a widely used approximation in statistical physics[@pottier_nonequilibrium_2014; @liboff_kinetic_2003; @goddard_unification_2013], namely the local-equilibrium approximation. It allows rewriting the kinetic stress in terms of the one-body density. With the definitions of local density, $\rho$, velocity, ${\boldsymbol{\mathfrak{v}}}$, and temperature, $T$, given by $$\begin{aligned}
    \int d{\boldsymbol{\mathfrak{p}}}\left\{\begin{array}{c}
             1\\
             {\boldsymbol{\mathfrak{p}}}\\
             ({\boldsymbol{\mathfrak{m}}}^{-1}\mathbf{c})\cdot\mathbf{c}
            \end{array}\right\}
          f^{(1)}({\boldsymbol{\mathfrak{r}}},{\boldsymbol{\mathfrak{p}}};t)=
          \left\{\begin{array}{c}
           1\\
           {\boldsymbol{\mathfrak{m}}}{\boldsymbol{\mathfrak{v}}}({\boldsymbol{\mathfrak{r}}};t)\\
           6k_BT
          \end{array}\right\}\rho({\boldsymbol{\mathfrak{r}}};t)
          \label{eq:20},\end{aligned}$$ with[@liboff_kinetic_2003] $\mathbf{c}\equiv\mathbf{c}({\boldsymbol{\mathfrak{r}}},{\boldsymbol{\mathfrak{p}}};t)={\boldsymbol{\mathfrak{p}}}-{\boldsymbol{\mathfrak{m}}}{\boldsymbol{\mathfrak{v}}}({\boldsymbol{\mathfrak{r}}};t)$ the deviation-from-the-mean microscopic momentum, the local-equilibrium approximation to the true probability distribution, $$\begin{aligned}
 f^{(1)}_{leq}({\boldsymbol{\mathfrak{r}}},{\boldsymbol{\mathfrak{p}}};t)=&\, \frac{|\mathbb{I}^{-1}|^{1/2}\rho({\boldsymbol{\mathfrak{r}}};t)}{m^{3/2}(2\pi k_BT)^{3}}\label{eq:21}
  \exp\left(-\frac{({\boldsymbol{\mathfrak{m}}}^{-1}\mathbf{c})\cdot\mathbf{c}}{2k_BT}\right)\end{aligned}$$ fulfils the normalisation conditions given in Equation (\[eq:20\]). While approximating $f^{(1)}$ with Equation (\[eq:21\]) can be reasonable as a first-order approach, there is in fact no actual reason to impose such a restriction. We instead introduce the expansion $f^{(1)}=f^{(1)}_{leq}\,\left[1+\varPsi \right]\equiv
f^{(1)}_{leq}+f^{(1)}_{neq}$, which takes into account both local-equilibrium and non-equilibrium effects. The non-equilibrium term must satisfy $$\begin{aligned}
 \int d{\boldsymbol{\mathfrak{p}}}\left\{\begin{array}{c}
             1\\
             {\boldsymbol{\mathfrak{p}}}\\
             ({\boldsymbol{\mathfrak{m}}}^{-1}\mathbf{c})\cdot\mathbf{c}
            \end{array}\right\}
          f^{(1)}_{neq}({\boldsymbol{\mathfrak{r}}},{\boldsymbol{\mathfrak{p}}};t)=
          \left\{\begin{array}{c}
           0\\
           \mathbf{0}\\
           0
          \end{array}\right\}.\label{eq:22}\end{aligned}$$ Hence, $f_{neq}^{(1)}$ only contributes to higher moments such as the stress, ${\overline{\overline{\boldsymbol{\Pi}}}}$, and the heat flux, $\boldsymbol{\mathfrak{Q}}$. Using Equations (\[eq:20\])–(\[eq:22\]) it is readily found that ${\boldsymbol{\mathfrak{j}}}({\boldsymbol{\mathfrak{r}}};t)={\boldsymbol{\mathfrak{j}}}_{leq}({\boldsymbol{\mathfrak{r}}};t)=\rho({\boldsymbol{\mathfrak{r}}};t){\boldsymbol{\mathfrak{v}}}({\boldsymbol{\mathfrak{r}}};t)$. Thus, recalling the definition given in Equation (\[eq:11\]), the kinetic-stress tensor satisfies the decomposition, $$\begin{aligned}
 {\overline{\overline{\boldsymbol{\Pi}}}}({\boldsymbol{\mathfrak{r}}};t)=& ({\boldsymbol{\mathfrak{v}}}({\boldsymbol{\mathfrak{r}}};t)\otimes{\boldsymbol{\mathfrak{v}}}({\boldsymbol{\mathfrak{r}}};t))\rho({\boldsymbol{\mathfrak{r}}};t)+
 \int d {\boldsymbol{\mathfrak{p}}}\left({\boldsymbol{\mathfrak{m}}}^{-1}{\boldsymbol{\mathfrak{p}}}\otimes {\boldsymbol{\mathfrak{m}}}^{-1}{\boldsymbol{\mathfrak{p}}}\right)f_{neq}^{(1)}({\boldsymbol{\mathfrak{r}}},{\boldsymbol{\mathfrak{p}}};t)\label{eq:23} ={\overline{\overline{\boldsymbol{\Pi}}}}_{leq}({\boldsymbol{\mathfrak{r}}};t) + {\overline{\overline{\boldsymbol{\Pi}}}}_{neq}({\boldsymbol{\mathfrak{r}}};t),\end{aligned}$$ where the first term stands for the local-equilibrium stress, which 