Onset of Collective and Cohesive Motion

We study the onset of collective motion, with and without cohesion, of groups of noisy self-propelled particles interacting locally. We find that this phase transition, in two space dimensions, is always discontinuous, including for the minimal model of Vicsek et al. [Phys. Rev. Lett. 75,1226 (1995)] for which a nontrivial critical point was previously advocated. We also show that cohesion is always lost near onset, as a result of the interplay of density, velocity, and shape fluctuations.

Figure 1

Onset of collective motion in cohesionless models (1) (original VM, circles) and (3) (vectorial noise, squares). Variation of order parameter $\phi$ (a) and Binder cumulant $G$ (b) with the noise strength $\eta$. ($v_0=0.5$, $L=32$, $\rho =2$, and equivalent statistics for both models.)

Figure 2

Discontinuous character of the onset of collective motion in the original VM at $\rho =\frac{1}{8}$. (a) $G$ vs $\eta$ at various system sizes. (b) Transverse density (bottom curve) and order-parameter profile (top curve) in the ordered phase ($L=1024$, $\eta =0.18$) (c) probability distribution function (PDF) of ${\phi }^t$ near the transition point, $t\in [\tau ;500\tau ]$; here the correlation time is $\tau \simeq {10}^5$ [13], $L=512$. (d) Snapshot of coarse-grained density field in disordered phase at threshold, $\rho =2$, $L=256$. The arrows indicate the direction of motion of dense, ordered regions.

Figure 3

Onset of motion of cohesive groups in model (4) with $\eta =1$, $v_0=0.05$. (a) $⟨\phi ⟩$ and $n/N$ (normalized size of largest connected cluster) vs $\alpha$ [$\rho =\frac{1}{16}$, $\beta =20$ (liquid phase): dashed lines, $N=4096$; solid lines, $N=16384$]. Inset: solid group ($\beta =84$) of $N=4096$ particles: dashed line, $⟨\phi ⟩$; solid line, relative diffusion of initially neighboring particles $\Delta \equiv ⟨\frac{1}{n_j}\sum _{k\sim j}[1-r_{jk}^2(t)/r_{jk}^2(t+T)]{⟩}_{j,t}$, where $T\approx 20N$ ($\Delta \simeq 1$ in the liquid phase, while $\Delta \simeq 0$ in the solid phase, see [12]). (b) Variation with $\alpha$ of the maximal absolute rotation angle $|\omega |$ averaged over 100 samples of 1000 vortices [$N=2048$, $\rho =\frac{1}{32}$, $\beta =30$ (liquid phase)]. Dashed line: $⟨\phi ⟩$ during the same runs. Inset: Distribution of rotation times at the transition with decay exponent $\sim 1.3$.

Figure 4

Typical shape of a liquid cohesive group of 16 384 particles [model (4), $\rho =\frac{1}{16}$, $\beta =20$ arrows indicate direction of motion]. (a) At onset before loss of cohesion, $\alpha =1.78$. (b) Static phase, round shape, $\alpha =0.5$. (c) In moving phase, typical triangular form (see 12), $\alpha =2.5$. (d) Closeup of a filament; no local order is apparent.