From LRO to Disorder via QLRO in Spatially Inhomogeneous Polar Flock

Abstract

We study the collective behavior of a polar flock in an inhomogeneous environment in two-dimensions. The inhomogeneity is modelled by introducing regions at random locations on the substrate with higher noise but accessible for the flock to move. Hence inside such regions the particles orientation get randomised. Such inhomogeneities are different from the physical disorder, which obstructs the space for the incoming particles. The study focuses on how the phase behavior of polar flock changes by tuning the packing fraction of inhomogeneity. As packing fraction increases, the system crosses over from long-range to quasi long range order and ultimately to a disordered phase, while the order disorder transition for flocking changes from discontinuous to continuous. The resultant phase behavior of polar flock patterns here is comparable to that exhibited in the presence of physical disorder.

Figures (8)

• Figure 1: (color online) Model picture: Self-propelled particles are moving in two-dimensional space. The red circles represent the high noise regimes, while the particle's color represents its orientation, as shown in the color bar. Inside the circles, all particles have varied colors because of high noise, while outside, all SPPs have the same color and exhibit coherent motion.
• Figure 2: (color online) Simulation snapshots of the system in the steady-state at $\eta = 0.2$ for four different values of $\rho_o$ as shown in (a),(b),(c) and (d). The orientation of the self-propelled particles (SPPs) is indicated by the color bar. Here $N = 10000$, and for clarity of the figure, the disorders are not shown. URL
• Figure 3: (color online) (a) Order parameter $\varphi$ vs. number of particle $N$ for various packing fractions $\rho_o$ on log-log scale, for $\eta = 0.2$. The symbol-color scheme: black circle ($\rho_{0} = 0.0$), red square ($\rho_{0} = 0.005$), green diamond ($\rho_{0} = 0.05$), blue upward triangle ($\rho_{0} = 0.1$), violet downward triangle ($\rho_{0} = 0.2$), magenta left-pointing triangle ($\rho_{0} = 0.3$), orange right-pointing triangle ($\rho_{0} = 0.4$), and brown plus sign ($\rho_{0} = 0.5$). The symbols are data points, and dashed lines are respective power-law fits: $\varphi\sim N^{-\zeta(\rho_o)}$. The size of the error bars is less than the size of the symbol. (b) Variation in the exponent $\zeta$ for different $\rho_o$. The red and black dashed lines represent $\zeta=0$ and $1/16$, respectively. (c) Two-point spatial correlation function of the orientation of polar particles $C(r)$ for several packing fractions $\rho_o$ on a log-log scale. Inset: y-axis rescaled via multiplication of expected critical exponent value $1/4$. For references, dashed green and maroon lines are drawn with slopes of $0.25$ and $0.0$, respectively. For clarity, the color and symbol scheme used here is the same as that introduced in Fig. \ref{['Phi_vs_N']}(a). Here $N=40000$ and $\eta=0.2$
• Figure 4: (color online) (a) Variation of order parameter $\varphi$ with respect to noise $\eta$ for five different values of $\rho_o$. Each curve is color-coded: black circle ($\rho_{0} = 0.0$), red square ($\rho_{0} = 0.0025$), green diamond ($\rho_{0} = 0.01$), blue upward triangle ($\rho_{0} = 0.05$), violet downward triangle ($\rho_{0} = 0.1$). Inset: The same plot zoomed near the transition point. Here $N = 10000$. Plots (b) and (c) illustrate system size analysis of Binder cumulant $G$ for two different $\rho_o = 0.0025$ and $0.05$. Three different colors-black, red, and green-represent three system sizes: $N = 2500$, $N = 4900$, and $N = 19600$, respectively.
• Figure 5: (colour online) This phase diagram in the $\rho_o-\eta$ plane, with semi-log x scale, represents the ordering of the Vicsek model in the presence of spatial inhomogeneity. The blue solid line with stars shows the discontinuous transition at low $\rho_o$, while the red dashed line with stars represents the continuous phase transition. Both blue solid and red dashed line meets at a black solid filled circle known as a tri-critical point at $\rho_o \approx 0.012$. The ordered phase has two regimes: LRO and QLRO, separated by the dash-dotted magenta line with a triangle up symbol.