Superdiffusion, large-scale synchronization and topological defects

TL;DR

This paper investigates how different types of motion, normal diffusion and superdiffusion, influence the emergence of synchronization and order in a system of noisy oscillators, revealing that superdiffusion promotes long-range order.

Contribution

It introduces a non-Hamiltonian field theory for oscillators with mobility, showing how superdiffusion suppresses topological defects and enhances synchronization.

Findings

Superdiffusion induces a transition to long-range order in two dimensions.

Normal diffusion leads to defect-mediated quasi long-range order.

Theoretical results are supported by particle-based simulations.

Abstract

We study an ensemble of random walkers carrying internal noisy phase oscillators which are synchronized among the walkers by local interactions. Due to individual mobility, the interaction partners of every walker change randomly, hereby introducing an additional, independent source of fluctuations, thus constituting the intrinsic nonequilibrium nature of the temporal dynamics. We employ this paradigmatic model system to discuss how the emergence of order is affected by motion of individual entities. In particular, we consider both, normal diffusive motion and superdiffusion. A non-Hamiltonian field theory including multiplicative noise terms is derived which describes the nonequilibrium dynamics at the macroscale. This theory reveals a defect-mediated transition from incoherence to quasi long-range order for normal diffusion of oscillators in two dimensions, implying a power-law dependence of all synchronization properties on system size. In contrast, superdiffusive transport suppresses the emergence of topological defects, thereby inducing a continuous synchronization transition to long-range order in two dimensions. These results are consistent with particle-based simulations.