two-body distributions. From a previous rigorous derivation of DDFT[@chan_time-dependent_2005], it is known that the one-body density determines the $N$-body distribution and, therefore, all the time-dependent properties of the system can be expressed as functionals of $\rho({\boldsymbol{\mathfrak{r}}};t)$. Thus, we make the approximation $$\begin{aligned}
f^{(2)}({\boldsymbol{\mathfrak{r}}}_1,{\boldsymbol{\mathfrak{r}}}_2,{\boldsymbol{\mathfrak{p}}}_1,{\boldsymbol{\mathfrak{p}}}_2;t) = f^{(1)}({\boldsymbol{\mathfrak{r}}}_1,{\boldsymbol{\mathfrak{p}}}_1;t)f^{(1)}({\boldsymbol{\mathfrak{r}}}_2,{\boldsymbol{\mathfrak{p}}}_2;t)g({\boldsymbol{\mathfrak{r}}}_1,{\boldsymbol{\mathfrak{r}}}_2;[\rho]),
\label{eq:32}\end{aligned}$$ although the exact form of the functional $g({\boldsymbol{\mathfrak{r}}}_1,{\boldsymbol{\mathfrak{r}}}_2;[\rho])$ is unknown, accurate approximations are known at equilibrium. However, the problem of obtaining an expression for $g$ is beyond the scope of the present study. From now on, we assume there exists at least a good approximation.

Substituting Equations (\[eq:31\]) and (\[eq:32\]) into (\[eq:30\]) yields $$\begin{aligned}
 \rho({\boldsymbol{\mathfrak{r}}}_1;t)\mathcal{D}_t{\boldsymbol{\mathfrak{v}}}({\boldsymbol{\mathfrak{r}}}_1;t)+&{\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}\cdot {\overline{\overline{\boldsymbol{\Pi}}}}_{neq}({\boldsymbol{\mathfrak{r}}}_1;t)+{\boldsymbol{\mathfrak{m}}}^{-1}\rho({\boldsymbol{\mathfrak{r}}}_1;t){\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}\frac{\delta \mathcal{F}[\rho]}{\delta \rho({\boldsymbol{\mathfrak{r}}}_1)}
 +({\boldsymbol{\Upsilon}}_{\parallel}+{\boldsymbol{\Upsilon}}_{\bot}{\boldsymbol{\epsilon}})\rho({\boldsymbol{\mathfrak{r}}}_1;t){\boldsymbol{\mathfrak{v}}}({\boldsymbol{\mathfrak{r}}}_1;t)\label{eq:33}\\
+&{\boldsymbol{\mathfrak{m}}}^{-1}\rho({\boldsymbol{\mathfrak{r}}}_1;t)\int d{\boldsymbol{\mathfrak{r}}}_2\left[
 \mathbf{Z}_1({\boldsymbol{\mathfrak{r}}}_1,{\boldsymbol{\mathfrak{r}}}_2){\boldsymbol{\mathfrak{m}}}{\boldsymbol{\mathfrak{v}}}({\boldsymbol{\mathfrak{r}}}_1;t)+\mathbf{Z}_2({\boldsymbol{\mathfrak{r}}}_1,{\boldsymbol{\mathfrak{r}}}_2){\boldsymbol{\mathfrak{m}}}{\boldsymbol{\mathfrak{v}}}({\boldsymbol{\mathfrak{r}}}_2;t)
 \right]\,\rho({\boldsymbol{\mathfrak{r}}}_2;t)\,g({\boldsymbol{\mathfrak{r}}}_1,{\boldsymbol{\mathfrak{r}}}_2;[\rho]) = 0. \notag\end{aligned}$$ Neglecting the non-local equilibrium term as discussed above, and dividing by the one-body density, we finally obtain our equation for the evolution of the flux, $$\begin{aligned}
 \mathcal{D}_t{\boldsymbol{\mathfrak{v}}}({\boldsymbol{\mathfrak{r}}}_1;t)+&{\boldsymbol{\mathfrak{m}}}^{-1}{\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}\frac{\delta \mathcal{F}[\rho]}{\delta \rho({\boldsymbol{\mathfrak{r}}}_1)}\label{eq:34}
 +({\boldsymbol{\Upsilon}}_{\parallel}+{\boldsymbol{\Upsilon}}_{\bot}{\boldsymbol{\epsilon}}){\boldsymbol{\mathfrak{v}}}({\boldsymbol{\mathfrak{r}}}_1;t)\\
 +&{\boldsymbol{\mathfrak{m}}}^{-1}\int d{\boldsymbol{\mathfrak{r}}}_2\left[
 \mathbf{Z}_1({\boldsymbol{\mathfrak{r}}}_1,{\boldsymbol{\mathfrak{r}}}_2){\boldsymbol{\mathfrak{m}}}{\boldsymbol{\mathfrak{v}}}({\boldsymbol{\mathfrak{r}}}_1;t)+\mathbf{Z}_2({\boldsymbol{\mathfrak{r}}}_1,{\boldsymbol{\mathfrak{r}}}_2){\boldsymbol{\mathfrak{m}}}{\boldsymbol{\mathfrak{v}}}({\boldsymbol{\mathfrak{r}}}_2;t)
 \right]\,\rho({\boldsymbol{\mathfrak{r}}}_2;t)\,g({\boldsymbol{\mathfrak{r}}}_1,{\boldsymbol{\mathfrak{r}}}_2;[\rho]) = 0\notag.\end{aligned}$$ Combining this equation with the conservation law resulting from Equations (\[eq:20\]) and (\[eq:22\]) in (\[eq:10\]), namely $$\begin{aligned}
 \partial_t\rho({\boldsymbol{\mathfrak{r}}}_1;t)+{\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}\cdot(\rho({\boldsymbol{\mathfrak{r}}};t){\boldsymbol{\mathfrak{v}}}({\boldsymbol{\mathfrak{r}}}_1;t))=0\label{eq:35},\end{aligned}$$ we finally have the generalised DDFT for systems of orientable particles.

We can readily check that for the special cases of point-like and smooth spherical particles (with perfect slip on the surface[@dickinson_brownian_1985]), these equations 