Flocking of two unfriendly species: The two-species Vicsek model

We consider the two-species Vicsek model (TSVM) consisting of two kinds of self-propelled particles, A and B, that tend to align with particles from the same species and to antialign with the other. The model shows a flocking transition that is reminiscent of the original Vicsek model: it has a liquid-gas phase transition and displays micro-phase-separation in the coexistence region where multiple dense liquid bands propagate in a gaseous background. The interesting features of the TSVM are the existence of two kinds of bands, one composed of mainly A particles and one mainly of B particles, the appearance of two dynamical states in the coexistence region: the PF (parallel flocking) state in which all bands of the two species propagate in the same direction, and the APF (antiparallel flocking) state in which the bands of species A and species B move in opposite directions. When PF and APF states exist in the low-density part of the coexistence region they perform stochastic transitions from one to the other. The system size dependence of the transition frequency and dwell times show a pronounced crossover that is determined by the ratio of the band width and the longitudinal system size. Our work paves the way for studying multispecies flocking models with heterogeneous alignment interactions.

Figure 1

Typical configurations of the TSVM (a) in a gas state at $(\rho ,\eta )=(0.5,0.3)$, (b) in a parallel flocking (PF) state at $(\rho ,\eta )=(0.5,0.24)$, (c) in an antiparallel (APF) flocking state at $(\rho ,\eta )=(0.5,0.2)$, and (d) in a liquid state at $(\rho ,\eta )=(2.0,0.2)$ (always APF). Configurations are for a square domain of size $2048\times 2048$ with $v_0=0.5$. Coarse-grained local densities of A and B particles are represented with red and blue pixels, respectively, color coded according to the color bar. In (b), (c), (d) particle species moves collectively in the direction indicated by an arrow.

Figure 2

Snapshots of the time evolution of the TSVM ($L_x=800, L_y=100, v_0=0.5$). A (B) particles are represented with red (blue) dots, and a local particle density is color coded according to the color bar. (a)–(b) Time evolution of a PF state ($\eta =0.3, \rho =1$). Bands of A and B species move in the same direction indicated by the arrows. (c)–(d) Comparison of a PF state at two different densities ($\eta =0.4$). There are two bands for $\rho =1.5$ and three bands for $\rho =2$. (e)–(h) Time evolution of an APF state ($\eta =0.3,\rho =1.4$). The two central bands approach each other ($t=10$ and $t=220$), collide ($t=300$), and pass by ($t=500$), as indicated by the arrows.

Figure 3

Probability distribution $P(v_s,v_a)$ at fixed $\rho =0.5$ with varying $\eta$ (top row) and at fixed $\eta =0.25$ with varying $\rho$ (bottom row). The velocity modulus is fixed to $v_0=0.5$. A single peak in (d) signifies the disordered gas phase while two peaks in (a), (b), (c), and (e) indicate stochastic switching between the PF and APF states. The ordered liquid phase is described in (f), which is characterized with a single peak with finite APF order parameter.

Figure 4

Order parameters in the restricted APF (square symbols) and PF (circular symbols) ensembles versus $\eta$ for $v_0=0.5, \rho =0.5, L_x=200$, and $L_y=25$.

Figure 5

Phase-separated (a) instantaneous and (b) time-averaged density profiles of species A for two values of $\rho$ with fixed $\eta =0.4$ and $v_0=0.5$. Black arrows indicate the direction of propagation.

Figure 6

(a) $\eta -{\rho }_0$ phase diagram for fixed $v_0=0.5$ and (b) $v_0-{\rho }_0$ phase diagram for fixed $\eta =0.4$ of the TSVM. The binodals ${\rho }_{\mathrm{liq}}$ and ${\rho }_{\mathrm{gas}}$ are the boundaries of the coexistence phase in the thermodynamic limit, obtained by extrapolating the finite-size boundaries to an infinite system size. The black dotted lines in the coexistence demarcates the boundary for the PF state and is also independent of system size: it indicates the boundary between the region of the coexistence phase where one observes PF-APF transition for any finite system size and the region where one does not.

Figure 7

Comparison of the time series of $v_s-v_a$ at two system sizes (with $\eta =0.2$) in (a), (b), and at two noise strengths (with $L_x=512, L_y=32$) in (c), (d). The thick solid line segments represent the time intervals during which the system dwells in the PF or APF states. $v_0=0.5$ and $\rho =0.5$.

Figure 8

(a) Average frequency $f$ of transitions between the APF and PF states and (b) average time ${\tau }_{\mathrm{PF}}$ spent by the system in the PF state as a function of linear system size $L_x$ (on a log-log scale), where $L_y=L_x/8$. The dashed lines with different slopes indicate a crossover in the scaling law at a crossover length $L_{x,c}$ depending of $\eta$ and $\rho$. Parameters: $v_0=0.5, \rho =0.35$ ($\eta =0.2$), $\rho =0.5$ ($\eta =0.22$), $\rho =0.6$ ($\eta =0.25$), and $\rho =1$ ($\eta =0.3$).

Figure 9

System-size dependence of the average dwell time of the APF state (empty symbols) and the PF state (filled symbols) with fixed aspect ratio $L_x/L_y=8$ in (a), fixed width $L_x=512$ in (b), and fixed height $L_y=24$ in (c). All measurements are done with $v_0=0.5, \rho =0.35$, and $\eta =0.2$.

Figure 10

Band width $W$ versus $L_y$ for various $(\eta ,\rho )$ combinations. In (a) $L_x$ is fixed to 400, and in (b) it is $L_x=8L_y$. In (a), the points (black cross) where $W\approx L_y$ [for each set of $(\eta ,\rho )$] are for $20<L_y<32$ whereas in (b) the $W\approx L_y$ points are around $L_y\approx 20$. The black dashed line represents $W=L_y$ and the shaded region is a guide to the eyes. The width is computed from the time- and ensemble-averaged density profile of bands and is defined as the distance between the two points in the density profile [cf. Fig. 5] where the density is equal to $({\rho }_{\mathrm{gas}}+{\rho }_{\mathrm{liq}})/2$.

Figure 11

(a) $V_a$ against $L$ for the ordered liquid phase: $\eta =0.1$ and $\rho =6$. For $v_0=0.5, V_a$ is constant over $L$ whereas for $v_0=0$, we expect $V_a$ decays to zero for $L\to \infty$. (b) Number fluctuations $\mathrm{\Delta }{n}^2=\langle n^2\rangle -{\langle n\rangle }^2$ versus average particle number $\langle n\rangle$ in a $200\times 200$ domain.

Figure 12

Zoomed in snapshots (keeping $L_y$ fixed) showing propagation of a single band of a particular species for (a) $L_y=10$ (longitudinal density wave) and (b) $L_y=50$ (transverse density wave). The red arrows represent the orientation vector of the particles and the black arrows signify the direction of propulsion. Parameters: $L_x=400, \eta =0.3, \rho =1$, and $v_0=0.5$.