Jammology
Physics of self-driven particles
Toward solution of all jams
Katsuhiro Nishinari
Faculty of Engineering, University of Tokyo
Outline

Introduction of “Jammology”
Self-driven particles, methodology

Simple traffic model for
ants and molecular motors

Conclusions
Jams Everywhere
School, Herd, Flock, etc.
What are self-driven particles (SDP)?

Vehicles, ants, pedestrians, molecular motors…

Non-Newtonian particles,
which do not satisfy three laws of motion.
ex. 1) Action  Reaction,
“force” is psychological
2) Sudden change of motion
D. Helbing, Rev. Mod. Phys. vol.73 (2001) p.1067.
D. Chowdhury, L. Santen and A. Schadschneider,Phys. Rep. vol.329 (2000) p.199.
Jammology=Collective dynamics of SDP
Text book of “Jammology”


Conventional mechanics,
or statistical physics cannot
be directly applicable.
Rule-based approach
(e.g., CA model)
Numerical computations
Exactly solvable models
(ASEP,ZRP)
Subjects of Jammology






Vehicles
car, bus, bicycle, airplane,etc.
Humans
Swarm, animals,
ant, bee, cockroach, fly, bird,fish,etc.
Internet packet transportation
Jams in human body
Blood, Kinesin, ribosome, etc.
Infectious disease, forest fire, money, etc.
Conventional theory of Jam
= Queuing theory
Out
In
Service
Breakdown of balance of in and out causes Jam.
What is NOT considered in Queuing theory
Exclusion effect of finite volume of SDP
ＡＳＥＰ model can deal
the exclusion!
ASEP=A toy model for jam
ＡＳＥＰ（Asymmetric Simple Exclusion Process）
Rule：move forward if the front is empty
t
0 1 0 1 1 0 0 1 0 1 1 1 0 0 0
t 1
0 0 1 1 0 1 0 0 1 1 1 0 1 0 0
This is an exactly solvable model, i.e., we can calculate
density distribution, flux, etc in the stationary state.
Who considered ASEP?
Macdonald & Gibbs, Biopolymers, vol.6 (1968) p.1.
Protein composition process of Ribosomes on mRNA

pr
This research has not been recognized until recently.

Fundamental diagram of ASEP
with periodic boundary condition
In the stationary state of TASEP,
flow J - density relation is
J

1
 v  1  1  4 q  (1   )
2
• Flow-density・ ・ ・ ・ ・Particle-hole symmetry
• Velocity-density・ ・ ・ ・ ・monotonic decrease
M.Kanai, K.Nishinari and T.Tokihiro,
J. Phys. A: Math. Gen., vol.39 (2006) pp.9071-9079..

Ultradiscrete method reveals the relation
between different traffic models!
J.Phys.A, vol.31 (1998) p.5439
ut  u xx  2uu x
Ultradiscrete method
ASEP(Rule 184)
Burgers equation
CA model
Macroscopic model
Euler-Lagrange transformation
Phys.Rev.Lett., vol.90 (2003) p.088701
OV model
Car-following model
Toward solution of all kind of jams!



Vehicular traffic
cars, bus, trains,…
Pedestrians
Jams in our body
Traffic in ant-trail
Ants drop a chemical (generically called pheromone) as
they crawl forward. Other sniffing ants pick up the smell of the
pheromone and follow the trail.
with periodic boundary conditions
Ant trail traffic models and experiments

Experiments and theory
1) M. Burd, D. Archer, N. Aranwela and D.J. Stradling, American Natur. (2002)
2) I.D.Couzin and N.R.Franks, Proc.R.Soc.Lond.B (2002)
3) A. Dussutour, V. Fourcassie, D.Helbing and J.L. Deneubourg, Nature (2004)

Differential equations
E.M.Rauch, M.M.Millonas and D.R.Chialvo, Phys.Lett.A (1995)
Ant
Langevin type equation
pheromonal field
 ( x, t )
d 2 x(t )
dx(t )
 
 U ( ( x))   (t )
2
dt
dt
 ( x, t )
 D 2 ( x, t )  f ( x, t )  g ( x)
t
f : evaporation rate
 (x ): ant density at x
Ant trail CA model
One lane, uni-directional flow
Dynamics: 1. Ants movement
2. Update Pheromone(creation & diffusion)
q
Parameters: q < Q, f
q
f
Q
f
f
D. Chowdhury, V. Guttal, K. Nishinari and A. Schadschneider,
J.Phys.A:Math.Gen., Vol. 35 (2002) pp.L573-L577.
Bus Route Model

