Activity induced synchronization: Mutual flocking and chiral self-sorting

Synchronization, the temporal coordination of coupled oscillators, allows fireflies to flash in unison, neurons to fire collectively, and human crowds to fall in step on the London millenium bridge. Here, we interpret active (or self-propelled) chiral microswimmers with a distribution of intrinsic frequencies as motile oscillators and show that they can synchronize over very large distances, even for local coupling in two dimensions (2D). This opposes canonical nonactive oscillators on static or time-dependent networks, leading to synchronized domains only. A consequence of this activity-induced synchronization is the emergence of a “mutual flocking phase,” where particles of opposite chirality cooperate to form superimposed flocks moving at a relative angle to each other, providing a chiral active matter analogue to the celebrated Toner-Tu phase. The underlying mechanism employs a positive feedback loop involving the two-way coupling between oscillators' phase and self-propulsion and could be exploited as a design principle for synthetic active materials and chiral self-sorting techniques.

Figure 1

(a) Trajectories of particles with opposite chirality. (b) Late-time snapshot of $N={10}^5$ mobile Kuramoto oscillators with a Gaussian frequency distribution, after relaxation from a completely synchronized state $(\text{Var}\left(\mathrm{\Omega }\right)=0.4$ and $g{\rho }_0=2.8)$. Colors represent oscillator phases. The dispersion of natural frequency spontaneously destabilizes the initially ordered state. (c) Mutual flocking: $N=3\times {10}^4$ chiral active particles with $\mathrm{\Omega }=\pm 0.6$ and $g{\rho }_0=40$ forming a globally polarized and synchronized phase; panel (d) shows a detailed view. Small arrows show particle orientations; large ones show the average polarization direction of both species represented in red and blue (arrow lengths are arbitrary). (e) Phase diagram of active oscillators. Symbols show simulation results for the phase boundaries (red symbols for two species, black ones for a normal distribution) using two different densities $({\rho }_0=10$, 20 in black crosses and squares); lines show analytical predictions $(g{\rho }_0=2[1+\text{Var}\left(\mathrm{\Omega }\right)]$ and $g{\rho }_0=2)$. The weak deviation between simulations and the $g{\rho }_0=2$ line comes from a reduction of the growth rate of unstable modes as $\mathrm{\Omega }$ increases (see the Supplemental Material [57]). Insets show snapshots in the chiral phase for two species (f) and a Gaussian distribution (g) both for $\text{Var}\left(\mathrm{\Omega }\right)=2;g{\rho }_0=2.8$; colors show intrinsic frequencies.

Figure 2

Mutual flocking phase. Detailed view of a late-time configuration for $\mathrm{\Omega }=\pm 1.5$ at $g{\rho }_0=5.6$ (a). Thick arrows show the average polarization of both species, while small arrows show particle orientations. Angle $\mathrm{\Delta }\mathrm{\Phi }\left(\right|\mathrm{\Omega }\left|\right)$ between the two flocks (b) and partial polarization $P\left(\right|\mathrm{\Omega }\left|\right)$ (c) of each species. Lines show Eq. (5) and points simulation results for $\mathrm{Pe}=2;{\rho }_0=200$. (d) $C\left(r\right)$ (in linear-log scale) of a system of active oscillators $(\mathrm{Pe}=2$, plain red line), static oscillators (in green), Brownian oscillators (in orange), and underdamped velocity-aligning oscillators (in blue) (in all cases $\text{Var}\left(\mathrm{\Omega }\right)=0.4$ and $g{\rho }_0=40)$ (see Ref. [57] for details). The dotted black line shows an exponential decay to the asymptotic value $P^2$.

Figure 3

Chiral phase: (a) spatial correlations for $g{\rho }_0=2.8,\text{Var}\left(\mathrm{\Omega }\right)=2$ and several system sizes $(N={10}^3,...,3.{10}^4$, from left to right). The inset shows the corresponding correlation length $\xi$ as a function of $\sqrt{N}$. We show for comparison a linear growth $\propto \sqrt{N}$ in dotted line. (b) Partial polarization $\tilde{P}$ as a function of $g{\rho }_0$ for several frequency dispersions.