Emergent Spatial Structures in Flocking Models: A Dynamical System Insight

We show that hydrodynamic theories of polar active matter generically possess inhomogeneous traveling solutions. We introduce a unifying dynamical-system framework to establish the shape of these intrinsically nonlinear patterns, and show that they correspond to those hitherto observed in experiments and numerical simulation: periodic density waves, and solitonic bands, or polar-liquid droplets both cruising in isotropic phases. We elucidate their respective multiplicity and mutual relations, as well as their existence domain.

Figure 1

Band patterns observed in agent-based simulations. (a) Smectic arrangement of polar bands in the Vicsek model with vectorial noise [5]. Speed $v=0.5$, noise intensity $\eta =0.6$, and density $\rho =1.1$. (b) A solitary band observed for the same model and same parameters as in (a). (c) Polar-liquid droplet in an isotropic phase observed in the active Ising model [10], with inverse temperature $\beta =6$, density $\rho =5$, hopping rate $D=1$, and bias $\epsilon =0.9$. More simulation details can be found in the Supplemental Material [16].

Figure 2

(a) Sketch of the motion of an oscillating point particle in the effective potential $H(W)$ for $P_0=1$, ${\rho }_{\mathrm{c}}=1$, $\lambda =0.5$, $\xi =1$, $c=0.9$, and ${\rho }^{\star }=0.7$. The system loses or gains energy $H$ when $F^{\prime }(W)>0$ or $F^{\prime }(W)<0$. (b) Band phase diagram for the same parameters. Nonlinear bands exist only in the gray region. The dashed lines correspond to the conditions $W_H>0$, and $W_F>0$. For $c<c^{*}$, the black solid line corresponds to the supercritical Hopf bifurcation. Polar-liquid droplet states are observed only at $c=c_h$.

Figure 3

(a) Dashed lines: dynamical system trajectories, for $P_0=1$, ${\rho }_{\mathrm{c}}=0.1$, $\lambda =0.5$, $\xi =4$, $D=0.1$, $c=0.9$. ${\rho }^{\star }$ was chosen such that $W_H\gtrsim W_F$. Thick line: stable limit cycle. Thin line: $Z=F(W)$ curve. (b) Polar smectic corresponding to the limit cycle shown in (a). (c) Homoclinic orbit; same parameters as in (a) but for a lower value of ${\rho }^{\star }$. (d) Solitonic band corresponding to the limit cycle shown in (c). (e) $H(W)$, solid line, plotted for $P_0=1$, ${\rho }_{\mathrm{c}}=1$, $\lambda =1$, $\xi =10$, and $D=50$. The dashed lines show the positions of $W_F$ and $W_H$. The values of $c$ and ${\rho }^{\star }$ give rise to a heteroclinic cycle [16]. (f) Polar-liquid droplet for the same values of the parameters as in (e); see also the Supplemental Material [16].