism of the faster-is-slowereffect, a nontrivial deceleration effect in pedestrian flow, is conducted with asimple analytical phenomenological model. With numerical simulations, it isshown that the rate of outflow from a bottleneck shows its maximum when theintensity of the driving force has a certain value. To explain the mechanismof the phenomenon, a phenomenological analytical model is proposed basedon physics-based intuitive mathematical modeling. It is confirmed that oursimplified model mimics fundamental properties of the original many-particlesystem. The model study shows that the faster-is-slower flow comes from thecompetition between the driving force and the nonlinear friction.

Chapter 5 explains the dynamic structure in pedestrian evacuation with

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the aid of image processing approach. it is shown that the dynamic arch-shape structure near bottlenecks correlates with the efficiency of pedestrianevacuation. It is known that the clogging occurs in front of exits when weconduct the social force model simulation which mimics the evacuation of aroom with a narrow exit since particles form arch-shape structure sponta-neously. Some previous studies have shown that the presence of obstaclesnear bottlenecks may improve the flow rate by experiments and numericalsimulations. In this chapter, we clarify a possible mechanism of the obstacleeffect with the aid of the concept of the dynamic arch. The time-averagedpictures generated by the snapshots of numerical simulations show us arch-shaped typical structure. In this chapter the relation between the shape ofthe dynamic arch and the efficiency of the outflow is discussed. From thisinvestigation, it is shown that the improvement of the flow rate is attainedby the manipulation of the emerging patterns in particle flow with obstacles.

Chapter 6 is the conclusion of this study. The studied presented here indi-cate that the mathematical modeling could be a useful method in the studiesof pattern formation in crowd dynamics. The combination of simulations,modeling and analysis contribute to the revelation of the fundamental mech-anism of crowd motion and the provision of the new insights into patterndynamics in discrete flow.

Contents

Abstract 2

1 Introduction 151.1 The purpose of the study . . . . . . . . . . . . . . . . . . . . . 151.2 What is crowd dynamics? . . . . . . . . . . . . . . . . . . . . 16

1.2.1 Crowd dynamics focuses on human flow . . . . . . . . 161.2.2 Crowd dynamics contribute to social safety . . . . . . . 161.2.3 Crowd dynamics provide new issues to mathematical

science . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.3 The importance of the study . . . . . . . . . . . . . . . . . . . 181.4 The methods of the study . . . . . . . . . . . . . . . . . . . . 19

1.4.1 Crowd flow . . . . . . . . . . . . . . . . . . . . . . . . 201.4.2 Biological entities . . . . . . . . . . . . . . . . . . . . . 211.4.3 Granular matter . . . . . . . . . . . . . . . . . . . . . 21

1.5 Historical review . . . . . . . . . . . . . . . . . . . . . . . . . 221.5.1 Prehistory of crowd flow . . . . . . . . . . . . . . . . . 221.5.2 Safety engineering . . . . . . . . . . . . . . . . . . . . . 231.5.3 Pattern formation in crowd flow . . . . . . . . . . . . . 241.5.4 Universality of crowd patterns . . . . . . . . . . . . . . 26

1.6 Critical discussion on crowd dynamics . . . . . . . . . . . . . . 261.6.1 Model dependence . . . . . . . . . . . . . . . . . . . . 271.6.2 The free will . . . . . . . . . . . . . . . . . . . . . . . . 28

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CONTENTS 7

1.6.3 Validation of mathematical models . . . . . . . . . . . 29

2 Pattern formation in pedestrian dynamics 302.1 Lane formation: particle-scale instability . . . . . . . . . . . . 302.2 Freezing-by-heating: noise-induced solidification . . . . . . . . 362.3 Oscillatory flow at bottlenecks: a rhythm in particle flow . . . 412.4 The faster-is-slower effect: the driving force v.s. the dragging

force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3 A mathematical model of oscillatory pedestrian flow 523.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.2 The outline of the oscillatory flow . . . . . . . . . . . . . . . . 563.3 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4 Analytical investigation of the faster-is-slower effect with asimplified phenomenological model 694.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.2 Review of the SF model . . . . . . . . . . . . . . . . . . . . . 724.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.5 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.6 Discussion and summary . . . . . . . . . . . . . . . . . . . . . 81

5 Dynamic structure in pedestrian evacuation: image process-ing approach 845.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.3 On the effect of the obstacle . . . . . . . . . . . . . . . . . . . 905.4 Possible mechanisms of the obstacle effect . . . . . . . . . . . 915.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

CONTENTS 8

6 Conclusion 95

Acknowledgement 98

Bibliography 100

List of Figures

1.1 A crowd (Shibuya, Tokyo, 2014). . . . . . . . . . . . . . . . . 171.2 Terada’s observation of commuter trains. . . . . . . . . . . . 23

2.1 Lane formation in self-driven particle systems. The bars inthe particles represent the direction of the driving force. Eachparticle has the driving force and the repulsive interaction Thegray particles move from left to right, whereas the white onesrun toward the opposite direction. The both ends of the cor-ridor are the periodic boundary. . . . . . . . . . . . . . . . . 31

2.2 Lane formation in high dense pedestrian counterflow (Nakano,Tokyo, 2014). The gray and white circles correspond to theoutgoing and incoming pedestrians. . . . . . . . . . . . . . . 32

2.3 Lane formation in low density pedestrians (Nakano, Tokyo,2014). Unstable multi-lanes are formed spontaneously. . . . . 32

2.4 Numerical simulation of freezing-by-heating. (Top) with largenoise, the system shows turbulent flow. (Middle) withoutnoise, the lanes are formed. (Bottom) with an appropriatenoise intensity, the system shows sudden transition to thefrozen state. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

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LIST OF FIGURES 10

2.5 Probability of transition. The graph shows which states thesystem tend to attain for each noise intensity σ. (circle) thefreezing state is attained. (square) lanes are formed. (triangle)no definite spacial structure is observed at 1000 s. Here 20calculations are conducted for each noise intensity. . . . . . . 38

2.6 Time series of the total kinetic energy of the system. Thesolid line represent the actual kinetic energy, and the dashedline means the possible maximum energy which correspond tothe completely ordered lanes. Here the noise is introduced as|ζi| = 0.01N(0, 1) for each time step. . . . . . . . . . . . . . . 40

2.7 Oscillatory pedestrian flow at the bottleneck. The directionof flow changes with time. . . . . . . . . . . . . . . . . . . . . 41

2.8 Momentum density in the vicinity of the bottleneck. The sys-tem shows the regular oscillation. Here the width of the bot-tleneck is set to 0.7 m. . . . . . . . . . . . . . . . . . . . . . . 43

2.9 The model of pedestrian evacuation of a room. The particleshave the self-driven force towards the exit. . . . . . . . . . . 46

2.10 Evacuation times as a function of the driving force. The solidline corresponds to the case in which the particles have thefriction force, whereas the dashed line for no friction. . . . . . 48

3.1 The pedestrian flow oscillator. The particles with oppositedriving force run through the bottleneck. The figures showthe time evolution of the system. The direction of the flowchanges with time. . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2 Schematic explanation of the interactions. . . . . . . . . . . . 57

LIST OF FIGURES 11

3.3 Boundary condition of the numerical simulation. The corri-dor has the length Lc and the width Lw, the bottleneck w, andthe periodic boundary condition. Each particle has the driv-ing force towards the center of the bottleneck. After passingthrough the bottleneck, they go to the boundaries and returnto the system from the opposite side. . . . . . . . . . . . . . 58

3.4 The oscillatory changes of the momentum of the system at thebottleneck. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.5 The Fourier amplitude as a function of the bottleneck width. 603.6 Model system. Those are the virtual particles which represent

the averaged motion of the original self-driven particles. Eachparticle has the driving force that correspond to the sum ofthe driving force of the SFM particles near the bottleneck.Besides, the representative particles have pressure from behindwhich originates the total force to the bottleneck from the bulkparticles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.7 numerical solutions of the model. The solid, dotted and chainline correspond to the initial conditions (x, v)=(0, 0.2), (0,0.6), and (0, 1.5) respectively. (Top-left) a2 = 3 is used. Thesolutions converge to zero fro all initial conditions. (Top-right)a2 = 12. Both the trivial solution and the limit cycle exists.(Bottom-left) a2 = 120. Most of the solution converges tothe limit cycle. (Bottom-right) The schematic bifurcation di-agram. In all cases a1 = 0 is used for simplicity. . . . . . . . . 66

3.8 function form of n(v). The solid and the dotted line corre-spond to n(v) and it polynomial approximation v3 − v5. . . . 67

LIST OF FIGURES 12

4.1 The numerical results for the stationary flow rate obtainedfrom the SF model simulation. A higher flow rate correspondsto a faster evacuation time. (a) v0- and N -dependence of theflow rate. The flow rate was not calculated for small values ofv0 and N . (b) The flow rate with a fixed N of 200. . . . . . . 73

4.2 Schematic phase diagram. The area below the rough addi-tional line is the “free state,” where the flow rate increases asthe driving force increases. Each point (x) shows the maxi-mum point for each N . The area above the line is the “jam-ming state,” where the “faster-is-slower” effect occurs. Thecritical v0 value depends on N , and this suggests that the ef-fect begins to slow down evacuation when N is relatively large. 74

4.3 Conceptual images of the model. (left) The typical formationof the arch near the exit. (right) The forces acting on thevirtual particles. . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.4 The force applied to the particle at the center of the semi-circlefor each N . The results are obtained from the SF simulationwith zero-width exit. Dashed lines represent Eq. (4.2) withthe dimensionless parameter b = 0.625. . . . . . . . . . . . . 76

4.5 The stationary flow velocity obtained from the model. (a)The v0- and N -dependence of the flow velocity. (b) The flowvelocity with a fixed N of 200. . . . . . . . . . . . . . . . . . . 79

4.6 The explicit representation of the function g(l(v0, N = 300)).(solid) the analytical expression based on Eq. (4.6), (dotted)the numerical result calculated directly from Eq. (4.4), (chain)the approximation curve g(l(v0, N)) ∼ vn

0 (n = 2). . . . . . . . 804.7 Numerical results of the SF model simulation corresponding

to each conjecture. Only (iv) shows a maximum. . . . . . . . . 82

LIST OF FIGURES 13

5.1 simulation of pedestrian evacuation passing through the bot-tleneck. The white and the black circle mean the pedestriansand the obstacle. . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.2 Details of the system. The self-driven particles rush to theexit and evacuate from the room. All particles have the samediameter (0.6 m) and the self-driven force (3.0 m/sec). Theygo out through the exit (1.0 m) and return to the room fromthe opposite side of the exit. The number of particles is 150.We observe this evacuation process up to 300 sec. . . . . . . . 87

5.3 Example of contact force network. The particles move fromleft to right side of the picture. When two particles have pen-etrated, the white line that connects two particles is depicted,and the system shows the network structure. In this calcula-tion, the thickness of the line does not depend on the magni-tude of force for simplicity. Note that the shape of the particles(circles in the picture) is depicted in this example picture forclarification, but it is not drawn in the actual calculation. . . 88

5.4 Example of time-averaged network. The white lines show theaveraged contact network in this many-particle system. Heregamma correction (γ = 2.0) is used and the additional cir-cles are depicted for visibility. The four-particle arch is themost frequently occurred structure. The picture also showsthe vague triangular lattice in the bulk region (behind thearch), and it reflects the mono-disperse property (all particleshave the same diameter) of the system. The thick white circleis the obstacle. . . . . . . . . . . . . . . . . . . . . . . . . . . 89

LIST OF FIGURES 14

5.5 Results of the numerical simulation. We calculate the flow rateby using the SFM with different configuration of the obstacle.The size of the obstacle is the same as the particles. (Left) Theposition of the obstacle considered here is shown as the latticepoint in the figure. The circle in the figure is the example ofthe obstacle. (Right) The color distribution shows the flowrate for each obstacle position. White area correspond to theflow rate without obstacle (340 person/300sec). . . . . . . . . 90

5.6 Morphological observation of the arch. In no-obstacle case(top-left), the shape of the arch is almost symmetric. On theother hand, the situation that attain the highest flow rate(top-right) shows asymmetric arch. Here gamma correctionis used for visibility (γ = 2.0). The obstacle covers the archin front of the exit and protect it from the particles behind.The obstacle makes a space, and it also allows the arch to beshifted and distorted (bottom). . . . . . . . . . . . . . . . . . 93

5.7 The correspondence of the position of the obstacle with theshape of the arches. Each gridpoint represents the position ofthe center of the obstacle, and each picture is the correspond-ing shape of the dynamic arches. Here gamma correction isused for visibility (γ = 2.0). . . . . . . . . . . . . . . . . . . . 94

Chapter 1

Introduction

The purpose of the study is to give mathematical descriptionfor self-organized phenomena in crowd dynamics and provide itspossible mechanisms. It is well known that a crowd shows vari-ous spontaneous collective patterns. The study of crowd dynam-ics would give new insight into collective phenomena and offeran opportunity of developing new mathematical description ofpattern dynamics. Also, in this chapter the historical review ofcrowd dynamics is provided.

1.1 The purpose of the study

In this study the investigation of self-organized pattern formation in crowddynamics is presented. Crowd systems that consist of active particles, showwide variety of collective motion, such as lane formation, the arch action,and oscillatory flows at bottlenecks. The importance of such phenomenais two fold: (i) these patterns are universal, since similar patterns are ob-served in colloid, plasma, granular and pedestrian systems (ii) analyzing thepatterns provides an opportunity of establishing new mathematical methodsto describe and understand discrete flow that has particle-scale instabilities.Based on these backgrounds, new mathematical description for self-organized

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1.2. WHAT IS CROWD DYNAMICS? 16

phenomena in crowd dynamics will be presented to provide possible mecha-nisms and deeper understanding of self-organization in crowd motion. Thiswork also intends to demonstrate the applicability of mathematical modelingto pattern formation in active particle flow. It can be a powerful tool andcontribute to understanding active discrete flow and the pattern formationin which the conventional fluid approximation does not hold.

1.2 What is crowd dynamics?

In this section we discuss the fundamental features of crowd dynamics. Thestudy of crowd motion have a very long history but is rather seemed as anunconventional science compare to the traditional physics and mathematics.Here we explain the characters, the rolls and the values of crowd dynamicswith the aid of a historical review. Moreover, interdisciplinary features andcritical discussion on crowd dynamics are presented.

1.2.1 Crowd dynamics focuses on human flow

Crowd dynamics is a discipline that treats collective motion of non-equilibriumparticles. The ultimate goal of the study is to clarify and predict the flowof human mathematically. Crowd dynamics is also referred to as pedestriandynamics. It is well known from our daily experience that a crowd showsflow-like behavior: they flows, clogs and diffuses in buildings, stations andstreets (Fig. 1.1). It would be possible to refer crowd motion as fluid or parti-cle flow. If this view holds, we are able to understand crowd motion with thelanguage of physics, and the analogy would promote deeper understandingof collective motion.

