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 $W(θ)=\cosθ+m\cos2θ$ 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 $m \leq 1/2,$ $K_{c}=1$, coinciding with that of the Kuramoto model. On the other hand, for $m \geq 2$, 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 (thedisordered phase''). Our approach extends the Dirichlet form method of Bertini et al. from the unimodal to bimodal setting. In particular, for $m \leq 1.590 \times 10^{-4}$ 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.