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 (the ``disordered 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.

Figures (1)

• Figure 1: The blue curve corresponds to the critical interaction strength $K_{c}(m),$ and the orange curve corresponds to the linear stability threshold for the uniform distribution, given by $K_{\#}(m) = \min\{1, m^{-1}\}.$ Theorem \ref{['phase theorem']} gurantees that these two curves coincide for $m \in [0, 0.5] \cup [m_{*}, \infty)$, and that the blue curve lies below the orange curve in the remaining region $m \in (0.5, m_{*}).$

Theorems & Definitions (49)

• Theorem 1.1
• Theorem 1.2
• Remark 1
• Theorem 1.3
• Remark 2
• Theorem 1.4
• Remark 3
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3