  Prior studies have shown that low symmetry particles such as micro-boomerangs
exhibit behaviour of Brownian motion rather different from that of high
symmetry particles because convenient tracking points (TPs) are usually
inconsistent with the center of hydrodynamic stress (CoH) where the
translational and rotational motions are decoupled. In this paper we study the
effects of the translation-rotation coupling on the displacement probability
distribution functions (PDFs) of the boomerang colloid particles with symmetric
arms. By tracking the motions of different points on the particle symmetry
axis, we show that as the distance between the TP and the CoH is increased, the
effects of translation-rotation coupling becomes pronounced, making the
short-time 2D PDF for fixed initial orientation to change from elliptical to
crescent shape and the angle averaged PDFs from ellipsoidal-particle-like PDF
to a shape with a Gaussian top and long displacement tails. We also observed
that at long times the PDFs revert to Gaussian. This crescent shape of 2D PDF
provides a clear physical picture of the non-zero mean displacements observed
in boomerangs particles.
