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

Noise-Induced Stop-and-Go Dynamics in Pedestrian Single-File Motion Andreas Schadschneider Institut für Theoretische Physik Universität zu Köln www.thp.uni-koeln.de/~as joined work with Antoine Tordeux ,Sylvain Lassarre, Jakob Cordes Universität zu Köln

Stop-and go waves • 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 Universität zu Köln

Stop-and go waves • role of inertia in pedestrian models ? • role of noise in pedestrian models ? • 1 st order stochastic (toy) model for pedestrian dynamics • new mechanism for stop-and-go waves Universität zu Köln

Stop-and-Go: Highway traffic jam: v ≈ 0 free flow: v ≈ v max separation into jams and free flow regions

Stop-and-Go: Pedestrian dynamics separation into jams with v ≈ 0 and slowly moving regions with v = v( ρ ) < v max different mechanism compared to highway traffic ? N=56 N=14 N=25 N=39 Universität zu Köln Wuppertal University FZ Jülich

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 Universität zu Köln

1 st order vs. 2 nd order models typically deterministic 2 nd order models force-based models optimal-velocity models 1 st order models Universität zu Köln

1 st order vs. 2 nd order models role of inertia: damped harmonic oscillator driving force inertial mass damping (friction) Universität zu Köln

1 st order vs. 2 nd order models role of inertia: damped harmonic oscillator 3 regimes damping dominated 4mk < b 2 inertia dominated Universität zu Köln

1 st order vs. 2 nd order models role of inertia: damped harmonic oscillator damping dominated: similar behavior to m = 0 � dynamics described by 1 st order equation Universität zu Köln

1 st order vs. 2 nd order models For pedestrian dynamics: almost instantaneous acceleration/stopping � motion not inertia-dominated! � 1 st order model !!! Universität zu Köln

1 st order vs. 2 nd order models Overdamped social-force model: (B. Maury, S. Faure 2019) dx i ∑ for τ → 0 : dt = U i + W ij j ≠ j U i = desired velocity W ij = corrections to desired velocity Universität zu Köln

Problems with 2 nd order models • “tunneling” (penetration) of particles • desired velocity can be exceeded • oscillations when obstacles are approached all related to inertia effects! inertia “too strong” in most 2 nd order pedestrian models (but: o.k. for vehicular traffic!) most cellular automata: no inertia, has to be implemented with transition rules Universität zu Köln

Problems with 2 nd order models other issues with 2 nd order models: • superposition principle does not hold in general (Seyfried, Sieben 2019) • how to incorporate “decisions”? (talk by Bailo yesterday) Universität zu Köln

Problems with 2 nd order models History never repeats? Universität zu Köln

Problems with 2 nd order models Problems in 2 nd 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 Universität zu Köln

Problems with 2 nd order models introduction of a new pressure term and second conservation law avoids problems Universität zu Köln

Stop-and-go waves in vehicular traffic Unstable models with inertia (2 nd order): • 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 t 0 Universität zu Köln

Stop-and-go waves for pedestrians Guiding principles: • 1 st order model (no inertia effects) • including stochasticity • shows stop-and-go waves • simple as possible Universität zu Köln

Role of stochasticity • additive noise – does not lead to a qualitative change of the underlying deterministic dynamics – „smears out“ trajectories – avoidance of unrealistic states • 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 Universität zu Köln

Nature of the noise (empirical) power spectrum of pedestrian speed ~ f -2 Brownian noise Data: ped.fz-juelich.de/database 23

1 st order model 1 st order Optimal-Velocity model (Pipes - Newell model) with additive Brownian noise (for 1d motion) (OV model with noise) (Langevin equation for the noise) linear OV function V(d) = (d –l)/T (only congested flow) W(t) = Wiener process (Brownian motion) l = pedestrian size α = noise amplitude T = time gap β = noise relaxation Universität zu Köln

1 st order model alternative derivation: Optimal-velocity model with reaction time and anticipation expansion: full-velocity difference model + add white noise ξ n (t) with : : Brownian noise Universität zu Köln

1 st order model alternative derivation: dx i ∑ Overdamped social-force model: dt = U i + W ij j ≠ j with corrections W ij to desired velocity U i � replace corrections by stochastic terms Universität zu Köln

1 st order model homogeneous solution (for n pedestrians): x j+1 (t) – x j (t) = L/n 1 /T Unstable Imaginary part λ 2 stability θ = 0 0 analysis: θ λ 1 − 1 /T − 2 /T − 1 /T − 1 / β 0 stable for all n !!! Real part Universität zu Köln

1 st order model: Trajectories n = 28 n = 45 n = 62 120 experiment: Real data 80 Time (s) 40 0 120 Simulation simulation: 80 Time (s) 40 0 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 Space (m) Space (m) Space (m) Universität zu Köln

1 st order model: Trajectories wave speed: c = - l/T pedestrian speed: v = (L/n- l)/T Universität zu Köln

1 st order model: Correlations autocorrelation of distance spacing deterministic model 0.8 Autocorrelation 1 /f = nT 0.4 0.0 -0.4 0 20 40 60 80 100 120 Time − t S (s) Period = nT: Oscillations at longest wavelength for any α , β > 0 nT = L/(v-c) Universität zu Köln

1 st order model: Correlations autocorrelation of distance spacing for various noise parameters β β (s) Autocorrelation 0.8 1.25 5 0.4 20 0.0 0 20 40 60 80 100 120 Time − t S (s) � frequency only depends on T and n for pedestrians: β = 5s is most realistic Universität zu Köln

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 Universität zu Köln

Summary • 1 st 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 Universität zu Köln