## 4 Discussion  
First of all, let us discuss the phenomenon of \"quantum tails\" of the MDFs and make some suggestions about its physical nature. As it follows from Figures 1-5, the single-particle MDF of the UEG starts to differ from the Fermi distribution when the system becomes strongly non-ideal. The value of the coupling parameter \(\Gamma\), at which the \"tail\" appears, depends on the degeneracy rate: in the case of \(\theta=0.5\), (significant degeneracy) it appears at \(\Gamma\) from 8 to 9 (\(r_{s}\approx 8\)), but in the case of \(\theta=4.0\), the \(\Gamma\approx 5\) is quite enough (\(r_{s}\approx 36\)). At the same time, the deviation of the MDFs from the Fermi distribution is always followed by the appearance of the visible maxima on the PCFs, and with increasing maxima, the deviation also grows. This observation points at a possible physical reason of \"quantum tails\": with increasing non-ideality, the UEG becomes more liquid-like and the short-range order appears. This results in the potential wells surrounding the electrons at the average distance \(r_{max}\), so the spatial localization of the electrons increases. According to the basic principles of quantum mechanics, the localization of the electrons in the momentum space is weakened and the MDFs become wider. When the coupling increases, the potential wells become deeper and the effect grows. Furthermore, this hypothesis can explain the fact that the \"quantum tail\" of the MDF appears at lower values of \(\Gamma\) when the degeneracy is low and at higher values in the opposite case. In fact, at low \(\theta\), the thermal wavelength \(\lambda\) becomes of the order of the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(r_{s}\) & \(\theta\) & \(E_{IFG0}\), Ha & \(E_{IFG}\), Ha & \(E_{kin}\), Ha & \(E_{pot}\), Ha \\ \hline
4.0 & 0.5 & 0.118 & 0.118(02) & 0.121(02) & \(-0.22386(12)\) \\
4.0 & 1.0 & 0.195 & 0.197(05) & 0.195(05) & \(-0.21480(10)\) \\
4.0 & 4.0 & 0.702 & 0.702(18) & 0.699(18) & \(-0.1782(14)\) \\
12 & 0.5 & 0.0131 & 0.0131(01) & 0.0141(03) & \(-0.081073(40)\) \\
12 & 1.0 & 0.0217 & 0.0216(03) & 0.0224(04) & \(-0.079157(35)\) \\
12 & 4.0 & 0.0780 & 0.0793(18) & 0.0789(18) & \(-0.070567(24)\) \\
16 & 0.5 & 0.00735 & 0.00735(05) & 0.00841(12) & \(-0.061467(25)\) \\
16 & 1.0 & 0.0122 & 0.0123(02) & 0.0126(02) & \(-0.060535(18)\) \\
16 & 4.0 & 0.0439 & 0.0438(10) & 0.0446(09) & \(-0.054942(17)\) \\
28 & 1.0 & 0.00399 & 0.00402(02) & 0.00464(04) & \(-0.035732(12)\) \\
28 & 2.0 & 0.00738 & 0.00735(08) & 0.00755(09) & \(-0.034815(10)\) \\
28 & 4.0 & 0.0143 & 0.0143(02) & 0.0149(02) & \(-0.033396(10)\) \\
36 & 1.0 & 0.00241 & 0.00240(00) & 0.00314(00) & \(-0.028246(08)\) \\
36 & 2.0 & 0.00446 & 0.00446(04) & 0.00480(04) & \(-0.027628(10)\) \\
36 & 4.0 & 0.00867 & 0.00869(11) & 0.00894(12) & \(-0.026520(05)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: \(E_{IFG0}\)—internal (kinetic) energy of the macroscopic IFG obtained from the analytical Fermi distribution; \(E_{IFG}\)—internal (kinetic) energy of the finite-size IFG obtained from the SMPIMC; \(E_{kin}\) and \(E_{pot}\)—kinetic and potential energy of the UEG. Two significant digits of the statistical error \(3\sigma\) are written in brackets. All energies are given per one electron. Figure 6: Left plot: average kinetic energy of the UEG at different values of \(r_{s}\) and \(\theta\) compared with the internal (kinetic) energy of the IFG. Right plot: average potential energy of the UEG. interparticle distance, so the electron can escape from the potential well more easily via the quantum tunneling and becomes less localized. Note that the definition of the coupling strength \(\Gamma\) is based on the classical value of the average kinetic energy \(kT\) and gives an overestimated value of the actual non-ideality in the degenerate case, which should be characterized by the quantum average kinetic energy between \(kT\) and \(E_{F}\).