Braiding a Flock: Winding Statistics of Interacting Flying Spins

When animal groups move coherently in the form of a flock, their trajectories are not all parallel, the individuals exchange their position in the group. In this Letter, we introduce a measure of this mixing dynamics, which we quantify as the winding of the braid formed from the particle trajectories. Building on a paradigmatic flocking model we numerically and theoretically explain the winding statistics and show that it is predominantly set by the global twist of the trajectories as a consequence of a spontaneous symmetry breaking.

Figure 1

(a) Instantaneous positions and orientations of the particles in a compact polar flock (250 particles), and definition of the parallel-transported frame (${\widehat{\mathit{\eta }}}_1,{\widehat{\mathit{\eta }}}_2,\widehat{\mathbf{\Pi }}$). (b) World lines of five particles in the same polar flock. $D=2.6\times {10}^{-2}$. (c) Definition of the crossing index. (d) Braid diagram associated to the worldlines drawn in (b).

Figure 2

(a) Probability distribution of the total winding $W(T)$, normalized by its standard deviation ($T={10}^3$). The different noise amplitudes correspond to different polarizations of the flock [17]. Solid line: Gaussian distribution. (b) Standard deviation, ${⟨W^2(T)⟩}^{1/2}$, as a function of time $T$, for different noise amplitudes. (c) Correlation function, defined by Eq. (4), as a function of $T$. (d) Variance of the total winding normalized by the variance of the pair winding, plotted versus the particle number. Open circles: $T={10}^{-1}$. Filled circles: $T={10}^3$. Black squares: random walkers confined in a circular box of radius 20, diffusivity: 10. Time step: 0.1. Dashed lines: slope $-1$.

Figure 3

(a) Standard deviations as a function of $T$: ${⟨W^2(T)⟩}^{1/2}$ (open circles), ${⟨W^{\star 2}(T)⟩}^{1/2}$ (diamonds), and ${⟨{\mathrm{Tw}}^2(T)⟩}^{1/2}$ (filled circles), ${⟨W^2(T)⟩}^{1/2}$ for confined random walkers (squares), same parameters as in Fig. 2. (b) Variance of the total winding normalized by the variance of the pair winding computed in the twisting frame for different particle numbers. Open circles: $T={10}^{-1}$. Filled circles: $T={10}^3$. Black squares: confined random walkers. Dashed lines: slope $-1$. (c) Probability distribution of the twist normalized by its standard deviation ($T={10}^3$). Solid line: Gaussian distribution. (d) The normalized twist diffusivity: ${\mathcal{D}}_{\text{Tw}}/({\mathcal{T}}^2{L}_{\text{flock}})$ decays quadratically with $N$. Solid line: slope $-2$.