e.g. [@goddard2012unification]) $$\begin{aligned}
\label{eq:excess_free_energy_equilibrium_sum}
\varrho(\vec{r})\nabla_{\vec{r}}\tfrac{\delta\mathcal{F}_{\text{ex}}}{\delta\varrho}[\varrho]=\sum_{n=2}^N\int\mathrm{d}\vec{r}^{n}\nabla_{\vec{r}}V_n(\vec{r}^n)\varrho_n(\vec{r}^n).\end{aligned}$$ where $\varrho_n(\vec{r}^n)$ is the standard $n-$particle configuration distribution function in equilibrium. Limiting the particle interactions to two-body, for example with the approximation $\varrho_2(\vec{r},\vec{r}') = \varrho(\vec{r})\varrho(\vec{r}')g(\vec{r},\vec{r}',[\varrho])$, we take the first term in the above series to obtain the equality $\nabla_{\vec{r}}\mathcal{F}_H[\varrho]= \nabla_{\vec{r}}\mathcal{F}[\varrho]$. In this way wee see that the density minimising $\mathcal{F}_H$ will minimise $\mathcal{F}$.

When $\boldsymbol{Z}_2\equiv 0 $, and by using the adiabatic approximation that holds out of equilibrium, we note that PDE simplifies to (cf. [@rex2009dynamical]) $$\begin{aligned}
\label{eq:ddft-eq}
\partial_{t}\varrho = \nabla_{\vec{r}}\cdot \left[\boldsymbol{D}(\vec{r},t)\varrho(\vec{r},t)\,\nabla_{\vec{r}}\tfrac{\delta \mathcal{F}}{\delta \varrho}[\varrho]\right].\end{aligned}$$ From we conclude that the dynamics under the choice $\bm{Z}_2 \equiv 0$ has a gradient flow structure. When $\bm{Z}_2$ is not necessarily zero, one cannot in general write the full dynamics as a gradient flow and, hence, the inclusion of HI introduces a novel perturbation away from the classical theory of gradient flow structure. Additionally, one sees how the free energy functional gives rise to the concept of a local pressure variation by the term inside the divergence of . In particular, the term $\tfrac{k_\text{B}\mathrm{T}}{m}\varrho(\vec{r},t)\,\nabla_{\vec{r}}\tfrac{\delta \mathcal{F}}{\delta \varrho}[\varrho]$ represents the spatial variation of the energy available to change particle configurations per unit volume at fixed particle number, in other words, it is 