Synchronization transition in the two-dimensional Kuramoto model with dichotomous noise

Abstract

We numerically study the celebrated Kuramoto model of identical oscillators arranged on the sites of a two-dimensional periodic square lattice and subject to nearest-neighbor interactions and dichotomous noise. In the nonequilibrium stationary state attained after a long time, the model exhibits a Berezinskii–Kosterlitz–Thouless (BKT)-like transition between a phase at a low noise amplitude characterized by quasi long-range order (critically ordered phase) and an algebraic decay of correlations and a phase at a high noise amplitude that is characterized by complete disorder and an exponential decay of correlations. The interplay between the noise amplitude and the noise-correlation time is investigated, and the complete, nonequilibrium stationary-state phase diagram of the model is obtained. We further study the dynamics of a single topological defect for various amplitudes and correlation time of the noise. Our analysis reveals that a finite correlation time promotes vortex excitations, thereby lowering the critical noise amplitude of the transition with an increase in correlation time. In the suitable limit, the resulting phase diagram allows one to estimate the critical temperature of the equilibrium BKT transition, which is consistent with that obtained from the study of the dynamics in the Gaussian white noise limit.

Synchronization in cilia carpets and the Kuramoto model with local coupling: breakup of global synchronization in the presence of noise

“…We show how the asymmetric coupling arising from a chiral cilia beat results in a dominant wave with non-trivial wave vector as long as the noise strength remains below a characteristic noise strength. Above this characteristic noise strength, global synchronization is lost, but dynamic domains of local synchronization may persist, analogous to the two-dimensional Kuramoto model [34]. Our work links an extensive literature on the Kuramoto model with local coupling and noise [34][35][36] (or quenched frequency disorder [37][38][39][40]) to more detailed models of cilia carpets with asymmetric oscillator coupling.…”

“…If a noise strength D satisfies D c (L 1 ) D D c (L 2 ), we may thus expect local synchronization on length scales L 2 , but not anymore on length scales L 1 . Indeed, previous simulations of the two-dimensional Kuramoto model showed examples of such local synchronization [34].…”

“…In contrast, the Kuramoto model with global, all-to-all coupling exhibits a conventional second-order phase transition as a function of noise strength [36,[49][50][51]. Thus, there is a stark difference between Kuramoto models with global coupling and two-dimensional models with local coupling, previously studied for quenched frequency disorder [37,40] and noise [34,35]. Models of cilia carpets [52] ignoring this difference have limited predictive power.…”

“…In the thermodynamic limit of infinite system size, Kuramoto models with local, short-range coupling exhibit a conventional order-disorder transition as a function of noise strength only in three-or higher-dimensional systems, but not in two-dimensional systems [34,35]. This is essentially a consequence of the famous Mermin-Wagner theorem from statistical physics, which rules out global order in two-dimensional systems with local coupling and continuous symmetries [48].…”

“…Comparison to Kuramoto model. As a reference, we recall the classical Kuramoto model with local coupling [34,36,37]…”