1.2.2 Crowd dynamics contribute to social safety

One of the important missions of crowd studies is to realize the safety so-ciety in our world. If we obtain useful knowledge of crowd motion, it will

1.2. WHAT IS CROWD DYNAMICS? 17

Figure 1.1: A crowd (Shibuya, Tokyo, 2014).

contribute to controlling the real human flow. Also, if it is possible to predictthe motion of pedestrians precisely, we are able to avoid catastrophic situ-ations that might result in tragic disasters. Therefore mathematical treat-ments of crowd motion have an important issues for safety engineering andarchitecture design.

1.2.3 Crowd dynamics provide new issues to mathe-matical science

Since crowd motion shows various spontaneous collective order, it is an in-teresting issue for statistical physicists and mathematicians. Real pedestriansystems sometimes show phenomena which we may call them phase transi-tion, the discontinuous changes of the status of systems. This implies thatcrowd systems could be seen as new objects to which the statistical-physicalmethods are applicable. If this is true, crowds provide new examples ofphysical phenomena and mathematical science may extend its applicabilityto social science.

1.3. THE IMPORTANCE OF THE STUDY 18

1.3 The importance of the study

Crowd flow offers us some physically and mathematically important issues.The primary importance of crowd dynamics is as follows: active particleflow shows various kinds of complex pattern formation that are characteris-tic of non-equilibrium dynamics. Some of them can be seen as dissipativestructures, that is, the space-time structures connected to the balance of en-ergy inflow and outflow. Crowd systems, examples of active particles in thereal world, show such structures spontaneously. Interestingly, some patternsin crowd motion show the phase transition, for instance, from fluid-like tosolid-like configuration, depending on the density, noise and the driving force.Some of them are specific to crowd motion, some others are universal sincesimilar patterns are observed in other systems. Comparison of the differ-ences and the similarities will give us new insights into the collective motionof active particle flow.

Another importance of studying crowd motion is that crowd dynamicsprovide a new issue: how should we describe and understand discrete parti-cle flow? In the above paragraph, we used the word “fluid”, and it would benatural to treat particle flow via fluid dynamics at least the zero-th order ap-proximation. However, the “flow” is just an analogy, so that the conventionalvariables (velocity, vorticity, etc.) in fluid dynamics becomes meaningless inthe study of particle flow. This difficulty comes from the concept of the“density”. Of course, when we have a large number of particles, we can usethe concept of the density according to the continuous approximation viacourse-graining methods. On the other hand, if we just have small numberof people, a fluid-like description breaks up since the discontinuity of systemsis evident. In the few-body systems, the concept of the density does not haveany definite meanings, then all of the concepts based on fluid dynamics be-comes meaningless. This holds when one consider, for example, particle flowpassing through narrow bottlenecks. In such cases, it is impossible to sep-arate the scale of particles from the scale of phenomena we are interested

1.4. THE METHODS OF THE STUDY 19

in, thus the conventional continuous approximation does not hold anymore.In general, it is difficult to formulate particle flow mathematically with theexception of the extreme situations (N = 2 or N → ∞).

Another powerful and conventional tools to handle particle flow is many-particle simulations, though they are not the ultimate resolution. So-calledmolecular dynamics (MD) and the discrete element method (DEM) are themajor methods of many-particle simulations. Recent developments of numer-ical environments have enable us to conduct easy-to-visualize investigationwith direct numerical simulations. Nevertheless, numerical many-particlecalculations do not necessarily lead deeper understanding of complex phe-nomena. It just mimics real phenomena using microscopic information anddoes not provide any holistic perspective or concepts to represent the entiresituation. Here we need to seek alternative methods to describe particle flow.The study of crowd dynamics would give an opportunity of developing newtype of mathematical description of pattern dynamics. Understanding struc-ture and stability of emerging patterns in active systems is an importantissue not only for the improvement of the human transportation, but alsothe development of non-equilibrium dynamics.

Of course, as a practical benefit of the studies, understanding collectivemotion in crowd systems could contribute to attaining more efficient trans-portation.

1.4 The methods of the study

In this study, we take three steps: simulations, modeling, and analysis. Theultimate interest is to model self-organized phenomena in real crowds. How-ever, because of its complexity, pattern formation in active particle flow is noteasy to describe rigorously. Moreover, observing self-organized patterns inparticle flow in the real world is challenging to conduct for technical reasons.Thus, it would be appropriate to observe such phenomena numerically, and


then we extract the essential features of phenomena. Note that numericalsimulations does not necessarily give wisdom and deep understanding, evenif simulations could reproduce pattern formations excellently. In this study,the word “understanding” means the identification of the causal relationship,the clarification of the correlation among parameters, and the provision ofthe reasonable physical pictures to phenomena. To attain this, in this studysome of the self-organized phenomena in crowd studies are modeled based onthe perspective of pattern formation. Through its analysis, their underlyingmechanisms are clarified.

The objects of the study is the active particles in the real and numericalworld, in particular a real crowd and numerical active particles that mimicscrowds. An active particle is a non-equilibrium entity, and its main featureis that the particle produces the kinetic energy by itself, so that no energyinjection from outside is needed for its motion. They are also called self-propelled particles or self-driven particles. In the real world, there are manytypes of entities that can be seen as active particles.

1.4.1 Crowd flow

Crowd flow is our primary interest in this study. We consider collectivemotion of pedestrians, evacuee and crowds. In our everyday life, we canobserve the collective transportation of human easily, and we even could beone of the crowd and the component of the transported stuff. In small scales,the discrete character of pedestrian flow is clearly observed, for example inbottleneck flow, narrow channel flow, and counterflow. On the other hand, wealso could have a crowd of large scale, like hundreds of thousands of crowdsgathering in a place for social events. Crowd dynamics handles such widevariety of situations from small to large scale. In this study, we restrict ourfocus on rather small scale phenomena, say the number of particles N ∼ 100,in which the discrete property of particle flow begins to appear.


1.4.2 Biological entities

Though this is not the main concern in the study, I would like to note therole of biological entities in the study of crowd motion. Biological entities areoften used to conduct experimental studies of collective motion. Of coursethey are totally different from human, but some research articles have showninteresting results that are reminiscent of crowd flow. The most famous an-imal of this line is ants, since it is easy to capture, set up experiments andcontrol parameters. We could understand the special and the universal char-acters of crowd dynamics by comparing the results to the observation in realhuman flow. Also, the numerical studies of flocking, schooling and swarminghave been developed considerably. One of the early numerical study of col-lective motion of biological stuff is the model of school behavior [Sakai1973].In this study, fish are represented as self-driven particles, and it has beenshown that they form some collective patterns, for instance torus and clus-ters, depending on parameters. When we see this study from today’s view,it is reminiscent of the so-called Vicsek model [Vicsek1995], the computa-tional model of collective motion inspired by ferromagnetism studies. Themost famous work of biological collective motion is the “boid” (bird droids),the model for computer graphics proposed by Reynolds [Reynolds1987]. Themodel imitates the realistic motion of birds and has given a strong impres-sion, and the possibility of describing collective motion via local rules, whichmeans they have no blue-print, no leader or no global communications.

1.4.3 Granular matter

It is worth mentioning that the granular media could be a model of crowdflow. Granular matter, the media which composed of a large number of dis-sipative particles, is also one of the famous model systems of non-equilibriumparticle flow. Interestingly, activated granular media by external forces showsvarious kinds of pattern formation, and they are astoundingly similar to thepatterns observed in other systems. An important example is the formation

1.5. HISTORICAL REVIEW 22

of lanes, the spontaneous segregation of particles with respect to the orien-tation of the driving force. In such systems, the scale of segregated lanes isalmost the same as the scale of particle size. The fluid-like treatment requiresthe separation of the fluid dynamic scale L of phenomena from the specificscale of particles D, that is L >> D. Clearly, the fluid dynamic approachfails in granular systems due to the condition L ∼ D. In this sense thesimilarity of crowd systems to granular systems is considerable, and granularmatter can be seen as a useful example for the investigation of the essentialsof dissipative particle systems.

1.5 Historical review

In this section a historical review of pedestrian studies is presented. Crowddynamics has a long history and variety of contributions. Actually, pedes-trian flow has been studied about one hundred years ago both in the Westand Japan. On top of the historical introduction, I would like to stress thescientific meanings of the crowd studies at each age.

In this review, I would like to introduce some research articles which areimportant contributions to the crowd science but have been forgotten andnot mentioned today. In the process of historical research, I have noticedthat there are many important contributions that have not been recognizedfor some reason. Therefore I would like to (re)introduce them and give theman appropriate context in the current study of crowd dynamics. This reviewis based on the standpoint of self-organization in crowd systems.

1.5.1 Prehistory of crowd flow

Perhaps the oldest scientific viewpoint to crowd transportation is given byTerada [Terada1922]. He was a physicist, but nowadays is rather famous forthe essays on his daily life through a scientific view. One of the objects tohis “daily science” is the commuters by train. He recorded the number of


Figure 1.2: Terada’s observation of commuter trains.

passengers on each train at a station (the Eulerian view) as a time series,then he found that it fluctuates considerably (Fig. 1.2). This means thatsome trains have high-dense crowd, other trains a few people on board. Fromtoday’s view, this phenomenon might be interpreted as the density waves, orthe spontaneous condensation of crowds. For him, however, this observationwas just a distraction and for his writing activity, and it seems that he didnot have any intention to formulate this results as a transportation scienceat that time. Thus, this work has no influence on today’s transportationscience but an scientific essay for fun.

1.5.2 Safety engineering

The first accessible research paper on crowd flow appeared in 1931. Yoshimuraet al. discussed the structure of building floors based on pedestrian evacu-ation [Yoshimura1931]. Their intention was to determine the appropriatewidth of stairs in department stores. To do this, they observed the travelingtime of pedestrians passing through stairs and established a relation betweenevacuation times and widths of stairs. The motivation of the work was toprovide knowledge to avoid disasters in department stores. In 1930’s, therewere no legitimate guidelines or studies that help them design safe high-rise buildings in Japan, so that the authors had concerned the possibility ofbuilding fire accidents. Actually, in 1932 a terrible fire disaster “Shiroki-yadepartment store fire” occurred in Tokyo. Consequently, the importance of


pedestrian studies was recognized, and most of the early studies were dedi-cated to safety engineering.

1.5.3 Pattern formation in crowd flow

Early mathematical studies of crowd flow was mainly based on fluid-dynamicanalogy. Because of the limitation of the numerical technology, models at theage assumed that there were definite relation between the boundary condi-tions (the geometry of the buildings) and evacuation times. Interestingly, anearly study of pedestrian evacuation referred to the relation between pedes-trian flow and emerging patterns. Even though their main concerns werepractical estimation of the efficiency of pedestrian flow, Togawa refereed tothe concept of the “arch action” as the main factor of the deceleration ofcrowd flow [Togawa1954].

Although the major interest in the studies of pedestrian flow was stillfor designing architecture that enhances the efficient pedestrian flow [Fruin1974], in 1970s the mathematical studies of pedestrian flow developed. Oneof the mathematical and physical approaches is given by Henderson whoimplied that there were similarities between crowds and fluid (or gas) me-dia from the viewpoint of physical property [Henderson1971, 1974]. On thecontrary to fluid-like treatment, many-particle simulations also developed inwhich each pedestrian are represented as independent particles. Nakamuraet al. proposed a particle model that can mimic the evacuation of pedes-trians inside buildings [Nakamura1974]. Perhaps this is the first work thatprovides the particle model based on the equation of motion. Besides, fromthe observational standpoint, Naka pointed out the formation of band-likepattern was observed in a real crowd at a train station [Naka1977]. At thesame year, Hirai et al. reproduced crowd patterns with a particle simulation[Hirai1977]. Perhaps these are the oldest research of the pattern formationin crowd systems. In Hirai’s paper, some numerical results were providedwhich shows “the lane formation in counterflow” and “the band formation


in an intersection” . However, their main focus was to establish the simula-tion model, so that the spontaneous order in the simulation results were notstressed. Moreover, the paper was published in the proceedings of an inter-national conference (1975) and published as a paper in Japanese (1977), sothat the study was difficult to access and did not have an striking influenceon the crowd researches in the western world.

The aspect of self-organization in crowd motion was not well recognizeduntil 1990s. Yamori et al. presented observational and numerical resultsof lane formation on a real pedestrian crossing, and they referred this phe-nomenon as self-organized patterns [Yamori1990, 1992a, 1992b]. They alsotried to quantify this spontaneous order with the concept of the band indexand discussed the difference between the ordered and disordered state. Herecrowd dynamics has became an example of spontaneous patterns to studycollective motion.

The study of crowd motion from the view of self-organization has beenstrongly proceeded by Helbing with the famous social force model [Hel-bing1995, 2000a]. This is a particle model based on the equations of mo-tion, in which any two agents interact though the psychological repulsiveinteraction, the social force. The model has been able to reproduce manytypes of spontaneous phenomena. Nowadays this model becomes one of themajor method of crowd simulations. By using this powerful method, he hasconducted thorough studies on pattern formation in crowd dynamics and hasfound various self-organized patterns, such as lane formation, arch formation,the behavior at intersections, freezing transition in counterflow, the faster-is-slower effect and oscillations at bottlenecks [Helbing2001]. It seems thatthe the frontier of pattern formation in crowd dynamics has already beeneradicated by him. However, note that the mechanisms and mathematicaldescriptions of the self-organized phenomena in crowd dynamics have notbeen given yet, though the phenomena itself are widely known.

1.6. CRITICAL DISCUSSION ON CROWD DYNAMICS 26

1.5.4 Universality of crowd patterns

Importantly, the spontaneous phenomena observed in crowd dynamics aresometimes very similar to the collective motion in other systems. Lane for-mation is such an instance, since it is observed in colloidal systems, andgranular media [Löwen2010]. Also, the formation of arches in the vicinityof bottlenecks are common in both pedestrian evacuation and the dischargeof granular hoppers (grain reservoirs). The formation of arches leads thedeceleration of outflow passing through bottlenecks, even in the worst casesthe clogging (the sudden freezing) occurs, causing catastrophic accidents. Asthese example shows, universal patterns exist in particle flow. The details ofsystems seem to be irrelevant to the resulting patterns.

On the contrary to the universal properties, active particle flow also showsstrange and unconventional phenomena. One of such examples is freezing-by-heating [Helbing2001]. The counterflow of active particles in a corridor showssudden freezing transition when particles have an appropriate level of noise.This noise-induced solidification seems very contradictory to the conventionalpicture of phase transition, since increasing internal energy results in thedisordered gaseous state. It is an interesting issue whether similar phenomenaexists in other systems, and the investigation of this line may give us a newinsight into the concept of phase transition.

