d{\boldsymbol{\mathfrak{p}}}_1({\boldsymbol{\mathfrak{m}}}^{-1}{\boldsymbol{\mathfrak{p}}}_1\otimes{\boldsymbol{\mathfrak{m}}}^{-1}{\boldsymbol{\mathfrak{p}}}_1)f^{(1)}({\boldsymbol{\mathfrak{r}}}_1,{\boldsymbol{\mathfrak{p}}}_1;t)\label{eq:11}\\
 &+{\boldsymbol{\mathfrak{m}}}^{-1}{\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}V_1({\boldsymbol{\mathfrak{r}}}_1;t)\rho({\boldsymbol{\mathfrak{r}}}_1;t)
 +{\boldsymbol{\mathfrak{m}}}^{-1}\sum_{n=2}^N\int d{\boldsymbol{\mathfrak{r}}}_2\dots d{\boldsymbol{\mathfrak{r}}}_n{\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}V_n({\boldsymbol{\mathfrak{r}}}^n;t)\rho^{(n)}({\boldsymbol{\mathfrak{r}}}^n;t)\notag\\
 &+{\boldsymbol{\Upsilon}}_{\parallel}{\boldsymbol{\mathfrak{j}}}({\boldsymbol{\mathfrak{r}}}_1;t)+{\boldsymbol{\Upsilon}}_{\bot}{\boldsymbol{\mathfrak{j}}}_{\bot}({\boldsymbol{\mathfrak{r}}}_1;t)+N{\boldsymbol{\mathfrak{m}}}^{-1}\sum_{j=1}^{N}\int d{\boldsymbol{\mathfrak{r}}}^{N-1}d{\boldsymbol{\mathfrak{p}}}^N\,\widetilde{{\boldsymbol{\Upsilon}}}_{1j}({\boldsymbol{\mathfrak{r}}}^N){\boldsymbol{\mathfrak{p}}}_j{f^{(N)}}({\boldsymbol{\mathfrak{r}}}^N,{\boldsymbol{\mathfrak{p}}}^N;t) = 0\notag\end{aligned}$$ where $\otimes$ is the dyadic product. Here we have defined ${\boldsymbol{\Upsilon}}_{\parallel}=\text{diag}(\gamma_{TT}\mathbf{1},\gamma_{RR}\mathbf{1})$, ${\boldsymbol{\Upsilon}}_{\bot}=\text{diag}(\gamma_{TR}\mathbf{1},\gamma_{RT}\mathbf{1})$ and ${\boldsymbol{\mathfrak{j}}}_{\bot}=\int
d{\boldsymbol{\mathfrak{p}}}_1({\boldsymbol{\mathfrak{m}}}^{-1}_{\bot}{\boldsymbol{\mathfrak{p}}}_1)f^{(1)}({\boldsymbol{\mathfrak{r}}}_1,{\boldsymbol{\mathfrak{p}}}_1;t)$, with ${\boldsymbol{\mathfrak{m}}}^{-1}_{\bot}=\text{adiag}(m^{-1}\mathbf{1},\mathbb{I}^{-1})$ the anti-diagonal block matrix with submatrices, $({\boldsymbol{\mathfrak{m}}}_{\bot}^{-1})_{1,2}=
m^{-1}\mathbf{1}$, $({\boldsymbol{\mathfrak{m}}}_\bot^{-1})_{2,1}=\mathbb{I}^{-1}$ and $({\boldsymbol{\mathfrak{m}}}_\bot^{-1})_{i,j}=\mathbf{0}$ otherwise. For particles where coupling cannot be neglected, such as screw-like colloids[@wittkowski_dynamical_2011], this first-momentum moment equation includes an additional term compared to that for spherical particles. This term, involving ${\boldsymbol{\mathfrak{j}}}_\bot({\boldsymbol{\mathfrak{r}}}_1;t)$, reflects the coupled translational and rotational forces. When such a coupling vanishes, as in the case of orthotropic-like particles[@wittkowski_dynamical_2011], the rotational and translational motions are completely decoupled and Equation (\[eq:11\]) can be split into two independent equations.

Equations (\[eq:10\])–(\[eq:11\]), are the basis of our DDFT. In the next section we introduce the closures which link the terms $f^{(1)}$, $\rho^{(n)}$ and ${f^{(N)}}$ of Eq (\[eq:11\]) to the one-body density, following the same reasoning as that applied for the spherical-particle case.

Extending the unified DDFT equation to non-spherical particles\[sec:extended-unified-DDFT\]
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The closure procedure will be performed in four steps: (i) the term involving the dyadic product will be rewritten using the definition of the kinetic stress, ${\overline{\overline{\boldsymbol{\Pi}}}}({\boldsymbol{\mathfrak{r}}}_1;t)$, and the equilibrium distribution function, $f_{eq}^{(1)}({\boldsymbol{\mathfrak{r}}}_1,{\boldsymbol{\mathfrak{p}}}_1)$; (ii) an adiabatic approximation will be used to substitute the $n$-body density 