Nonequilibrium criticality driven by Kardar-Parisi-Zhang fluctuations in the synchronization of oscillator lattices

Abstract

The synchronization of oscillator ensembles is pervasive throughout nonlinear science, from classical or quantum mechanics to biology, to human assemblies. Traditionally, the main focus has been the identification of threshold parameter values for the transition to synchronization as well as the nature of such transition. Here, we show that considering an oscillator lattice as a discrete growing interface provides unique insights into the dynamical process whereby the lattice reaches synchronization for long times. Working on a generalization of the Kuramoto model that allows for odd or non-odd couplings, we elucidate synchronization of oscillator lattices as an instance of generic scale invariance, whereby the system displays space-time criticality, largely irrespective of parameter values. The critical properties of the system (like scaling exponent values and the dynamic scaling Ansatz which is satisfied) fall into universality classes of kinetically rough interfaces with columnar disorder, namely, those of the Edwards-Wilkinson (equivalently, the Larkin model of an elastic interface in a random medium) or the Kardar-Parisi-Zhang (KPZ) equations, for Kuramoto (odd) coupling and generic (non-odd) couplings, respectively. From the point of view of kinetic roughening, the critical properties we find turn out to be quite innovative, especially concerning the statistics of the fluctuations as characterized by their probability distribution function (PDF) and covariance. While the latter happens to be that of the Larkin model irrespective of the symmetry of the coupling, in the generic non-odd coupling case the PDF turns out to be the Tracy-Widom distribution associated with the KPZ nonlinearity. This brings the synchronization of oscillator lattices into a remarkably large class of strongly-correlated, low-dimensional (classical and quantum) systems with strong universal fluctuations.