Synchronization transition with finite-size fluctuations — dθᵢ/dt = ωᵢ + (K/N)Σ sin(θⱼ−θᵢ)
Kuramoto (1975): N coupled oscillators with random natural frequencies ωᵢ ~ g(ω).
Critical coupling: K_c = 2/(πg(0))
For K > K_c, a synchronized cluster forms: r → finite as N→∞.
r ~ √((K−K_c)/K) near transition.
Finite-N: r fluctuates even below K_c; amplitude ~ N⁻¹/² (central limit). The phase circle shows individual oscillator positions; the arrow is the mean-field phasor R·e^(iΨ).