we will make use of the adiabatic approximation. This approximation identifies, at any time $t$, the one-body density, $\rho({\boldsymbol{\mathfrak{r}}}_1;t)$, with the equilibrium one, $\rho_{eq}({\boldsymbol{\mathfrak{r}}}_1)$. We now consider the Helmholtz free-energy functional[@evans_nature_1979; @lutsko_recent_2010] $$\begin{aligned}
 \mathcal{F}[\rho_{eq}]={\textcolor{black}{k_BT}}\int_{\mathfrak{V}}d{\boldsymbol{\mathfrak{r}}}_1\,\rho_{eq}({\boldsymbol{\mathfrak{r}}}_1)\left(\ln\left(\Lambda^{3}\rho_{eq}({\boldsymbol{\mathfrak{r}}}_1)-1\right)\right)
 +\mathcal{F}_{exc}[\rho_{eq}]+\int_{\mathfrak{V}}d{\boldsymbol{\mathfrak{r}}}_1\,\rho_{eq}({\boldsymbol{\mathfrak{r}}}_1)V_1({\boldsymbol{\mathfrak{r}}}_1)
 \label{eq:15}\end{aligned}$$ where the first term on the right-hand side accounts for the ideal-rotator gas component, the last term contains the external potential contribution to the free energy, $\mathcal{F}_{exc}$ is an excess over ideal gas contribution and $\Lambda$ is the de Broglie wavelength (which will turn out to be irrelevant). From classical DFT[@lutsko_recent_2010] we know that there is a unique external potential, $\mathfrak{W}({\boldsymbol{\mathfrak{r}}}_1)=w({\boldsymbol{\mathfrak{r}}}_1)+V_1({\boldsymbol{\mathfrak{r}}}_1)$, for which $\rho_{eq}({\boldsymbol{\mathfrak{r}}}_1)$ is the equilibrium density. $${\textcolor{black}{
 \frac{k_BT{\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}\rho({\boldsymbol{\mathfrak{r}}}_1)}{\rho({\boldsymbol{\mathfrak{r}}}_1)}+{\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}\mathfrak{W}({\boldsymbol{\mathfrak{r}}}_1) =
 -\left.{\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}\frac{\delta \mathcal{F}_{exc}[\rho]}{\delta \rho({\boldsymbol{\mathfrak{r}}}_1)}\right|_{\rho=\rho_{eq}}}}
 \label{eq:16-a}$$ $$ {\textcolor{black}{{\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}w({\boldsymbol{\mathfrak{r}}}_1)=-\left.{\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}\frac{\delta \mathcal{F}[\rho]}{\delta \rho({\boldsymbol{\mathfrak{r}}}_1)}\right|_{\rho=\rho_{eq}}\equiv
 -{\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}\mu({\boldsymbol{\mathfrak{r}}}_1)
 }}
 \label{eq:16}$$

Furthermore, under the same circumstances Equation (\[eq:11\]) becomes the first equation of Yvon-Born-Green (YBG) hierarchy[@barrat_basic_2003; @liboff_kinetic_2003] $$k_BT{\boldsymbol{\mathfrak{m}}}^{-1}{\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}\rho_{eq}({\boldsymbol{\mathfrak{r}}}_1)+{\boldsymbol{\mathfrak{m}}}^{-1}{\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}\mathfrak{W}({\boldsymbol{\mathfrak{r}}}_1;t)\rho_{eq}({\boldsymbol{\mathfrak{r}}}_1)=
 {\boldsymbol{\mathfrak{m}}}^{-1}\sum_{n=2}^N\int d{\boldsymbol{\mathfrak{r}}}_2\dots d{\boldsymbol{\mathfrak{r}}}_n{\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}V_n({\boldsymbol{\mathfrak{r}}}^n)\rho^{(n)}_{eq}({\boldsymbol{\mathfrak{r}}}^n)
 \label{eq:17}$$ Making the approximation that Equations (\[eq:15\])–(\[eq:17\]) hold out of equilibrium, i.e. substitute $\rho_{eq}^{(n)}$ by $\rho^{(n)}(t)$ for all $n$, yields, $$\begin{aligned}
 \rho({\boldsymbol{\mathfrak{r}}}_1;t){\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}\frac{\delta \mathcal{F}[\rho]}{\delta \rho({\boldsymbol{\mathfrak{r}}}_1)}=&\,
 \rho({\boldsymbol{\mathfrak{r}}}_1;t){\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}V_1({\boldsymbol{\mathfrak{r}}}_1;t)+k_BT{\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}\rho({\boldsymbol{\mathfrak{r}}}_1;t)\label{eq:18}\\
 &+\sum_{n=2}^N\int d{\boldsymbol{\mathfrak{r}}}_2\dots d{\boldsymbol{\mathfrak{r}}}_n{\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}V_n({\boldsymbol{\mathfrak{r}}}^n)\rho^{(n)}({\boldsymbol{\mathfrak{r}}}^n;t). \notag\end{aligned}$$ Substituting Equations (\[eq:13\]) and (\[eq:18\]) into Equation (\[eq:11\]) produces, $$\begin{aligned}
 \partial_t{\boldsymbol{\mathfrak{j}}}({\boldsymbol{\mathfrak{r}}}_1;t)+&{\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}\cdot{\overline{\overline{\boldsymbol{\Pi}}}}({\boldsymbol{\mathfrak{r}}}_1;t)+{\boldsymbol{\mathfrak{m}}}^{-1}\rho({\boldsymbol{\mathfrak{r}}}_1;t){\boldsymbol{\nabla}}_{{\boldsymbol{\mathfrak{r}}}_1}\frac{\delta \mathcal{F}[\rho]}{\delta \rho({\boldsymbol{\mathfrak{r}}}_1)}
 +{\boldsymbol{\Upsilon}}_{\parallel}{\boldsymbol{\mathfrak{j}}}({\boldsymbol{\mathfrak{r}}}_1;t)+{\boldsymbol{\Upsilon}}_{\bot}{\boldsymbol{\mathfrak{j}}}_{\bot}({\boldsymbol{\mathfrak{r}}}_1;t)\label{eq:19}\\
 +&N{\boldsymbol{\mathfrak{m}}}^{-1}\sum_{j=1}^{N}\int d{\boldsymbol{\mathfrak{r}}}^{N-1}d{\boldsymbol{\mathfrak{p}}}^N\,\widetilde{{\boldsymbol{\Upsilon}}}_{1j}({\boldsymbol{\mathfrak{r}}}^N){\boldsymbol{\mathfrak{p}}}_j{f^{(N)}}({\boldsymbol{\mathfrak{r}}}^N,{\boldsymbol{\mathfrak{p}}}^N;t) = 0.\notag\end{aligned}$$

Local-equilibrium approximation and beyond\[subsec:local-equilibrium\]
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The next step of the derivation is the introduction of 