圆上平均场相互作用扩散的亚稳定性与混沌传播的尖锐阈值
Metastability and Sharp Propagation-of-Chaos Thresholds for Mean-Field Interacting Diffusions on the Circle
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中文总结 AI 辅助
本文研究圆上平均场相互作用扩散的亚稳定性和混沌传播，确定了尖锐的指数时间阈值，并给出了临界轨道有限性的可检验准则，应用于Kuramoto模型和Transformer交互。
中文摘要 AI 辅助
我们研究了圆上具有多个局部自由能极小值的平均场相互作用扩散的亚稳定性和长期混沌传播。在模旋转的意义下，我们确定了亚稳定性的尖锐指数时间尺度以及McKean-Vlasov近似的有效性和失效性。假设存在有限多个临界旋转轨道且局部极小能量互不相同，我们证明，对于非全局极小值吸引域中的初始数据，亚稳定性和混沌传播在确定性时间至$e^{N(D-\varepsilon)}$内一致成立，并在任何增长快于$e^{N(D+\varepsilon)}$的确定性序列处失效。这里$N$是粒子数，$D$是以该极小值为底的Freidlin-Wentzell能量循环的最大深度。同样的阈值以更强的路径意义控制平方近似误差的时间平均，并对适当对齐的时间平均经验测度给出相应的亚稳定性和混沌传播界。当全局极小值轨道唯一时，我们在其吸引域内建立一致时间的混沌传播，而不需要假设有限多个临界轨道。证明发展了经验测度过程在旋转商空间上的Freidlin-Wentzell循环分析。关键步骤将Dawson和Gärtner的固定时间水平大偏差估计扩展到亚稳态井之间无界持续时间的游荡，从而得到尖锐的循环退出估计和确定性时间的局部化。我们给出了临界轨道有限性的可检验的傅里叶准则，以及具有多个井的显式例子，并讨论了在噪声Kuramoto模型中的应用以及对噪声Transformer相互作用的条件性结果。我们还研究了自由能景观中随噪声强度变化的相变如何影响亚稳定性和混沌传播。
英文摘要
We study metastability and long-time propagation-of-chaos for mean-field interacting diffusions on the circle with multiple local free-energy minima. Working modulo rotations, we identify sharp exponential time scales for metastability and the validity and breakdown of the McKean-Vlasov approximation. Assuming finitely many critical rotation orbits and distinct local-minimum energies, we show that, for initial data in the attraction basin of a nonglobal minimum, metastability and propagation-of-chaos hold uniformly over deterministic times up to $e^{N(D-\varepsilon)}$ and fail at every deterministic sequence growing faster than $e^{N(D+\varepsilon)}$. Here $N$ is the particle number and $D$ the depth of the maximal Freidlin-Wentzell energy cycle having that minimum as its bottom. The same threshold governs running averages of the squared approximation error in a stronger pathwise sense, with corresponding metastability and propagation-of-chaos bounds for suitably aligned time-averaged empirical measures. When the global-minimum orbit is unique, we establish uniform-in-time propagation-of-chaos in its attraction basin without assuming finitely many critical orbits. The proofs develop Freidlin-Wentzell cycle analysis for the empirical measure process on the rotation quotient. The key steps extend Dawson and Gärtner's fixed-horizon large deviation estimates to excursions of unbounded duration between metastable wells, yielding sharp cycle-exit estimates and localization at deterministic times. We give a checkable Fourier criterion for finiteness of the critical orbits and explicit examples with multiple wells, and discuss applications to the noisy Kuramoto model and conditional consequences for noisy transformer interactions. We also investigate how phase transitions in the free-energy landscape, as noise strength varies, affect metastability and propagation-of-chaos.
发表机构
- University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校）
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