Noise-Induced Stop-and-Go Dynamics in Pedestrian Single-File Motion - - PowerPoint PPT Presentation

Universität zu Köln

Andreas Schadschneider

Institut für Theoretische Physik Universität zu Köln www.thp.uni-koeln.de/~as

Noise-Induced Stop-and-Go Dynamics in Pedestrian Single-File Motion

joined work with Antoine Tordeux,Sylvain Lassarre, Jakob Cordes

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• observed in vehicular, bicycle and pedestrian motion
• succession of braking (shock) and acceleration

(rarefaction) sequences

• self-organized collective phenomenon
• have negative impact on safety, comfort, environment

Stop-and go waves

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• role of inertia in pedestrian models ?
• role of noise in pedestrian models ?

Stop-and go waves

• 1st order stochastic (toy) model for pedestrian dynamics
• new mechanism for stop-and-go waves

Stop-and-Go: Highway traffic

jam: v ≈ 0 free flow: v ≈ vmax separation into jams and free flow regions

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N=56 N=14 N=39 N=25

Stop-and-Go: Pedestrian dynamics

Wuppertal University FZ Jülich

separation into jams with v ≈ 0 and slowly moving regions with v = v(ρ) < vmax different mechanism compared to highway traffic ?

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Pedestrian models

Classification of models:

• description: microscopic ↔ macroscopic
• dynamics: stochastic ↔ deterministic
• variables: discrete ↔ continuous
• interactions: rule-based ↔ force-based
• fidelity: high ↔ low
• concept: heuristic ↔ first principles

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1st order vs. 2nd order models

typically deterministic 2nd order models force-based models

• ptimal-velocity

models 1st order models

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1st order vs. 2nd order models

role of inertia: damped harmonic oscillator

inertial mass damping (friction) driving force

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1st order vs. 2nd order models

role of inertia: damped harmonic oscillator inertia dominated damping dominated

4mk < b2

3 regimes

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1st order vs. 2nd order models

role of inertia: damped harmonic oscillator damping dominated: similar behavior to m = 0 dynamics described by 1st order equation

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1st order vs. 2nd order models

For pedestrian dynamics: almost instantaneous acceleration/stopping motion not inertia-dominated! 1st order model !!!

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1st order vs. 2nd order models

Overdamped social-force model:

(B. Maury, S. Faure 2019)

τ →0 :

dxi dt = Ui + Wij

j ≠ j

∑

for

Ui = desired velocity Wij = corrections to desired velocity

Problems with 2nd order models

• “tunneling” (penetration) of particles
• desired velocity can be exceeded
• oscillations when obstacles are approached

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all related to inertia effects! inertia “too strong” in most 2nd order pedestrian models (but: o.k. for vehicular traffic!) most cellular automata: no inertia, has to be implemented with transition rules

Problems with 2nd order models

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• ther issues with 2nd order models:
• superposition principle does not hold in general
• how to incorporate “decisions”?

(Seyfried, Sieben 2019) (talk by Bailo yesterday)

Problems with 2nd order models

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History never repeats?

Problems with 2nd order models

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Problems in 2nd order fluid models:

• Isotropy: no distinction between interactions with

following and preceding cars

• characteristic speed can be larger than the

average velocity (flow in front of a car is influenced by the traffic behind)

• Wrong-way travel: negative velocities possible
• Form of jam fronts unrealistic

Problems with 2nd order models

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introduction of a new pressure term and second conservation law avoids problems

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Stop-and-go waves in vehicular traffic

• stop-and-go in certain parameter

regimes

• homogeneous solution becomes

unstable

• phase transition from

homogeneous to heterogeneous configurations

• periodic solutions (limit-cycle)
• metastable or even chaotic

dynamics; hysteresis, capacity drop

single perturbation at time t0

Unstable models with inertia (2nd order):

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Stop-and-go waves for pedestrians

Guiding principles:

• 1st order model (no inertia effects)
• including stochasticity
• shows stop-and-go waves
• simple as possible

Role of stochasticity

• intrinsic stochasticity

– essential for dynamics – deterministic limit usually not realistic – persons act differently even in the same situation – reflects lack of knowledge about the true interactions

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• additive noise

– does not lead to a qualitative change of the underlying deterministic dynamics – „smears out“ trajectories – avoidance of unrealistic states

Nature of the noise

23

Data: ped.fz-juelich.de/database

(empirical) power spectrum of pedestrian speed ~ f-2 Brownian noise

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1st order Optimal-Velocity model (Pipes - Newell model) with additive Brownian noise (for 1d motion)

linear OV function V(d) = (d –l)/T (only congested flow) W(t) = Wiener process (Brownian motion)

1st order model

l = pedestrian size T = time gap α = noise amplitude β = noise relaxation

(Langevin equation for the noise) (OV model with noise)

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1st order model

alternative derivation: Optimal-velocity model with reaction time and anticipation expansion: full-velocity difference model + add white noise ξn(t) with : : Brownian noise

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1st order model

alternative derivation: Overdamped social-force model:

dxi dt = Ui + Wij

j ≠ j

∑

with corrections Wij to desired velocity Ui replace corrections by stochastic terms

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1st order model

homogeneous solution (for n pedestrians):

Real part Imaginary part

Unstable

λ2 λ1

θ −2/T −1/T −1/β −1/T 1/T θ = 0

stable for all n !!! xj+1(t) – xj(t) = L/n stability analysis:

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1st order model: Trajectories

n = 28

Time (s)

40 80 120

n = 45 n = 62

Real data 1 2 3 4

Space (m) Time (s)

40 80 120 1 2 3 4

Space (m)

1 2 3 4

Space (m)

Simulation

experiment: simulation:

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1st order model: Trajectories

wave speed: c = - l/T pedestrian speed: v = (L/n- l)/T

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1st order model: Correlations

Period = nT: Oscillations at longest wavelength for any α,β > 0

nT = L/(v-c)

autocorrelation of distance spacing

20 40 60 80 100 120

• 0.4

0.0 0.4 0.8

Time − tS (s) Autocorrelation 1/f = nT deterministic model

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1st order model: Correlations

autocorrelation of distance spacing for various noise parameters β

20 40 60 80 100 120 0.0 0.4 0.8

Time − tS (s) Autocorrelation β (s) 1.25 5 20

frequency only depends on T and n for pedestrians: β = 5s is most realistic

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Noise-induced stop-and-go

Noise-induced stop-and-go:

• homogeneous solution stable for

ALL parameter values

• Oscillation of the system at own

deterministic frequency (longest wavelength) due to stochasticity Linear stochastic system having unique stationary distribution No instability/metastability, non-linearity, inertia (or reaction time), neither as phase transition New mechanism for stop-and-go

Summary

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• 1st order pedestrian model
• unique homogeneous stationary state,

stable for all parameter values

• correlated noise “kicks” system out of

stable state

• stop-and-go not induced by instability
• no phase transition
• mechanism different from that in other

continuous models

• closer to the mechanism in stochastic

cellular automata models

• clarify role of noise and inertia in

pedestrian dynamics