On the trend to global equilibrium for Kuramoto Oscillators

TL;DR

This paper analyzes the convergence to global equilibrium in Kuramoto oscillators, providing quantitative estimates on the rate of synchronization for large coupling strength and oscillator populations, using advanced mathematical tools.

Contribution

It introduces new quantitative estimates for convergence rates of Kuramoto oscillators, employing entropy methods, stability analysis, and transportation inequalities in a non-gradient flow setting.

Findings

Convergence rate estimates improve with increasing oscillator number.

Oscillators concentrate around equilibrium at a rate quantified probabilistically.

New inequalities and stability results enhance understanding of Kuramoto dynamics.

Abstract

In this paper, we study the convergence to the stable equilibrium for Kuramoto oscillators. Specifically, we derive estimates on the rate of convergence to the global equilibrium for solutions of the Kuramoto-Sakaguchi equation in a large coupling strength regime from generic initial data. As a by-product, using the stability of the equation in the Wasserstein distance, we quantify the rate at which discrete Kuramoto oscillators concentrate around the global equilibrium. In doing this, we achieve a quantitative estimate in which the probability that the oscillators will concentrate at the given rate tends to one as the number of oscillators increases. Among the essential steps in our proof are: 1) An entropy production estimate inspired by the formal Riemannian structure of the space of probability measures, first introduced by F. Otto in [35]; 2) A new quantitative estimate on the instability of equilibria with antipodal oscillators based on the dynamics of norms of the solution in sets evolving by the continuity equation; 3) The use of generalized local logarithmic Sobolev and Talagrand type inequalities, similar to the ones derived by F. Otto and C. Villani in [36]; 4) The study of a system of coupled differential inequalities, by a treatment inspired by the work of L. Desvillettes and C. Villani [13]. Since the Kuramoto-Sakaguchi equation is not a gradient flow with respect to the Wasserstein distance, we derive such inequalities under a suitable fibered transportation distance.