Metastability of Constant-Density Flocks

We study numerically the Toner-Tu field theory where the density field is maintained constant, a limit case of “Malthusian” flocks for which the asymptotic scaling of correlation functions in the ordered phase is known exactly. While we confirm these scaling laws, we also show that such constant-density flocks are metastable to the nucleation of a specific defect configuration, and are replaced by a globally disordered phase consisting of asters surrounded by shock lines that constantly evolves and remodels itself. We demonstrate that the main source of disorder lies along shock lines, rendering this active foam fundamentally different from the corresponding equilibrium system. We thus show that in the context of active matter also, a result obtained at all orders of perturbation theory can be superseded by nonperturbative effects, calling for a different approach.

Figure 1

Left: $\overline{v}(L)$ for different $\lambda$ values ($a=0.7$, $\eta =0.5$ and initial ordered state). Error bars correspond to two standard errors of the mean computed over five independent samples. For large $\lambda$ values, $\overline{v}(L)$ is reasonably well fitted by $\overline{v}-{\overline{v}}_{\infty }=A{L}^{-\omega }$ with $\omega =2/3$ (${\overline{v}}_{\infty }=0.67$, $A=0.77$, for $\lambda =0.5$, orange curve, and ${\overline{v}}_{\infty }=0.68$, $A=0.57$, for $\lambda =1.0$, red curve). Right: $S_{\perp }$ versus $q_{\parallel }$ for $q_{\perp }=0$ (red squares) and versus $q_{\perp }$ for $q_{\parallel }=0$ (blue dots) ($a=5$, $\lambda =1$, $\eta =0.5$, $L=800$). Dashed lines: scaling predicted at small wave vectors [6], which arises only beyond a crossover scale for $S_{\perp }(q_{\perp })$. Averages are taken over a time ${10}^6$ small enough that the nucleation events described in Fig. 2 are not observed.

Figure 2

(a)–(d) Snapshots taken during a run starting from an ordered state showing the nucleation of a first defect and the following evolution ($a=0.48$, $\lambda =1.0$, $\eta =0.5$, $L=3600$). (e) Probability distribution $\mathcal{P}$ of the lifetime of the ordered phase $\tau$, defined as the first time for which the nucleation of an aster decreases $\overline{v}$ by more than 20% ($a=0.45$, $\lambda =1.0$, $\eta =0.5$). Data obtained at three different $L$ values, rescaled by a factor $s^2$ proportional to $L^2$.

Figure 3

Left: variation of correlation length $\xi$ with $a$ for different sizes $L$. Data taken in the active foam steady state ($\lambda =1.0$, $\eta =0.5$). Error bars correspond to 2 standard errors of the mean, computed over 10 independent time intervals, each much longer than the correlation time of the global velocity. At a given $a$ value, small size data may not be reliable since $\xi$ can then be of the order of $L$. At $L=900$ in particular, data for $a>2.0$ are not shown since only very few or even no asters are present. Right: typical configuration taken during coarsening in the deterministic limit $\eta =0$ [$a=1$, $\lambda =1.0$, $L=900$, $dx=0.5$, $dt=0.01$, colors as in Fig. 2, from Movie 3 in Ref. [28] ]. Apart from the five clearly visible asters, which have $+1$ topological charge, nine shock-line embedded $-1$ defects are present (white circles), as well as four vortexlike $+1$ defects present at some shock line vertices (black diamonds). Labels A, B, and C in the main panel point to the defects shown more clearly in the small lower panels.

Figure 4

Study of the interface $h(y)$ separating two half domains with different bulk orientation ($a=1$, $\lambda =1$, $\eta =0.1$, periodic boundary condition in $y$). (a) Typical snapshot showing $h(y)$ (red line) defined by the points along $x$ where $v_x$ changes sign ($300\times 100$ system). Velocity is fixed symmetrically on lateral boundaries at an angle $\alpha =\pi /3$ (blue arrows). (b) Mean horizontal profiles of $v_x$ and $v_y$ measured relative to the interface position $h(y)$, averaged over $y$ and time. (c) Variation with $\alpha$ of the width of the interfacial region, measured either by fitting the average profile $⟨v_x⟩$ [as in (b)] by a tanh function (magenta squares) or as the roughness of the interface $w=\sqrt{⟨⟨h^2{⟩}_y-⟨h{⟩}_y^2{⟩}_t}$ (blue circles). System size $100\times 100$. (d) Same as (c) but showing the propagation speed of fluctuations along the interface obtained either as the maximum of $⟨v_y⟩$ in (b) (magenta squares) or from the space-time Fourier transformed correlation function $⟨|h(q_y,\omega {)}^2|{⟩}_t$ as the position of the peaks $\omega =v{q}_y$ at small wave vector $q_y$.