Then the following thermodynamic limit exists $$\boxed{ \lim_{N\to{\infty}}\frac{E \big(\rho_0{{\ensuremath {\mathds 1} }}_{\Omega_N}\ast\chi_{\delta_N}\big)}{|\Omega_N|}
=e_{\rm UEG}\big(\rho_0\big) }
 \label{eq:thermodynamic_limit_quantum}$$ where the function $e_{\rm UEG}$ is independent of the sequence $\{\Omega_N\}$, of $\delta_N$ and of $\chi$.

For more properties of the UEG energy $e_{\rm UEG}$ we refer to  [@LewLieSei-18] and the references therein. In [@LewLieSei-18 Thm. 5.1] we rather optimized over the values of $\rho$ in the transition region around $\partial\Omega_N$. We were able to prove the simple limit  only when $\Omega_N$ is a tetrahedron. Using an upper bound on $E(\rho)$ that will be derived later in Proposition \[prop:upper\_bound\], we are now able to treat more reasonable limits in the form of . The proof of Theorem \[thm:UEG\_thermo\_limit\_quantum\] is provided in Section \[sec:proof\_thermo\_limit\] below.

The function $\chi_{\delta_N}$ is used to regularize the function $\rho_0{{\ensuremath {\mathds 1} }}_{\Omega_N}$ which cannot be the density of a quantum state, since its square root is not in $H^1({\mathbb{R}}^3)$ [@Lieb-83b]. The first condition $\delta_N/|\Omega_N|^{1/3}\to0$ implies that the smearing happens in a neighborhood of the boundary $\partial\Omega_N$ which has a negligible volume compared to $|\Omega_N|$. The second condition $\delta_N|\Omega_N|^{1/3}\to{\infty}$ ensures that the kinetic energy in the transition region stays negligible in the thermodynamic limit.

The same result holds under the 