kohta suzuno

Pattern formation in Crowd dynamics

Kohta SUZUNO

MIMS, Meiji Univ.

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Contents

(1) Introduc-on to crowd dynamics (2) Pa6ern forma-on in crowd dynamics (3) Mathema-cal modeling of crowd dynamics

Observa-on

Par-cle simula-on

Dynamical system

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What is crowd dynamics?

(C)NHK (C)NHK

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Examples of crowd dynamics

-‐ Queuing -‐ One-‐way flow

-‐ Crossing -‐ Counter flow

-‐ Conges-on -‐ Turbulent

Crowd = Self-‐driven par-cles with physical and social interac-on

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The importance (1)

Crowd flow shows collec-ve pa6erns

-‐ Turbulent pa6ern -‐ Lane forma-on -‐ Freezing transi-on

-‐ Dissipa-ve structures -‐ Universality -‐ Fluid-‐Par-cle correspondence

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The importance (2)

Crowd dynamics contributes to social safety

(dys)func-on of collec-ve mo-on control -‐ avoid crowd disasters -‐ flow op-miza-on -‐ efficient transporta-on

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The importance (3)

-‐ Crowd mo-on has par-cle-‐scale instability. -‐ Crowd system refuse the con-nuous approxima-on. -‐ Need an alterna-ve descrip-on!

How should we describe and understand the discrete flow?

Fluid?

Par-cle?

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Application (1)

cf. Lexus Interna-onal, "Amazing in Mo-on -‐ SWARM" (2013) and others.

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Application (2)

cf. worldwarzmovie.com (2013).

zombi

wall

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Methods of crowd dynamics (1)

(1) Real crowds

Observa-on Experiments

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(2) Biological en--es

[Soria et al., Safety Science 50, 1584 (2012)]

[安倍北夫, パニックの心理(1974)] [Zuriguel et al., Scien-fic Reports 4, Ar-cle no.7324 (2014)]

obst-‐acle

sheep

sheep

sheep

sheep

sheep

sheep

sheep

s-mula-on

ant

(3) Simula-ons

(i) Rule-‐based Baer, report -‐ Carnegie-‐Mellon Univ. (1974) Kirchner et al., PRE 67, 056122 (2003)]

(ii) Physics-‐based 中村他, 人間工学10 (3), 93 (1974) Alonso-‐Marroquin et al., arxiv.org (2013)

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Mathematical description

Stochas-c models

Fluid models

Self-‐driven par-cles

Phenomenological ODEs

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simple real

macro

micro

CA Mul- agent

Fluid mode

Par-cle simula-on

Neteork model

Dynamical system model

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Lane formation

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Lane formation: formulation

The social force model (Helbing2000)

perio

dic

the self-‐driven force

N=50

m=80 kg, tau=0.5 s, v0=1 m/s, ri=0.3 m, A=2000 N, B=0.08 m,

15 m

5 m

the two-‐body interac-on

The B. C.

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Lane formation: observation

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Lane formation: observation

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Lane formation: properties

(1) A popular collec-ve phenomenon -‐ possibility (1974) -‐ observa-on and simula-on (1992) (2) Counter driving force + Social repulsive force (3) "par-cle-‐resolved instability" (4) Universality

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Lane formation: similar phenomena

Granular stra-fica-on

[新屋他, JSSI & JSSE Joint Conference (2012)]

[Dzubiella et al., PRE 65, 021402 (2002)]

colloid

[Makse et al., Nature 386, 27 (1997)]

Granular Rayreigh-‐ Taylor instability

Electric field

sand

sand

g

g

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Freezing-‐‑‒by-‐‑‒heating

small noise

large noise

noise-‐induced crystalliza-on [Helbing(2000)]

no noise

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Freezing-‐‑‒by-‐‑‒heating: formulation

The social force model

perio

dic

driving force noise

The B. C.

N=20

m=80 kg, tau= 0.5 s, v0=1 m/s, ri = 0.3 m, A=2000 N, B=0.08 m

15 m

2 m

interac-on

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Freezing-‐‑‒by-‐‑‒heating: results

Noise induces the freezing !?

noise intensity

transi-

on probability

*data from 20 realiza-ons

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Freezing-‐‑‒by-‐‑‒heating: time series

almost lanes

The -me series of the total energy

perfect lanes

-me (s)

total kine-

c E (J)

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Freezing-‐‑‒by-‐‑‒heating: scenario

kine-c energy

noise intensity freezing

laning

small noise -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ bistable intermediate noise -‐-‐-‐ laning is prohibited large noise -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ all possible states break up

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Freezing-‐‑‒by-‐‑‒heating: properties

(1) Noise-‐induced order Increasing energy leads solid state (not gaseous!)

