P^{[\alpha]}_{2,\mathbf{e}_{j}+\mathbf{e}_{k}}(\mathbf{x})=\,\frac{x_jx_k}{\alpha}-\delta_{jk}.\end{aligned}$$ Setting now $\alpha= \sqrt{k_BT}$, the one-body distribution function can be expanded as $$\begin{aligned}
 f^{(1)}({\boldsymbol{\mathfrak{r}}},{\boldsymbol{\mathfrak{p}}};t)=f^{(1)}_{leq}({\boldsymbol{\mathfrak{r}}},{\boldsymbol{\mathfrak{p}}};t)\left(1+\sum_{n,\boldsymbol{\ell}}B_{n,\boldsymbol{\ell}}({\boldsymbol{\mathfrak{r}}};t)\,
 P^{[\alpha]}_{n,\boldsymbol{\ell}}\left({\boldsymbol{\mathfrak{m}}}^{-1/2}\mathbf{c}\right)\right).\label{eq:29}\end{aligned}$$ with $B_{n,\boldsymbol{\ell}}({\boldsymbol{\mathfrak{r}}};t)$ the moments of the distribution $f_{neq}^{(1)}$. From Equation (\[eq:22\]), we then find that $B_{0,\mathbf{0}}=B_{1,\mathbf{e}_j}=0$ and $\sum_{j}{B}_{2,2\mathbf{e}_j}=0$. Finally, the non-equilibrium term in Equation (\[eq:26\]) can be expressed as $$\begin{aligned}
\boldsymbol{\Pi}_{jk}^{neq}({\boldsymbol{\mathfrak{r}}};t) = k_B T \rho({\boldsymbol{\mathfrak{r}}};t) B_{2,\mathbf{e}_j+\mathbf{e}_k}({\boldsymbol{\mathfrak{r}}};t),
\label{eq:30}\end{aligned}$$ which represents an extension of the analogous relationship found for the spherical case (Section 3.2 of Ref. ). This relationship does not necessarily imply the non-local kinetic pressure to be small, since the functions $B_{2,\mathbf{e}_j+\mathbf{e}_k}({\boldsymbol{\mathfrak{r}}};t)$ do not depend on $\rho,\, T$ or ${\boldsymbol{\mathfrak{v}}}$. However, at this point we make the assumption that $f_{neq}^{(1)}$ can be either approximated or neglected, inspired by previous rigorous analyses for spherical particles[@goddard_overdamped_2012; @goddard_unification_2013]. Therefore, to obtain a DDFT, we need to deal with the last term in Equation (\[eq:26\]).

Two-body interactions and closure equation\[subsec:enskog-closure\]
-------------------------------------------------------------------

With the aim of closing Equation (\[eq:26\]) in $\rho$, we restrict our attention to systems where the HIs are given by linear combinations of two-body interactions. In particular, we assume the general structure[@goddard_unification_2013; @goddard_overdamped_2012] $$\begin{aligned}
 \widetilde{{\boldsymbol{\Upsilon}}}_{jk}({\boldsymbol{\mathfrak{r}}}^N)=\delta_{jk}\sum_{r\neq j}\mathbf{Z}_1({\boldsymbol{\mathfrak{r}}}_j,{\boldsymbol{\mathfrak{r}}}_r)+(1-\delta_{jk})\mathbf{Z}_2({\boldsymbol{\mathfrak{r}}}_j,{\boldsymbol{\mathfrak{r}}}_k)\label{eq:31}.\end{aligned}$$ The integral of Equation (\[eq:26\]) depends only on $f^{(2)}({\boldsymbol{\mathfrak{r}}}_1,{\boldsymbol{\mathfrak{r}}}_2,{\boldsymbol{\mathfrak{p}}}_1,{\boldsymbol{\mathfrak{p}}}_2;t)$ and we now need a closure relation between the one- and 