Coupled Oscillators
The Kuramoto model and synchronization phase transitions
N oscillators, each with its own natural frequency, interact through sine coupling. Below a critical coupling strength K_c, the population remains incoherent -- phases scatter uniformly around the circle. Above K_c, a macroscopic order parameter r emerges: oscillators lock into collective synchronization. This is a continuous phase transition -- the same mathematics governs firefly synchronization, neural oscillations, and power grid stability.
The Kuramoto model: N oscillators with different natural frequencies, coupled by sine interaction. Below a critical coupling K_c, oscillators are incoherent (r near 0). Above K_c, a phase transition occurs and the population synchronizes (r near 1).