Kuramoto Model — Mean-Field Synchronization Transition

N coupled oscillators: dθᵢ/dt = ωᵢ + (K/N)Σⱼsin(θⱼ−θᵢ). Each oscillator has a natural frequency drawn from a Lorentzian distribution (half-width γ). When K exceeds the critical coupling Kc = 2γ, the order parameter r = |⟨eⁱθ⟩| jumps from 0 to a nonzero value — a second-order phase transition to collective synchrony.

r = 0.00 | Kc = 2.00