Small Obstacle in a Large Polar Flock

We show that arbitrarily large polar flocks are susceptible to the presence of a single small obstacle. In a wide region of parameter space, the obstacle triggers counterpropagating dense bands leading to reversals of the flow. In very large systems, these bands interact, yielding a never-ending chaotic dynamics that constitutes a new disordered phase of the system. While most of these results were obtained using simulations of aligning self-propelled particles, we find similar phenomena at the continuous level, not when considering the basic Toner-Tu hydrodynamic theory, but in simulations of truncations of the relevant Boltzmann equation.

Figure 1

Typical flow reversals ($\eta =0.5$, $L=1024$). (a) Sketch of collision rules: postcollision orientation ${\mathit{e}}_{\alpha }^{t+1}$ interpolates between the precollision one ($\alpha =0$, blue arrow) and that of the reflected trajectory ($\alpha =1$, red arrow). (b) Time series of the global polarization $|\mathbf{\Psi }|$, in black, and its orientation $\cos \theta$, in red; shaded regions indicate the time windows for the snapshots series in (d), and (e); the empty and filled triangles indicate reversals triggered respectively from the homogeneous flock state [such as in (d)] and from the passage of a dense band (such as in (e) ($\alpha =0$, $\sigma =22$). (c) Distributions of inter-reversal times $P(\tau )$; dashed lines indicate exponential decays (green line: $\alpha =0$, $\sigma =22$; purple line: $\alpha =0.325$, $\sigma =120$). (d),(e) Snapshots of the system shown in (b) during an isolated reversal (d), and a reversal triggered by the passage of the dense band (e). On the color wheel on the right, color and intensity represent, respectively, the angle of the locally averaged polarity and the locally averaged density (with black for near-empty regions). The obstacle (at the center of the images) is in gray. Time marks are indicated at the top of each panel in units of 1000 time steps. For the first 2 panels, enlargements are provided showing the birth of the counterpropagating front around the obstacle. See Movies S1–S4 and S1z–S2z in [45].

Figure 2

(a) Average inter-reversal time, $⟨\tau ⟩$ vs $\eta$ (purple triangles $\alpha =0$, $\sigma =35$, $L=256$; blue circles $\alpha =0.325$, $\sigma =120$, $L=1024$); the shaded region indicated the band phase of the pure system. (b) Average global polarization vs system size with obstacle (red squares) and without (dashed line) ($\alpha =0$, $\eta =0.5$, $\sigma =35$). (c) Snapshot of a $L=4096$ system [$\alpha =0$, $\eta =0.5$, $\sigma =35$, colors as in Fig. 1]; at these parameters the system almost never recovers the homogeneous polar state. See related Movie S5 in [45].

Figure 3

Reversals triggered by the introduction of a finite blob of counterpropagating particles ($L=1024$). (a) Reversal probability $P_{\mathrm{rev}}$ vs $\eta$ at different $N_b$ for ${\rho }_b=180$, magenta squares indicate the estimated value for ${\eta }^{*}$. (b) ${\eta }^{*}$ for different ${\rho }_b$ as a function of the number of particles in the blob $N_b$.

Figure 4

Simulations of the Boltzmann equation [Eq. (3)] truncated at $K=4$ at system density ${\rho }_0=2$ and blob radius ${\sigma }_b=2$. (a) Snapshots showing the evolution of the momentum field ($f_1=p_x+i{p}_y$) at noise $\eta =0.5$ after introduction of a blob with density ${\rho }_b=8{\rho }_0$ in the ordered solution at time $t=0$. Note that the bivariate color map leads dilute disordered regions to appear darker. (b) Fate of blob in the $({\rho }_b,\eta )$ plane: black circles mark parameters leading to a reversal, while white squares indicate where no reversal occurs. More details about the numerical protocol in [45].