Phase transitions in Doi-Onsager, Noisy Transformer, and other multimodal models

We study phase transitions for repulsive-attractive mean-field free energies on the circle. For a $\frac{1}{n+1}$-periodic interaction whose Fourier coefficients satisfy a certain decay condition, we prove that the critical coupling strength $K_c$ coincides with the linear stability threshold $K_\#$ of the uniform distribution and that the phase transition is continuous, in the sense that the uniform distribution is the unique global minimizer at criticality. The proof is based on a sharp coercivity estimate for the free energy obtained from the constrained Lebedev--Milin inequality. We apply this result to three motivating models for which the exact value of the phase transition and its (dis)continuity in terms of the model parameters was not fully known. For the two-dimensional Doi--Onsager model $W(\theta)=-|\sin(2\pi\theta)|$, we prove that the phase transition is continuous at $K_c=K_\#=3\pi/4$. For the noisy transformer model $W_\beta(\theta)=(e^{\beta\cos(2\pi\theta)}-1)/\beta$, we identify the sharp threshold $\beta_*$ such that $K_c(\beta) = K_\#(\beta)$ and the phase transition is continuous for $\beta \leq \beta_*$, while $K_c(\beta)<K_\#(\beta)$ and the phase transition is discontinuous for $\beta > \beta_*$. We also obtain the corresponding sharp dichotomy for the noisy Hegselmann--Krause model $W_{R}(\theta) = (R-2\pi|\theta|)_{+}^2$ .