1.6 Critical discussion on crowd dynamics

Finally, let us make a critical consideration of crowd dynamics. The majorproblems in crowd dynamics are as follows: (i) the dependence on models,(ii) how to model the free will, and (iii) how to validate the results frommathematical crowd models. The relation between a modeled crowd and areal crowd is similar to that between the theory of an ideal gas and a real gas.The model is useful as far as we investigate the essential physical propertiesof crowd dynamics. On the other hand, it is less effective when we require the


realistic description abilities to modeled crowds. Also, we have to carefullycheck whether phenomenon we are interested can be fully explained by theconcept of self-organization.

1.6.1 Model dependence

First, we focus on the model dependence of crowd studies. There are widevariety of models for collective motion of crowds, i.e. particle models, CA,fluid models and so on. The problem is that some models reproduce the realcrowd motion whereas others do not. It is therefore not easy to concludethat results obtained from numerical simulations holds in the real world.

For the most simple example, particle models allow pedestrians to move ina continuous manner, whereas the motion is discretized in CA models. Thisdifference would be minor when considering, for instance, lane formation,since various models show qualitatively the same results. The discontinuitydoes not have any curtail role in lane formation. However, if we are interestedin complex and subtle configuration of particles, such as the formation of archin front of bottlenecks, the discrete model would be not suitable because suchmodels always show the same shape of arches, so that we cannot discuss thearch action or the effect of friction on particle flow. In this cases, somespecial modeling would be needed to reproduce the arch formation, and wehave to take great care not to manipulate models to obtain desired resultsbut construct models based on physically acceptable backgrounds.

Another simplification that may cause critical effect on models is theshape of particles. In most simulations, each person is represented as a two-dimensional circle, though it is evident that the real human body is ratherelliptic. This simplification is valid only when we investigate situations withlow density where no contacts among particles are assumed. On the contrary,in closely-packed situations, the effect of the particle shape is not negligible.This would be critical when the friction force between particles has an crucialeffect, such as bottleneck flow, the clogging in counterflow. The simplification


is acceptable only when we are interested in the essential features. For morepractical discussion, the effect of particle shape should be in our mind. Hope-fully, the technical difficulty of handling non-spherical particles has becameminor due to the sophistication of the numerical model with non-sphericalparticles [Thompson1995, Kiyono2004, Langston2006, Chraibi2010, Alonso-Marroquin2014].

To avoid model dependence, we need to compare the results from variousmodels. By this, the prediction ability of models becomes clear, and theessential features of the phenomenon concerned is extracted. One of the effortof this line are provided by Duives in which the detailed comparison of modeloutputs are listed [Duives2013]. Also, it is extremely important to designatethe independent and dependent variables properly to explain phenomenawe are interested in. What aspects of systems we want to clarify? Whichparameters would be irrelevant to our interest? What is the primary purposeof the numerical simulation we are trying to conduct? When we successfulygive reasonable explanations to the results based on physical intuition, thenit will be possible to provide a new perspective, not just giving a case study.

1.6.2 The free will

The second problem in pedestrian studies is how we should quantify the be-havior of human, the existence which has the free will. This is the largestdifference between physical and social entities. It is clear that crowds arenot physical particles like granular media. Apparently, they do not obey theequations of motion. Is it meaningful to model the behavior of crowds viaphysics-based concepts? The answers for this questions are still yes, withthe limitation of the applicability. In some cases, collective motion of humancould be seen automatic and materialistic, as Togawa said “the materializa-tion of a crowd” [Togawa1970]. The examples for such situations are: (i)panic situations, (ii) each person in a crowd have their definite destinationbut the way to reach there is less conscious, and (iii) the size of crowds is


tremendously large. In such situations, the difference of each people dimin-ishes, and the averaged behavior could be treated statistically. Only in suchcases physics-based crowd dynamics do hold. It is indispensable to checkwhether we can assume these conditions when considering crowd modeling.One should recognize the limitation of models, estimate the applicability,and discuss the correspondence between the numerical and the real behaviorof crowds.

1.6.3 Validation of mathematical models

The third difficulty of pedestrian studies is the validation of mathematicalmodels. If we are interested in an extreme situation, for instance people inpanic try to evacuate via a narrow exit in a competitive way, it is impossibleto conduct real experiments due to the safety of crowds. Numerical modelscould be powerful tools in this case. Here the problem is how to confirm thevalidity of models. To avoid this problem, some researchers have proposedexperimental procedure using biological entities, like ants [Shiwakoti2012],sheep [Zuriguel2013], mice [Abe1973, Abe1974, Saloma2003]. By establish-ing the scaling relation between human and other animals, the experimentswith biological entities could provide useful information, without dangerousexperiments of pedestrian evacuation. Recent developments of observationaltechnique are also helpful. For example, observations of a quite large scaleof crowd and its analysis via image processing technique has been conductedand has revealed a new phenomena called “crowd turbulence” [Helbing2007].The development of observation and analysis technique should be needed andbecomes more important in crowd flow studies.

Chapter 2

Pattern formation inpedestrian dynamics

In this chapter some of the spontaneous phenomena observedin crowd dynamics are reviewed. It is well known that the crowdmotion is accompanied by self-organized order. The emergenceof phenomena sometimes enhances the crowd flow, sometimescauses inefficiency on the other hand. The study of self-organizedphenomena in crowd dynamics (i) leads to the attainment of theefficient flow, (ii) clarifies the mechanism of collective motion, and(iii) may offer new concepts of the non-equilibrium physics. Herethe numerical examples of crowd patterns, fundamental features,and detailed historical reviews are presented. The discussion pro-vided here is mainly based on the social force model.

2.1 Lane formation: particle-scale instability

Lane formation is a typical spontaneous pattern in particle flow. The notableproperty of this phenomenon is that the emergence of lanes are observed inwide variety of systems, such as pedestrian, granular, colloidal, and biologicalsystems, both in experiments and numerical simulations. Let us consider

30

2.1. LANE FORMATION: PARTICLE-SCALE INSTABILITY 31

Figure 2.1: Lane formation in self-driven particle systems. The bars in theparticles represent the direction of the driving force. Each particle has thedriving force and the repulsive interaction The gray particles move from leftto right, whereas the white ones run toward the opposite direction. The bothends of the corridor are the periodic boundary.

counterflow in a corridor with two species of particles (Fig. 2.1). At anearly stage, the system shows turbulent-like flow, and no definite structureis observed. However, they gradually form stable lanes spontaneously andattain the stationary flow. The elapsed times needed to form lanes and thestability of the lanes depend on the number of particles, the intensity of thedriving force, and other physical properties of particles.

Realistic examples showing lane formation are in Fig. 2.2. In the realworld, lanes are frequently observed in pedestrian counterflow in streets,stations, and buildings. The example shows highly dense counterflow ofcrowd in a narrow corridor, here two wide bands are formed. This segregation


Figure 2.2: Lane formation in high dense pedestrian counterflow (Nakano,Tokyo, 2014). The gray and white circles correspond to the outgoing andincoming pedestrians.

Figure 2.3: Lane formation in low density pedestrians (Nakano, Tokyo, 2014).Unstable multi-lanes are formed spontaneously.


is spontaneous, since no instruction or indicator exists to tell them “keeplanes”. The laned state shown here is stable and has a long lifetime. Onthe other hand, in the case of the low density, the situation is different. Theexamples in Fig. 2.3 are such situations, as small amount of people are inthe large space. In this case we can observe the relatively unstable lanesthat are created and destroyed repeatedly. Here the state of two lanes aredominant, whereas sometimes the short-lived multi-lanes are formed. Againthe formation of multi-lanes are automatic.

Here the explanation of the numerical treatment of the crowd lane for-mation is given. Let us consider counterflow of N self-driven particles. Halfof them have the driving force towards the positive direction, and the rest ofthem have the opposite driving force. They are confined in a narrow channelwhere both ends (±x) are connected by the periodic boundary condition. Att = 0, they are distributed randomly in the region. Here the particles obeythe following equations of motion called the social force model [Helbing1995,Helbing2000a]

mdvi

dt= m

τ

[(−1)iv0ex − vi

]+

∑i̸=j

Ae−dij −rij

B nij, (2.1)

here the first term is the self-driven force, and the second term representsthe repulsive two-body force. m is the mass, τ is the time constant, v0 is thedesired velocity (the intensity of the driving force), ex = (1, 0)t is the unitvector which directs to +x direction, vi is the velocity of the particle i, A andB is the coupling constants of the repulsive force, dij is the distance betweeni and j, rij = ri + rj is the sum of radii of i and j, and nij is the normalvector from j to i. The parameters used here are the following: m=80 kg,τ=0.5 s, v0=1 m/s, A=2000 N, B=0.08 m, ri=0.3 m. In this system, with ansufficiently high density, the particles begin to segregate according to theirmoving direction and finally form lanes. The number of lanes depend on aninitial condition. The stability of the lanes also depend on the density andthe intensity of the driving force v0. For low density, the correlation of the


motion among the particles are small, and they do not form definite lanes.On the other hand, for high density we can observe lanes or bands whichwidths are the order of the particle size. When the system establishes thelaned state, each lane is stable and never destroyed. If the system is at highdensity but the driving force is large, the lanes become unstable.

It would be no doubt that the formation of lanes in pedestrian flow areknown long ago, possibly from ancient civilizations since they establishedgreat cities and commercial distribution networks. However, the scientificrecords have been conducted relatively recent years. From the standpoint ofmathematical sciences, lane formation in crowd systems have been studiedfor the past four decades. In the paper by Nakamura et al., lane formation innumerical pedestrian counterflow are implied, even though the authors do notstress the emergence of pattern [Nakamura1974]. They proposed a particlemodel based on the equations of motion, and showed the model is able toreproduce realistic maneuvers of pedestrians. (It is notable that the model atthis time includes the two-body repulsive interaction based on psychologicalpressure, which will be proposed independently by Helbing in about 20 yearsafter.) Nowadays, by focusing the crowd motion in real intersections atcities, lane formation is observed easily. Yamori et al. recorded the real laneformation in crowds and analyzed the lanes as an example of spontaneousorder by quantifying lanes using the concept of band index, a kind of orderparameter [Yamori1990]. The results were reconfirmed by numerical model[Yamori1992a, b]. Independently from these works, Helbing et al. showsthe formation of lanes as an example of spontaneous order in numerical self-driven particle systems [Helbing1995] then it becomes widely known.

It is important to point out that various type of simulation model is ableto reproduce lane formation. For example, the CA model [Burstedde2001,Weng2006, Nowak2012], the lattice gas model [Tajima2002], the OV (opti-mal velocity) model [Nakayama2005] , DEM (the discrete element method)[Kiyono1996] also show the similar laning patterns. Also, recent development


of numerical environment enhances a new type of sophisticated descriptionof human crowd based on macroscopic continuum theory which reproducesdynamic lane formation [Hoogendoorn2014].

Moreover, from the experimental aspects, the formation of lanes has beenstudied vigorously. The studies have been conducted for the comparison ofexperiments and simulations [Isobe2004], an analysis of fundamental diagramin counterflow [Zhang2012], the behavior of crawling human on the floor[Nagai2005], the pedestrian segregation in circular corridor [Moussaïd2012].

The formation of lanes are quite general feature in pedestrian flow, sinceit just requires only two physical properties: the repulsive interaction amongparticles and the oppositely directed driving forces. The spontaneous forma-tion of lanes are insensitive to the details of particle characters, such as theshape, material properties or the function form of the interaction.

More importantly, the mechanism of lane formation is universal. Thewidth of lanes are of the order of the particle size, thus the scale of symmetrybreaking is almost the same as the scale of emergent pattern. This type ofinstability is called particle-resolved instability [Löwen2010], which is familiarin granular and colloidal systems. It implies that the fluid approximation isimpossible, since the fluid approximation only holds when we can assume thefluid-dynamic scale is large enough to the scale of the system elements. Theformation of lanes are perhaps the most common but still challenging issuesto describe mathematically.

Finally, note that the formation of lanes in crowd systems originate fromtwo factors, the physical and the social factor. It is more evident when wecarefully observe the structures of lanes. Concretely, the lanes in Fig. 2.2may include both physical and social effects. When we conduct a long timeobservation, we can see that the lanes are stable and the pedestrians alwayskeep left. Physically speaking, the “keep right” state is also possible and theselection is determined by how the system break the symmetry. Therefore itis plausible that the pattern presented here would include the effect of social

2.2. FREEZING-BY-HEATING: NOISE-INDUCED SOLIDIFICATION36

common sense. On the other hand, the situation in the Fig. 2.3 is ratherunstable and they do not always keep the common rule. The lesson fromthis is we should be careful that all of the spontaneous collective phenomenaare not the result of self-organization. We should carefully consider theapplicability of the mathematical and physical concepts to the phenomenain the real world.

2.2 Freezing-by-heating: noise-induced solid-ification

The freezing-by-heating is solidification of particle flow by noise. Let usconsider the counterflow of self-driven particles in a narrow channel or cor-ridor. As expected, they show stable lanes after some time. However, if astochastic force is introduced with an appropriate level, the system showstransition from the fluid-like state to the frozen state. The existence of noiseincreases the freezing probability. This seemingly inconsistent transition iscalled freezing-by-heating, since increasing internal energy of the system re-sults in solidification.

Here we consider N self-driven particles. They are confined in a narrowcorridor which ends are connected by the periodic boundary condition. Theparticles are governed by the Langevin-type social force model

mdvi

dt= F self

i +N∑

i̸=j

Fij + Fiw + ζi(t), (2.2)

F selfi = m

τ[(−1)iv0ex − vi], (2.3)

Fij = Ae−dij −rij

B nij, (2.4)

Fiw = Ae− diw−riB niw, (2.5)

< ζi(t) > = 0, (2.6)

< ζi,x(t1)ζj,y(t2) > = Cδ(t1 − t2)δijδxy, (2.7)


Figure 2.4: Numerical simulation of freezing-by-heating. (Top) with largenoise, the system shows turbulent flow. (Middle) without noise, the lanesare formed. (Bottom) with an appropriate noise intensity, the system showssudden transition to the frozen state.

which means that N/2 particles have the self-driven forces to the positivedirection, and others have the opposite ones. They also have two-particlerepulsive interactions Fij, the wall-particle interaction Fiw, and the randomforces ζi(t) with mean 0. C is the noise intensity (it can be interpreted asthe temperature). The meaning of other symbols is presented in the sectionof lane formation. Fig. 2.4 shows the numerical results. Here the followingparameters are used: the width and the length of the corridor W = 2 m,L = 15 m, N=20, m=80 kg, τ = 0.5 s, v0=1 m/s, ri = 0.3 m, A=2000N, B=0.08 m. The random force is applied to the equations of motion foreach time step as the stochastic term σN(0, 1), where N(0, 1) is the Gaussianrandom number with the mean 0 and the variance 1. When the noise intensityis sufficiently small, the system shows both stable lanes and the frozen states.The selection is probabilistic. On the contrary, introducing lager fluctuation


Figure 2.5: Probability of transition. The graph shows which states thesystem tend to attain for each noise intensity σ. (circle) the freezing state isattained. (square) lanes are formed. (triangle) no definite spacial structureis observed at 1000 s. Here 20 calculations are conducted for each noiseintensity.

leads turbulent-like flow, and it never shows any spatial structure. However,if the particles have an appropriate level of noise, we can observe the suddentransition from turbulent motion to frozen state frequently. The freezingprobability is maximized at a certain value of noise intensity (Fig. 2.5). Thisis the unexpected results in the sense that increasing the thermal energyleads the state with lower kinetic energy.

