## 3 Simulation Results  
We studied the unpolarized UEG in the range of densities and temperatures \(0.2\leq r_{s}\leq 36,0.5\leq\theta\leq 4\), that corresponds to the coupling parameter \(0.03\lesssim\Gamma\lesssim 20\) and the degeneracy parameter \(0.2\lesssim\chi\lesssim 4\). Therefore, a wide range of states has been covered: from almost ideal to strongly non-ideal and, simultaneously, from an almost classical to degenerate system. We carried out our simulations for \(N=66\) electrons in the cubic cell with PBCs. Such number of electrons has been chosen for conformity with the papers [34; 36; 41] and is quite enough for \(\theta\gtrsim 0.5\). Furthermore, we simulated the unpolarized ideal Fermi gas (IFG) under the same conditions, i.e., the UEG with \"turned-off\" Coulomb interactions at the same \(\theta\) and \(r_{s}\). Comparison of these results with the analytical Fermi distributions allows us to take into account the finite-size effects and control the related systematic errors in the MDF normalization factors. The influence of the finite-size effects on the MDFs manifests itself in the deviation of the calculated MDF for the IFG from the analytical Fermi distribution, which appears at some high value of momentum \(p_{FS}\), so the interval \(p\lesssim p_{FS}\) contains the reliable data. We estimated roughly the value of \(p_{FS}\) from the plot of \(F(p)\) in the logarithmic scale. One can find the values of \(p_{FS}\) for each plot in Table 1. This table also contains the approximate positions \(r_{max}\) and \(r_{min}\) of the first maximum and minimum of the PCFs \(g_{uu}\), \(g_{dd}\), and \(g_{ud}\). For low values of \(r_{s}\) (\(r_{s}\lesssim 1\)), the UEG is characterized by relatively weak non-ideality (\(\Gamma\lesssim 1\) in the considered range of \(\theta\)), and the results are quite trivial and do not deserve to \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(\mathbf{r_{s}}\) & \(\theta\) & \(\mathbf{\Gamma}\) & \(\chi\) & \(\mathbf{\lambda},\mathbf{a_{0}}\) & \(\mathbf{L},\mathbf{a_{0}}\) & \(\mathbf{p_{FS}},\mathbf{h/a_{0}}\) & \(\mathbf{r_{max}},\mathbf{a_{0}}\) & \(\mathbf{r_{min}},\mathbf{a_{0}}\) \\ \hline
4.0 & 0.5 & 4.3 & 4.3 & 10 & 26 & 0.85 & — & — \\
4.0 & 1.0 & 2.2 & 1.5 & 7.4 & 26 & 1.2 & — & — \\
4.0 & 4.0 & 0.54 & 0.19 & 3.7 & 26 & \(>\)1.9 & — & — \\
12 & 0.5 & 13 & 4.3 & 31 & 78 & 0.28 & 20–22 & 30–32 \\
12 & 1.0 & 6.5 & 1.5 & 22 & 78 & 0.42 & 20–22 & — \\
12 & 4.0 & 1.6 & 0.19 & 11 & 78 & 0.65 & — & — \\
16 & 0.5 & 17 & 4.3 & 42 & 100 & 0.21 & 26–28 & 40–42 \\
16 & 1.0 & 8.7 & 1.5 & 30 & 100 & 0.30 & 26–28 & — \\
16 & 4.0 & 2.2 & 0.19 & 15 & 100 & 0.60 & — & — \\
28 & 1.0 & 15 & 1.5 & 52 & 180 & 0.17 & 45–50 & 70–75 \\
28 & 2.0 & 7.6 & 0.53 & 37 & 180 & 0.26 & 45–50 & 70–75 \\
28 & 4.0 & 3.8 & 0.19 & 26 & 180 & 0.38 & 50 & — \\
36 & 1.0 & 20 & 1.5 & 66 & 230 & 0.14 & 60–65 & 90–95 \\
36 & 2.0 & 9.8 & 0.53 & 47 & 230 & 0.21 & 60–65 & — \\
36 & 4.0 & 4.9 & 0.19 & 33 & 230 & 0.28 & 60–65 & — \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Some parameters and data describing the MDFs and PCFs from the Figures 1–5. The values are shown with two significant digits. be shown in figures. The MDFs completely coincide with the related Fermi distribution for \(p\lesssim p_{FS}\). The PCFs are also similar to the ideal case: \(g_{uu}\) and \(g_{dd}\), and are determined mainly by the exchange repulsion and slowly rise from 0 to 1 with growing \(r\), while \(g_{ud}\) is determined only by relatively weak Coulomb repulsion and rises much faster. The results for \(r_{s}=4.0\) are shown in Figure 1. In cases of \(\theta=0.5\) and \(\theta=1\), the UEG is significantly non-ideal (\(\Gamma\approx 4.3\) and \(\Gamma\approx 2.2\)), while at \(\theta=4\) it remains weakly coupled (\(\Gamma\approx 0.54\)).