身体-姿态对齐：一级相变，通过四元数与棒状聚合物连接，以及稳定性
Body-attitude alignment: first order phase transition, link with rodlike polymers through quaternions, and stability
摘要：
我们提出了一个简单的模型，大量的刚体（旋转矩阵建模）的内部旋转噪声的对齐。数值模拟结果表明，在取向强度的变化过程中，存在一级相变现象，在两个阈值处出现突变。在第一个阈值以下，系统在大时间内是无序的：旋转矩阵均匀分布。在第二阈值以上，系统的长时间行为是集中在给定的旋转矩阵周围。当强度在两个阈值之间时，两种情况都可能发生。然后，我们研究了这个模型的平均场极限，作为粒子的数量趋于无穷大，这需要一个非线性福克-普朗克方程的形式。我们描述了这个方程的稳态，这符合数值实验的完整分类。这种分类是由Degond，Diez，Merino-Aceituno和作者在以前的工作中获得的，这要归功于该模型与通过Maier-Saupe势相互作用的棒状聚合物悬浮液的Doi-Onsager方程的四维推广之间的联系。以前的研究涉及一个类似的BGK型方程，其稳态是相同的。我们利用在这个框架中得到的稳定性结果，并能够证明两个家庭的稳定状态的指数稳定性：无序的均匀分布时，对齐的强度小于第二个阈值，和一个家庭的非各向同性的稳定状态（一个为每个可能的旋转矩阵，集中在它周围），当强度是大于的第一个阈值。我们还表明，其他家庭的稳定状态是不稳定的，在协议的数值观测。
英文摘要：
We present a simple model of alignment of a large number of rigid bodies (modeled by rotation matrices) subject to internal rotational noise. The numerical simulations exhibit a phenomenon of first order phase transition with respect the alignment intensity, with abrupt transition at two thresholds. Below the first threshold, the system is disordered in large time: the rotation matrices are uniformly distributed. Above the second threshold, the long time behaviour of the system is to concentrate around a given rotation matrix. When the intensity is between the two thresholds, both situations may occur. We then study the mean-field limit of this model, as the number of particles tends to infinity, which takes the form of a nonlinear Fokker--Planck equation. We describe the complete classification of the steady states of this equation, which fits with numerical experiments. This classification was obtained in a previous work by Degond, Diez, Merino-Aceituno and the author, thanks to the link between this model and a four-dimensional generalization of the Doi--Onsager equation for suspensions of rodlike polymers interacting through Maier--Saupe potential. This previous study concerned a similar equation of BGK type for which the steady-states were the same. We take advantage of the stability results obtained in this framework, and are able to prove the exponential stability of two families of steady-states: the disordered uniform distribution when the intensity of alignment is less than the second threshold, and a family of non-isotropic steady states (one for each possible rotation matrix, concentrated around it), when the intensity is greater than the first threshold. We also show that the other families of steady-states are unstable, in agreement with the numerical observations.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
