Spontaneous emergence of counterclockwise vortex motion in assemblies of pedestrians roaming within an enclosure

Abstract

The emergence of coherent vortices has been observed in a wide variety of many-body systems such as animal flocks, bacteria, colloids, vibrated granular materials or human crowds. Here, we experimentally demonstrate that pedestrians roaming within an enclosure also form vortex-like patterns which, intriguingly, only rotate counterclockwise. By implementing simple numerical simulations, we evidence that the development of swirls in many-particle systems can be described as a phase transition in which both the density of agents and their dissipative interactions with the boundaries play a determinant role. Also, for the specific case of pedestrians, we show that the preference of right-handed people (the majority in our experiments) to turn leftwards when facing a wall is the symmetry breaking mechanism needed to trigger the global counterclockwise rotation observed.

Figure 1

Counterclockwise vortex motion in pedestrian experiments. (a) Snapshot of 18 pedestrians moving “fast” within the arena. The tails show 2-s trajectories. (b) Temporal evolution of the instantaneous angular momentum $L(t)$ averaged over all pedestrians for the experiment shown in A. The time intervals when the movement is “free” (the ones analyzed in this work) are shown in green. (c,d) Distributions of the instantaneous angular momentum $L(t)$ for 12, 18 and 24 pedestrians walking slow (S) in C, and fast (F) in D. (e) Temporal average of $L(t)$ for each free motion interval vs. experimental condition. (f–h) Average density fields for 12, 18, and 24 moving fast pedestrians respectively. (i–k) The corresponding fields of the velocity modulus $⟨\left|\overrightarrow{v}\right|⟩$. The value of $⟨\overrightarrow{v}⟩$ obtained at each location is indicated by arrows (the black arrow on the top right of figure I corresponds to 1 m/s). (l) Distribution of Voronoi areas for three experimental scenarios (“Exp” in the legend), along with the corresponding simulations (“Sim”) when using γ = 1.5 and no turning preference. (m) Distribution of Voronoi areas for the experimental case of 18 pedestrians moving fast and the four different scenarios simulated: with (D) and without (ND) damping, with (TP) and without (NTP) turning preference. Also, the Voronoi areas obtained in an elastic gas of spheres are reported for reference.

Figure 2

Pedestrian simulations without turning preference. (a–c) Temporal evolution of the instantaneous angular momentum $L(t)$ for 12 (panel A), 18 (panel B) and 24 (panel C) agents interacting dissipatively with the boundary (γ = 1.5). The time intervals considered in our analysis to avoid the inclusion of a possible initial transient (grey) are indicated with solid colours. (d) Bifurcation diagram in which the temporal averaged value of $L(t)$ obtained for each simulation run is represented as a function of the number of pedestrians in the arena (100 data points for each). Different colours correspond to simulations for γ = 1.5 and γ = 0. (e–g) Distributions of the instantaneous angular momentum $L(t)$ for γ = 0 (panel E), γ = 0.75 (panel F) and γ = 1.5 (panel G), for different number of simulated pedestrians (see legend of panel E). (h,i) Average density fields obtained for 12 (panel H), and 24 (panel I) pedestrians. (j,k) The corresponding velocity fields $⟨\left|\overrightarrow{v}\right|⟩$. Arrows show the value of $⟨\overrightarrow{v}⟩$ obtained at each location (the arrow on the top right of panel J corresponds to 1 m/s). (l) Vortex-like motion phase space. The absolute values of the angular momentum averaged for all frames and the 100 runs for each simulated condition are represented with different colours as a function of the damping parameter and number of pedestrians.

Figure 3

Pedestrian simulations with turning preference. (a) Sketch of a single experimental measurement of the turning preference. The trajectory of the analysed pedestrian is shown by circles of different colours (blue for the present position and red, orange and pink for positions after 1, 1.5 and 2 s respectively). The shadowed area corresponds to the region that must be empty to take into account this event (see methods). (b,c) Temporal evolution of the instantaneous angular momentum $L(t)$ averaged over all pedestrians in the arena for simulations of 12 (panel B), and 24 (panel C) pedestrians with a preferred turning preference and dissipative boundary interaction. The time lapses considered in our analysis to avoid the inclusion of a possible initial transient are shown in solid colours. (d) Bifurcation diagram in which the temporal averaged value of $L(t)$ obtained for each simulation run is represented as a function of the number of pedestrians in the arena (100 data points for each simulated condition). All simulations include turning preference and different colours correspond to cases with and without pedestrian-boundary damping, as indicated in the legend. (e,f) Distributions of the instantaneous angular momentum $L(t)$ when γ = 0 (panel E), and γ = 1.5 (panel F), for different numbers of simulated pedestrians (see legend of panel E). (g,h) Average density fields obtained for 24 agents when γ = 0 (panel G) and γ = 1.5 (panel H). The corresponding fields of the velocity modulus averaged for each case $⟨\left|\overrightarrow{v}\right|⟩$ are shown in (i,j). The value of $⟨\overrightarrow{v}⟩$ obtained at each location is indicated by arrows (the black arrow on the top right of panel I corresponds to 1 m/s). (k) Vortex-like motion phase space. The absolute values of the angular momentum averaged for all frames and the 100 runs for each simulated condition are represented with different colours (see colour bar for reference) as a function of the damping parameter and number of pedestrians.