Figure 4. Collective-dynamics simulations.

(a) The numerical phase diagram of the confined population is composed of three regions: isotropic gas (low φ₀, small b), swarm coexisting with a gaseous phase (intermediate φ₀ and b) and vortex state (high φ₀ and b). R_c=0.5 mm. (b) Snapshot of a vortex state. Numerical simulation for φ₀=0.1 and b=5a. (c) Snapshot of a swarm. Numerical simulation for φ₀=4.5 × 10⁻² and b=2a. (d) Variation of the density correlation length as a function of R_c. Above R_c=1 mm, ξ plateaus and a vortex is reached (φ₀=3 × 10⁻², b=3a). (e) Four numerical snapshots of rollers interacting via: alignment interactions only (A), alignment interactions and repulsive torques (A+B, where the magnitude of B is five times the experimental value), alignment and excluded volume interactions (A+b, where the repulsion distance is b=5a), alignment and the C-term in equation 3 (A+C). Polarized vortices emerge solely when repulsive couplings exist (A+B and A+b).