Fig. 3: Hyperuniformity in ensembles of point vortices and rotors.

A Snapshots of 10,000 point vortices initially (left) and at steady state (right). Insets show the structure factor, S(q) with a distinct cavity at steady state. B Angular average of the structure factor shown in (A), in a log-log scale with solid line showing a q^1.3 scaling. Error bars are standard deviation over ten well-separated timesteps. Inset shows the structure factor of the rotors shown in (C) with increasing hue corresponding to increased concentration ϕ = (0.14, 0.24, 0.37, 0.54). Solid line is the same α ~ 1.3 scaling. C Steady-state configurations of 2000 membrane rotors with the corresponding structure factors, showing a transition from disordered hyperuniformity to a hexagonal lattice. Particles are colored according to their local bond orientation parameter ψ₆. For particle j, \({\psi }_{6}^{j}={\sum }_{i}{e}^{6i{\theta }_{ij}}\), with the sum taken over nearest neighbors as found by a Voronoi diagram. The table gives ensemble-averaged values, \({N}^{-1}{\sum }_{j}{\psi }_{6}^{j}\). D A plot of the relative deviation for each particle, with the relative deviation of particle i defined by how far it is displaced from its position at the previous cycle, i.e., ∣r_i(t + t_cyc) − r_i(t)∣. The cycle time is calculated at steady state as the average time it takes the system to rotate by 2π. Particles are colored by their relative deviation, from blue to yellow with increasing deviation. The plot at the bottom shows the strobed position of four particles during a time interval of Δt ≈ 115 cycles; the particles in the strobed frame move along Brownian-like trajectories.