Fluctuation-Induced Phase Separation in Metric and Topological Models of Collective Motion

We study the role of noise on the nature of the transition to collective motion in dry active matter. Starting from field theories that predict a continuous transition at the deterministic level, we show that fluctuations induce a density-dependent shift of the onset of order, which in turn changes the nature of the transition into a phase-separation scenario. Our results apply to a range of systems, including models in which particles interact with their “topological” neighbors that have been believed so far to exhibit a continuous onset of order. Our analytical predictions are confirmed by numerical simulations of fluctuating hydrodynamics and microscopic models.

Figure 1

Simulations of the 2D generalization of Eqs. (1) and (3) detailed in [52]. (Top left) Average magnetization as $\alpha$ is varied. The transition occurs at ${\alpha }_c<0$, shifted from the mean-field prediction (green line). At the onset of order, inhomogeneous profiles (black squares) separate homogeneous ordered and disordered phases (blue dots). Parameters: $D=v=\gamma =\sigma =1$, $dx=0.5$, $dt=0.01$, $L_x=400$, $L_y=40$, $\overline{\rho }\equiv N/(L_x{L}_y)=1.1$. (Bottom) A snapshot close to the transition shows an ordered traveling band in a disordered background. (Top right) The corresponding density and magnetization fields averaged along $y$. Parameters: same as before up to $L_y=100$, $dx=0.1$, $\alpha =-0.9$.

Figure 2

(Top left) Phase diagram of the microscopic topological model defined in Eqs. (13) and (14). The homogeneous ordered (blue) and disordered (green) regions are separated by a coexistence phase (red). The mean-field critical temperature is $\beta =1$ (dashed line). The red lines are visual guides that show how the transitions shift as the mean density varies. (Bottom) Snapshot of a propagating band corresponding to the black triangle in the phase diagram. Blue and red particles correspond to positive and negative spins, respectively. The corresponding density and magnetization fields, averaged over $y$ and time, are shown in the upper right panel. Parameters: $D=8$, $L_y=400$, $L_x=2000$, $k=3$, $\mathrm{\Gamma }=0.5$, $v=0.9$.

Figure 3

Simulations of the topological Vicsek model in 2D. At small noise, the system is disordered at low enough densities. Increasing the density then leads to an onset of order accompanied by propagating bands. Particles align with their $k=3$ nearest neighbors. Parameters: $L_x=2000$, $L_y=400$, $\sigma =0.08$, $k=3$, $v_0=0.2$, $\mathrm{\Delta }t=1$, $\overline{\rho }=0.25$ (top), and $\overline{\rho }=0.4$ (bottom).

Figure 4

Magnetization vs inverse temperature $\beta$ for the RAM (left), the LAM (center), and the topological model [(13) and (14)] (right). Only the latter exhibits traveling bands (black squares). Blue crosses and red dots correspond to mean densities $\overline{\rho }=0.25$ and $\overline{\rho }=0.5$, respectively. Parameters: $L_x=2000$, $L_y=400$, $D=8$, $v=0.9$, $\mathrm{\Gamma }=0.5$, and $k=3$. For LAM, $k=4$.