Kardar-Parisi-Zhang fluctuations in the nonequilibrium critical dynamics of synchronization in one dimension

The study of synchronous dynamics has traditionally focused on the transition to synchronization from a static perspective, the dynamical process whereby systems of oscillators synchronize at long times having been much less studied. While one might expect it to be strongly system-dependent, on the contrary it displays the robust universal features of nonequilibrium critical dynamics, as suggested by a mathematical connection between oscillator models and equations of surface kinetic roughening. By means of detailed numerical studies of 1D systems of phase and limit-cycle oscillators in the presence of quenched and time-dependent noise, we show that the synchronization process in these systems is characterized by forms of generic scale invariance associated with kinetically rough interfaces, such as the Kardar-Parisi-Zhang (KPZ) universality class. In fact, the large-scale behavior and the role of symmetries and randomness can be analytically understood by a combination of continuum approximations and phase-reduction methods. The phase fluctuations generically follow a Tracy-Widom distribution, associated with the KPZ nonlinearity. Synchronization and surface growth appear much more closely related than previously anticipated, making the experimental observation of the nonequilibrium criticality of synchronization an enticing possibility.