Synchronization and random long time dynamics for mean-field plane rotators

TL;DR

This paper analyzes the long-time stochastic dynamics of mean-field plane rotators, revealing how finite-size effects cause the synchronization phase to perform a Brownian motion around the stable manifold, deviating from the PDE limit.

Contribution

It characterizes the finite-size induced stochastic behavior of the synchronization phase in mean-field plane rotators, including the Brownian motion of the synchronization center.

Findings

Empirical measure reaches a neighborhood of the stable manifold rapidly.

Synchronization center performs a Brownian motion with a computed diffusion coefficient.

Finite size effects cause deviations from the PDE behavior over long times.

Abstract

We consider the natural Langevin dynamics which is reversible with respect to the mean-field plane rotator (or classical spin XY) measure. It is well known that this model exhibits a phase transition at a critical value of the interaction strength parameter K, in the limit of the number N of rotators going to infinity. A Fokker-Planck PDE captures the evolution of the empirical measure of the system as N goes to infinity, at least for finite times and when the empirical measure of the system at time zero satisfies a law of large numbers. The phase transition is reflected in the fact that the PDE for K above the critical value has a stable manifold of stationary solutions, that are equivalent up to rotations. These stationary solutions are actually unimodal densities parametrized by the position of their maximum (the synchronization phase or center). We characterize the dynamics on times of order N and we show substantial deviations from the behavior of the solutions of the PDE. In fact, if the empirical measure at time zero converges as N goes to infinity to a probability measure (which is away from a thin set that we characterize) and if time is speeded up by N, the empirical measure reaches almost instantaneously a small neighborhood of the stable manifold, to which it then sticks and on which a non-trivial random dynamics takes place. In fact the synchronization center performs a Brownian motion with a diffusion coefficient that we compute. Our approach therefore provides, for one of the basic statistical mechanics systems with continuum symmetry, a detailed characterization of the macroscopic deviations from the large scale limit -- or law of large numbers -- due to finite size effects. But the interest for this model goes beyond statistical mechanics, since it plays a central role in a variety of scientific domains in which one aims at understanding synchronization phenomena.