N coupled phase oscillators — emergence of synchrony
Order parameter r = 0.00 (r→1 = synchronized, r→0 = incoherent)
Kuramoto (1984): Each oscillator has a natural frequency ω drawn from a Lorentzian distribution.
The coupling term K/N · Σ sin(θⱼ − θᵢ) pulls oscillators toward the mean phase.
At coupling K > K_c = 2/πg(0) (where g is the frequency distribution), a phase transition occurs:
the order parameter r = |⟨e^{iθ}⟩| jumps from 0 to a nonzero value — spontaneous synchronization.
Analogous to: firefly flashing, cardiac pacemakers, power grid stability, neural oscillations.