布朗刚体上多个跟踪点的平均轨迹：收敛、对齐和扭曲
Mean Trajectories of Multiple Tracking Points on A Brownian Rigid Body: Convergence, Alignment and Twist
摘要：
我们考虑在一个外力场存在和不存在的情况下进行布朗运动的刚体上的多个跟踪点的平均轨迹。基于忽略流体动力学相互作用的刚体-多边形和多面体的朴素表示，研究了这些布朗多边形和多面体的朗之万动力学.以常动力、简谐动力和指数衰减动力为例进行了研究。在二维空间中，根据外力的大小和形式以及物体的各向同性和各向异性，这些跟踪点的平均轨迹可以表现出三种相互作用状态：收敛，其中平均轨迹收敛到一个点或单个轨迹;对齐，其中平均轨迹平行并置;扭曲，其中平均轨迹扭曲和交错，形成辫状结构。此外，我们已经表明，一般来说，一个刚体可以从这些制度和它们之间的过渡采样。在这种转变过程中，其布朗行为会发生改变。值得注意的是，从二维空间中的多边形到三维空间中的多面体，由于三维空间中的两个更多的旋转自由度，对准和扭曲状态消失，并且仅存在会聚状态。
英文摘要：
We consider mean trajectories of multiple tracking points on a rigid body that conducts Brownian motion in the absence and presence of an external force field. Based on a na\"ıve representation of rigid body - polygon and polyhedron where hydrodynamic interactions are neglected, we study the Langevin dynamics of these Brownian polygons and polyhedra. Constant force, harmonic force and an exponentially decaying force are investigated as examples. In two dimensional space, depending on the magnitude and form of the external force and the isotropy and anisotropy of the body, mean trajectories of these tracking points can exhibit three regimes of interactions: convergence, where the mean trajectories converge to either a point or a single trajectory; alignment, where the mean trajectories juxtapose in parallel; twist, where the mean trajectories twist and intertwine, forming a plait structure. Moreover, we have shown that in general a rigid body can sample from these regimes and transit between them. And its Brownian behavior could be modified during such transition. Notably, from a polygon in two dimensional space to a polyhedron in three dimensional space, the alignment and twist regimes disappear and there is only the convergence regime survived, due to the two more rotational degrees of freedom in three dimensional space.
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