Traffic Jams, Pedestrian Streams, Escape Panics, and Supply Chains

New Insights from Many-Particle-Physics

Prof. Dr. rer. nat. habil. Dirk Helbing

Institute for Economics and TrafficFaculty of Traffic Sciences

Dresden University of Technology

www.helbing.org

Literature:

73 (4), 1067 – 1141 (2001).D.H., Traffic and related self-driven many particle systems, Reviews of Modern Physics

Some Analogies between Pedestrian Crowds and Fluids

Footprints in snow look similar to streamlines of fluids.There are gaseous (free), fluid (obstructed), and solid (immobile) states.Passing through a standing pedestrian crowd leads to river-like streams. In pushy pedestrian crowds one can observe shock waves.Similarities with granular media are dominating at high pedestrians densities.

Are there Regular Phenomena, i.e. “Laws of Nature”?

Fluid-Induced Particle Size Segregation in Sheared Granular Media

S. B. Santra, S. Schwarzer, and H. J. Herrmann, Phys. Rev. E 54, 5066 (1996).

Social Force Concept

Pedestrians are confronted with standard situations.The reactions to these are rather automatic.We assume a more ore less optimized behavior regarding the avoidance of collision and time delays. This allows us to describe the average behavior in a mathe-matical way.In addition, we have to consider fluctuations.We model the systematic contribution to the acceleration by a superposition of several (Non-Newtonian) forces reflectingdifferent motivations and influence factors.

Experiments Regarding the Superposition of Forces in Conflict Situations

Social Force Model

Acceleration Equation for Pedestrians:

( ){)()()()()()()(

)(

0 ttFtFtFtvtevmdt

tvdm ik

attik

bi

ij

wwijiii

i

iii ξ

τ

r

43421321

r

43421

r

444 3444 21

rr

321

r++++−= ∑∑

≠

Direction Desired Velocity; Desired Time; onAccelerati Mass; Speed; Time; Place;

=

======

i

iiiii

evτmvtx

r

rr 0

Equation of Motion:

)()( tvdt

txdi

i rr

=

Driving ForceAcceleration InteractionsBorders,

FireFluctuations

Attractions

321

r

321

r

321

rr)()()()( tFtFtFtF att

ijph

ijpsy

ijww

ij ++=

Psychological Repulsion

Physical Interactions

Attractions between People

Social Force Model

Specification of the Interaction Forces:

( )

||||)(

)(

)()()(

]/)exp[()()],()([)],()([)(

ij

ijattij

ijtjiijijijijij

phij

ijijijijijìpsy

ijpsy

ij

xxxx

CtF

tvdrndrktF

nBdrAtvtvtvtxtxFtF

rr

rrr

44 344 21

r

4434421

rr

rrrrrrrr

−−

=

∆−Θ+−Θ=

−=−−=

e.g.

e.g.

e.g.

κ

Constants Large Velocity; Tangential Relative

; otherwise , for ; Distance; ||

Radii; of Sum Strength; Attraction n;Interactio of Range Repulsion;

==−=

=⊥−==−=

=+====

∆

>Θ

κ,)(00)(,/)(||

ktvvvxxxntdxxnxxd

rrrCBA

ijijtji

ijijijjiijjiij

jiij

rrr

rrrrrrr

Compression Friction

Self-Organization of Pedestrians

Optimization of Pedestrian Facilities

Conventional Improved

Evolutionary Optimization

Original Multiplication

Selection Mutation(Random Variations)

EvaluationTest

(Simulation)Performance Criterion:

Efficiency

∑ ⋅=

α α0)(1

vetv

NE αα

rr

Evolutionary Optimization of a Bottleneck

Noise-Induced Ordering

Small Noise

Medium Noise

Large Noise

Role of Fluctuations

Small Fluctuations: Lane Formation

Large Fluctuations: ‘‘Freezing by Heating‘‘

Ensemble-Averaged Efficiency

D.H., I. Farkas, T. Viscek, Phys. Rev. Lett. 84, 1240 (2000).