The freezing-by-heating is found by Helbing et al. in the study of pedes-trian simulation based on the social force model [Helbing2000b]. They dis-cuss the strange transition from fluid (lane) to gaseous state (freely moving)via solid state (freezing). The frozen state is induced by increasing noise(thermal energy). By investigating the transportation efficiency and the me-chanical energy, they pointed out that the transition is continuous, and the


frozen state has higher potential energy than the laned state [Helbing2000b].This means that the system does not like the lowest energy state. In thissense, the freezing-by-heating offers an interesting novel type of transition.Recently, similar noise induced order has been studied in other physical sys-tems [Sagués2007]. This phenomenon might also be crucial when consideringpanicking crowds in emergency situations, since the freezing could cause dis-astrous accidents if it occurs in the real pedestrian flow.

Here we discuss the mechanism of the phenomenon by observing the timeseries of the total kinetic energy of the particles (Fig. 2.6). The first im-portant fact is that the system is bi-stable in the case of no noise. Fromits boundary structure and the shape of the particles, the system consideredhere has two stationary states: the laned state and the frozen state. Whenmost of the particles collide simultaneously by chance, the sudden transitionfrom fluid to frozen state is attained, otherwise the particles form lanes. Theselection of the stationary state is totally probabilistic. In the case of no fluc-tuation, both states can be observed, though the probability of establishingthe laned state is larger than that of the frozen state, since there is very littlechance of massive collision.

The second crucial factor for the freezing-by-heating is the robustnessof the stationary states. Importantly, the laned state is more vulnerable tonoise than the frozen state. The oppositely running lanes have an instabilityfor the perturbation which is perpendicular (±y) to the moving direction(±x). Let us consider the particles moving in a row for x-direction withlarge momentum. By the velocity fluctuation of x-component, the distanceof the particles in the lane is not uniform. On top of that, the y-directionalfluctuation gives the particles the momentum for y-direction, then the prob-ability of the collision of two oppositely running lanes increases drastically.The fluctuation destroys the lanes, and the system returns to the turbulentstate. This picture is confirmed when we observe the time evolution of thesystem. The peaks of the Fig. 2.6 means the highly ordered flowing state i.e.


Figure 2.6: Time series of the total kinetic energy of the system. The solidline represent the actual kinetic energy, and the dashed line means the pos-sible maximum energy which correspond to the completely ordered lanes.Here the noise is introduced as |ζi| = 0.01N(0, 1) for each time step.

the laned state, but the lifetime is short. On the contrary to the laned state,the frozen state is a bit more stable. This is due to the hexagonal structurewith high packing ratio (the freezing transition is similar to crystallization).Therefore the system finally attains the stable frozen state.

In conclusion, (i) when the fluctuation is absent, the lane formation ap-pears with higher probabilities, whereas (ii) if the particles have an appro-priately adjusted intensity of noise, then the lane formation is prohibited,therefore the other state, the frozen state, is selected. (iii) Of course, if thefluctuation is large enough, all possible structures will be destroyed then thesystem always show the unstable turbulent state. This is the fundamentalpicture of the freezing-by-heating, the noise-induced solidification.

2.3. OSCILLATORY FLOW AT BOTTLENECKS: A RHYTHM INPARTICLE FLOW 41

Figure 2.7: Oscillatory pedestrian flow at the bottleneck. The direction offlow changes with time.

2.3 Oscillatory flow at bottlenecks: a rhythmin particle flow

Oscillatory flow at bottlenecks is a self-excited oscillation in pedestrian bot-tleneck flow. Placing a narrow door in pedestrian counterflow, the directionof the flow changes periodically.

Let me consider a narrow corridor with the length Lc and the width Wc.At the center of the corridor, the bottleneck with the width W exists. Theboth ends of the corridor are connected by the periodic boundary condition.Let us consider N self-driven particles confined in the corridor. Half of themhave the driving force towards left to right, and the rest of them have thedriving force of the same intensity but direct to the opposite direction. They


are governed by the equations of motion

mdvi

dt= F self

i +N∑

i ̸=j

Fij + Fiw, (2.8)

F selfi = m

τ((−1)iv0ex − vi), (2.9)


B nij, (2.10)

Fiw = Ae− diw−riB niw, (2.11)

where the meaning of each symbol is presented in the former section. Herethe particles do not interact through the wall. The driving force directs tothe center of the bottleneck before passing, whereas directs to infinity afterpassing. The particles are distributed randomly without overlaps at t = 0.Here the parameters are as follows: W=0.7 m, L=45 m, N=100, m=80 kg,τ=0.5 s, v0=1 m/s, A=2000 N, B=0.08 m, ri=0.3 m. The resulting pattern isshown in Fig. 2.7. The direction of the flow passing through the bottleneckalters as time goes. The oscillation is more evident when we observe themomentum of the particles in the vicinity of the bottleneck. Fig. 2.8 clearlyshows the regular oscillation. Since there are no external forces in the system,this oscillatory flow is self-induced one and does not depend on the details ofinitial conditions. Also, the periodic boundary does not take an essential rolefor the ignition of the oscillation, as it just contributes to the conservationof the number of particles. This is easily justified by the observation of thesystem without the periodic boundary.

Although when it was discovered is unknown, the oscillatory flow at bot-tlenecks is very plausible to occur in real crowd flow, such as the entranceof buildings and railway stations. In the context of self-organized patternformation, it was presented as one example of spontaneous collective mo-tion in self-driven particles [Helbing1995]. After that, the mechanism of thephenomenon was given qualitatively [Helbing2001]. Many types of particlemodels reproduce the oscillatory flow, and the qualitative character of the os-cillation is insensitive to the choice of the interaction potential [Corradi2012].


Figure 2.8: Momentum density in the vicinity of the bottleneck. The systemshows the regular oscillation. Here the width of the bottleneck is set to 0.7m.

It is reported that a CA model also shows the similar phenomenon, and itsbehavior has been investigated with statistical methods [Kretz2006].

From the mathematical view, this phenomenon is related to the Hopfbifurcation. By analyzing numerical data via “the equation free approach”,it is shown that the emergence of the oscillation is understood as the Hopfbifurcation, where the bifurcation parameter is the width of the bottleneck[Hjorth2011, Corradi2012]. The narrow bottleneck interferes with the flowthen the clogging (frozen state) occurs, whereas the wide bottleneck desta-bilize the clogging and allow oscillation. Although the governing equationbehind the phenomenon is not given, this destabilization of the trivial equi-librium state has also been confirmed with other technique of data analysis[Marschler2014a, Marschler2014b]. The oscillatory flow could be a nice in-terface between crowd dynamics and dynamical system, and offers us pos-sibilities to apply the scheme of dynamical system to pattern formation indiscrete flow.


The oscillatory flow was also confirmed in the experiments of human flowpassing through a bottleneck, though that presents rather irregular oscillation[Helbing2005]. The oscillatory flow can be observed, for instance, in the theprocess of boarding train. However, in this case the social custom is ratherdominant, then the specific modeling is needed to investigate the oscillationin reality [Dai2013, Guo2014].

Importantly, this Oscillatory flow is in a sense a universal example ofnonlinear rhythm phenomena. It is pointed that the system considered hereshows the transition from oscillatory state to lane formation[Corradi2012].This transition is very similar to that in saltwater oscillators [Yoshikawa1991],and plastic bottle oscillators [Kohira2007, Kohira2012]. It is interesting thatthe discrete particle systems also show the same type of the phenomenon ob-served in fluid systems, and the investigation of the similarity may offer a newinsight into the study of nonlinear oscillators. As a rhythm phenomena, itwould be natural to ask whether the pedestrian flow oscillators have synchro-nization phenomena. The partial answer for the question is that there existsa correlation between two pedestrian flow oscillators, although its coherenceis different from other nonlinear oscillators [Helbing1998, Helbing2001]. It isshown that when two parallel bottlenecks with a certain distance exists atthe center of a corridor, and if its distance is sufficiently near, then each bot-tleneck is occupied by just one species. This means no oscillation is observed.Of course, increasing in the distance of two bottlenecks, the correlation re-duces so that the oscillators become independent. How pedestrian oscillatorsinteract with each other is an interesting issue, but the detailed propertiesare still not revealed. Finally, without saying that, the knowledge from thisinvestigation will contribute to designing architecture which realizes the effi-cient and safe pedestrian flow.

The qualitative mechanism of the phenomenon was given in [Helbing2001].When the first person passes the bottleneck, the same species of particles areeasy to follow the forgoing one due to the less resistance from the counter

2.4. THE FASTER-IS-SLOWER EFFECT: THE DRIVING FORCE V.S.THE DRAGGING FORCE 45

crowd. Therefore there exists a tendency of successive avalanche-like flowwhich consist of the same species. In the process of this “burst”, the numberof particles of the one side decreases, then the pressure from the oppositeside becomes strong, then the flow halts. At this time the difference of thenumber of particles is large, therefore the particle flow begin to reverse. Theflow is reinforced by the same mechanism of the “burst”, then the oscillatoryflow is sustained.

The mechanism of the self-excited flow can be understood physically withthe aid of plasma and granular physics. When observing the collision of op-positely charged flows, it is know that density clusters grows rapidly due tothe two-stream instability (c.f. [Birdsall2004]). It is pointed out that thesame kind of instability is observed in granular systems [Löwen2010]. Thismight be seen as a minimal version of the granular Rayleigh-Taylor instabil-ity, the particle-scale density instability [Vinningland2007]. This instabilityis also the essential factor of forming lanes therefore the stabilities of thiskind is also called laning instability [Löwen2010]. In our oscillatory system,the motion of the particles is restricted due to the bottleneck-type bound-ary, then the two-stream instability grows just at the bottleneck. Thus therepetition of the unstable alternative growth of the density is observed, andshown as the oscillatory flow.

2.4 The faster-is-slower effect: the drivingforce v.s. the dragging force

The faster-is-slower effect means a counter-intuitive relation between evac-uation times and the driving force in a self-driven particle system. Whenwe observe particle flow through a bottleneck and plot evacuation times asa function of the driving force, the results do not show the monotone de-creasing manner. Contrary to out intuition, egress times are minimized at acertain intermediate strength of the driving force.


Figure 2.9: The model of pedestrian evacuation of a room. The particleshave the self-driven force towards the exit.

Let us consider particle flow passing through a bottleneck (Fig. 2.9).Here the motion of the particles is described by the social force model [Hel-bing2000],

mdvi

dt= F self

i +N∑

i ̸=j

Fij + Fiw, (2.12)

F selfi = m

τ(v0ei − vi), (2.13)


B nij + kg(rij − dij)nij + κg(rij − dij)(tij · (vj − vi))tij,(2.14)

Fiw = Ae− diw−riB niw + kg(ri − diw)nij + κg(ri − diw)(tij · (0 − vi))tij.

(2.15)

The model consists of the self-driven force F selfi , two-body interaction Fij

and the particle-wall interaction Fiw. The interaction includes the socialforce (the repulsive two-body force), the elastic and the friction force. Herem is the mass, τ is the time constant, vi is the velocity of the particle i, v0

is the intensity of the driving force, ei is the unit vector which directs to theexit from the position of the particle i, N is the total number of particles,A is the coupling constant of the social force, B is the range of the socialforce, dij = |xi −xj| is the distance between the particle i and j, rij = ri +rj


is the sum of the radii of the particles i and j, nij = (nij,x, nij,y)t = xi−xj

dij

is the normal vector from j to i, k is the elastic constant, κ is the frictionconstant, tij = (−nij,y, nij,x)t is the tangential vector, g(x) is the contactfunction which satisfy g(x ≥ 0) = x and g(x < 0) = 0, and w means theposition of the nearest wall from the particle i. The particles are inside asquare room with a single narrow exit and distribute randomly as an initialcondition. The particles have the self-driven force which directs towards theexit, then they rush to the exit when starting evacuation. After passing thebottleneck, they run away to infinitely. Under this condition, we measureevacuation times as a function of the driving force v0. The resulting plotsare shown in Fig. 2.10. Here the parameters are: the width of the exitW = 1 m, N = 150, m = 80 kg, τ = 0.5 s, A = 2000 N, B = 0.08 m,ri = 0.3 m, k = 120000 kg/s2, κ = 240000 kg/(m s). As is expected, theegress times decreases as the driving force becomes larger for the region of lowdriving forces. On the contrary, sufficiently strong driving force increases theevacuation times. This means that the faster motion of the particles resultsin the slower evacuation, and called the faster-is-slower effect.

This seemingly strange phenomenon is first reported by Helbing et al.[Helbing2000]. They showed the nontrivial relation between the driving forceand the evacuation times in the social force model (SFM) simulations whenthe effect of friction is large enough. The motivation of the study was toinvestigate the behavior of crowds under panic. If the faster-is-slower effectexists in the real world, it may cause crowd disasters when accidents hap-pened and the large number of people began to evacuate via a narrow exit.Therefore we should investigate the phenomena and find answers to the fol-lowing questions: (i) what is the primary factor of the faster-is-slower effect?(ii) how could we avoid the phenomena? (iii) in the first place, can the faster-is-slower effect be observed in the real crowds? To clarify the properties ofthe phenomena, the wide variety of studies have been conducted, includingnumerical, theoretical, experimental and observational ways.


Figure 2.10: Evacuation times as a function of the driving force. The solidline corresponds to the case in which the particles have the friction force,whereas the dashed line for no friction.

After the discovery of the faster-is-slower effect, Parisi et al. investigatedthe properties of the faster-is-slower effect in detail by using the modifiedSFM, and showed that the blocking structures formed by particles near bot-tlenecks are related to the efficiency of outflows [Parisi2005]. They also clar-ified that the competition between the driving force and the granular force(the effect of exclusive volume and friction) is the critical cause of the effect[Parisi2006, Parisi2007a, Parisi2007b]. Also, Frank et al. directly demon-strated that the absence of the friction force results in the faster-is-fasterflows. The importance of the roll of friction is also confirmed analyticallythrough the investigation via a phenomenological model [Suzuno2013].

