terms with an expression involving the free-energy functional, $\mathcal{F}[\rho]$; (iii) the $N$-body distribution function will be expressed as a sum of a local-equilibrium and non-equilibrium distributions, yielding an extension of the result obtained for spherical-colloidal fluids, and finally; (iv) we restrict our attention to the case of two-body HIs and use an approximation.

Equilibrium distribution, kinetic stress and adiabatic approximation \[subsec:equilibrium-kinetic-stress-adiabatic-approximation\]
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At equilibrium the one-body distribution function has a Maxwell-Boltzmann structure[@condiff_brownian_1966] $$\begin{aligned}
 f_{eq}^{(1)}({\boldsymbol{\mathfrak{r}}}_1,{\boldsymbol{\mathfrak{p}}}_1)=\frac{\sqrt{|\mathbb{I}^{-1}|}}{m^{3/2}(2\pi k_BT)^3}\rho_{eq}({\boldsymbol{\mathfrak{r}}}_1)\exp\left(-\frac{({\boldsymbol{\mathfrak{m}}}^{-1}{\boldsymbol{\mathfrak{p}}}_1)\cdot{\boldsymbol{\mathfrak{p}}}_1}{2k_BT}\right)
 \label{eq:12},\end{aligned}$$ with $|\mathbb{I}^{-1}|=\prod_{i}(I_{ii})^{-1}$. Hence, one can easily check that $
 \int d{\boldsymbol{\mathfrak{p}}}_1 ({\boldsymbol{\mathfrak{m}}}^{-1}{\boldsymbol{\mathfrak{p}}}_1\otimes {\boldsymbol{\mathfrak{m}}}^{-1}{\boldsymbol{\mathfrak{p}}}_1)f_{eq}^{(1)}({\boldsymbol{\mathfrak{r}}}_1,{\boldsymbol{\mathfrak{p}}}_1)=  k_BT{\boldsymbol{\mathfrak{m}}}^{-1}\rho_{eq}({\boldsymbol{\mathfrak{r}}}_1)$. By adding and subtracting ${\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}\cdot \left(
k_BT{\boldsymbol{\mathfrak{m}}}^{-1}\rho({\boldsymbol{\mathfrak{r}}}_1)\right)$ on the left-hand side of Equation (\[eq:11\]) we can write, $$\begin{aligned}
 k_BT{\boldsymbol{\mathfrak{m}}}^{-1}{\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}\rho({\boldsymbol{\mathfrak{r}}}_1;t)+{\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}\cdot{\overline{\overline{\boldsymbol{\Pi}}}}({\boldsymbol{\mathfrak{r}}}_1;t)&=\,{\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}\cdot\int d{\boldsymbol{\mathfrak{p}}}_1({\boldsymbol{\mathfrak{m}}}^{-1}{\boldsymbol{\mathfrak{p}}}_1\otimes{\boldsymbol{\mathfrak{m}}}^{-1}{\boldsymbol{\mathfrak{p}}}_1)f^{(1)}({\boldsymbol{\mathfrak{r}}}_1,{\boldsymbol{\mathfrak{p}}}_1;t) \label{eq:13}\end{aligned}$$ with the kinetic-stress tensor defined as[@goddard_unification_2013] $$\begin{aligned}
 {\overline{\overline{\boldsymbol{\Pi}}}}({\boldsymbol{\mathfrak{r}}}_1;t)&=\,\int d{\boldsymbol{\mathfrak{p}}}_1\left(({\boldsymbol{\mathfrak{m}}}^{-1}{\boldsymbol{\mathfrak{p}}}_1\otimes{\boldsymbol{\mathfrak{m}}}^{-1}{\boldsymbol{\mathfrak{p}}}_1)-k_BT{\boldsymbol{\mathfrak{m}}}^{-1}\right)f^{(1)}({\boldsymbol{\mathfrak{r}}}_1,{\boldsymbol{\mathfrak{p}}}_1;t)\label{eq:14}.\end{aligned}$$ According to this definition, the term involving the one-body distribution function and the dyadic product in Equation (\[eq:11\]) can be identified as the remainder of the spatial and angular divergence of the kinetic stress. Although at this stage Equation (\[eq:13\]) only seems to be a more convenient way of expressing the second term on the left-hand side of Equation (\[eq:11\]), the advantage will become clear in Section \[subsec:local-equilibrium\] where we discuss how to deal with out-of-equilibrium distributions.

But before this, 