{\boldsymbol{\mathfrak{m}}}_j^{-1}{\boldsymbol{\mathfrak{p}}}_j(t)\label{eq:2},\\
 \dot{{\boldsymbol{\mathfrak{p}}}}_j(t)=&\ {\boldsymbol{\mathfrak{F}}}_j-\sum_{k=1}^N{\boldsymbol{\Gamma}}_{jk}({\boldsymbol{\mathfrak{r}}}^N){\boldsymbol{\mathfrak{p}}}_k + \sum_{k=1}^N\mathbf{A}_{jk}\boldsymbol{\xi}_k(t)\notag,\end{aligned}$$ with ${\boldsymbol{\mathfrak{m}}}_j = \text{diag}(m\mathbf{1},\mathbb{I})$, $\boldsymbol{\mathfrak{r}}_j=(\mathbf{r}_j,\boldsymbol{\Phi}_j)$, $\boldsymbol{\mathfrak{p}}_j=(\mathbf{p}_j,\mathbf{L}_j)$ and ${\boldsymbol{\mathfrak{F}}}_j=(-\frac{\partial V}{\partial {\mathbf{r}}_j},-\frac{\partial V}{\partial {\boldsymbol{\Phi}}_j}-(\mathbb{I}^{-1}{\mathbf{L}}_j)\times{\mathbf{L}}_j)$, where $V({\boldsymbol{\mathfrak{r}}}^N)$ is the solvent-averaged interaction-potential of mean force[@snook_langevin_2006]. The gradient operators are defined by $\boldsymbol{\nabla}_{\mathfrak{r}_j}=(\frac{\partial}{\partial
\mathbf{r}_j},\frac{\partial }{\partial \boldsymbol{\Phi}_j})^\top$, with $\frac{\partial}{\partial {\boldsymbol{\Phi}}}=\mathbf{e}_x\frac{\partial}{\partial
\Phi_x}+\mathbf{e}_y\frac{\partial}{\partial
\Phi_y}+\mathbf{e}_z\frac{\partial}{\partial \Phi_z}$ the angular gradient \[see Equations (\[eq:a-17\])–(\[eq:a-18\])\], and $\boldsymbol{\nabla}_{\boldsymbol{\mathfrak{p}}_j}= (\frac{\partial
}{\partial \mathbf{p}_j},\frac{\partial }{\partial \mathbf{L}_j})^\top$. Here ${\boldsymbol{\xi}}_j=(\mathbf{f}_j,\mathbf{t}_j)^\top$ is a 6-dimensional Gaussian white noise vector where $\mathbf{f}_j$ and $\mathbf{t}_j$ are random forces and torques acting upon the $j$-th particle, respectively, such that $\langle
\xi_j^a(t)\rangle=0$ and $\langle \xi_j^a(t)\xi_k^b(t')\rangle =
2\delta_{jk}\delta^{ab}\delta(t-t')$. As for the spherical case, the motion of the colloidal particles induces flows in the bath which in turn result in forces on the colloids, what we have already referred to in Section \[intro\] as HIs. These HIs are represented by the $6\times6$ friction tensors, $
 {\boldsymbol{\Gamma}}_{jk}=({\boldsymbol{\Gamma}}_{jk}^{\mu\nu})
$ where the superscripts $\mu,\nu\in\{T,R\}$ denote the coupling between the translational and angular momenta. Furthermore, the interaction between bath and colloidal particles has a random component representing unpredictable collisions of the bath particles with the colloidal particles. The strength of these random forces is given by the $6\times 6$ tensors $\mathbf{A}_{jk}=(\mathbf{A}_{jk}^{\mu\nu})$ which are coupled to the friction tensors via the fluctuation-dissipation relation,[@wolynes_dynamical_1977; @dickinson_brownian_1985] $$\begin{aligned}
 {\boldsymbol{\mathfrak{m}}}\,{\boldsymbol{\Gamma}}_{jk}({\boldsymbol{\mathfrak{r}}}^N) 