The crowd bottleneck flow has also been investigated with various typeof numerical methods. Perez et al. showed that an agitated crowd showsslower stream compare to a non-agitated crowd with a cellular automata(CA) model [Perez2002]. The effect of friction is also investigated with CAby Kirchner et al. [Kirchner2002]. They reported that the plot of evac-


uation times versus the moving speed have its minimum when the frictioncoefficient is large. Song et al. presented a CA model called CAFE re-produces the faster-is-slower results [Song2005, Song2007]. Yamamoto et al.proposed the real-coded CA (RCA) and applied to the bottleneck flow, show-ing that evacuation times increases considerably if the number of particlesin the system exceeds a threshold value. From the viewpoint of game the-ory, Heliovaara et al. showed the existence of inpatient agents reduce theresulting outflow [Heliovaara2013].

On the contrary to the numerical investigations, conducting experimentalstudies of pedestrian flow passing through a bottleneck is rather difficult forsome safety reasons. In spite of that, recent experimental developments ofthe studies on the faster-is-slower effect are considerable.

One of the major methods is experiments with self-driven biological en-tities. Perhaps ants are the most widely-used model organism. They hasbeen used to investigate the evacuation of a narrow exit in many boundaryconditions. However, the interpretation of the experimental results from antsis rather controversial. Shiwakoti et al. compared the bottleneck flow of antsand a CA simulation, and stated the model for the the ant system are appli-cable to human systems via a scaling relation [Shiwakoti2011]. Soria et al.performed experiments in which ants agitated by chemical repellent evacuatefrom a bottleneck [Soria2012]. By changing the concentration of the chem-ical, they showed the faster-is-slower flow in the real system, although theypointed out that the mechanism of the deceleration of outflow in the experi-ments is different from the one in pedestrians. Besides, another experimentwith ants conducted by Boari et al. indicates that the egress of ants ignitedby thermal heat results in the faster-is-faster behavior, and clogging at abottleneck which is well-known in pedestrian systems is absent [Boari2013].Also, Parisi et al. confirmed that the ants do not show any clogging orthe dense region in the vicinity of a bottleneck even in case of emergency,then concluded that it is not suitable we treat ants as the model organ-


ism for pedestrian studies [Parisi2015]. Anyway, these efforts have stronglypropelled the study of crowd evacuation. In addition to that, other animals,such as mice [Abe1984, Saloma2003] and sheeps [Zuriguel2013], has also beenemployed to investigate bottleneck flows.

Another experimental possibilities are the utilization of granular media.Garo et al. performed the experiment in which study they showed thegranular faster-is-slower effect using 2D vibrated granular layer [Gago2013].

Very recently, Garcimartin et al. conducted an experiment of bottleneckevacuation with a real pedestrians [Garcimartin2014]. They compared thecompetitive flow (the participants are moderately pushy) to the cooperativeflow of pedestrians. The result was that the cooperative flow is faster than thecompetitive one, and they concluded this is the first experimental evidenceof the faster-is-slower effect, the competition-induced deceleration.

The origin of the faster-is-slower effect is the effect of friction. Fig. 2.10includes the direct evidence of this statement. The simulation results for thecase of no friction (κ = 0) shows the absence of the faster-is-slower character.In the case that the driving force is weak enough, the density of the particlesaround the bottleneck is low due to the repulsive force, hence the smoothevacuation is possible. In this state the system is faster-is-faster, the higherenergy leads the higher outflow. However, if the driving force is enough tobreak the repulsive potential barrier, the particles are packed closely, thus theeffect of contact friction affects the particle motion and decelerates the flowvelocity. In this sense the repulsive interaction takes a role of the lubricationeffect. Without the lubrication effect, the strong driving force does not leadthe increase of flow rate at the bottleneck. From these, we can say that thethreshold of the driving force above which the faster-is-slower effect emergesis determined by the balance between the driving force and the friction force.In the real world, the effect of fiction is derived from not only the physicalcontact friction, but also the pushy behavior of agitated people. Recent ex-periment indicates that the competition mode results in the slow evacuation


compare to the cooperative mode in human system [Garcimartin2014].

Chapter 3

A mathematical model ofoscillatory pedestrian flow

In this study the investigation of the oscillatory pedestrianflow via mathematical modeling is performed.1. This “pedestrianflow oscillator” has been observed both numerically and exper-imentally and known as an example of self-organized order incrowd systems. In this chapter, a model that describes the os-cillatory flow is proposed. Through the analysis of the model,it is shown that the oscillatory flow is understood as a van derPol-type self-excited oscillator.

3.1 Introduction

Crowd dynamics is now an important example of dissipative particle flow anddraws much attention from the view of collective motion and transportation.Though the main stream of the study was to contribute to safety engineering,

1This chapter is the improved author-created version of the following publication: K.Suzuno, A mathematical model of oscillatory pedestrian flow, Proceedings of the 20thSymposium on Simulation of Traffic flow, pp. 79-82 (2014) (in Japanese), The Mathemat-ical Society of Traffic Flow. The reproduction is in accordance with the copyright policyof The Mathematical Society of Traffic Flow.

52

3.1. INTRODUCTION 53

recent development of non-equilibrium physics casts a new light on crowddynamics.

One of the importance of studying crowd systems is that they show var-ious kinds of spontaneous patterns. It is known that pedestrian flow formslanes, arches, and clusters automatically from the study of numerical simu-lations [Helbing1995, Helbing2000] and observations [Helbing2005]. Interest-ingly, similar patterns are also observed in other systems, such as colloidalsystems and granular media. It implies that these patterns have universalcharacters, although a real crowd is different from physical particles. Bycomparing these systems, we could be able to clarify the mechanism of col-lective motion and then attain new understanding of pattern formation indiscrete systems.

Another importance of the study is that crowd motion provide a newissue to be resolved: how should we describe and understand the characterof discrete flow? The conventional method of pedestrian studies have de-veloped based on the analogy from fluid dynamics [Henderson1971, 1974].However, when we handle situations in which discrete character is evident(c.f. particle flow through bottlenecks), the conventional method fails be-cause of the difficulty of the continuous approximation. In such cases, it isimpossible to decompose the typical scale of patterns L from the scale ofparticles D, i.e. L ∼ D, so that the assumption of the continuity breaksup. Another possible description method for particle flow is numerical sim-ulations. Nowadays, we have high-performance computers which enable usto conduct direct many-particle simulations. However, this is also not suffi-cient for the holistic understanding of complex phenomena. Crowd studiesrequires the establishment of the understanding mesoscale discrete flow withalternative mathematical methods.

Here we use an alternative method, the phenomenological mathematicalmodeling, to describe and reveal the mechanism of a phenomenon in dis-crete systems. As an example, this study focuses on oscillatory pedestrian


Figure 3.1: The pedestrian flow oscillator. The particles with opposite driv-ing force run through the bottleneck. The figures show the time evolution ofthe system. The direction of the flow changes with time.

flow at bottlenecks [Helbing1995]. The outline of the phenomenon is as fol-lows (Fig. 3.1). First, set up a channel (a corridor) which has a bottleneckat the center of the system. Then, from the both side of the bottleneck,inject particles which have the driving force towards the center of the bottle-neck. Particles show jamming at the bottleneck, and after some time particleflow begin to oscillate almost periodically. The change of the flow direction isspontaneous. This “pedestrian flow oscillator” has became a famous exampleof a spontaneous pattern in pedestrian flow since Helbing et al. have shown itwith the numerical simulation of the social force model [Helbing1995]. Notethat the presented explanation so far has been just a qualitative one and the


mechanism of the phenomena has not been fully explained [Helbing1998].We should also add that the oscillatory flow has been observed in real ex-periments [Helbing2005]. It has been shown that there exists non-trivialcorrelation between the left and the right flow in the real crowd experiments.From the viewpoint of data analysis, Kretz et al. have tried to characterizethe flow oscillation via a statistical method [Kretz2006]. They refer the oscil-lation to an reconciliation of two extreme cases, i.e. the stationary one-wayflow and the zipper alternation (the flow direction changes one-by-one). Fromthe mathematical aspects, recent studies have clarified that the emergenceof the oscillation is understood as the Hopf bifurcation. This new mathe-matical insight has been provided by Corradi et al. by using the method ofequation-free analysis [Corradi2012]. Simply put, their study is based on theobservation of many-particle numerical calculations. The implication of theresult is considerable since it means that the mechanism of pattern forma-tion in crowd motion could be explained by the scheme of dynamical system.However, their main focus is to establish the method of equation-free ap-proaches so that they have not given the detailed discussion on the physicalmechanism of the oscillation.

Based on these backgrounds, this study present a mathematical modelof the pedestrian flow oscillation. Through the analysis of the model, themechanism of the oscillation is clarified. To model the phenomenon, wefocus on the motion of particles just around the bottleneck, extract twoessential features from that, and build a simple analytical model. Next, itis shown that the model has oscillatory solutions, and the solutions indicatebifurcation depending on the bottleneck width. Finally, we interpret theresults from physical view and identify the origin of the phenomenon.

One of the importance of the study is the pedestrian oscillation can beseen as a discrete version of a spontaneous rhythm phenomena. It is knownthat some fluid systems show the periodic flow, such as saltwater oscillators[Yoshikawa1991] and plastic bottle oscillators [Kohira2007]. The pedestrian

3.2. THE OUTLINE OF THE OSCILLATORY FLOW 56

flow oscillator may be a counterpart of them in discrete media. Anotherimportance is that the study presented here demonstrates that the patternformation in discrete flow can be understood via phenomenological mathe-matical modelings. Though it is rather unconventional approaches in thisfield, mathematical modeling could be an alternative method to understandthe discrete flow where the continuous approximation does not hold.

3.2 The outline of the oscillatory flow

First, we observe fundamental characters of the pedestrian flow oscillation bynumerical simulations. Here a simplified microscopic particle model is usedto describe the dependence of the flow rate on the bottleneck width. Themodel used here is based on the social force model (SFM) [Helbing2000a],but the effect of exclusion volume is omitted. Consider the system with N

self-driven particles. The particle i has the self-driven force F selfi , the two-

body interaction Fij and the wall-particle interaction Fiw. The motion ofparticles are governed by the equations of motion (Fig. 3.2)

mdvi

dt= F self

i +N∑

i̸=j

Fij + Fiw (3.1)

F selfi = m

τ(v0ei − v) (3.2)

Fij = Ae−dijB nij (3.3)

where m is the mass, τ is the relaxation time, v0 is the desired speed (theintensity of the driving force), ei is the unit vector which represent the direc-tion of the driving force, A and B are the parameters which characterize theinteraction, vi is the velocity, dij is the distance between i and j, nij is thenormal vector from j to i, and w means the nearest wall from the particle i.The function form of Fiw is the same as Fij. In this system, the interactionbetween two particles separated by the wall is absent. The parameters usedhere are as follows: m = 80 kg, τ = 0.5 s, A = 573 N, B = 0.08 m, N = 100,


Figure 3.2: Schematic explanation of the interactions.

Lc = 45 m, Lw = 5 m.The boundary condition is shown in Fig. 3.3. The calculation area has the

shape of the narrow corridor with the length Lc and the width Lw, and theboth ends of the corridor is connected by the periodic boundary condition.At the center of the corridor, a bottleneck with the width w is installed. Thedriving force of the particles directs to the center of the bottleneck beforepassing the bottleneck, while they run to the boundary after passing thebottleneck.

The initial condition is the random distribution of the particles with zerovelocity and no overlaps in the corridor. Note that the fluctuation of theinitial condition does not have a critical effect for the resulting oscillatorystate.

To describe the state of the system quantitatively, we observe the mo-mentum density in the vicinity of the bottleneck. Here the definition of “thevicinity” is the area which has the width w and the length 4 m. The centerof the measurement area is coincide with the center of the bottleneck. Thetotal momentum of the particles in the area is monitored for each time stepand divide them by the area 4w. This value is the indicator of the oscillation


Figure 3.3: Boundary condition of the numerical simulation. The corridorhas the length Lc and the width Lw, the bottleneck w, and the periodicboundary condition. Each particle has the driving force towards the centerof the bottleneck. After passing through the bottleneck, they go to theboundaries and return to the system from the opposite side.

in this study.The numerical results are presented in Fig. 3.4. The graph shows the

time series of the momentum density of the particles near the bottleneck. Inthe graph, the state at t = 0 corresponds to the initial condition. Here thebottleneck width w=0.7 m and the driving force v0=1 m/s are used. The fig-ure clearly shows the regular oscillation of the particle flow. The fluctuationof the time series originates from the discrete addition and the subtractionof the particles in the measurement area. Importantly, the oscillation stilloccurs when we remove the periodic boundary condition, although the totalnumber of particles decreases with time. Therefore it is concluded that theperiodic boundary is not an essential factor to the emergence of the oscilla-tion.

Next, we investigate the dependence of the oscillation on the bottleneckwidth. We observed the oscillatory flow as a function of the width w. The


Figure 3.4: The oscillatory changes of the momentum of the system at thebottleneck.

range of the bottleneck width considered here is 0.58 m ≤ w ≤ 0.72 m. Thenwe calculate the amplitudes of the oscillation by the Fourier transform. Theresults are shown in Fig. 3.5. In the region of the small w, the oscillation isabsent. On the other hand, when w becomes larger we observe the transitionto the oscillatory states. Between the both limits there exist bistable statesin which both the trivial equilibrium state and the limit cycle coexist.

Now we consider the background mechanism of the results. The numer-ical simulation shows the existence of the oscillatory flow depending on thebottleneck width. Since there are no periodic external forces in the sys-tem, the oscillation is spontaneous. In general, the self-excitation requiresthe positive and the negative feedbacks that enhance and suppress instabili-ties. The mission here is to identify the corresponding entities in the system.When we observe the system with this in mind, we recognize that the particleflow through the bottleneck is similar to the avalanche-like burst flow. Theparticles of one-side tend to go through the bottleneck successively, and they


Figure 3.5: The Fourier amplitude as a function of the bottleneck width.

never show the so-called zipper effect, the one-by-one passing of the both sideof the particles. This tendency seems to come from an density instability.When a particle of species 1 passes the bottleneck, particles that belong tospecies 2 are pushed out so that other particles of species 1 can pass throughthe bottleneck more easily then burst flow realizes. The burst can be under-stood as the result of special form of lane formation, although it is spatiallyrestricted by the bottleneck. The same type of density instability (the ag-gregation of the same kined of particles) is observed in plasma, colloidal andgranular systems and known as the two-stream instability [Löwen2010] (alsocalled the laning instability). Its fundamental feature is characterized by thesimple formula

dn

dt∝ n, (3.4)

where n is the density. We assume that this type of instability also holds inour system and take a roll of the positive feedback. On the other hand, amechanism that suppresses the instability should exist to sustain the oscil-lation. After the burst, the difference of the number of particles of species 1and 2 grows, therefore the force that direct the opposite orientation to theflow is applied to the system. This pressure difference can be seen as the

3.3. MODELING 61

negative feedback of the system which suppresses the growth of the burst. Inconclusion, we assume that the combination of the positive and the negativefeedback explained above results in the oscillatory particle flow.

