From Phase to Microphase Separation in Flocking Models: The Essential Role of Nonequilibrium Fluctuations

We show that the flocking transition in the Vicsek model is best understood as a liquid-gas transition, rather than an order-disorder one. The full phase separation observed in flocking models with $Z_2$ rotational symmetry is, however, replaced by a microphase separation leading to a smectic arrangement of traveling ordered bands. Remarkably, continuous deterministic descriptions do not account for this difference, which is only recovered at the fluctuating hydrodynamics level. Scalar and vectorial order parameters indeed produce different types of number fluctuations, which we show to be essential in selecting the inhomogeneous patterns. This highlights an unexpected role of fluctuations in the selection of flock shapes.

Figure 1

Top: Microphase separation in the Vicsek model. $\eta =0.4$, $v_0=0.5$, ${\rho }_1=1.05$ (left), ${\rho }_2=1.93$ (right). Bottom: Phase separation in the active Ising model. $D=1$, $ϵ=0.9$, $\beta =1.9$, ${\rho }_1=2.35$ (left), ${\rho }_2=4.7$ (right). System sizes $800\times 100$. Red arrows indicate the direction of motion.

Figure 2

(a) Phase diagram of the Vicsek model. The binodals ${\rho }_{\ell }(\eta )$ and ${\rho }_h(\eta )$ mark the limit of the coexistence region. (b) Number of bands vs $L_{\parallel }({\rho }_0-{\rho }_{\text{gas}})$ varying either the excess density for $Lx\times Ly=2000\times 100$ (squares) or the system size along the direction of propagation (dots) for ${\rho }_0=0.6$ ($\eta =0.3$) or ${\rho }_0=1.2$ ($\eta =0.4$). The straight black lines are guides for the eyes. (c) Density profiles of Fig. 1 averaged along the transverse direction ${\mathbf{e}}_{\perp }$, $\overline{\rho }(x_{\parallel })=⟨\rho (x_{\perp },x_{\parallel }){⟩}_{x_{\perp }}$. (d) Time average of the band profiles shown in (c). A threefold increase of the excess density changes the number of bands but not the gas density or the shape of the bands. $v=0.5$, $\eta =0.4$, ${\rho }_0=1.05$ (red/light grey lines), ${\rho }_0=1.93$ (blue/dark grey lines).

Figure 3

(a) Polarization and (b) mean velocity vs ${\rho }_0$ for $L=2048$ (squares) and $L=1024$ (lines). Red, green, blue, cyan, and magenta correspond to solutions with 1 to 5 bands. (c) Hysteresis loop between gas and microphase states. (d) Hysteresis loop between microphase and liquid states. The variance $\mathrm{\Delta }{\rho }_{\parallel }^2$ quantifies inhomogeneity along the direction of motion. 100 runs are used for (c) and (d), with $\eta =0.4$, system size $400\times 400$.

Figure 4

Density field in the PDEs after integration over $t={10}^5$. Left: scalar PDE, ordered initial condition with a periodic perturbation. Right: vectorial PDE, disordered initial condition. Parameters: $\lambda ={\rho }_c=D=P_0=1$, ${\rho }_0=1.2$. System size $800\times 100$.

Figure 5

Left: Number fluctuations $\mathrm{\Delta }n=\sqrt{⟨n^2⟩-⟨n{⟩}^2}$ where $n$ is the number of particles in boxes of different sizes. Measures are done in the homogeneous liquid phase. Parameters: size $400\times 400$ (all), ${\rho }_0=5$, $\beta =2.4$ (AIM), ${\rho }_0=5$, $\eta =0.4$ (VM), $\lambda ={\rho }_c=D=P_0=1$, ${\gamma }^2=0.4$, ${\rho }_0=3$ (sSDE and vSDE). Right: Numerical integration of sSDE (top) and vSDE (bottom). Parameters: ${\rho }_c=\lambda =D=P_0=1$, ${\gamma }^2=0.4$, system size $2000\times 100$.