LL}$, hence to consider the energy $$\boxed{E(\rho):=F_{\rm LL}(\rho)-\frac12\int_{{\mathbb{R}}^3}\int_{{\mathbb{R}}^3}\frac{\rho(x)\rho(y)}{|x-y|}dx\,dy.}
\label{eq:def_E_quantum}$$ The (grand-canonical) *exchange-correlation energy* is defined by $E_{\rm xc}(\rho):=E(\rho)-T(\rho)$ where $$T(\rho):=\inf_{\substack{\Gamma_n=\Gamma_n^*{\geqslant}0\\ \sum_{n=0}^{\infty}{{\rm Tr}\,}(\Gamma_n)=1\\ \sum_{n=1}^{\infty}\rho_{\Gamma_n}=\rho}} \Bigg\{\sum_{n=1}^{\infty}{{\rm Tr}\,}_{{\mathfrak{H}}^n}\Bigg(-\sum_{j=1}^n\Delta_{x_j}\Bigg)\Gamma_n\Bigg\}=\inf_{\substack{0{\leqslant}\gamma=\gamma^*{\leqslant}1\\ \rho_\gamma=\rho}}{{\rm Tr}\,}(-\Delta)\gamma
\label{eq:def_T}$$ is the lowest possible kinetic energy. We will study the functional $T(\rho)$ in Section \[sec:kinetic\] below.

The Uniform Electron Gas
------------------------

In [@LewLieSei-18 Section 5], we have defined the uniform electron gas, which is obtained in the limit when $\rho$ approaches a constant function in the whole space. This is believed to be the same as the ground state energy of Jellium, where the density is not necessarily constant but the electrons instead evolve in a constant background [@LieNar-75]. This has recently been proved in the classical case in [@LewLieSei-19b; @CotPet-19b] and the same is expected in the quantum case.

The following result is a slight improvement of [@LewLieSei-18 Thm. 5.1].

\[thm:UEG\_thermo\_limit\_quantum\] Let $\rho_0>0$. Let $\{\Omega_N\}\subset{\mathbb{R}}^3$ be a sequence of bounded connected domains with $|\Omega_N|\to{\infty}$, such that $\Omega_N$ has a uniformly regular boundary in the sense that $$|\partial\Omega_N+B_r|{\leqslant}Cr|\Omega_N|^{2/3},\qquad\text{for all $r{\leqslant}|\Omega_N|^{1/3}/C$,}$$ for some constant $C>0$. Let $\delta_N>0$ be any sequence such that $\delta_N/|\Omega_N|^{1/3}\to0$ and $\delta_N|\Omega_N|^{1/3}\to{\infty}$. Let $\chi\in L^1({\mathbb{R}}^3)$ be a radial non-negative function of compact support such that $\int_{{\mathbb{R}}^3}\chi=1$ and $\int_{{\mathbb{R}}^3}|\nabla\sqrt\chi|^2<{\infty}$. Denote $\chi_\delta(x)=\delta^{-3}\chi(x/\delta)$. 