We study stationary solutions of McKean-Vlasov equations on the circle. Our main contributions stem from observing an exact equivalence between solutions of the stationary McKean-Vlasov equation and an infinite-dimensional quadratic system of equations over Fourier coefficients, which allows explicit characterization of the stationary states in a sequence space rather than a function space. This framework provides a transparent description of local bifurcations, characterizing their periodicity, and resonance structures, while accommodating singular potentials. We derive analytic expressions that characterize the emergence, form and shape (supercritical, critical, subcritical or transcritical) of bifurcations involving possibly multiple Fourier modes and connect them with discontinuous phase transitions. We also characterize, under suitable assumptions, the detailed structure of the stationary bifurcating solutions that are accurate upto an arbitrary number of Fourier modes. At the global level, we establish regularity and concavity properties of the free energy landscape, proving existence, compactness, and coexistence of globally minimizing stationary measures, further identifying discontinuous phase transitions with points of non-differentiability of the minimum free energy map. As an application, we specialize the theory to the Noisy Mean-Field Transformer model, where we show how changing the inverse temperature parameter $\beta$ affects the geometry of the infinitely many bifurcations from the uniform measure. We also explain how increasing $\beta$ can lead to a rich class of approximate multi-mode stationary solutions which can be seen as `metastable states'. Further, a sharp transition from continuous to discontinuous (first-order) phase behavior is observed as $\beta$ increases.

翻译：本文研究圆周上McKean-Vlasov方程的稳态解。我们的核心贡献源于发现稳态McKean-Vlasov方程的解与傅里叶系数构成的无限维二次方程组之间存在精确等价关系，这使得我们能够在序列空间而非函数空间中显式刻画稳态。该框架为局部分岔现象提供了透明描述，刻画了其周期性及共振结构，同时兼容奇异势函数。我们推导了表征分岔现象（涉及多个傅里叶模式）产生机制、形式与类型（超临界、临界、亚临界或跨临界）的解析表达式，并将其与不连续相变建立联系。在适当假设下，我们进一步刻画了稳态分岔解的精细结构，其精度可达任意阶傅里叶模式。在全局层面，我们建立了自由能景观的正则性与凹性，证明了全局极小稳态测度的存在性、紧致性与共存性，并将不连续相变与最小自由能映射的不可微点相关联。作为应用，我们将理论具体应用于噪声平均场Transformer模型，展示了逆温度参数$\beta$如何影响均匀测度处无穷多分岔的几何结构。同时阐明了增大$\beta$值如何产生丰富的近似多模态稳态解，这类解可视为"亚稳态"。此外，随着$\beta$增大，我们观察到从连续相变到不连续（一级）相变行为的急剧转变。