The Kuramoto model: N=100 phase oscillators on the unit circle, each with its own natural frequency ωi drawn from a Gaussian, coupled all-to-all via θ˙i=ωi+NKj=1∑Nsin(θj−θi).
The order parameter reiψ=N1∑jeiθj (gold arrow + bottom strip) measures coherence: r=0 is incoherent, r=1 is full lock. Below a critical Kc≈2σω/πg(0) the dots smear around the circle; above it a synchronized cluster spontaneously condenses out of the noise — a continuous phase transition with no leader. Move your cursor left→right to scrub K∈[0,6] and watch the order parameter snap on.