Next, let us investigate the bifurcation structure given by the numericalsimulation. The numerical result shows the transition from trivial equilib-rium to the oscillatory states when increasing the bottleneck width w. Fromthe bistable structure shown in Fig. 3.5, the phenomenon seems to show thesubcritical bifurcation. Here note that a previous study suggests that theoscillation is the supercritical type [Corradi2012], whereas our system showsthe subcritical one. The reason of the contradiction might come from thedifference of the particle model, since in their study the different type ofthe two-body interaction and the effect of collision avoidance are introduced.Moreover, they quantify the oscillation by using the center of mass of thesystem. However, except the bifurcation type the fundamental features ofthe oscillation presented here is in accordance with the previous study, sothat we proceed the investigation with our particle model.

3.3 Modeling

Based on the above observation, we construct a model that represent theessentials of the oscillatory flow. Here we call the particles from left to right“species 1” and others “species 2”. Let the direction from left to right bepositive. Now we model the original many-particle system as a one dimen-sional oscillator which is affected by the driving force of the particles nearthe bottleneck and the pressure from the bulk particles. To represent thesystem behavior more clearly, we assume that the oscillation is the result ofthe competition of two species around the bottleneck(Fig. 3.6). In this sim-plified picture, each species is represented by a representative particles whichmotion is interpreted as the averaged motion of the original many-particlesystem. Now the oscillatory flow of the original system is translated to the

3.3. MODELING 62

Figure 3.6: Model system. Those are the virtual particles which representthe averaged motion of the original self-driven particles. Each particle hasthe driving force that correspond to the sum of the driving force of theSFM particles near the bottleneck. Besides, the representative particles havepressure from behind which originates the total force to the bottleneck fromthe bulk particles.

motion of the model oscillator that consist of two representative particles.First, we formulate the driving force. We assume that the driving force in

the model system F1 includes the same type of self-driven force in the SFM,

m

τ(v0ei − vi). (3.5)

The driving force in the model system should also depends on the densityof the particles in the vicinity of the bottleneck. Here we notate the localdensity of species i around the bottleneck as ni. On top of that we requirethe following properties: the restriction 0 ≤ ni ≤ 1 and the normalizationcondition n1 + n2 = 1. Here n1 = 1 means the situation that only species 1occupy the vicinity of the bottleneck exclusively. If the velocity of the particleflow is positive, then species 1 exists dominantly around the bottleneck. Sothat n1 should be an increasing function of v and n2 a decreasing functionaround v = 0, where v is the velocity of the model system. Note thatn2(v) = n1(−v) due to the symmetry of the system. By using these notations,

3.3. MODELING 63

the driving force of the oscillator should be represented by

F1 = m

τ(v0 − v)n1(v) + m

τ(−v0 − v)n2(v)

= −m

τv + m

τv0n(v). (3.6)

Here the normalization condition n1(v) + n2(v) = 1 is used, and the functionn1(v) − n2(v) ≡ n(v) is defined.

Next, we formulate the pressure term. we assume that the pressurearound the bottleneck F2 is proportional to the intensity of the driving forcev0 and the difference of the number of particles on both side of the bottleneckN1 − N2,

F2 = c1v0(N1 − N2), (3.7)

here c1 is a positive constant. The number of particles N1(N2) decreaseswhen the flow velocity is positive (negative), therefore we assume that thetime evolution of Ni is represented as

dNi

dt= (−1)ic2v (i = 1, 2), (3.8)

where c2 is a positive constant. Though N1 increases if the velocity is negativein this formulation, this behavior can be interpreted as the supplement ofparticles by the periodic boundary. By integrating the above formula, andset the constant of integration to zero with the condition N1 = N2 for v = 0,the pressure effect is written as

F2 = −kx, (3.9)

where k ≡ 2c1c2v0, and x means the position of the center of mass of themodel system.

Based on these assumptions, the equation of motion of the model systemis given by

τdv

dt= −v + v0n(v) − kτ

mx, (3.10)

dx

dt= v. (3.11)

3.3. MODELING 64

Each term corresponds to the dumping, the positive feedback, and the neg-ative feedback of the system. The remaining problem is to give the functionn(v). By definition, n(v) is the weakly increasing odd function that satisfy−1 ≤ n(v) ≤ 1. n = 1 corresponds to the state in which the vicinity ofthe bottleneck is occupied only by species 1, whereas n = 1 for species 2.n(0) = 0.5 means that the same number of species 1 and 2 coexist equallyaround the bottleneck. For now, as one of the possible realization of such afunction, we assume n(v) as the following function form (Fig. 3.8)

n(v) =

+1 (v > vc)a1v + 1

3a2v3 (−vc ≤ v ≤ vc)

−1 (v < −vc),(3.12)

where vc is the number which satisfy n(vc) = 1, a1 > 0 and a2 are theconstants. The verification of the above assumption will be given later. Theabove formula and the equation of motion are the final form of our model.In the model, a1 and a2 correspond to the bottleneck width in the originalmany-particle system. Increasing a1 and a2 results in the faster growth ofn(v) (the steep gradient of n with respect to v), which corresponds to thesituation of the massive flow of particles due to the wide bottleneck.

As a next step, we find the solution of the model numerically. For sim-plicity, we choose the parameter a1 = 0, and a2 is treated as the bifurcationparameter. Now we calculate the time evolution of the model system withthree initial conditions, i.e. (x(0), v(0)) = (0, 0.2), (0, 0.6) and (0, 1.5). Theother parameters are fixed as m=80 kg, τ=0.5 s, c=1000 kg/m, v0=1 m/s.Fig. 3.7 shows the numerical results. As it can be seen, the solution showsthe bifurcation according to a2. When we have small a2, in all cases thesolution converges to zero, namely no oscillation is observed. On the otherhand, large a2 leads the stable oscillation. In the intermediate values of a2,bifurcation occurs. These results are shown schematically in Fig. 3.7, whichindicates the amplitude of the oscillation as a function of a2. As you cansee, it has the bistable structure. By comparing the original many-particle

3.4. ANALYSIS 65

picture, the case of small a2 correspond to the blocking situation, whereasthe case of large a2 is the counterpart of the oscillatory flow with the largebottleneck. The fundamental character of Fig. 3.7 mimics the bifurcationstructure shown in Fig. 3.5, therefore the model captures the essential factorof the oscillatory phenomenon.

3.4 Analysis

We further investigate the model to identify the mechanism of the self-excitedoscillation. The origin of the oscillation in the model is n(v), the local densityof the particles around the bottleneck. The function n(v) is composed of thepolynomial with odd power and the cutoff |n(|v| > vc)| = 1. This cutoffbrings the higher order nonlinearity. As shown in Fig. 3.8, the main featureof n(v) can be roughly approximated by the polynomial

n(v) ∼ v3 − v5. (3.13)

Since n(v) is the odd function, it is natural to approximate n(v) as a poly-nomial with odd powers around zero. Combining the above representationto our model, the equation of motion has the form

dv

dt∼ −v + v3 − v5 − x, (3.14)

this is identical with the van der Pol equation with the fifth order nonlinearterm.

The oscillator is excited by the positive feedback, the instability of thenumber of particles near the bottleneck. This mechanism is understood asfollows: the combination of the time derivative

dn(v)dt

= dn(v)dv

dv

dt, (3.15)

and the equation of motion

dv

dt= −v

τ+ v0n(v)

τ− k

mx, (3.16)

3.4. ANALYSIS 66

Figure 3.7: numerical solutions of the model. The solid, dotted and chainline correspond to the initial conditions (x, v)=(0, 0.2), (0, 0.6), and (0, 1.5)respectively. (Top-left) a2 = 3 is used. The solutions converge to zero froall initial conditions. (Top-right) a2 = 12. Both the trivial solution and thelimit cycle exists. (Bottom-left) a2 = 120. Most of the solution convergesto the limit cycle. (Bottom-right) The schematic bifurcation diagram. In allcases a1 = 0 is used for simplicity.

3.5. CONCLUSION 67

Figure 3.8: function form of n(v). The solid and the dotted line correspondto n(v) and it polynomial approximation v3 − v5.

includes the the density instability at the bottleneck

dn

dt∝ n. (3.17)

The above character takes a role of the positive feedback and the cause ofthe oscillation.

Note that the model is also able to show the supercritical-type bifurcationwhen we choose the parameters a1, a2 differently. The choice of a1, a2 inn(v) depends on the character of the original many-particle system, i.e. theparameter of the particles and the function form of the interaction. So thatthere is no critical contradiction to the previous study in which the oscillationis understood as the supercritical [Corradi2012].

3.5 Conclusion

In this study, a mathematical model is given that represent the spontaneousoscillatory phenomenon observed in pedestrian flow through bottlenecks. By

3.5. CONCLUSION 68

analyzing the model, we reveal the mechanism of the phenomenon which aresummarized as follows: (i) the pedestrian flow oscillator can be interpretedas the van der Pol-type self-excited oscillator, (ii) the oscillation shows thebifurcation depending on the width of the bottleneck, (iii) the oscillation isthe result of the competition between the local density instability that issimilar to the two-stream instability and the global pressure that suppressthe instability.

The remaining important issue is the identification of the bifurcationtype. More detailed observation of the phenomenon is needed to determinethe free parameters in our model. Also, the comparison of the pedestrianoscillator to the rhythm phenomena in fluid media would be an interestingfuture work. It is known that the oscillation of particle flow presented here isqualitatively very similar to some other self-excited oscillatory phenomena,such as the saltwater oscillator [Yoshikawa1991] and the plastic bottle oscil-lator [Kohira2007]. The comparison may clarify the specific and universalfeatures of the self-excited oscillation in discrete flow.

Chapter 4

Analytical investigation of thefaster-is-slower effect with asimplified phenomenologicalmodel

We investigate the mechanism of the phenomenon called thefaster-is-slower effect in pedestrian flow studies analytically witha simplified phenomenological model 1 . It is well known thatthe flow rate is maximized at a certain strength of the drivingforce in simulations using the social force model when we considerthe discharge of self-driven particles through a bottleneck. Inthis study, we propose a phenomenological and analytical modelbased on a mechanics-based modeling to reveal the mechanismof the phenomenon. We show that our reduced system, withonly a few degrees of freedom, still has similar properties to the

1This chapter is the improved author-created version of our following publication:K. Suzuno, A. Tomoeda,and D. Ueyama, Analytical investigation of the faster-is-slowereffect with a simplified phenomenological model, Phys. Rev. E 88, 052813 (2013).This reprint is in accordance with the copyright policy of the American Physical Soci-ety. Copyright 2013 American Physical Society. The final publication is available athttp://dx.doi.org/10.1103/PhysRevE.88.052813

69


original many-particle system and that the effect comes from thecompetition between the driving force and the nonlinear frictionfrom the model. Moreover, we predict the parameter dependenceson the effect from our model qualitatively, and they are confirmednumerically by using the social force model.

4.1 Introduction

Collective motion of self-driven particles has captured much interest for thepast few decades from the perspective of non-equilibrium dynamics. Theconcept of self-driven particles has been widely used in many fields (e.g.,traffic flow [Helbing2001, Nagatani2002, Chowdhury2000, Schad2010], ac-tive matter [Ramaswamy2010, Toner2005, Vicsek2012], and granular media[Aranson2006, Ikeda2012]) and has enhanced interdisciplinary collaboration.Pedestrian dynamics has drawn the attention of physicists as a collectivemotion of self-driven particles since it shows a wide variety of self-organizedphenomena [Helbing2001], such as lane formation, segregation, and flow os-cillations at a bottleneck. Studies of these phenomena are also expected tocontribute to the safe evacuation of people in an emergency in the real world.In particular, the flow of particles through a bottleneck has been studiedintensively with many approaches, such as the social force (SF) model [Hel-bing1995], the lattice gas model [Tajima2001], cellular automata (CA) [Kirch-ner2003, Yanagisawa2009], and real experiments [Kretz2006, Seyfried2009].

The “faster-is-slower” effect [Helbing2000a] is a well-known phenomenonobserved in certain systems of self-driven particles. Let us consider the dis-charge of self-driven particles from a square room through a narrow exit. Insuch a system, the discharge flow rate is a monotonically increasing functionof the self-driven force when the force is relatively weak. However, counter-intuitively, the flow rate begins to decrease when the driving force exceeds acritical value. This “faster-is-slower” effect was shown for the first time in a


numerical study using the SF model [Helbing2000a]. The SF model is basedon ODEs, and detailed numerical studies have been performed using thismodel and its variations [Parisi2005, Parisi2006, Parisi2007a, Parisi2007b,Frank2011]. It is also known that the CA model shows a similar effect [Kirch-ner2003]. In this model, each particle moves in a discrete manner with givenprobabilities that include the effect of friction between two particles. The CAmodel gives some insights into the role of the friction and the effect of an ob-stacle on the evacuation problem. Both numerical approaches indicate thatthe friction is crucial in this phenomenon, but a theoretical understanding ofthe phenomenon has not been attained.

We also should note that we have a gap between the SF model and a realsystem. Although the SF model shows the “faster-is-slower” effect, we haveno definite evidence that shows the existence of the phenomenon in a realpedestrian system. Some systematic experiments have been performed so far[Helbing2006, Seyfried2009], they do not show any supportive evidence withthe exception of a biological experiment [Soria2012]. A better strategy tounderstand the reason of the gap is to know which factors included in the SFmodel generate “faster-is-slower” effect through the model-aided analysis.This approach allows us to discuss the effect of each parameter and therelation among them more clearly. In the present chapter, we propose asimplified phenomenological model to clarify the reason why the original SFmodel proposed by Helbing et al. shows the “faster-is-slower” effect, or moregenerally, what are the physical causes of the phenomenon. An analyticalinvestigation of our model is conducted, and give comparisons to the result ofSF model. From this study, We expect to gain a better understanding of themicroscopic mechanism of the phenomenon. Describing a possible scenarioof the phenomenon would be helpful to understand the gap between the SFsimulations and the reality.

