Katsuhiro Nishinari Faculty of Engineering, University of Tokyo Jammology Physics of self-driven...

Katsuhiro NishinariKatsuhiro NishinariFaculty of Engineering, University of Faculty of Engineering, University of

TokyoTokyo

JammologyJammology Physics of self-driven particlesPhysics of self-driven particles

Toward solution of all jamsToward solution of all jams

Outline

Introduction of “Jammology”

Self-driven particles, methodology

Simple traffic model for

ants and molecular motors

Conclusions

Jams Everywhere

School, Herd, Flock, etc.

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

What are self-driven particles (SDP)?

D. Helbing, Rev. Mod. Phys. vol.73 (2001) p.1067.D. Chowdhury, L. Santen and A. Schadschneider,Phys. Rep. vol.329 (2000) p.199.

Conventional mechanics,

or statistical physics cannot

be directly applicable. Rule-based approach

(e.g., CA model)

Numerical computations

Exactly solvable models

(ASEP,ZRP)

Jammology=Collective dynamics of SDP Text book 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.

Subjects of Jammology

Conventional theory of Jam 　　　 = Queuing theory

In Out

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 jamＡＳＥＰ（ Asymmetric Simple Exclusion

Process ）

t

1t

0 1 0 1 1 0 0 1 0 1 1 1 0 0 0

0 0 1 1 0 1 0 0 1 1 1 0 1 0 0

Rule ： move forward if the front is empty

This is an exactly solvable model, i.e., we can calculatedensity 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

rp

This research has not been recognized until recently.

Fundamental diagram of ASEP

• Flow-density ・ ・ ・ ・ ・ Particle-hole symmetry

• Velocity-density ・ ・ ・ ・ ・ monotonic decrease

with periodic boundary condition

)1(q4112

1 vJJIn the stationary state of TASEP,flow - density relation is

M.Kanai, K.Nishinari and T.Tokihiro,J. Phys. A: Math. Gen., vol.39 (2006) pp.9071-9079..

Ultradiscrete method

Euler-Lagrange transformation

Macroscopic model

Burgers equation

ASEP(Rule 184)

OV model

CA model

Car-following model

Phys.Rev.Lett., vol.90 (2003) p.088701

J.Phys.A, vol.31 (1998) p.5439

xxxt uuuu 2

Ultradiscrete method reveals the relation between different traffic models!

Toward solution of all kind of jams! Vehicular traffic

cars, bus, trains,… Pedestrians Jams in our body

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

Traffic in ant-trail

Ant trail traffic models and experiments Experiments and theory

Differential equations

)())(()()(

2

2

txUdt

tdx

dt

txd

)(),(),(),( 2 xgtxftxD

t

tx

Ant

pheromonal field

Langevin type equation

),( tx: ant density at x)(xf : evaporation rate

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)

E.M.Rauch, M.M.Millonas and D.R.Chialvo, Phys.Lett.A (1995)

Ant trail CA model

q q Q

1. Ants movement

2. Update Pheromone(creation & diffusion)

Dynamics:

f f fParameters: q < Q, f

One lane, uni-directional flow

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

ｆ　　　　 ｆ　　　 f f

Loose cluster formation = buses bunching up together

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　　　

D.Helbing, I.Farkas and T.Vicsek, Nature (2000).

Floor field model (CA model)

Social force model (Continuous model)

C.Burstedde, K.Klauck, A.Schadschneider,J.Zittartz, Physica A (2001)

Floor field CA Model

Idea ： Footprints = Feromone

Long range interaction is

imitated by local interaction

through „memory on a floor“.

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?

Details of FF model

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 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

ijp

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 ・・・

Herding behaviour = choose the cell that has more footprints Store global information to local cells

4

2

1

1

Static FF (SFF) = Dijkstra metric Distance to the destination is recorded at each cell

This is done by Visibility Graph and Dijkstra method.Two exits with four obstacles

0 20 40 60 80 1000

20

40

60

80

100

One exit with a obstacle

K. Nishinari, A. Kirchner, A. Namazi and A. Schadschneider, IEICE Trans. Inf. Syst., Vol.E87-D (2004) p.726.

Problem of “Zone partition”

By using SFF

Application of SFF

Which door is the nearest?

The ratio of an area to the totalarea determines the number ofpeople who use the door in escaping from this room.

Probability of movement

)exp()exp()exp( ijIijsijDij IkSkDkp

ijI

ijS

ijDDistance 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

1. Update DFF (dissipation & diffusion)

2. Calculate and determine the target cell

3. Resolution of conflict 　　　　　　　　　　　　

4. Movement

5. Add DFF +1

ijp

Parameter ]1,0[

1

All of them cannot move.

One of them can move.

Resolution of conflict

Initial: Calculate SFF

Meanings of parameters in the model kS kD 　 　 　　　　　　　　　　　

kS large: Normal (kS small: Random walk)

kD large: Panic

　　　 kD / kS ・・・ Panic degree (panic parameter)

　 large: competition

small: coorporation

)exp()exp( ijsijDij SkDkp

Simulations using inertia effect There is a minimum in the evacuation time when

the effect of inertia is introduced.

People becomeless flexible toform arches.(Do not care others!) People become flexible

to avoid congestion.

SFF is strongly disturbed.

Simulation Example: Evacuation at Osaka-Sankei Hall

Jams near exits.

Hamburg airport in Germany

Influence of Obstacle3.0,0,10 DS kkPlaced an obstacle near exit.

1offset

Center

None

2offset

Intensive competition.

A.Kirchner, K.Nishinari, and A.Schadschneider,Phys. Rev. E, vol.67 (2003) p.056122.D.Helbing, I.Farkas and T.Vicsek, Nature, vol.407 (2000) p.487.

If an obstacle is placed asymmetrically, total evacuation time is reduced ！

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

• Jammology = Math. , Physics and Engineering

Conclusions

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