Fore-aft asymmetric flocking

We show that fore-aft asymmetry, a generic feature of living organisms and some active matter systems, can have a strong influence on the collective properties of even the simplest flocking models. Specifically, an arbitrarily weak asymmetry favoring front neighbors changes qualitatively the phase diagram of the Vicsek model. A region where many sharp traveling band solutions coexist is present at low noise strength, below the Toner-Tu liquid, at odds with the phase-separation scenario well describing the usual isotropic model. Inside this region, a “banded-liquid” phase with algebraic density distribution coexists with band solutions. Linear stability analysis at the hydrodynamic level suggests that these results are generic and not specific to the Vicsek model.

Figure 1

Phase diagram of fore-aft asymmetric Vicsek model with front preference ($\alpha >0$) in $({\rho }_0,\eta )$ plane. Cyan region: stability of the one-band solution. Red region: Toner-Tu liquid phase. Blue-stripes region: banded-liquid phase. Black line: order-disorder transition. In the white regions, one observes solutions with more than one band. Blue line: upper limit of “banded-liquid” phase. Cyan line: limit of stability of the one-band solution. Magenta line: limit of stability of the two-band solution [only determined in (a), (b)]. Red line: upper limit of the Toner-Tu liquid phase. (a)–(d) finite-size systems ($L=256$) with $\alpha =1,0.6,0.3$, and 0.1. (e), (f) asymptotic ($L\to \infty$) diagrams for $\alpha =0$ (usual Vicsek model, data from [25]) and $\alpha =0.3$.

Figure 2

Banded-liquid phase observed for $\alpha =1,{\rho }_0=1,\eta =0.055$. (a), (b) Snapshots of density field calculated over square boxes of linear size $\ell =4$ in a system of size $L=512$ (logarithmic color scale: (a) pure front model $\alpha =1$ and (b) Toner-Tu liquid of symmetric Vicsek model $\alpha =0$. (c) Giant number fluctuations: rms $\mathrm{\Delta }n$ of number of particles $n$ contained in boxes of various linear sizes. Top three curves (shifted up for clarity): pure front model ($\alpha =1$). Bottom three curves: symmetric Vicsek model ($\alpha =0$). Dashed cyan (resp. magenta) line is a power law of exponent 0.7 (resp. 0.75). The system size is indicated near each curve. (d) Probability distribution function of coarse-grained density (black and red lines: $\alpha =1,L=256,512$; green and blue lines: $\alpha =0,L=256,512$). Dashed magenta line is a power law of slope $-1.7$.

Figure 3

Robustness of the one-band solution ($\alpha =1,\eta =0.1$). (a) Time-averaged density profiles at various ${\rho }_0$ values. From bottom to top: ${\rho }_0=0.06$, 0.1, 0.3, 1.2, 10, 20, 50 ($L=256$). (b) Peak density (top) and gas density (bottom) vs ${\rho }_0$ ($L=256$). (c) Time-averaged profiles for $L=128,256,512,1024,2048$ from bottom upward (${\rho }_0=0.25$). (d) Same as (b) but for profiles of (c).

Figure 4

(a) Sketch of the geometry of interaction between particles where particle 1 is taken as the reference particle. (b) Phase diagram in the $({\rho }_0,\sigma )$ plane showing the linear stability analysis of the homogeneous ordered solution ${\left|\mathbf{w}\right|}^2=\mu /\xi$ of Eqs. (3) and (4). The color coding indicates the direction of the most unstable mode compared to that of the polar order. In the white region, the homogeneous order is stable; the gray region denotes the stable disordered phase.

Figure 5

(a) Convergence with system size of the transition point ${\eta }^{*}\left(L\right)$ delimiting the lower part of the stability region of the one-band solution ($\alpha =0.3,{\rho }_0=0.5$). The red solid line is a fit ${\eta }^{*}\left(L\right)={\eta }_{\infty }^{*}-aexp(-bL)$ with ${\eta }_{\infty }^{*}\simeq 0.158$. (b)–(e) Unstable modes for the homogeneous solution ${\left|\mathbf{w}\right|}^2=\mu /\xi$ at ${\rho }_0=1.5$ in the $(q_{\parallel },q_{\perp })$ plane ($q_{\parallel },q_{\perp }\in [-0.025,0.025]$). Color codes for the growth rate only if it is positive, from blue $=0$ to red $={10}^{-3}$: (b) $\sigma =0.267$, (c) $\sigma =0.265$, (d) $\sigma =0.263$, and (e) $\sigma =0.261$.