Bus operation system＝In fact the ant CA！
The dynamics is the same as the ant model
Q
Q
q
Loose cluster formation = buses bunching up together
ｆ
ｆ
f
f
Modeling of pedestrians

Basic features of collective behaviours of pedestrians
1) Arch formation at exit
2) Oscillation of flow at bottleneck
3) Lane formation of counterflow at corridor

Models for evacuation
Social force model (Continuous model)
D.Helbing, I.Farkas and T.Vicsek, Nature (2000).
Floor field model (CA model)
C.Burstedde, K.Klauck, A.Schadschneider,J.Zittartz, Physica A (2001)
Floor field CA Model
C.Burstedde, K.Klauck, A.Schadschneider,J.Zittartz, Physica A, vol.295 (2001) p.507.
Pedestrians in evacuation
= herding behavior
= long range interaction
For computational efficiency, can we describe
the behavior of pedestrians by using local
interactions only?
Idea： Footprints = Feromone
Long range interaction is
imitated by local interaction
through „memory on a floor“.
Details of FF model

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Floor is devided into cells (a cell=40*40 cm2)
Exclusion principle in each cell
Parallel update
A person moves to one of nearest cells with
the probability pij defined by „floor field(FF)“.
Two kinds of FF is introduced in each cell:
1) Dynamic FF・・・footprints of persons
2) Static FF・・・Distance to an exit
Dymanic FF (DFF)
Number of footprints on each cell
 Leave a footprint at each cell whenever a
person leave the cell
 Dynamics of DFF
dissipation＋diffusion
dissipation・・・ 
diffusion・・・ 
4
1


Herding behaviour =
1
choose the cell that has more footprints
Store global information to local cells
2
Static FF (SFF)

= Dijkstra metric
Distance to the destination is recorded at each cell
100
80
60
40
20
0
0
20
40
60
80
One exit with a obstacle
100
Two exits with four obstacles
This is done by Visibility Graph and Dijkstra method.
K. Nishinari, A. Kirchner, A. Namazi and A. Schadschneider,
IEICE Trans. Inf. Syst., Vol.E87-D (2004) p.726.
Application of SFF
Problem of “Zone partition”
Which door is the nearest?
By using SFF
The ratio of an area to the total
area determines the number of
people who use the door in
escaping from this room.
Probability of movement
pij  exp( k D Dij ) exp( k s Sij ) exp( k I I ij )
S ij
Dij
I ij
Distance between the cell (i,j) and a door.
Number of footprints at the cell (i,j).
Set I=1 if (i,j) is the previous direction of motion.
Update procedure
Initial: Calculate SFF
1.
2.
3.
4.
5.
Update DFF (dissipation & diffusion)
Calculate pij and determine the target cell
Resolution of conflict
Movement
Add DFF +1
Resolution of conflict
Parameter

1 
  [0,1]
All of them cannot move.
One of them can move.
Meanings of parameters in the model
kS kD

pij  exp( k D Dij ) exp( k s Sij )

kS large: Normal (kS small: Random walk)
kD large: Panic
kD / kS ・・・Panic degree (panic parameter)

 large: competition

small: coorporation
Simulations using inertia effect

There is a minimum in the evacuation time when
the effect of inertia is introduced.
SFF is strongly
disturbed.
People become
less flexible to
form arches.
(Do not care others!)
People become flexible
to avoid congestion.
Simulation Example:
Evacuation at Osaka-Sankei Hall
Jams near exits.
Hamburg airport in Germany
Influence of Obstacle
Placed an obstacle near exit.
k S  10, k D  0,   0.3
2offset
None
Center
1offset
If an obstacle is placed asymmetrically,
total evacuation time is reduced！
Intensive
competition.
D.Helbing, I.Farkas and T.Vicsek, Nature, vol.407 (2000) p.487.
A.Kirchner, K.Nishinari, and A.Schadschneider,Phys. Rev. E, vol.67 (2003) p.056122.
Conclusions
• Traffic Jams everywhere
= Jammology is interdisciplinary research
among Math. , Physics and Engineering!
Examples
• Ant trail CA model is proposed by extending
ASEP. The model is well analyzed by ZRP.
• Non-monotonic variation of the average speed of
the ants is confirmed by robots experiment.
• Traffic jam in our body is related to diseases.
• Modelling molecular motors
Conclusions


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Ant trail CA model is proposed by extending
ASEP. The model is well analyzed by ZRP.
Non-monotonic variation of the average speed
of the ants is confirmed by robots experiment.
FF model is a local CA model with memory,
which can emulating grobal behavior.
FF model is quite efficient tool for simulating
pedestrian behavior.
Jammology = Math. , Physics and Engineering