Aversion of Pedestrians to Face-to-Face Situations Eases Crowding
Sho Yajima
1
, Kiwamu Yoshii
2+
, and Yutaka Sumino
1†
1
Department of Applied Physics, Graduate School of Science, Tokyo University of Science, Katsushika, Tokyo 125-8585, Japan
2
Earthquake Research Institute, The University of Tokyo, Bunkyo, Tokyo 113-0032, Japan
(Received April 1, 2020; accepted May 7, 2020; published online June 19, 2020)
We conducted a numerical simulation of a crowd of pedestrians. Each pedestrian, modeled with three circles, has a
shape whose long axis is perpendicular to the anteroposterior axis and is designed to move in a fixed desired direction,
i.e., +x or −x. Pedestrians have friction at their surface, soft repulsion, and a back force to resist backward motion. In this
study, we newly introduced an active rotation that captures the psychological effect of aversion to face-to-face situations.
The numerical simulation revealed that active rotation induces the fluidization of the system, leading to the higher flux of
pedestrians. To reveal the mechanism of the fluidization, we calculated the minimum principal stress for each pedestrian.
Furthermore, we visualized force chains observed in the system by connecting the minimum principal stresses. We
confirmed that the fluidization of pedestrians is due to the fragmentation of the force chains induced by the active
rotation.
1. Introduction
Active matter is a system composed of many elements that
transduce energy into motion locally. Such active matter is a
novel type of nonequilibrium system showing rich dynamical
patterns. Bird flocks, fish schools, and bacterial colonies
are well-known examples of active matter.
1–4)
A crowd of
pedestrians is also considered as active matter and forms
various patterns;
5–8)
for example, lane formation, avalanche
behavior, and clogging.
We here make a model of a high-density crowd of
pedestrians. It is known that a high-density crowd of
pedestrians may lead to fatal accidents.
9–11)
Such a critical
situation is due to the formation of force chains within the
population, which leads to stress concentration at a small
number of pedestrians. Thus, understanding the crowd of
pedestrians is required to avoid such a dangerous situation.
However, the situation is not experimentally accessible, so
numerical simulation is a relevant measure to understand
the behavior of pedestrians in a high-density crowd. Many
numerical simulations have been conducted to understand the
behavior of a high-density crowd of pedestrians.
5,12–15)
In
such numerical simulations, a pedestrian can be modeled as
a particle that follows Newton’s laws of motion, where
each individual walks to a fixed destination. A crowd of
pedestrians is a type of “dry active matter ”,
3)
where the
momentum of the system is not conserved as the pedestrians
move on a rigid substrate. Furthermore, the essential feature
of these models is that the physical contact force plays an
important role because of the crowded condition. For such
reasons, the behavior of a crowd of pedestrians is described
as a type of active granular system.
Notably, a pedestrian has anisotropy in its function and
shape: it has a frontal part directing its motion and is wider in
the direction perpendicular to the anteroposterior axis.
16)
In
this study, we newly adopt an active rotation induced by a
psychological effect. The front direction is the preferred
direction of movement. Interestingly, however, people have a
psychological aversion to face-to-face situations with others.
We adopt this effect via the active rotation. Our numerical
simulation reveals that the coupling between the anisotropic
shape and the active rotation eases crowded conditions for
pedestrians.
2. Simulation Setup
As shown in Fig. 1, we consider a rectangular system of
L
x
L
y
and impose the periodic boundary condition in the x
direction. In this study, we model a pedestrian i with a central
circle whose radius is ‘
i
and side circles whose radius is
‘
i
=2. ‘
i
is fixed for each pedestrian and has a homogeneous
distribution around the mean. Each pedestrian has a desired
direction e
i
¼ ð1; 0Þ, a front direction p
i
¼ðcos
i
; sin
i
Þ
(
i
2 ½%;%Þ), and a velocity v
i
. A pedestrian tries to walk in
the desired direction at a speed v
0
i
, and m
i
is the mass of the
pedestrian. v
0
i
and m
i
also have homogeneous distributions
around the mean. The position of the center of the pedestrian
is denoted by r
i
¼ðx
i
;y
i
Þ. The centers of the side circles lie
on the circumference of the central one, with the line joining
the center perpendicular to p
i
. The center of the right circle
is, r
r
i
¼ðx
r
i
;y
r
i
Þ¼ðx
i
þ
‘
i
2
sin
i
;y
i
‘
i
2
cos
i
Þ and that of the
left circle is r
l
i
¼ðx
l
i
;y
l
i
Þ¼ðx
i
‘
i
2
sin
i
;y
i
þ
‘
i
2
cos
i
Þ. We
can define the velocity of the central circle as v
i
¼
_ r
i
¼ðv
x;i
;v
y;i
Þ, and those of the left and right circles as
v
r
i
¼ðv
r
x;i
;v
r
y;i
Þ¼ðv
x;i
þ
‘
i
!
i
2
cos
i
;v
y;i
þ
‘
i
!
i
2
sin
i
Þ and v
l
i
¼
ðv
l
x;i
;v
l
y;i
Þ¼ðv
x;i
‘
i
!
i
2
cos
i
;v
y;i
‘
i
!
i
2
sin
i
Þ, respectively.
!
i
represents the angular velocity of the rotation.
The equation of motion for the ith pedestrian is given by
m
i
dv
i
dt
¼ m
i
v
0
i
e
i
v
i
(
þ
X
j≠i
f
ij
þ
X
W
f
iW
þ F
b
i
; ð1Þ
3ℓ
p
e
v
desired
front
motion
x
y
L
x
L
y
2ℓ
Direction of motion
Fig. 1. (Color online) Schematic illustration of our pedestrian model and
the initial conditions of numerical simulation. Pedestrians are distributed in a
rectangular system of L
x
L
y
.A fixed ratio of pedestrians R walk in the
positive x direction, whereas the others walk in the negative x direction. The
desired direction of a pedestrian is denoted by e. Each pedestrian has a shape
composed of three circles and a front direction p. The central circle is twice as
large as the other ones with the line joining the centers. The centers of the side
circles lie on the circumference of the central one, with perpendicular to p. Each
pedestrian also has a velocity vector v. The color represents the direction of p.
Journal of the Physical Society of Japan 89, 074003 (2020)
https://doi.org/10.7566/JPSJ.89.074003
074003-1
©
2020 The Physical Society of Japan
J. Phys. Soc. Jpn.
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