Swarming and Flocking Unified Through Aggregation

Abstract

Natural flocks (aligned) and swarms (non-aligned) both exhibit features of near-criticality, challenging their treatment as two ends of the same phase transition. We present a model for the aggregation of active individuals, in which their velocities align as a byproduct of achieving stable cohesion. In our framework, individuals move in open space and possess differing self-propelling velocities. Furthermore, velocity fluctuations are triggered by individual errors when following the aggregation rules. Notably, the system exhibits scale invariance, which is shown to be rooted in the model's definition -- a feature that we label as structural criticality. Finally, we show the emergence of a striking regime where spatial and orientational coherence decouple. That is, the system can achieve states of high and low polarization while maintaining spatial homogeneity.

Figures (5)

• Figure 1: System's long-term linear size as a function of different coupling strengths (left column) and noise amplitudes (right column). The self-propelling velocities were chosen according to Table \ref{['tab:rang']}: $f^{(1)}$ (top row) and $f^{(2)}$ (bottom row). Simulations corresponding to the left panels were performed for systems composed of $50$ (squares) and $500$ (circles) individuals. For the right panels, $\alpha$ was fixed at $1$; for their respective insets, however, the coupling strengths ($\alpha$) were chosen within $[1, 5]$. In both, figures and insets (right column), the number of individuals ($N$) is uniformly distributed between $50$ and $500$. The dashed lines are positioned at the same values of $L$ in each panel. See SM1 and SM2 in SupMat for movies (detailed descriptions are presented in Appendix \ref{['app:movs']}).
• Figure 2: Average polarization of the system as a function of different coupling strengths and noise amplitudes. The self-propelling velocities were drawn using $f^{(1)}$ for (a), and $f^{(2)}$ for (b). All the simulations were performed for populations of $N = 50$ (squares) and $N = 500$ (circles) individuals. The inset complements (b) by extending its domain to include $D_\alpha$ values from $0.1$ to $5$; the abrupt decay occurs at $D_\alpha^{-1}\approx0.2$.
• Figure 3: Scale invariance in the system for events with $\alpha= 1$ (left column), and $\alpha \in [1, 5]$ (right column). The self-propelling velocities were set following $\vec{f}^{(1)}$ (top row) and $\vec{f}^{(2)}$ (bottom row). For panels on the left, simulations were performed for $50$ (dashed lines) and $500$ (solid lines) individuals, and the $\epsilon$-axis is cropped. For those on the right, the population sizes were chosen as: $N\in[50,500]$. For the sake of clarity, the cases for $D_\alpha = 80$ are shown only in panels on the right. In these, moreover, the axes fully cover only the points corresponding to cases with $D_\alpha = 0.1$ and $D_\alpha = 40$. For larger sizes ($L$), the trend of the curve corresponding to $D_\alpha=80$ is to bend.
• Figure 4: Static crossover in the transition between the system's states of high and low polarization. The coupling strength $\alpha$ is fixed at $1$, and the population sizes range from $N = 50$ to $N = 10^4$. The increment of $N$ is represented by the blue curves becoming bolder.
• Figure 5: Susceptibility of the system considering the mean-interparticle distance as the control parameter ($\tau = \ell$). All simulations were performed for $\alpha = 1$ and population sizes between $N=50$ and $N = 10^4$. The self-propelling velocities were chosen using ${f}^{(1)}$ for the top panels [(a), (b), and (c)], and ${f}^{(2)}$ for the bottom ones [(d), (e), and (f)]. Blue lines characterize the susceptibility calculated for intermediate values of $D_\alpha$, between $0.1$ and $80$.