it exists) satisfies the nonlinear equation $$\varrho = Z^{-1}e^{-\tfrac{(V_1+(gV_2)\star\varrho)}{k_BT}}.$$ The existence of a probability density $\varrho$, and therefore a probability measure $\mu$ in , is obtained by Lemma \[thm:exis\_fix\_point\]. The functional $\mathcal{F}$ gives rise to the density minimising the free energy associated to the system - as $\gamma \to \infty$, which will be shown in Theorem \[thm:association \_of\_free\_energy\].

To make the connection between the free energy functional $\mathcal{F}$ in and the theory of non-uniform classical fluids, one may consider the Helmholtz free energy functional, which is the central energy functional of DFT [@evans1979nature] $$\begin{aligned}
\label{eq:ddft-helmholtz_func}
\mathcal{F}_{H}[\varrho] = \int_U\mathrm{d}\vec{r}\,\varrho(\vec{r},t)V_1(\vec{r},t)+ k_BT\int_U\mathrm{d}\vec{r}\,\varrho(\vec{r},t)[\log(\Lambda^3\varrho(\vec{r},t))-1]+\mathcal{F}_\text{ex}[\varrho]\end{aligned}$$ where $\mathcal{F}_{\text{ex}}$ is the excess over ideal gas term and $\Lambda$ the de Broglie wavelength, which turns out to be superfluous. The term $\mathcal{F}_{\text{ex}}$ is not in general known, the exception being for one dimensional hard rods [@percus1976equilibrium]. Using the free energy functional $\mathcal{F}_H$, the corresponding Euler-Lagrange equation is $$\begin{aligned}
\label{eq:chemical_potential_euler_lagrange_eqn}
\mu_{c}=V_1(\vec{r}) + k_BT [\log (\Lambda^3\varrho(\vec{r}))-1]+\tfrac{\delta\mathcal{F}_{\text{ex}}}{\delta\rho}[\varrho]\end{aligned}$$ where $\mu_{c}$ is the chemical potential which is constant at equilibrium. Note that $\mu_c$ should not be confused with the measure $\mu$ defined in . After taking the gradient of and multiplying by $\varrho$ we obtain $$\begin{aligned}
0=\varrho(\vec{r})\nabla_{\vec{r}}\tfrac{\delta\mathcal{F}}{\delta\rho}[\varrho]=k_BT\nabla_{\vec{r}}\varrho+\varrho(\vec{r})\nabla_{\vec{r}}\Big(V_1(\vec{r})+\tfrac{\delta\mathcal{F}_{\text{ex}}}{\delta\rho}[\varrho]\Big).\end{aligned}$$ At equilibrium, the sum rule holds (see, 