Dry Active Matter Exhibits a Self-Organized Cross Sea Phase

The Vicsek model of self-propelled particles is known in three different phases: a polar ordered homogeneous phase, also called the Toner-Tu phase, a phase of polar ordered regularly arranged high density bands with surrounding low density regions without polar order, and a homogeneous phase without polar order. Here, we show that the standard Vicsek model has a fourth phase for large system sizes: a polar ordered cross sea phase. We demonstrate that the cross sea phase is not just a superposition of two waves, but it is an independent complex pattern with an inherently selected crossing angle.

Figure 1

Snapshots of the two dimensional standard Vicsek model in its four phases: (i) Toner-Tu phase also called polar ordered liquid ($\eta =0.22$). (ii) Cross sea phase ($\eta =0.30$). (iii) Band phase ($\eta =0.37$). (iv) Disordered phase ($\eta =0.45$). Dark color represents high particle density. Red arrows indicate the average direction of motion. Simulations have been started at random initial conditions. The snapshots have been taken after a thermalization time of $T=2\times {10}^5$. System parameters: $R=1$, $v=1$, $\mathrm{\Delta }t=1$, particle number $N={10}^6$, system size $L=1253.3$. On average, there are two particles within a unit circle. Periodic boundary conditions have been used.

Figure 2

Structural order parameter $C_2$ (blue line) and polar order parameter $p=1/N|{\sum }_i{\mathbf{v}}_i|$ (black line) averaged over 15 realizations [29] and ${10}^4$ time steps after a thermalization time of $T=2\times {10}^5$ [30] (blue solid line). For the noise values $\eta =0.31,\dots ,0.40$ we find several realizations in the cross sea phase as well as in the band phase, see Supplemental Material [31]. Averaging only over realizations that are identified (by hand) as clearly in the cross sea phase, results in the red upper line. Analogously, averaging over the band phase realizations only, results in the green lower line. We see that the correlation parameter $C_2$ increases from the Toner-Tu phase (i) to the cross sea phase (ii) and then decreases to the band phase (iii) and even more to the disordered phase (iv). The transition lines from (i) to (ii) and (ii) to (iii) (vertical dashed and dotted lines) have been obtained as dips of the Binder cumulant, see Fig. 3. The transition towards disorder (dash-dotted vertical line) was obtained by hand as all realizations are bands for $\eta =0.44$ and all realizations are disordered for $\eta =0.45$, see Supplemental Material [31]. Parameters are as in Fig. 1.

Figure 3

Binder cumulant of the structural order parameter $C_2$ obtained from the same simulations as used in Fig. 2. We can identify two clear dips indicating the transitions from phases (i) to (ii) (dashed line) and from phases (ii) to (iii) (dotted line). In the inset, we show some additional points [29] close to the transition from phase (i) to (ii). In principle, there should be a third dip between phases (iii) and (iv) (dash-dotted line). However, this is not observed here due to its expected sharpness.

Figure 4

(a) Average of local particle density $\rho$ and local momentum density over ${10}^3$ time steps after a thermalization time of $T=2\times {10}^5$ within the comoving frame [35] of the cross sea pattern. The particle density is displayed by the color map. One sees that a lot of mass accumulates at the crossing points of the pattern. The densities of the crossing points are approximately four to five times as large as the density along the single fronts. The black arrows display the momentum density, where thick arrows correspond to large momentum. We see that mass is mainly transported along the high density bands and, on average, in the opposite directions of the mean velocity of the particles in the lab frame (red arrow). Parameters are as in Fig. 1, $\eta =0.29$. (b) Illustration of the mass transport mechanism within the pattern. See text for explanations.