4.2. REVIEW OF THE SF MODEL 72

4.2 Review of the SF model

First, a brief review of the SF model is given [Helbing2000, Helbing1995]. Itis widely used in the study of pedestrian flow due to its notable ability toreproduce many interesting phenomena [Helbing2001a, Helbing2001b, Hel-bing1995]. In the SF model, each particle has a self-driven force, a socialforce, linear elasticity, and friction. The equations of motion for N particlesare given by

mdvi

dt= m

τ(v0ei − vi)

+N∑j

(Ae−

dij −rijB + kg(rij − dij)

)nij

+N∑j

κg(rij − dij) ((vj − vi) · tij) tij + Fiw, (4.1)

where vi is the velocity of the particle i, dij is the distance between particlesi and j, rij = ri + rj is the sum of the radii of the particles, nij is the normalunit vector from j to i, tij is the tangential unit vector, and v0ei is the desiredvelocity, which is always directed toward the exit. The particle i has mass m,time constant τ , social force parameters A and B, elasticity k, and frictioncoefficient κ. The function g is defined as g(x) = 0 if x < 0 and as g(x) = x

if x ≥ 0. The interaction with the nearest wall Fiw has the same form as thetwo-body interaction.

Let us solve Eq. (4.1) to obtain the flow rate as a function of v0 and N .As a boundary condition, we consider a large square room that has a narrowexit in the center of one of its walls. In this calculation, we consider thestationary state. Since the flow rate depends on N [Gaw2012], we need tokeep N constant. For this purpose, particles are appropriately supplied fromthe side of the room opposite the exit [Parisi2006, Parisi2007a]. Here, we usethe following parameters: the width of the exit is 1.2 m, ri = 0.3 ± 0.05 m,m = 80 kg, τ = 0.5 s, A = 2000 N, B = 0.08 m, k = 120000 kg/s2,and κ = 240000 kg/(m · s). Basically, the parameters are based on Helbing


0 1 2 3 4 5v0 (m/s)

0

50

100

150

200

250

300

N

0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3

flo

w r

ate

(1

/s)

(a)

0

0.6

1.2

1.8

2.4

3

0 1 2 3 4 5

flo

w r

ate

(1

/s)

v0 (m/s)

(b)

Figure 4.1: The numerical results for the stationary flow rate obtained fromthe SF model simulation. A higher flow rate corresponds to a faster evacua-tion time. (a) v0- and N -dependence of the flow rate. The flow rate was notcalculated for small values of v0 and N . (b) The flow rate with a fixed N of200.


Figure 4.2: Schematic phase diagram. The area below the rough additionalline is the “free state,” where the flow rate increases as the driving forceincreases. Each point (x) shows the maximum point for each N . The areaabove the line is the “jamming state,” where the “faster-is-slower” effectoccurs. The critical v0 value depends on N , and this suggests that the effectbegins to slow down evacuation when N is relatively large.

et al. [Helbing2000a]. Although it is known that these parameters lead tounrealistic behavior [Lakoba2005], we use these values in this study since ourintention is to identify the cause of the “faster-is-slower” effect rather thanto seek more suitable model parameters. The resulting stationary flow rateis given by Fig. 4.1, where the flow rate is calculated from the time neededfor 500 evacuation events. When v0 increases under fixed N , there existsa critical point above which the flow rate decreases. The schematic phasediagram is given in Fig. 4.2, which shows that (v0, N) space is divided intoqualitatively different regions.

4.3. MODEL 75

Figure 4.3: Conceptual images of the model. (left) The typical formation ofthe arch near the exit. (right) The forces acting on the virtual particles.

4.3 Model

To investigate the origin of the phenomenon analytically, we introduce a newqualitative phenomenological model inspired by the SF model to describe thedischarge velocity of the many-particle system. The derivation of the modelis as follows (see Fig. 4.3). To simplify the original many-particle system, wemade the following assumptions. The discharge property is determined by themotion of particles in the immediate vicinity of the exit. The flow is governedby the radial motion toward the exit [Helbing2006] and has no significantangular dependence on long-time average. The total number of particlesinside the room is fixed. We introduce virtual particles for each directionthat obey the above assumptions and have the same type of interactions inthe SF model (the social force, elasticity and friction). The particles aroundthe exit are also affected by the force from behind

h(v0, N) = bv0√

N, (4.2)

4.3. MODEL 76

Figure 4.4: The force applied to the particle at the center of the semi-circlefor each N . The results are obtained from the SF simulation with zero-widthexit. Dashed lines represent Eq. (4.2) with the dimensionless parameterb = 0.625.

where b is the dimensionless constant. An intuitive interpretation of theabove formula is as follows. In the original many-particle system, the par-ticles rush into the exit and form a semi-circular shape around the exit dueto the clogging. The force driving the particles is proportional to v0. Weassume that the pressure around the exit is proportional to the radius ofthe semi-circle, which is dependent on

√N . As a result, we obtain Eq. (4.2).

The validity of this assumption, at least as a first approximation, can be con-firmed by performing a numerical calculation using the SF model in whichthe width of the exit is nearly zero (see Fig. 4.4). Next, we assume that therepetitive formation and breakup of the four-particle arch in front of the exitis the most frequently observed dynamic structure and the major cause ofthe clogging. According to an experimental study of granular media [Garci-martin2010] in which the discharge of particles from a rectangular-shapeorifice is investigated, the size of the typical arch depends on the width ofthe exit, η = 1.41 + 1.15R, where η is the number of particles in the arch,

4.3. MODEL 77

R is the width of the exit measured by the diameter of particles. Using thisrelation here would be valid since the original SF model is very similar to themodel of granular media. By applying this relation to our system (R = 2),we obtain η ∼ 4 and assume that the four-particle arch is the dominantformation in the process of evacuation considered here. Finally, we simplifythe process of the breakup of the arch. Namely, one of the particles in thearch moves toward the exit and other particles in the arch stand still duringthe motion. It is not important which particle in the arch moves as far aswe consider long-time averaged flow. The equation of motion of the virtualparticles that obey the above assumption is given by

dvr

dt= (v0 − avr) − 2κg(l(v0, N))vr + h(v0, N). (4.3)

Besides, by constructing the equation of the balance of the forces along thearch, we obtain

(v0 + h(v0, N)) sin θ

2= Ael(v0,N) + kg(l(v0, N)). (4.4)

Where vr is the radial flow velocity, a is the deceleration parameter whichrepresent the effect of collisions, θ = π/η is the angle between two particles,l(v0, N) is the characteristic overlap length between particles, which is as-sumed to be a monotonically increasing function of v0 and N . The functionh(v0, N) represents an external force driving the particles from behind. Theother symbols have the same meanings as in the SF model. Note that theseequations are scaled and dimensionless: for example, A∗ = τ2

BmA and write A∗

as A for simplicity. The other parameters and variables are also scaled in thesame manner. The parameters need to satisfy the condition a(k +A) < 2κA,the meaning of which is that the friction coefficient is sufficiently large.

The equations describe the motion of virtual particles that represent theoriginal many-particle system. In the stationary state, the average dischargevelocity can be interpreted as being its velocity. We assume that the dischargeflow in the original system is characterized by the stationary solution ofEq. (4.3).

4.4. ANALYSIS 78

Note that our simplified model is based on the flow velocity, not theflow rate. Nevertheless, our model was able to describe the “faster-is-slower”effect. It is known that the maximum bulk kinetic energy corresponds to themaximum discharge flow rate when numerical experiments are performedwith various values for the desired speed v0 [Parisi2007a]. Therefore, weassume that the maximum discharge velocity in the model corresponds tothe maximum discharge flow rate.

4.4 Analysis

Let us consider the stationary state of Eqs. (4.3) and (4.4). We set v̇r = 0.The solution is given by

vr(v0, N) = v0 + h(v0, N)a + 2κg(l(v0, N))

, (4.5)

l(v0, N) =

ln (v0+h(v0,N)) sin θ2

A(v0 < vc)

(v0+h(v0,N)) sin θ2 −A

k+A(v0 ≥ vc),

(4.6)

where vc = A/[(1+ b√

N) sin θ2 ]. Here, the Taylor expansion is used to derive

Eq. (4.6). The approximation error of the expansion is less than 4% (seeFig. 4.6) in the case of v0 ≤ 5 and N ≤ 300, so that the analytical expressionabove is valid in the discussion presented here. The solution is shown inFig. 4.5. In this study, we choose θ = 45◦,a = 10 and b = 0.625. Theparameter a is chosen to realize appropriately small outflow velocity, and b

is determined from the numerical data fitting shown in Fig. 4.4. The otherparameters used here are the same as those in Eq. (4.1) but scaled. As canbe seen, the model presented here has a solution that shows the “faster-is-slower” effect, and it mimics the v0- and N -dependence of the flow ratequalitatively.

The existence of the “faster-is-slower” solution can be understood in termsof the competition between the driving force and the nonlinear friction. Forsimplicity, we fix the number of particles to, e.g., N = 200. The flow velocity

4.4. ANALYSIS 79

Figure 4.5: The stationary flow velocity obtained from the model. (a) Thev0- and N -dependence of the flow velocity. (b) The flow velocity with a fixedN of 200.

4.4. ANALYSIS 80

Figure 4.6: The explicit representation of the function g(l(v0, N = 300)).(solid) the analytical expression based on Eq. (4.6), (dotted) the numericalresult calculated directly from Eq. (4.4), (chain) the approximation curveg(l(v0, N)) ∼ vn

0 (n = 2).

is given by Eq. (4.5), which means that the flow velocity is determined bythe driving force and the drag force. From the model, the total driving forcev0+h(v0, N) is linear with respect to v0. On the other hand, the friction termis a nonlinear function of v0. The concrete representation of g in Eq. (4.5) isgiven by the following nonlinear function

g(l(v0, N)) =

0 (v0 < vc)(v0+h(v0,N)) sin θ

2 −A

k+A(v0 ≥ vc),

(4.7)

which is shown in Fig. 4.6. Equation (4.7) behaves like a nonlinear functionof v0 around vc, and it could be approximated by g(l(v0, N)) ∼ vn

0 , for n > 1.The competition between the linear driving force and friction that has sucha nonlinearity causes the solution (4.5) to have a maximum value.

The nonlinearity of the friction term originates from the coupling of thesocial force with the linear friction. The critical point in Eq. (4.7) is de-termined by the coupling constant of the social force A. Below the criticaldriving force, there is no friction because no overlap between particles existsdue to a weak driving force. Note that the friction becomes linear if A = 0.

4.5. SIMULATION 81

From these points, we can say that the social force plays the role of a barrier.A packing force that is strong enough to break the barrier is needed for theemergence of contact friction. The piecewise function in Eq. (4.7) shows sucha barrier effect. The convexity of Eq. (4.7) comes from the combination ofthe linear behavior of the friction and the existence of the social force.

4.5 Simulation

If the above analysis is correct, we can immediately make the following state-ment. The outflow shows no “faster-is-slower” property (i) if κ = 0, (ii) ifri is small, or (iii) if A = 0. If the size of the particle is small, the overlapbetween particles does not exist (that is, l(v0, N) < 0), then g(l(v0, N)) = 0in Eq. (4.5) so that (ii) holds. We can also state that (iv) if A = 0 butthe friction term maintains its nonlinearity for some reason, the “faster-is-slower” effect still exists. One of the ways of realizing such a condition isthe introduction of a shift parameter c only in the friction term in Eq. (4.1),such that it becomes g(ri + rj − dij − c). The argument of the g in the elasticterm in Eq. (4.1) remains unchanged. The procedure is equivalent to giving africtionless soft skin to the particles and causes an effect similar to the socialforce barrier. Under this implementation, such a result is expected even ifthere is no social force.

Now we perform the numerical simulation using the SF model (Eq. (4.1))to test the above predictions. Here ri = 0.1 ± 0.016 m in (ii) and c = 0.015m in (iv). The results are show in Fig. 4.7. As can be seen, all the abovepredictions are confirmed numerically.

4.6 Discussion and summary

From the simplified model and its theoretical analysis, we conclude that the“faster-is-slower” effect comes from the competition between the driving forceand the nonlinear friction. The result is consistent with a previous study,

4.6. DISCUSSION AND SUMMARY 82

Figure 4.7: Numerical results of the SF model simulation corresponding toeach conjecture. Only (iv) shows a maximum.

which dealt with this phenomenon numerically [Parisi2007a, Parisi2007b,Frank2011]. The nonlinearity is the result of the coupling of the repulsiveforce with the linear friction, which is based on the effect of the exclusivevolume and the surface friction.

This phenomenon would be observed not only in pedestrian flow but alsoin any system that realizes such a mechanism. The origin of the repulsiveforce seems less critical. The effect of collisions is also not important, at leastqualitatively.

Of course, our model is so simple that some issues remain. Though weintroduced many assumptions in the derivation of the model, further investi-gation is needed to obtain more evidence that supports the assumptions usedin our model. Another issue is that the results presented here are qualitativebecause of the simplicity of the model. Note that the flow velocity obtainedfrom the model is overestimated compared to that of the SF simulation dueto non-strict treatment of the collision effect.

We have provided an analytical investigation of the “faster-is-slower” ef-fect by using a simple phenomenological model. We have shown that our

4.6. DISCUSSION AND SUMMARY 83

novel analytical model is capable of describing the essentials of the phe-nomenon and that the competition between the driving force and the nonlin-ear friction is crucial for this phenomenon. Furthermore, the effects of eachparameter in the original many-particle system were predicted analyticallyby our model and confirmed numerically by using the SF model. The sim-plified technique presented here may be useful for understanding the otherphenomena originating from competition of flows in dissipative self-drivenparticle systems, such as lane formation, flow oscillations at bottlenecks, andthe freezing by heating [Helbing2001a].

Chapter 5

Dynamic structure inpedestrian evacuation: imageprocessing approach

We show that there exists a typical dynamic arch-shape struc-ture in pedestrian evacuation system governed by the social forcemodel 1. It is well known that the simulation of pedestrian evac-uation from a square room using the social force model showsarch-shape formation and clogging in front of the exit. It is alsoknown experimentally and numerically that an obstacle near theexit could improve the flow rate, but detailed mechanism of thiseffect is not clear. In this study, we show the existence of the “dy-namic arch,” the typical structure in the long term, by using thesocial force model and the image processing. The time-averagedimage of the system shows us the existence of the typical struc-

1This chapter is the improved author-created version of our following publication:Kohta Suzuno, Akiyasu Tomoeda, Mayuko Iwamoto, and Daishin Ueyama, DynamicStructure in Pedestrian Evacuation: Image Processing Approach, in Traffic and Gran-ular Flow ’13, pp. 195-201, edited by M. Chraibi, M. Boltes, A. Schadschneider,and A. Seyfried, Springer (2015). This reprint is in accordance with the copyrightpolicy of Springer. Copyright 2015 Springer. The final publication is available atlink.springer.com or http://link.springer.com/chapter/10.1007%2F978-3-319-10629-8_23, see also http://www.springer.com/978-3-319-10628-1

84


ture in the system and it can be interpreted as the probabilitydistribution of the arch formation. With this method, we dis-cuss the possible physical mechanism of the effect of an obstaclein the pedestrian system. From the observation of the morpho-logical feature of the arch obtained by the simulation and imageprocessing, we show that the obstacle affects the structure of thearch in three ways. These effects could lead the easy-to-breakarch that enhances the flow rate of the system.

