= \beta\sum_{l=1}^N\mathbf{A}_{jl}({\boldsymbol{\mathfrak{r}}}^N)\mathbf{A}_{kl}({\boldsymbol{\mathfrak{r}}}^N)\label{eq:3}.\end{aligned}$$ Here $\beta=1/k_BT$ with $k_B$ the Boltzmann constant and $T$ the temperature, is assumed to be constant. We define the $6N\times6N$ total-mass tensor ${\boldsymbol{\mathfrak{m}}}_{_N}=\text{diag}({\boldsymbol{\mathfrak{m}}}_j)$, along with the $6N\times 6N$ tensors ${\boldsymbol{\Gamma}}=({\boldsymbol{\Gamma}}_{jk})$, and $\mathbf{A}=(\mathbf{A}_{jk})$, transforming Equation (\[eq:3\]) to ${\boldsymbol{\mathfrak{m}}}_{_N}{\boldsymbol{\Gamma}}= \beta\mathbf{A}\mathbf{A}^\top$. Due to the asymmetry of the particles, the friction and noise matrices are not generally symmetric. However, as discussed by @brenner_stokes_1964-2 and @condiff_brownian_1966, the asymmetric part of ${\boldsymbol{\Gamma}}_{ij}$ is exclusively due to nondissipative gyrostatic forces and torques acting on the particles. If such effects can be neglected[@condiff_brownian_1966], then one can assume ${\boldsymbol{\Gamma}}_{jk}^{TR}=({\boldsymbol{\Gamma}}_{jk}^{RT})^\top$. The fluctuation-dissipation relation is then $\mathbf{A} = \sqrt{k_BT {\boldsymbol{\mathfrak{m}}}_{_N}
{\boldsymbol{\Gamma}}}$, analogous to that for spherical particles.

Starting from the Smoluchowski equation, a fluctuating DDFT equation was derived recently[@donev_dynamic_2014]. The authors of this study argued that any equation of motion that accounts for inertial effects of the colloidal particles must include the inertia of the fluid bath. Moreover, they claim that there are only two consistent ways of dealing with this problem: either *fluctuating hydrodynamics*[@hauge_fluctuating_1973] or the overdamped LEs based on the work of @hinch_application_1975 and @roux_brownian_1992. They justify this claim using the estimate obtained by @roux_brownian_1992, who also refers to @hinch_application_1975, of the typical relaxation time of the 