Phase transitions and linear stability for the mean-field Kuramoto-Daido model

Abstract

We consider the mean-field noisy Kuramoto-Daido model, which is a McKean-Vlasov equation on the circle with bimodal interaction $W(\theta )=\cos \theta +m\cos 2\theta$ for $m\ge 0$ and interaction strength $K$, generalizing the celebrated noisy Kuramoto model corresponding to $m=0$. Our first contribution is to characterize the phase transition threshold $K_c$ by comparing it to the linear stability threshold $K_{\#}=\min (1,m^{-1})$ of the uniform distribution. When $m\le 1/2,$ $K_c=1$, coinciding with that of the Kuramoto model. On the other hand, for $m\ge 2$, we show $K_c=m^{-1}$. We also classify the regimes in which the phase transition is continuous or discontinuous. Our second contribution is to analyze the linear stability of a global minimizer $q$ (the ``ordered phase'') of the mean-field free energy in the supercritical regime $K>1$. This stationary solution of the Kuramoto-Daido equation is unique up to translation invariance and distinct from the uniform distribution (the ``disordered phase''). Our approach extends the Dirichlet form method of Bertini et al. from the unimodal to bimodal setting. In particular, for $m\le 1.590\times 10^{-4}$ and $K>1$, we show an explicit lower bound on the spectral gap of the linearized McKean-Vlasov operator at $q$. To our knowledge, this is the first rigorous stability analysis for this class of models with bimodal interactions.

arXiv comment41 pages, 1 figure