Emergent Structural Mechanisms for High-Density Collective Motion Inspired by Human Crowds

Collective motion of large human crowds often depends on their density. In extreme cases like heavy metal concerts and black Friday sales events, motion is dominated by physical interactions instead of conventional social norms. Here, we study an active matter model inspired by situations when large groups of people gather at a point of common interest. Our analysis takes an approach developed for jammed granular media and identifies Goldstone modes, soft spots, and stochastic resonance as structurally driven mechanisms for potentially dangerous emergent collective motion.

Figure 1

Large dense groups of people give rise to emergent collective motion. (a) Attendees gather near the stage at a heavy metal concert. Credit: Ulrike Biets. (b) Customers gather for black Friday sale to purchase low-cost consumer goods. Credit: Jerry Bailey. (c) Simulated trajectories of SPPs aggregating near a point of interest $\mathcal{P}$ located at the right-most edge of a simulation box. (d) Closeup of trajectories show SPPs self-organize into a densely packed disordered aggregate.

Figure 2

Eigenmode analysis of asocial model for high-density human crowds. (a) Eigenvalue spectrum ${\lambda }_m$ of the displacement correlation matrix exhibits scaling properties between ${\lambda }_m\sim m^{-1}$ and $\sim m^{-2}$ (black dashed lines). Low $m$ eigenmodes in both $x$ (blue) and $y$ (orange) directions are larger than a random matrix model (${\mathrm{RM}}_{\sigma }$), and thus describe correlated motion. (b) Snapshot of instantaneous displacements $d\overrightarrow{r}$ and example vector fields for various eigenmodes. Lower $m$ eigenmodes are more spatially correlated than higher $m$. (c) A heatmap of the polarization correlation function for the first 10 eigenmodes as a function of distance $d$ between SPPs. Black line is where the correlation function decays to 0 demonstrating a long-range highly correlated mode for $m=1$.

Figure 3

Soft spots within the crowd undergo large displacements. (a) SPPs are shown as disks. Soft spots near the core of the aggregate colocalize with SPPs that displace the most in real dynamics. This region is subject to large structural rearrangements when the system is perturbed, and is likely a region where injury can occur. Apparent soft spots along the periphery are artifacts due to underconstrained edge effects. (b) Averaging over all simulation runs show soft spots generally occur near the core of the aggregate at radial distance $\rho \approx (2\pm 1)$ away from $\mathcal{P}$. (c) Structure factor $g(\mathcal{r})$ measures the pair-wise SPP distribution of distances between particles and reveals structural features distinguishing SPPs in soft spots that suggest why they are subject to large displacements.

Figure 4

Introducing a fraction $f$ of agitated SPPs with variance ${\sigma }_a$ in ${\overrightarrow{F}}_i^{\text{noise}}$ probes structural origins of collective motion. Each heat map is the polarization fluctuation correlation function for the first 10 eigenmodes as a function of distance $d$ [same as Fig. 2]. Low fluctuations (white background) preserve the long-range highly correlated Goldstone mode near $m=1$. High fluctuations (dark gray background) destroy long-range correlated modes. Intermediate fluctuations (light gray background) add new modes with long-range correlations, indicating stochastic resonance.