5.1 Introduction

Congestion of particles at a bottleneck is one of the major problems in gran-ular and pedestrian systems. Let us consider the discharge of dissipativeparticles from a square box through a single narrow exit. Such a systemsometimes shows clogging when many particles rush to the exit simultane-ously. We can see such phenomena in the granular systems (the silo, the glasshour) and the pedestrian systems (the evacuation from a room or through acorridor). In many cases, this clogging leads undesirable results so that themechanism should be clarified to improve the particle flow.

One possible cause for the phenomena is the existence of the dynamicarch, the arch-shaped dynamic structure [Carlevaro2012]. The arch is formedand broken repeatedly, and this structure decreases the flow rate.

On the other hand, we also have some hopeful solutions for that situa-tion. Some experimental and numerical studies have shown that one possi-ble solution is to place an obstacle near the exit [Helbing2005, Zuriguel2011,Yanagisawa2010] (Fig. 5.1). It is said that the obstacle absorbs pressure[Helbing2005] and enhances smooth evacuation. But the effect of the obsta-cle depends on the configuration of the obstacle (the shape, the position, andothers) so that we need to clarify how the obstacle works.

In this study, we discuss this obstacle effect from the viewpoint of the


Figure 5.1: simulation of pedestrian evacuation passing through the bottle-neck. The white and the black circle mean the pedestrians and the obstacle.

dynamic arch by using the social force model (SFM) [Helbing2000] and theimage processing technique. Inspired by the work that visualize the staticclogging of granular media [Garcímartin2010], we show the existence of thelong-term structure of the dynamic arch in the evacuation system by usingthe SFM and the image processing. By this, we give the visualization of thedynamic arch and investigate the relation between the position of the obstacleand the structure of the arch. In this study, we use quite simple geometryand focus only on the effect of the position of the obstacle. Generally, thepedestrian system has many parameters and they affect the efficiency ofthe evacuation in a complicated way. For example, the relation among thedriving force, the position of the obstacle, and the size of the obstacle is alsoimportant aspect of the system [Matsuoka2013]. But the main purpose ofthis study is to discuss the relation between the structure of the arch and theflow rate, so that we just focus on the effect of the obstacle position, and wealso pay attention to the small area just in front of the exit.

5.2. METHODS 87

Figure 5.2: Details of the system. The self-driven particles rush to the exitand evacuate from the room. All particles have the same diameter (0.6 m)and the self-driven force (3.0 m/sec). They go out through the exit (1.0 m)and return to the room from the opposite side of the exit. The number ofparticles is 150. We observe this evacuation process up to 300 sec.

5.2 Methods

Let us consider the discharge of the self-driven particles from a box-shaperoom through a single narrow exit. Let the motion of the particles be gov-erned by the SFM [Helbing2000]. The situation of the system is described inFig. 5.2. The size of the particles are uniform, and the self-driven force foreach particle is set to 3 m/sec. The number of particles is 150. The particlesdistribute randomly in the room as an initial condition. The SFM parame-ters used here are based on Helbing et al. [Helbing2000]. In this study, weuse the periodic boundary to keep the number of particles inside the roomconstant to remove the N dependence of the flow rate. The numerical schemeis RK2 with dt = 0.001 s.

From this simulation, we generate many snapshots and investigate them.

5.2. METHODS 88

Figure 5.3: Example of contact force network. The particles move from left toright side of the picture. When two particles have penetrated, the white linethat connects two particles is depicted, and the system shows the networkstructure. In this calculation, the thickness of the line does not depend onthe magnitude of force for simplicity. Note that the shape of the particles(circles in the picture) is depicted in this example picture for clarification,but it is not drawn in the actual calculation.

But what we are interested in here is the dynamic formation of the particles,thus we depict the contact network of the particles instead of the shape of theparticles. If any two particles have the overlap above a given threshold, wedraw a line that connects two particles. Here the threshold is set to 0.01 mto visualize the dominant connection in the system and to remove the “lighttouch” connection between particles. As a result, we obtain the pictures thatshow the connection of the particles for each time step (Fig. 5.3).

Next, we perform the following image processing technique. Let us denotethe pixel value in the position (i, j) on the picture taken at time t as Iij(t).Then we generate the time-averaged image

< I >ij=1N

N∑n=1

Iij(ndt), (5.1)

where N is the number of images generated from the simulation, and dt isthe time step. The resulting image means the probability distribution of the

5.2. METHODS 89

Figure 5.4: Example of time-averaged network. The white lines show the av-eraged contact network in this many-particle system. Here gamma correction(γ = 2.0) is used and the additional circles are depicted for visibility. Thefour-particle arch is the most frequently occurred structure. The picture alsoshows the vague triangular lattice in the bulk region (behind the arch), andit reflects the mono-disperse property (all particles have the same diameter)of the system. The thick white circle is the obstacle.

contact network. We assume that the structure presented in the resultingimage can be interpreted as the dynamic arch of the system. The imageobtained by the above method visualizes the possible dynamic structure andits frequency in the system. The typical result is given in Fig. 5.4.

As we can see, it shows many types of arch-like structures in front ofthe exit. It implies that the arch that consist of four particles is the mostfrequently observed structure. Around the arch, many faint triangles areobserved and they are the results of mono-disperse property of the system.

5.3. ON THE EFFECT OF THE OBSTACLE 90

Figure 5.5: Results of the numerical simulation. We calculate the flow rateby using the SFM with different configuration of the obstacle. The size ofthe obstacle is the same as the particles. (Left) The position of the obstacleconsidered here is shown as the lattice point in the figure. The circle in thefigure is the example of the obstacle. (Right) The color distribution showsthe flow rate for each obstacle position. White area correspond to the flowrate without obstacle (340 person/300sec).

5.3 On the effect of the obstacle

By using the above technique, we investigate visually the effect of the obstacleon the structure of the arch. We place the obstacle near the exit in our systemwhich shape is the same as the particles. Then we conduct the simulationswith different position of the obstacle and obtain the distribution of the flowrate as a function of the obstacle position. We place the obstacle inside thefollowing region: 0.9 m≤ X ≤1.9 m and 0.0 m≤ Y ≤ 1.0 m, where X and Y

are the distance from the center of the exit. The resulting flow rate is givenin Fig. 5.5.

The distribution of the flow rate shows that the obstacle improves theflow rate in many cases. But we also have some “spots” where the flow rate

5.4. POSSIBLE MECHANISMS OF THE OBSTACLE EFFECT 91

decreases compare to the no-obstacle case. As we can see, these spots aredistributed in a triangular manner, and their positions correspond to thecase of the most packing situation. In such cases, the obstacle enhances theclosely-packed dense configuration of the particles. Another notable featurein the figure is the top-right area where the flow rate improves considerablydue to the obstacle. This means that the shifted obstacle (not just in frontof the exit) leads better flow rate.

5.4 Possible mechanisms of the obstacle ef-fect

We performed simulations with different positions of the obstacle, and com-pared the results of the shape of the arch. Fig. 5.6 shows the case of no-obstacle and the case that attain the highest flow rate. Comparing theseresults, we can extract three morphological factors of the shape of the arch(see also Fig. 5.7). The flow rate increases when (i) the arch has a space todeform, (ii) the position of the arch is shifted, and (iii) the arch is distorted.If there is a enough space around the exit, the arch can be deformed easilyand the symmetry is broken. The obstacle could take a role as a barrier fromthe particles behind, protect the arch, and make a room to deform. Theexistence of the obstacle also breaks the symmetry of the force from behindso that the center-of-mass of the arch is shifted to one side and one of theleg of the arch is on the edge of the exit. Furthermore, asymmetric forceleads the distorted arch, and the combination of these factors result in aneasy-to-break arch. The improvement of the flow rate occurs when we placethe obstacle at an appropriate position where the above three factors arerealized.

5.5. CONCLUSION 92

5.5 Conclusion

We investigated the dynamic arch by combining the simulation with the im-age processing. By using these techniques, we qualitatively clarified the effectof the obstacle in the pedestrian simulation. The morphological observationof the result from the SFM and the time-averaged image is performed, andthe results imply that the flow improvement by the obstacle comes from (i)the space around the arch (these might be equivalent to the pressure absorp-tion [Helbing2005] or dilatancy), (ii) the shift of the center of the arch, and(iii) the distortion of the arch. Note that the study presented here has manysimplifications. We focus only on the physical properties of the system, andneglect some social aspects. The contact force network we investigate heredoes not reflect the magnitude of force (so called force chain). We need toinclude these factors for further quantitative investigation of the effect of theobstacle.

5.5. CONCLUSION 93

Figure 5.6: Morphological observation of the arch. In no-obstacle case (top-left), the shape of the arch is almost symmetric. On the other hand, thesituation that attain the highest flow rate (top-right) shows asymmetric arch.Here gamma correction is used for visibility (γ = 2.0). The obstacle coversthe arch in front of the exit and protect it from the particles behind. Theobstacle makes a space, and it also allows the arch to be shifted and distorted(bottom).

5.5. CONCLUSION 94

Figure 5.7: The correspondence of the position of the obstacle with theshape of the arches. Each gridpoint represents the position of the center ofthe obstacle, and each picture is the corresponding shape of the dynamicarches. Here gamma correction is used for visibility (γ = 2.0).

Chapter 6

Conclusion

In this study, the investigation of the spontaneous phenomena in crowd sys-tems are conducted with the aid of mathematical modeling. Through the in-vestigation, It is showen that the physics-based mathematical modeling canbe an alternative method to the other descriptions (fluid dynamical treat-ments, direct numerical simulations) when describing pattern formation indiscrete active particle flow.

In chapter 1, the general introduction has been given. The fundamentalcharacter, approaches, problematic in crowd dynamics has been discussed. Inthe following chapter 2, a review of self-organized phenomena in self-drivenparticle systems has been provided. Some of the interesting spontaneousorder have been explained with the aid of numerical examples and historicalreviews. In the following chapter 3, we have focused on the crowd flowoscillations, an example of self-organized self-excited phenomenon in discreteflow, and have clarified its mechanism via mathematical modeling. Anotherexample of nontrivial collective phenomenon is the faster-is-slower effect,discussed in chapter 4, which is related to the pedestrian evacuation of a roomwith a narrow exit. The comparison of the numerical simulations with theproposed phenomenological mathematical model has given the clear pictureof the phenomenon and allowed us to predict some fundamental charactersin the bottleneck particle flow. Interestingly, the efficiency of bottleneck

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flows are strongly affected by the presence of an obstacle in the vicinity ofthe bottleneck, as shown in chapter 5. With the new technique (combiningmany-particle simulations with the image processing of the contact forcenetwork), the qualitative mechanism of the “obstacle effect” has been given,which has provided the intuitive understanding of the effect of an obstacle.

Finally, I would like to note the implication of the studies. As you can see,these studies are based on new simplified approaches to grasp the essentialfeatures of the bottleneck flow. The common viewpoint of the simplificationmethods used in each chapter (3, 4 and 5) are a reduction of systems basedon specific character of the boundary conditions. In the studies, we haveassumed that the main features of the bottleneck flow is characterized by theparticle motion just around the bottleneck, and the bulk particles (far fromexits) do not have critical effects on the outflow. By focusing the particular-ity of the bottleneck-shape boundaries and the averaged motion of particlespassing through bottlenecks, it is possible to reduce a given bottleneck flow toa simplified system with a few degree-of-freedom. Actually, in chapter 3 and4, the original many-particle systems that have large number of 2D particleshave been reduced to simple systems that can be treated analytically. Also,in chapter 5, the complexity of the particle flow is summarized to the mor-phology of the simple spatio-temporal structures. The reduction introducedhere allow us to attain the physical understanding of crowd motion based onclassical mechanics and the analytical understanding of discrete flow with theaid of dynamical system. The generalization of the simplification approachespresented here would lead to new useful methods that contribute to betterunderstanding of crowd dynamics. Most of the previous studies have beenbased on case studies using numerical simulations and dedicate to find newself-organized phenomena. As a result, the clarification of the underlyingmechanism draws less attention of physicists and engineers. One of the rea-sons of this drawback is the absence of appropriate description methods toinvestigate the collective motion of spontaneous order in dissipative particle

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systems. Here I would like to stress that the mathematical modeling couldbe a useful method in crowd dynamics, as this study shows. The combi-nation of simulations, modeling and analysis offers us the revelation of thefundamental mechanism of crowd motion and new insights into the patterndynamics in discrete flow, as shown in this study.

Acknowledgement

First of all, I would like to thank Professor Daishin Ueyama of Meiji Uni-versity. He gave many chances to develop my research activities. ProfessorH. Ninomiya, Professor R. Kobayashi and Professor H. Takayasu are thesub-supervisors in my Ph.D. course. Professor H. Nishimori reviewed thisdissertation carefully and gave valuable comments. Associate Professor A.Tomoeda provides insights into the study of traffic flow, and his feedback im-proves my work considerably. Lecturer M. Iwamoto helped me with practicalinformation that is necessary to conduct the numerical study. I also thankProfessor M. Mimura, Director of MIMS (Meiji Institute for Advanced Studyof Mathematical Sciences), and the members of MIMS. I am really inspiredby their viewpoints, ideas, scientific attitude and the self-driven forces towardbetter studies. This work is supported by the Meiji University Global COEProgram “Formation and Development of Mathematical Sciences Based onModeling and Analysis”, and the Graduate School of Advanced Mathemati-cal Sciences, Meiji University.

Some chapters in this dissertation are based on the author-created ver-sion of the following papers, which comes from my Ph.D. course activities:(Chapter 3) K. Suzuno, A mathematical model of oscillatory pedestrian flow,Proceedings of the 20th Symposium on Simulation of Traffic flow, pp. 79-82(2014), (Chapter 4) K. Suzuno, A. Tomoeda,and D. Ueyama, Analytical in-vestigation of the faster-is-slower effect with a simplified phenomenologicalmodel, Phys. Rev. E 88, 052813 (2013), (Chapter 5) Kohta Suzuno, AkiyasuTomoeda, Mayuko Iwamoto, and Daishin Ueyama, Dynamic Structure in

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Pedestrian Evacuation: Image Processing Approach, in Traffic and GranularFlow ’13, pp. 195-201, edited by M. Chraibi, M. Boltes, A. Schadschneider,and A. Seyfried, Springer (2015).

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A study on pattern formation in crowd dynamics via ... · Kohta SUZUNO Graduate School of Advanced Mathematical Sciences, Meiji University, Japan January 2015 (ver. 150204) 1. Abstract