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 1 1 This is an exactly solvable model, i.e., we can calculate density distribution, flux, etc in the stationary state.

This research has not been recognized until recently. Who considered ASEP? Macdonald & Gibbs, Biopolymers, vol.6 (1968) p.1. Protein composition process of Ribosomes on mRNA This research has not been recognized until recently.

Fundamental diagram of ASEP with periodic boundary condition In the stationary state of TASEP, flow - density relation is 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 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 Differential equations 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 Langevin type equation pheromonal field : evaporation rate : ant density at x

Ant trail CA model f f f q q Q One lane, uni-directional flow Dynamics: Ants movement Update Pheromone(creation & diffusion) q q Q Parameters: q < Q, f f 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 ｆ ｆ f f Bus operation system＝In fact the ant CA！ Q Q q 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 Idea： Footprints = Feromone 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 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

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

Static FF (SFF) = Dijkstra metric Distance to the destination is recorded at each cell One exit with a obstacle 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.

Problem of “Zone partition” Application of SFF Problem of “Zone partition” By using SFF Which door is the nearest? 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 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 Update DFF (dissipation & diffusion) Initial: Calculate SFF Update DFF (dissipation & diffusion) Calculate and determine the target cell Resolution of conflict　　　　　　　　　　　　 Movement Add DFF +1 Resolution of conflict Parameter All of them cannot move. One of them can move.

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

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 If an obstacle is placed asymmetrically, Placed an obstacle near exit. 2offset None Center 1offset Intensive competition. If an obstacle is placed asymmetrically, total evacuation time is reduced！ 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 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