Figure 1

The figures show the development of systems in space-time plots from random initial density and velocity distributions on systems with $L=128$ sites. The greyscale measures density $\rho (x,t)$ on a logarithmic scale. In this and the following figures parameter sets are indicated [thus] (a) $[\eta =0.2,\beta =1$, $\kappa =0.5$, $⟨\rho ⟩=1$, $C\to \infty ]$ A disordered system showing no sustained correlations in density or velocity. (b) $[\eta =0.02,\beta =8$, $\kappa =0.5$, $⟨\rho ⟩=1$, $C\to \infty ]$ An alternating flock exhibiting a nonzero velocity between regular reversals. (c) $[\eta =0.125,\beta \to \infty$, $\kappa =0$, $⟨\rho ⟩=8]$ A homogeneous flock having homogeneous density and fixed global velocity, emerging from transient alternations. (d) $[\eta =0.02,\beta =8$, $\kappa =0.75$, $⟨\rho ⟩=1,C\to \infty ]$ A dipole system consisting of several dipoles separated by low density homogeneous domains undergoes a slow coarsening process towards a single large dipole. (For the definition of a dipole see text.)Reuse & Permissions

Figure 2

Diagram in the $\kappa \text{−}\beta ∕(1+\beta )$ plane showing the regimes present in a system with $L=128$, $⟨\rho ⟩=4$, $\eta =0.05$ with alignment and centering effects only $(C\to \infty )$. The dipoles, alternating flock, and homogeneous flock all occupy characteristic regions. Intermediate states at the boundaries of these regions are characterized by mixed behaviors: (a) Flocks behave for sustained periods as both homogeneous flocks and alternating flocks, (b) dipoles begin to traverse large distances in single motions. The model of [22] is case (c), a homogeneous flock for this parameter set and system size. For comparison, properties of the homogeneous solutions of the continuun mean field equations, discussed in Sec. 4, are also shown by numerical solution of the stability equations for $\eta =0.05$. The solid line is the boundary separating the linearly stable homogeneous flocking and disordered regimes (20). The dashed line is found from (40,41) and encompasses a region in which both disordered and homogeneous flock solutions are found to be linearly unstable. In this region dipole solutions exist (25) and are anticipated to be the unique stable solutions for these parameter sets.Reuse & Permissions

Figure 3

(a) Breakdown times for homogeneous flocks and (b) reversal times for alternating flocks vs $\ln L$ where $L$ is the system size (number of lattice sites). Data is averaged over 100 and 1000 breakdowns and reversals respectively. (a) $H1$: $[\eta =0.0413,\beta =1,\kappa =0,⟨\rho ⟩=10]$, $H2$: $[\eta =0.125,\beta =\infty ,\kappa =0,⟨\rho ⟩=8]$. The straightline is a least squares fit to $T=L∕4+A{L}^{-B}$ (see text). The exponent is $B=1.03$ for $H2$, and $0.89$ for $H1$. (b) $A1$: $[\eta =0.005,\beta =2,\kappa =0,⟨\rho ⟩=1]$, $A2$: $[\eta =0.02,\beta =8,\kappa =0.25,⟨\rho ⟩=1,C=\infty ]$. The reversal times in alternating flocks are logarithmic and the straight lines are least squares fits to the data.Reuse & Permissions

Figure 4

The homogeneous flock of Fig. 1c undergoes a rapid change of behavior. (a), (b), (c) Density and $G$ are plotted as a function of position at fixed times. (a) $t=58014$: values are consistent with a homogeneous flock throughout except near site 32 where a localized momentum fluctuation exists. (b) $t=58046$: The interface between the two opposing domains created moves leftwards and the fluctuation gathers momentum. (c) $t=58078$: Within a short space of time the entire flock is reversed, and the density and $G$ profiles are characteristic of an alternating flock. Following this event an alternating flock is temporarily established, before an homogeneous flock reemerges. (d), (e) A measure of the spatial distribution of boids and the global velocity plotted against time from before the breakdown, through the temporary alternating flock until the emergence of a homogeneous flock.Reuse & Permissions

Figure 5

$[\eta =0.1,\beta =4$, $\kappa =0.75$, $C\to \infty ]$. Density profiles (upper panels) and profiles of the expected updated velocity, $G$, (lower panels) after $50000$ time steps. (a) At high density $⟨\rho ⟩=2000$ domains, with exponentially decaying densities and fixed velocity, bounded by discontinuities in $G$ are clear. (b) At density $⟨\rho ⟩=1$ dipoles occupy a small portion of the lattice, but contain the majority of boids. There are large portions devoid of boids with $G=0$.Reuse & Permissions

Figure 6

(Color online) System with the same parameter set as Fig. 1d on a larger system of $L=1024$. (a) A typical space-time plot starting from random initial conditions. After an initial transient period when the dipoles form, an intricate coarsening process takes place wherein small dipole eventually dissolve and large dipoles grow. (b) and (c) Statistical data on the remaining dipoles as a function of time averaged from 100 runs similar to (a). (b) The number of dipoles remaining of size greater than the values indicated in the legend: Eventually the number of dipoles of intermediate size decreases to zero (curves $B,C,D,E$ converge to 1); After ${10}^7$ time steps a single large dipole emerges alongside approximately $2$ small transient dipoles. (c) The probability that a boid is contained within a dipole greater than the sizes indicated in the legend as a function of time: Only a small number of boids ever exist outside dipoles $\left(A\right)$ and after sufficient time the majority of boids are contained in a single large dipole $\left(E\right)$.Reuse & Permissions

Figure 7

$[\eta =0.02,\beta =8$, $\kappa =0.85$, $⟨\rho ⟩=1,C=20]$ This system contains two large fixed-density dipoles alongside small dipoles and solitary boids. In small dipoles centering is the dominant effect and wandering is limited, however in the two fixed-density dipoles large translations are observed. See text for explanation of the term fixed-density dipole.Reuse & Permissions