Revisiting the Flocking Transition Using Active Spins

We consider an active Ising model in which spins both diffuse and align on lattice in one and two dimensions. The diffusion is biased so that plus or minus spins hop preferably to the left or to the right, which generates a flocking transition at low temperature and high density. We construct a coarse-grained description of the model that predicts this transition to be a first-order liquid-gas transition in the temperature-density ensemble, with a critical density sent to infinity. In this first-order phase transition, the magnetization is proportional to the liquid fraction and thus varies continuously throughout the phase diagram. Using microscopic simulations, we show that this theoretical prediction holds in 2D whereas the fluctuations alter the transition in 1D, preventing, for instance, any spontaneous symmetry breaking.

Figure 1

Top left: Phase diagram in 2D with ordered liquid ($L$), disordered gas ($G$), and coexistence region ($G+L$). The red and blue lines correspond to low and high densities of phase separated profiles; they enclose the region where such profiles can be seen. $D=1$, $ϵ=0.9$, $L=300$, ${\rho }_0=N/L$. Bottom: Snapshots of the different profiles averaged over the transverse direction. Top right: Phase diagram predicted by the RMFM. In addition to ${\rho }_h$ and ${\rho }_{\ell }$, black and green dashed spinodal lines signal the loss of linear stability of the homogeneous profiles. $D=v=r=1$.

Figure 2

Left: Fraction of the ordered liquid phase when ${\rho }_0$ is either increased or decreased for the RFMF (top) and in 2D microscopic simulations (bottom). Right Corresponding profiles of the system. Parameters: RMFM $L=100$, $v=D=1$, $r=1.6$, $\beta =1.75$, $\Delta {\rho }_0={10}^{-2}$ every $\Delta t=15000$; 2D lattice model $L=250$, $\beta =2$, $D=1$, $\epsilon =0.9$, $\Delta {\rho }_0={10}^{-2}$ every $\Delta t=500$.

Figure 3

Reversal of a 1D cluster due to a localized fluctuation. $v_2$ is greater than $v_1$ until $\rho (x)={\rho }_h$ in the whole cluster. (See movies in 42.) ${\rho }_0=5$, $D=1$, $\epsilon =0.9$, $\beta =1.7$.

Figure 4

Left: Cluster length as a function of time, showing a linear spreading between reversals. $D=1$ $\epsilon =0.9$ $\beta =2$ ${\rho }_0=3$. Center and Right: $P(|m|)$ for ${\rho }_0=4$; ${\chi }_m(\rho )/L$; $\beta =1.538$, $D=1$, $\epsilon =0.9$.

Figure 5

Histograms and Binder cumulant of the total magnetization for ${\rho }_0=3$, $D=1$ (left) and ${\rho }_0=0.2$, $D=10$ (right). $\epsilon =0.9$. $L=8000$ for $P(m)$.