Collective Turns in Spinless Flocks

Abstract

Using a minimal aggregation-based model, we address the efficient information transfer observed in natural flocks during collective turns. Specifically, we demonstrate that this feature can arise solely from the non-reciprocal nature of local interactions. Through a perturbative analysis, moreover, we find that velocity fluctuations (in the continuum) can be described by a Born approximation. We then show that a wave propagating across the flock undergoes scattering. Our model provides testable predictions and can be extended to study other physical contexts exhibiting polar order.

Figures (9)

• Figure 1: Collective behavior of a flock over time. (a) Individuals start positioned in the red dots; the black curve describes the trajectory followed by the individual initiating the turn. (b) The asymmetry index, ranging from $0$ (perfect asymmetry) to $1$ (perfect symmetry), is calculated for the event in (a) via $\psi_A = \|0.5(A+A^T)\|_F/ \| A\|_F$. The flock follows a sequence of aggregation (blue arrow), straight flight (orange arrow), turning (green arrow), and straight flight again. (c) During the turn, the radial acceleration, $a_i$, is computed for the initiator (dashed line) and random neighbors. The simulation was performed for $N = 100$ individuals, each influenced by $\eta = 6$ neighbors, and coupling strengths $(\alpha, \gamma) = (1, 10)$. Time ($t$) is reported in simulation units ($\Delta\tau = 0.01$; see SM for computation details and additional simulations).
• Figure 2: Normalized spectra of velocity correlations during a collective turn. The Fourier transform of $C(\vec{k}, t)$ was evaluated for (a) $\vec{k} = (k, 0, 0)$, (b) $\vec{k} = (0, k, 0)$, (c) $\vec{k} = (0, 0, k)$, and (d) $\vec{k} \rightarrow k$ (isotropic averaging). In each panel, the spectra were calculated for wave numbers starting at their respective minima: (a) $k_0 = 0.067$, (b) $k_0 = 0.059$, (c) $k_0 = 0.069$, and (d) $k_0 = 0.045$. The frequency axes are cropped.
• Figure 3: Power spectral density for the system's velocity correlations. All computations were performed considering the individual velocity fluctuations during collective turns. The turning events occurred over 1000 time steps for panels (a) and (b), and 100 time steps for (c) and (d). The evaluation wave vectors were oriented along the $x$-axis, $\vec{k} = (k_x, 0, 0)$, for (a) and (c), and along the $y$-axis, $\vec{k} = (0, k_y, 0)$, for (b) and (d). The pixel intensities represent the power spectral density for each ($\omega, \vec{k}$) pair, expressed in decibels. The estimated velocities of propagation, in [distance/time] units, are (a) $c_s \approx 8$, (b) $c_s \approx 9$, (c) $c_s \approx 27$, and (d) $c_s \approx 8$. See SM1 and SM2 in SM for movies of the collective turns.
• Figure S1: Time-varying Laplacian as a small perturbation. The evaluated turning events lasted $1000$ (left panels) and $100$ (right panels) time steps. The number of entry changes (top row) is computed as the Frobenius norm of the difference between consecutive Laplacian matrices. The relative magnitude of the perturbation (bottom row) is given as $\epsilon_F = \|L _t - L_0\|_F/\| L_0\|_F$, where $L_0$ is the temporal average of Laplacian matrices during the collective turn: $\langle L_t \rangle$.
• Figure S2: Dispersion diagrams calculated using the Fourier transform for a turning event lasting 100 time steps. The wave vectors are defined as (a) $\vec{k} = (k_x, 0, 0)$ and (b) $\vec{k} = (0, k_y, 0)$.