Flocking Beyond One Species: Novel Phase Coexistence in a Generalized Two-Species Vicsek Model

Abstract

A hallmark in natural systems, self-organization often stems from very simple interaction rules between individual agents. While single-species self-propelled particle (SPP) systems are well understood, the behavior of binary mixtures with general alignment interactions remains largely unexplored with a few scattered results hinting at the existence of a rich emergent phase behavior. Here, we investigate systematically a generalization of the two-species Vicsek model with reciprocal intra- and interspecies (anti-)alignment couplings, uncovering a rich phenomenology of emergent states. Notably, we show that rather than destroying polar order, anti-aligning interactions can promote phase separation and the emergence of global polar order. In doing so, we uncover a novel mechanism for microphase separation. We further find these coexistence patterns can be generalized to multi-species systems with cyclic alignment interactions.

Figures (5)

• Figure 1: Numerical phase diagram obtained from particle simulations. Here, we use the following parameters $N=2\times10^4$, $\rho=100$, and $D_r=0.2$. Phases were classified numerically, with data averaged over 10 independent realizations for each data point. Here, we list the phases observed and the parameters used to obtain the associated example snapshots: (a) Nematic stripes, demixed (dark-green plus), $J_{AA}=-30$, $J_{AB}=16$; (b) Flocking stripes (dark-blue star), $J_{AA}=-30$, $J_{AB}=30$; (c) Parallel flocking (purple upward triangle), $J_{AA}=-5$, $J_{AB}=20$; (d) Independent flocking (pink square), $J_{AA}=10$, $J_{AB}=0$; (e) Nematic stripes, mixed (light-green hexagon), $J_{AA}=-30$, $J_{AB}=-20$; (f) Antiparallel flocking stripes (light-blue diamond), $J_{AA}=-20$, $J_{AB}=-24$; (g) Antiparallel flocking (pink downward triangle), $J_{AA}=-5$, $J_{AB}=-20$; (h) Independent nematic ordering (khaki cross), $J_{AA}=-45$, $J_{AB}=0$. Finally, we also observe a disordered hyperuniform phase (orange circles). Phase boundaries were added manually as a visual guide. See accompanying supplementary movies S1--S9 and their descriptions in Note3.
• Figure 2: Phase diagrams when varying noise strength and density. (a) Phase behavior in the $(J_{AB}, D_r)$-plane at fixed density $\rho=100$, intraspecies interaction strength $J_{AA}=-25$. (b) Phase behavior in the $(J_{AB}, \rho)$-plane at fixed rotational diffusion $D_r=0.2$ and intraspecies interaction strength $J_{AA}=-25$. Here again, the system size was taken to be $N=2\times 10^4$. Symbols used here are the same as those introduced in Fig. \ref{['fig:phase_diagram']}; Phase boundaries were added manually as a visual guide.
• Figure 3: (a) Sketch of interactions between species in the flocking stripes phase. The interspecies couplings are aligning, whereas the intraspecies coupling is anti-aligning. (b) Snapshot of a simulation at low density for $J_{AA}=-J_{AB}=-20$ (parallel flocking with phase separation) with zoomed in snapshot showing the microphase separation. The inset shows a polar histogram of the particle angles. These simulations were performed with $N=10^4$, $\rho=1$, and $D_r=0.02$. (c) Snapshot of a high density simulation in the traveling band phase ($N=10^4$, $\rho=200$, $D_r=0.2$, $J_{AA}=-J_{AB}=-20$) and (d) its time-averaged density profile, projected along the direction of travel $x_\parallel$. (e) Simplified schematic to explain the stability mechanism.
• Figure 4: Multi-species ($m>2$) flocking. Snapshots of systems with alignment interactions governed by Eq. (\ref{['eq:cyclic_multispecies']}) with (a) $m=3$ and (c) $m=4$ species. (b)--(d) Time-averaged density profiles for the systems in (a)--(c), respectively. Simulation parameters were: $N=m\times 5\times 10^4$, $\rho=200$, $D_r=0.2$, $J=10(m+1)$.
• Figure 5: Numerical order parameters (polar order, nematic order and demixing) against $J_{AB}$ for fixed (a) $J_{AA}=-15$ and (b) $J_{AA}=-25$. Vertical gray dashed lines show the phase boundaries between different phases, whose symbols plotted above correspond to those in Fig. \ref{['fig:phase_diagram']}.