Breakdown of Ergodicity and Self-Averaging in Polar Flocks with Quenched Disorder

We show that spatial quenched disorder affects polar active matter in ways more complex and far reaching than heretofore believed. Using simulations of the 2D Vicsek model subjected to random couplings or a disordered scattering field, we find in particular that ergodicity is lost in the ordered phase, the nature of which we show to depend qualitatively on the type of quenched disorder: for random couplings, it remains long-range ordered, but qualitatively different from the pure (disorderless) case. For random scatterers, polar order varies with system size but we find strong non-self-averaging, with sample-to-sample fluctuations dominating asymptotically, which prevents us from elucidating the asymptotic status of order.

Figure 1

Stylized finite-size phase diagrams drawn from the data presented in [52]. Top: RC case. (a) $({\rho }_0,ϵ)$ plane for $\eta =0$; (b) $(\eta ,ϵ)$ for ${\rho }_0=1$. Bottom: RS case. (c) $({\rho }_0,\eta )$ plane at fixed $ϵ=0.03$ (the inset shows the small-${\rho }_0$, small-$\eta$ region). (d) $(\eta ,ϵ)$ for ${\rho }_0=1$. The blue dashed line in (b) and (d) marks ergodicity breaking at the system size considered. The pure VM ($ϵ=0$) lies on the $x$ axis in (a), (b), and (d).

Figure 2

Ergodicity breaking in the RC (left, $\eta =0$, $ϵ=0.2$) and RS (right, $\eta =0.18$, $ϵ=0.035$) ordered phases [$L=2048$ in (a)–(f)]. (a),(b) ${\theta }_m(t)$ observed on a single sample from initial conditions ordered along eight (left) and six (right) different directions. Time increases radially outward (log scale) for $2\times 10^6$ time steps. Gray curves: pure case (VM, $ϵ=0$). (c),(d) long-time-averaged momentum field ${\mathbf{m}}_{\infty }(\mathbf{r})$ of two of the RC-case attractors shown in (a) (color map in top row). (e),(f) Same as (c),(d) but for RS-case attractors of (b). (g),(h) Fraction of nonsteady and 1-, 2-, 3-, 4-attractor samples vs system size.

Figure 3

Ergodicity and memory in the disordered phase (RS case, $\eta =0.18$, $ϵ=0.055$, $L=2048$). (a) $⟨|{\mathbf{m}}_T(\mathbf{r})-{\mathbf{m}}_{\infty }(\mathbf{r}){|}^2{⟩}_{\mathbf{r}}$ vs $T$ for different initial conditions, either ordered (orange curves) or taken in the steady state (blue curves). All converge like $1/T$ to ${\mathbf{m}}_{\infty }(\mathbf{r})$. (b) Long-time-averaged momentum field ${\mathbf{m}}_{\infty }(\mathbf{r})$ used in (a) (color map as in Fig. 2). (c) $Q(T)$ vs $T$ in the steady state (gray curve: pure case $ϵ=0$).

Figure 4

Fluctuations and order in the ordered phases (same parameters as in Fig. 2). (a) PDF($m$) for two attractors of the same sample (blue and orange) and $\mathrm{PDF}(⟨m{⟩}_t)$ over samples and attractors (green) in the RC case ($L=512$). (b) Same as (a) but for the RS case ($L=512$) (c) ${\chi }_{\mathrm{con}}$, ${\chi }_{\mathrm{dis}}$, ${\chi }_{\mathrm{att}}$, and ${\chi }_{\mathrm{tot}}$ vs $L$ for the RS case. (d) $M$ vs $L$ for RC, RS, and VM ($ϵ=0$). (e) Local exponent $\sigma$ vs $L$, extracted from data in (d), calculated as $\sigma (\sqrt{L_n{L}_{n+1}})=-\log [M(L_{n+1})/M(L_n)]/\log (L_{n+1}/L_n)$, where $L_{n,n+1}$ are two consecutive system sizes.

Figure 5

RS case at ${\rho }_0=1$, $\eta =0.18$. (a)–(c) Local slope $\sigma$, ${\chi }_{\mathrm{tot}}$, and ${\chi }_{\mathrm{dis}}$ vs $L$ for various $ϵ$ values [see legends in (a) and (b)]. (d) ${\chi }_{\mathrm{tot}}\times L^2$ vs $ϵ$ across the order-disorder transition at various sizes. The estimated peaks are indicated by the black open squares. (e) Scaling of the peaks detected in (d): heights (black squares, left scale) and locations ${ϵ}^{*}(L)$ (blue circles, right scale). The solid blue line is a fit ${ϵ}^{*}(L)-{ϵ}_{\infty }^{*}\sim L^{-1/\nu }$ with $\nu \simeq 1.66$ and ${ϵ}_{\infty }^{*}\simeq 0.0453$.