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Mean Trajectories of Multiple Tracking Points on A Brownian Rigid Body: Convergence, Alignment and Twist

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.