(2) A novel type of phase transi-on? (3) Model?

liquid solid gas

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Oscillatory flow

the periodic change of the flow direc-on

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Oscillatory flow: history

-‐ numerically found Helbing et al., PRE 51, 4282 (1995)

-‐ experimentally confirmed

Helbing et al., Transporta-on sci. 39 1 (2005) -‐ empirically plausible　

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Oscillatory flow

-‐ Numerically iden-fied as the Hopf bifurca-on (Corradi2012) -‐ The physical mechanism is s-ll unknown -‐ A similar phenomenon: saltwater oscillator (Yoshikawa1991)

bo6leneck width

center of m

ass

water

saltwater

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Simulation: setup

Model: the SFM

B.C. : a periodic channel 45 m

5 m w

4 m

Parameters: N=150, m=80 kg, 𝜏=0.5 s, v0=1.0m/s A=573 N, B=0.08 m

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Simulation: results

The -me evolu-on of the momentum density The Fourier amplitude v.s. bo6leneck width

-me[s]

m

omen

tum

bo6leneck width [m]

amplitu

de

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Oscillatory flow: open questions

-‐ A type of nonlinear self-‐excitable oscillator? -‐ Mathema-cal model? -‐ The rela-on to the fluid oscillator? -‐ Synchroniza-on?

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The faster-‐‑‒is-‐‑‒slower effect

Faster mo-on results in slower evacua-on [Helbing(2000)}

driving force (m/s)

evacua-o

n -m

e (s)

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the microscopic many-‐par-cle model (the social force model, SFM)

self-‐driven force

repulsive force

elas-c force fric-on

wall

exit

The faster-‐‑‒is-‐‑‒slower effect: detail

m dvi (t)dt

= fself + fijj≠i∑

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The faster-‐‑‒is-‐‑‒slower effect: detail

mechanism?

modeling!

driving force

driving force driving force

Suzuno et al., Phys. Rev. E 88, 052813 (2013).

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We just consider the par-cle near the exit and its equa-on of mo-on.

NAnaly-c expression of the flow velocity

The outline of the modeling

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-‐ the eq. of mo-on

h v0

kg(l)+Ae κg(l)vr

-‐ balance of force

x

g(x)

We focus on the arch forma-on of the par-cles.

l

v0

Note: dimensionless. a means the collision effect.

[ ]

The modelSuzuno et al., Phys. Rev. E 88, 052813 (2013).

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(1) The discharge property is determined by the par-cles in the vicinity of the exit. (2) The flow has radial symmetry. (3) N is fixed. (4) The flow rate is propor-onal to the velocity of the model par-cle. (5) The parameters sa-sfy . (This means that fric-on is appropriately large.)

N

The model assumptions

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Model

Sta-onary situa-on

l

the analy-cal expression of the velocity!

[ ]

The model analysis

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Our model reproduces the simula-on results. our model simula-on

The model results

"faster" is "slower"


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1+g(v0) (contact fric-on) v0 (driving force)

coupling const. of the social force

linear elas-city

faster-‐is-‐slower

ouulow ~

The solu-on has the form

The mechanism of the phenomenon


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If our model is correct, the original system shows:

(a) If fric-on is linear, then no "faster-‐". (b) If no fric-on, then no "faster-‐".

Validation

linear fric-on

no fric-on

correct predic-ons!


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(1) We proposed a simplified model for the "faster-‐is-‐slower" effect. (2) We clarify that the "faster-‐" comes from the compe--on between driving force and nonlinear fric-on.

(3) This work gives an example of the study of collec-ve discrete flow via mathema-cal modeling.

Summary of "faster-‐‑‒is-‐‑‒slower"

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Summary of the talk

Crowd dynamics offers: (i) examples of spontaneous pa6ern forma-on

(ii) insights into the efficient transporta-on

(iii) mathema-cal issues: how to describe the mesoscale, transient and discrete flow?

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Critical discussion (1)

cri-cism 1. How do you believe you can describe mathema-cally the crowd mo-on, which is related to the free will?

cri-cism 2.

You can reproduce any results from simula-ons. cri-cism 3.

Is crowd mo-on the result of self-‐organiza-on truly?

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Mathema-cal (physical) studies of crowd dynamics only hold for:

(i) panic situa-ons, (ii) each person have their definite des-na-ons but the ways to reach there are less conscious, (iii) the size of crowds is large,

that is, the absence of sophis-cated intelligent ac-on.

Many types of model is available:

To avoid the arbitrariness of results, we should take the following steps:

(i) assume physically-‐acceptable mechanisms, (ii) reproduce the phenomenon by minimal models, and (iii) iden-fy the necessary condi-on.

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circle ellip-c non-‐spherical

rather social

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The concept of self-‐organiza-on is NOT omnipotent

rather physical

-‐ Crowd mo-on includes social factors. -‐ We have to no-ce that all crowd mo-on should not be reduced to Mathema-cal models.

END

Thank you for your a6en-on

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