## 3 Simulation Results
Although the difference between the MDFs and the related Fermi distribution does not exceed the statistical error in all cases, the PCFs demonstrate completely non-ideal behavior: \(g_{ud}\) rises only slightly faster than \(g_{uu}\) and \(g_{dd}\) (especially at \(\theta=4.0\)) because of the same order of the Coulomb and exchange interactions. The results for \(r_{s}=12.0\) are shown in Figure 2. In the case of \(\theta=0.5\), the coupling is very strong (\(\Gamma\approx 13\)), and the MDF significantly differs from the Fermi distribution demonstrating an extensive \"tail\" at high momenta, which is not camouflaged by the finite-size effects. All PCFs have maxima exceeding 1 at \(r_{max}\), and \(g_{ud}\) grows almost with the same rate as \(g_{uu}\) and \(g_{dd}\), because the Coulomb repulsion plays a major role. In the case of \(\theta=1.0\), the coupling is quite strong (\(\Gamma\approx 6.5\)). The difference between MDF and the  
Figure 1: Left plots: single-particle momentum distribution functions of the unpolarized UEG at \(r_{s}=4.0\) and \(\theta=0.5,1,4\) calculated with the SMPIMC method. The results are compared with the SMPIMC results for the IFG in the same conditions and with the analytical Fermi distributions. Right plots: the pair correlation functions \(g_{uu}\), \(g_{dd}\), and \(g_{ud}\) of the UEG. Fermi distribution does not exceed the statistical error. However, there are weak maxima on the PCFs, and \(g_{ud}\) almost coincides with \(g_{uu}\) and \(g_{dd}\). At \(\theta=4.0\), the UEG is significantly non-ideal (\(\Gamma\approx 1.6\)), the MDF does not differ from the Fermi distribution, and all the PCFs are the same and do not have any maxima. The results for \(r_{s}=16.0\) are shown in Figure 3. In the case of \(\theta=0.5\) (\(\Gamma\approx 17\)), the MDF is quite similar to the previously considered one with \(r_{s}=12\), but the difference between it and the Fermi distribution is more significant. The PCFs are also analogous to the case of \(r_{s}=12\), but the maxima are more pronounced and the minima (at \(r_{min}\)) begin to appear. In the case of \(\theta=1.0\), the coupling is also strong (\(\Gamma\approx 8.7\)), and the MDF begins to differ from the Fermi distribution. The PCFs have distinct maxima at \(r\approx 28a_{0}\). In the case of \(\theta=4.0\), the coupling strength \(\Gamma\approx 2.2\), and the MDF coincides with the Fermi distribution within the statistical error. The PCFs do not have any maxima yet, and there are no differences between \(g_{uu}\), \(g_{dd}\), and \(g_{ud}\), so the exchange interaction is negligible in comparison with the Coulomb one. Figure 2: Left plots: single-particle MDFs of the unpolarized UEG at \(r_{s}=12\) and \(\theta=0.5,1,4\) calculated with the SMPIMC method. The results are compared with the SMPIMC for the IFD at the same conditions and with the analytical Fermi distributions. Right plots: PCFs \(g_{uu}\), \(g_{dd}\) and \(g_{ud}\) of the UEG. The results for \(r_{s}=28.0\) are shown in Figure 4. In the case of \(\theta=1\), the UEG is strongly non-ideal (\(\Gamma\approx 15\)), and the MDF differs from the Fermi distribution very distinctly. All PCFs are almost the same and have maxima and minima. In the case of \(\theta\approx 2\), the coupling strength \(\Gamma\approx 7.6\) and non-ideality is quite strong. The MDF begins to deviate from the Fermi distribution, while the PCFs \(g_{uu}\), \(g_{dd}\), and \(g_{ud}\) are exactly the same and have distinct maxima. In the case of \(\theta\approx 4.0\), the coupling strength \(\Gamma\approx 3.8\), and the MDF, slightly differ from the Fermi distribution. The maxima of the PCFs are small but distinguishable. Figure 3: Left plots: single-particle MDFs of the unpolarized UEG at \(r_{s}=16\) and \(\theta=0.5,1,4\) calculated with the SMPIMC method. The results are compared with the SMPIMC for the IFG at the same conditions, and with the analytical Fermi distributions. Right plots: PCFs \(g_{uu}\), \(g_{dd}\), and \(g_{ud}\) of the UEG.