Kuramoto Model: Finite-Size Order Parameter

Synchronization transitions & finite-size fluctuations

Kuramoto Model

N coupled phase oscillators synchronize when coupling K exceeds a critical threshold K_c. The order parameter R measures coherence.

dθᵢ/dt = ωᵢ + (K/N)Σⱼ sin(θⱼ−θᵢ) R·e^(iψ) = (1/N)Σⱼ e^(iθⱼ) K_c = 2/[π·g(0)] (Lorentzian: K_c = 2γ)

Finite-size effects: for small N, fluctuations scale as 1/√N even below K_c, creating apparent partial synchrony.

⟨R²⟩ - ⟨R⟩² ~ 1/N
Order Parameter R
0.00
K_c ≈ 1.00