Kuramoto Synchronization

N coupled phase oscillators spontaneously synchronize above critical coupling K_c = 2γ/π

Order r = |⟨e^{iθ}⟩|—
K_c = 2γ/π—
K/K_c—
dθᵢ/dt = ωᵢ + (K/N)Σⱼsin(θⱼ−θᵢ)

ωᵢ ~ Lorentzian(0,γ)

r→0: incoherent
r→1: fully synchronized
The Kuramoto model (1975) describes N coupled phase oscillators with natural frequencies drawn from a distribution g(ω). Below the critical coupling K_c = 2/(πg(0)), oscillators drift independently (r≈0). Above K_c, a macroscopic fraction locks together — the order parameter r = |⟨e^{iθ}⟩| jumps to a nonzero value. For the Lorentzian distribution g(ω) = γ/(π(ω²+γ²)), K_c = 2γ/π exactly. Near criticality: r ∝ √(K−K_c). This is a second-order phase transition — synchronization is a collective phenomenon analogous to ferromagnetism, with applications from fireflies to power grids to neural oscillations.