平均场平面旋转体的同步和随机长时间动力学
Synchronization and random long time dynamics for mean-field plane rotators
摘要：
我们考虑自然朗之万动力学是可逆的平均场平面旋转（或经典自旋XY）措施。众所周知，该模型在相互作用强度参数K的临界值处表现出相变，在旋转子数目N趋于无穷大的极限中。一个福克-普朗克偏微分方程捕捉到系统的经验测度的演化，因为N趋于无穷大，至少在有限的时间内，并且当系统在时间零点的经验测度满足大数定律时。相变反映在以下事实中：K高于临界值的PDE具有稳定的固定解的流形，其等同于旋转。这些稳态解实际上是由其最大值（同步相位或中心）的位置参数化的单峰密度。我们的动力学的顺序N的时间上的特征，我们表现出很大的偏差的PDE的解决方案的行为。事实上，如果当N趋于无穷大时，时间0的经验测度收敛到一个概率测度（它远离我们所刻画的薄集），如果时间以N加速，经验测度几乎瞬间到达稳定流形的一个小邻域，然后它粘在那里，在那里发生非平凡的随机动力学。事实上，同步中心执行一个布朗运动与扩散系数，我们计算。因此，我们的方法提供了一个基本的连续对称的统计力学系统，从大尺度极限-或大数定律-由于有限尺寸效应的宏观偏差的详细表征。但对这个模型的兴趣超出了统计力学，因为它在各种旨在理解同步现象的科学领域中起着核心作用。
英文摘要：
We consider the natural Langevin dynamics which is reversible with respect to the mean-field plane rotator (or classical spin XY) measure. It is well known that this model exhibits a phase transition at a critical value of the interaction strength parameter K, in the limit of the number N of rotators going to infinity. A Fokker-Planck PDE captures the evolution of the empirical measure of the system as N goes to infinity, at least for finite times and when the empirical measure of the system at time zero satisfies a law of large numbers. The phase transition is reflected in the fact that the PDE for K above the critical value has a stable manifold of stationary solutions, that are equivalent up to rotations. These stationary solutions are actually unimodal densities parametrized by the position of their maximum (the synchronization phase or center). We characterize the dynamics on times of order N and we show substantial deviations from the behavior of the solutions of the PDE. In fact, if the empirical measure at time zero converges as N goes to infinity to a probability measure (which is away from a thin set that we characterize) and if time is speeded up by N, the empirical measure reaches almost instantaneously a small neighborhood of the stable manifold, to which it then sticks and on which a non-trivial random dynamics takes place. In fact the synchronization center performs a Brownian motion with a diffusion coefficient that we compute. Our approach therefore provides, for one of the basic statistical mechanics systems with continuum symmetry, a detailed characterization of the macroscopic deviations from the large scale limit -- or law of large numbers -- due to finite size effects. But the interest for this model goes beyond statistical mechanics, since it plays a central role in a variety of scientific domains in which one aims at understanding synchronization phenomena.
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