reduce to the unified DDFT recently derived by @goddard_unification_2013. Section \[subsec:wl-eq\] will be devoted to showing how Equation (\[eq:34\]) turns into the equation derived by @rex_dynamical_2007 and @wittkowski_dynamical_2011 for systems of arbitrary-shape colloids by setting $\mathbf{Z}_1=\mathbf{Z}_2=0$, and showing that the generalised Einstein relationships[@dickinson_brownian_1985; @peters_smoluchowski_2000] can be recovered in such a case. Finally, we note that the overall effect of the last two terms in Equation (\[eq:34\]) will be a friction-like retardation of the translational and angular velocities ${\boldsymbol{\mathfrak{v}}}({\boldsymbol{\mathfrak{r}}}_1,t)$, akin to what happened for spherical particles[@goddard_unification_2013].

Connection with previous DDFTs\[subsec:wl-eq\]
----------------------------------------------

Here we connect our DDFT with the DDFTs obtained by @rex_dynamical_2007 and @wittkowski_dynamical_2011 for systems of arbitrary-shape colloids with overdamped dynamics, neglecting HIs. Within the context of our study, this is equivalent to setting $\mathbf{Z}_1=\mathbf{Z}_2=0$. We then have $$\begin{aligned}
  ({\boldsymbol{\Upsilon}}_{\parallel}+{\boldsymbol{\Upsilon}}_{\bot}{\boldsymbol{\epsilon}}){\boldsymbol{\mathfrak{v}}}({\boldsymbol{\mathfrak{r}}}_1;t) = -\mathcal{D}_t{\boldsymbol{\mathfrak{v}}}({\boldsymbol{\mathfrak{r}}}_1;t)-&{\boldsymbol{\mathfrak{m}}}^{-1}{\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}\frac{\delta \mathcal{F}[\rho]}{\delta \rho({\boldsymbol{\mathfrak{r}}}_1)}\label{eq:36}.\end{aligned}$$ Using this result in the continuity equation (\[eq:35\]) gives $$\begin{aligned}
 \partial_t\rho({\boldsymbol{\mathfrak{r}}}_1;t)={\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}\cdot\left(\rho({\boldsymbol{\mathfrak{r}}}_1;t)\boldsymbol{\zeta}^{-1}\left({\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}\frac{\delta \mathcal{F}[\rho]}{\delta \rho({\boldsymbol{\mathfrak{r}}}_1)}+{\boldsymbol{\mathfrak{m}}}\mathcal{D}_t{\boldsymbol{\mathfrak{v}}}({\boldsymbol{\mathfrak{r}}}_1;t)\right)\right)\label{eq:37}\end{aligned}$$ where we defined $$\begin{aligned}
({\boldsymbol{\Upsilon}}_{\parallel}+{\boldsymbol{\Upsilon}}_{\bot}{\boldsymbol{\epsilon}}){\boldsymbol{\mathfrak{m}}}=\boldsymbol{\zeta}\equiv\,\begin{pmatrix}
                                                           \boldsymbol{\zeta}_{\text{TT}} & \boldsymbol{\zeta}_{\text{TR}}\\
                                                           \boldsymbol{\zeta}_{\text{RT}} & \boldsymbol{\zeta}_{\text{RR}}
                                                          \end{pmatrix}\label{eq:38}
 \end{aligned}$$ along with $\boldsymbol{\zeta}_{\text{TT}} =\, m\gamma_{TT}\mathbf{1},\ \boldsymbol{\zeta}_{\text{TR}} =\, m^{-1}\gamma_{TR}\mathbb{I}^2,\ \boldsymbol{\zeta}_{\text{RT}} =\, m^2\gamma_{RT}\mathbb{I}^{-1},$ and $\boldsymbol{\zeta}_{\text{RR}} =\, \gamma_{RR}\mathbb{I}$. In the overdamped regime, inertial forces and unsteady acceleration are neglected, i.e. ${\boldsymbol{\mathfrak{m}}}\,\mathcal{D}_t{\boldsymbol{\mathfrak{v}}}\rightarrow0$. Therefore, Equation (\[eq:38\]) becomes $$\begin{aligned}
\partial_t\rho({\boldsymbol{\mathfrak{r}}}_1;t)={\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}\cdot\left(\rho({\boldsymbol{\mathfrak{r}}}_1;t)\beta\mathbf{D}({\boldsymbol{\mathfrak{r}}}_1;t)\left({\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}\frac{\delta \mathcal{F}[\rho]}{\delta \rho({\boldsymbol{\mathfrak{r}}}_1)}\right)\right)\label{eq:39}\end{aligned}$$ where $\beta\mathbf{D}=\boldsymbol{\zeta}^{-1}$ is the 