Synchronization for discrete mean-field rotators

TL;DR

This paper studies a non-reversible mean-field jump process for discrete q-valued rotators, demonstrating synchronization and the existence of a locally attractive periodic orbit, with insights into global attractivity via Lyapunov functions.

Contribution

It introduces a novel analysis of non-reversible mean-field dynamics for discrete rotators, showing synchronization and periodic behavior.

Findings

Existence of a locally attractive periodic orbit.

Synchronization in the mean-field jump dynamics.

Discussion of global attractivity using Lyapunov functions.

Abstract

We analyze a non-reversible mean-field jump dynamics for discrete q-valued rotators and show in particular that it exhibits synchronization. The dynamics is the mean-field analogue of the lattice dynamics investigated by the same authors in [26] which provides an example of a non-ergodic interacting particle system on the basis of a mechanism suggested by Maes and Shlosman [32]. Based on the correspondence to an underlying model of continuous rotators via a discretization transformation we show the existence of a locally attractive periodic orbit of rotating measures. We also discuss global attractivity, using a free energy as a Lyapunov function and the linearization of the ODE which describes typical behavior of the empirical distribution vector.