Reminder: The temperature is proportional to

the velocity variance.

Escape Panics

Physical Interactions and Friction Effects due to Uncontrolled Rush and Pushy Behavior

Application to the Simulation of Evacuation

Scenarios

Faster-is-Slower Effect

Phantom Panics

Optimal Escape Strategy and Collective Problem Solving

Mixture of Individualistic Behavior and Herding:

Inefficient Usage of Doors due to Herding Effect

D. H., I. Farkas, and T. Viscek, Nature 407, 487 (2000).

Emergence of a “Phantom Traffic Jam“

J. Treiterer et al. (1966, 1970, 1974).

Velocity-Density-Relation

Speed Limit

1-min-Averages

Mean Values

Fit Function

Instability of Traffic Flow

Sub- and Supercritical Perturbations

J.M. de Castillo, TGF’97 (1998).

subcritical

supercritical

Metastability of Traffic Flow

Breakdown of Traffic due to a Supercritical Reduction of Traffic Flow

Phase Diagram of Traffic States at Bottlenecks

D. H. et al., Science 282, 2001 (1998); Physical Review Letters 82, 4360 (1999).

Empirical Representatives of Congested Traffic States

Avoidance of Traffic Breakdowns- Intelligent Speed Limits- On-Ramp Control

Breakdown of Traffic With Control

Illustration of an On-Ramp-Control

- Driver-Assistance Systems

Model Ingredients

Many similar units (particles, pedestrians, vehicles, individuals, …).

The units are externally or internally driven, i.e. there is some energy input, e.g. they can move.

Units are non-lineary interacting, i.e., under certain conditions, small variations can have large effects. The system behavior is then dominated by the interactions rather than by the boundary conditions (the external control).

There is a competition for limited ressources such as space, time, energy, or money.

Each unit has a certain extension in space or time.

When units come too close, they have frictional effects and/or obstruct each other.

Conclusions from traffic models are relevant for the functionality, stability, reliability, and efficieny of societies, organizations, administrations, companies, production processes, etc.

Paradoxical Phenomena

Phantom traffic jamsPhantom panicsFaster-is-slower effectNoice-induced self-organization and orderingFreezing by heatingAttraction effects in driven systems with repulsive forcesMore efficiency with less ressources

Similar breakdown, jamming and herding effects can be observed in biological systems, organizations, companies, markets, administrations, societies, politics, economy and science.

Conclusions: The theoretical understanding of traffic dynamics is a good starting-point for studying elementary human interactions under experimental conditions.

It is also suitable for managing logistic problems (e.g. optimizing the throughput in the production of computer chips).

Physical Properties and Phenomena

(Auto-)Solitons

(Bose-Einstein-)Condensation

Catalysis

Chaos

Clogging

Complex Dynamics

Crystallization

Dislocations

Granular Flows

History-/Path-Dependence

Hysteresis

Indirect Interactions

Meta-/Multi-Stability

Noise-Induced Ordering

Non-Equilibrium States

Non-Linear Waves

Optimal Self-Organization

Pair Interactions

Phase Transitions

Power Laws

Reaction-Diffusion-Dynamics

Rotation

Scaling Laws

Segregation

Self-Organization

Shock Waves

Synchronization

Universality

Physical Concepts

Active Brownian Particles

Boltzmann Equation

Cellular Automata

Driven Many-Particle Systems

Fokker-Planck Equation

Ginzburg-Landau Equation

Korteweg-de-Vries Equation

Langevin Equation

Master Equation

Micro-Macro-Link

Molecular Dynamics

Phase Diagram

Self-Organized Criticality

Spin-Systems

Physical Fields

Fluid Dynamics

Kinetic Gas Theory

Mechanics

Non-Equilibrium Thermodynamics

Non-Linear Dynamics

Quantum Theory

Soft Matter

Solid State Physics

Statistical Physics