Nonquenched rotators ease flocking and memorize it

We introduce a minimal model for a two-dimensional polar flock with nonquenched rotators and show that the rotators make the usual macroscopic long-range order of the flock more robust than the clean system. The rotators memorize the flock-information which helps in establishing the robustness. Moreover, the memory of the rotators assists in probing the moving flock. We also formulate a hydrodynamic framework for the microscopic model that makes our study comprehensive. Using linearized hydrodynamics, it is shown that the presence of such nonquenched heterogeneities increases the sound speeds of the flock. The enhanced sound speeds lead to faster convection of information and consequently the robust ordering in the system. We argue that similar nonquenched heterogeneities may be useful in monitoring and controlling large crowds.

Figure 1

Schematic of interaction among flocks and NR. (a) Top—A test SPP (“1”) interacts with its neighbors (“2, 3, 4”) within a distance $R$. Bottom—Vectorial representation of the orientation update rule of the SPP (“1”). The SPP-NR (“1”–“4”) interaction is tuned by $\mu$. The dashed line shows the resultant orientation and the black arrow on it represents the corresponding unit vector. (b) Similar presentation for the orientation update of a test NR (“5”). The NR-SPP interactions are tuned by $\alpha$. (c) Two subflocks (A—broken square; B—solid square) of SPPs and a NR (broken hexagon) are considered in an arbitrary 2D space at time $t_1$. The patterns (striped, checkered, and solid) inside the geometries indicate information therein. All the information is called pure at $t_1$. As A and NR are nearby at $t_1$, they interact mutually through $\mu$ and $\alpha$ terms, and meanwhile B remains reclusive. During interactions, A and NR exchange information as per the values of $\mu$ and $\alpha$. A passes by the NR outside its interaction range by $t_2$. By that time, B comes close to NR and interact mutually. Consequently, NR receives information of B, and transfer its current information (which also contains information of A) to B, as depicted for a late time $t_3$. Other than direct interactions among A and B, this extra means of information transfer through the inertia of NR establishes ordering robust than the clean system. Also, NR acts as probe to the flocks as it memorizes information of the flocks.

Figure 2

Robust ordering due to NRs. The data are shown for $\mu =200$. (a) The SPPs show monotonic order-disorder transition with ${\eta }_{\theta }$ in the presence of the NRs ($c_r=0.01,\alpha =1$), similar to the clean system ($c_r=0$). Unlike that, the quenched model ($c_r=0.004,\alpha =0$) shows nonmonotonic transition. The curves are zoomed near the transition on the right panel of (a). (b) Critical noise ${\eta }_{\theta }^c$ increases with $L$. The numerics on the left margin indicate $c_r$, and the dashed lines show respective quadratic fits. The thermodynamic-limit value ${\eta }_{\theta }^c(L\to \infty )$ of the critical noise is obtained from the extrapolation. (c) ${\eta }_{\theta }^c\left(L\right)$ increases quadratically with density $c_r$. The broken lines show respective fits. Order and disorder states of the SPPs are indicated for the thermodynamic limit.

Figure 3

LRO in the presence of NRs. The data are shown for $\mu =200$ and ${\eta }_{\theta }=0.15$. (a) The SPP-order parameter does not vary with system size for the clean system ($c_r=0$) or the system with the NRs ($c_r=0.01,\alpha =1$). However, in the presence of the quenched rotators ($c_r=0.004,\alpha =0$), the order parameter decreases algebraically. The lines show respective algebraic fits. (b) The orientation distribution of the SPPs do not depend on the system size in the presence of the NRs, whereas the distribution broadens with $L$ for the quenched model.

Figure 4

Flock information is stored in the NRs. The data are shown for $\mu =200$. (a) Autocorrelation function $C_{\varphi }$ of the NRs ($c_r=0.01,\alpha =1,L=300$) are shown at the late time, $t_0$ being the reference steady-state time. $C_{\varphi }$ shows modulations over its saturated value $V_r^2$, and these modulations are prominent near the critical point (${\eta }_{\theta }=0.45$) where band emerges. The solid line shows Gaussian fit to these modulations, and the labels correspond to the snapshots shown in panels (b–g). The snapshots are shown for a zoomed substrate of dimension $300\times 100$ for clarity. The black circles indicate the NRs (only few shown), and the colors represent orientations of the SPPs and the NRs. Separation in the black vertical lines in panel (b) gives an estimate of the bandwidth $W_b$. $C_{\varphi }$ assumes a finite value only if the current position of the band overlaps to that in panel (b). (h) $W_b$ and $\sigma v_f{V}_s/V_r$ decreases with ${\eta }_{\theta }$ with the same slope. Here the flock speed $v_f=L/\tau$ where $\tau$ is the peak-to-peak separation of $C_{\varphi }$ modulations.

Figure 5

NRs increase sound-mode speeds. (a) The effective diffusivity $D_{\parallel }$ decreases and (b) the factor $X$ increases with increasing density ${\rho }_r$ of heterogeneity. The changes are more dominant for smaller ${\alpha }_1\left(\overline{\rho }\right)$, i.e., near the transition. Legends are the same in panels (a) and (b). (c) $c_{+}\left(\phi \right)$ is plotted in polar coordinates for various ${\alpha }_1\left(\overline{\rho }\right)$ and ${\rho }_r$, and its zoomed version is shown in panel (d). The numerics adjacent to the radial and the angular grids indicate respective values, and help to realize the magnification in panel (d) as compared to panel (c). Panels (c) and (d) share the same legends. The mode speed increases with ${\rho }_r$ along the flock direction, which is more prominent for small ${\alpha }_1\left(\overline{\rho }\right)$.

Figure 6

Critical noise ${\eta }_{\theta }^c$ increases with $L$. Binder cumulants are shown for $c_r=0.008$, $\mu =200$, and $\alpha =1$, and zoomed near their minima for the sake of clarity.

Figure 7

${\eta }_{\varphi }$-term introduces fluctuations in the system. (a) $V_s$ versus ${\eta }_{\theta }$ plot is shown for $c_r=0.01,\alpha =1$, and various ${\eta }_{\varphi }$. The SPPs obtain optimal ordering at a finite ${\eta }_{\theta }$ in the presence of the ${\eta }_{\varphi }$- term. The curves are obtained near the optimal point only, and the maximum $V_s$ suggests respective ${\eta }_{\theta }^{opt}$. (b) ${\eta }_{\theta }^{opt}$ varies linearly with ${\eta }_{\varphi }$. The dashed line represents the extrapolation which verifies ${\eta }_{\theta }^{opt}=0$ for ${\eta }_{\varphi }=0$. (c) Variation in $V_s$ with system size is shown on semilog scale for $c_r=0.01,\alpha =1$, and ${\eta }_{\varphi }=1$. The solid lines show fits quadratic in $1/N_s$ and have finite intercepts in the thermodynamic limit.