6. 6
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
7. 7
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?
16. 16
Lane formation: formulation
The
social
force
model
(Helbing2000)
periodic
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.
19. 19
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
20. 20
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
25. 25
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
28. 28
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
29. 29
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
mass
water
saltwater
30. 30
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
31. 31
Simulation: results
The
-me
evolu-on
of
the
momentum
density
The
Fourier
amplitude
v.s.
bo6leneck
width
-me[s]
momentum
bo6leneck
width
[m]
amplitude
32. 32
Oscillatory flow: open questions
-‐
A
type
of
nonlinear
self-‐excitable
oscillator?
-‐
Mathema-cal
model?
-‐
The
rela-on
to
the
fluid
oscillator?
-‐
Synchroniza-on?
34. 34
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 + fij
j≠i
∑
35. 35
The faster-‐‑‒is-‐‑‒slower effect: detail
mechanism?
modeling!
driving force
driving force
driving force
Suzuno
et
al.,
Phys.
Rev.
E
88,
052813
(2013).
36. 36
We
just
consider
the
par-cle
near
the
exit
and
its
equa-on
of
mo-on.
N
Analy-c
expression
of
the
flow
velocity
The outline of the modeling
37. 37
-‐
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 model
Suzuno
et
al.,
Phys.
Rev.
E
88,
052813
(2013).
38. 38
(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
39. 39
Model
Sta-onary
situa-on
l
the
analy-cal
expression
of
the
velocity!
[ ]
The model analysis
40. 40
Our
model
reproduces
the
simula-on
results.
our
model
simula-on
The model results
"faster"
is
"slower"
Suzuno
et
al.,
Phys.
Rev.
E
88,
052813
(2013).
41. 41
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
Suzuno
et
al.,
Phys.
Rev.
E
88,
052813
(2013).
42. 42
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!
Suzuno
et
al.,
Phys.
Rev.
E
88,
052813
(2013).
43. 43
(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"
44. 44
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?
45. 45
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?
46. 46
Critical discussion (2)
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.
47. 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.
47
Critical discussion (3)
circle
ellip-c
non-‐spherical
48. rather
social
48
Critical discussion (4)
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.
