论文标题
身体姿势比对：一阶相变，通过四元素与棒状聚合物链接，稳定性
Body-attitude alignment: first order phase transition, link with rodlike polymers through quaternions, and stability
论文作者
论文摘要
我们提出了一个由内部旋转噪声的大量刚体（由旋转矩阵建模）的简单模型。数值模拟表现出与对齐强度的一阶相变现象，并在两个阈值下突然过渡。在第一个阈值以下，该系统在很大程度上是无序的：旋转矩阵均匀分布。在第二个阈值之上，系统的长时间行为是集中在给定的旋转矩阵周围。当强度是两个阈值之间的强度时，两种情况可能会发生。 然后，我们研究了该模型的平均场限制，因为颗粒的数量倾向于无穷大，这采用了非线性fokker-planck方程的形式。我们描述了该方程式的稳态的完整分类，该分类符合数值实验。由于该模型与DOI的四维概括之间的联系，DeGond，Diez，Merino-Aceituno和作者在先前的工作中获得了这种分类，该分类是通过Maier-Saupe潜力相互作用的DOI-Onsager方程的四维概括。 这项先前的研究涉及类似的BGK类型方程，稳态相同。我们利用在此框架中获得的稳定结果，并能够证明两个稳态家族的指数稳定性：当对齐强度小于第二个阈值的强度时，无序的均匀分布，一个非异位性稳态的家族（一个可能围绕IT的每个可能的旋转矩阵，一个强度比第一个强度更大的旋转量），一个比第一个seption稳定的稳定状态。我们还表明，与数值观察一致的其他稳态家庭是不稳定的。
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.
