Note: Brownian motion of colloidal particles of arbitrary shape.

In general, the diffusion and mobility matrices of a particle depend on the choice of the reference center which is used to describe the rotational and translational motion. The requirement that the rotational-translational mobility matrix μ is symmetric defines uniquely the so-called center of mobility. In Ref. [1], this center was used to analyze the Brownian motion of a particle with arbitrary shape, and the corresponding analytical expressions for the time-dependent cross-correlations of the translational and rotational Brownian displacements were derived. In this note, we derive the analogical analytical expressions for the cross-correlations of the translational and rotational displacements of an arbitrary reference center. We denote the time-dependent position of this center as R(t), and we introduce three mutually perpendicular unit vectors u(t), p = 1, 2, 3, to describe the particle orientation at time t [1, 2]. The translational and rotational Brownian displacements are,