## 2 Theoretical Part
### Single Momentum Approach  
To obtain the expressions for a single-particle MDF and PCFs required for the numerical calculations, we use the \"single momentum approach\" presented in [34]. Thisapproach is based on the pseudoprobability density \(W(p,q)\) in the phase space known as the Wigner function:  
\[W(p,q)=\int\mathrm{d}^{3N}\xi\,\mathrm{e}^{\frac{\mathrm{i}\,(p,\xi)}{\hat{\xi}} \,\langle q-\xi/2|\hat{\rho}|q+\xi/2\rangle}, \tag{6}\]  
where \(\rho=\exp(-\beta H)\), \(\beta=1/(k_{B}T)\), \((p,\xi)=\sum_{a=1}^{N}\mathbf{p}_{a}\mathbf{\xi}_{a}\). The quantum states \(|q\rangle=|\mathbf{q}_{1},\mathbf{q}_{2},\dots,\mathbf{q}_{N}\rangle\) and \(|p\rangle=|\mathbf{p}_{1},\mathbf{p}_{2},\dots,\mathbf{p}_{N}\rangle\) are the \(N\)-particle states with certain coordinates and momenta, antisymmetrized according to the Fermi-Dirac statistics:  
\[|q\rangle =\frac{1}{N!}\sum_{P_{a},P_{d}}(-1)^{P_{a}+P_{d}}\,|\mathbf{q}_{p _{1}}\rangle|\mathbf{q}_{p_{2}}\rangle\dots|\mathbf{q}_{p_{N}}\rangle,\] \[|p\rangle =\frac{1}{N!}\sum_{P_{a},P_{d}}(-1)^{P_{a}+P_{d}}\,|\mathbf{p}_{a _{1}}\rangle|\mathbf{p}_{a_{2}}\rangle\dots|\mathbf{p}_{a_{N}}\rangle. \tag{7}\]  
Here, the sums are taken over all permutations \(P_{u}\) and \(P_{d}\) of the electrons with the positive and negative spin projections having the factor \(+1\) for each even and \(-1\) for each odd permutation. The Weyl symbol can be associated with a quantum operator \(\hat{A}\):  
\[A(p,q)=(2\pi\hbar)^{-3N}\int\mathrm{d}^{3N}\xi\,\mathrm{e}^{\frac{\mathrm{i}} {\hbar}(p,\xi)}\,\langle q+\xi/2|\hat{A}|q-\xi/2\rangle, \tag{8}\]  
so the average value of \(\hat{A}\) over the canonical ensemble can be calculated via the classical-like expression:  
\[\langle\hat{A}\rangle=\iint\mathrm{d}^{3N}p\,\mathrm{d}^{3N}q\,A(p,q)\,W(p,q). \tag{9}\]  
The \(N\)-particle momentum and coordinate distribution functions can be obtained from the Wigner function via the integration over coordinates and momenta, respectively [37]:  
\[F(\mathbf{p}_{1},\dots,\mathbf{p}_{N})=\int\mathrm{d}^{3N}q\,W(p,q),\quad g( \mathbf{q}_{1},\dots,\mathbf{q}_{N})=\int\mathrm{d}^{3N}p\,W(p,q). \tag{10}\]  
The integration over \(\mathbf{p}_{2},\dots,\mathbf{p}_{N}\) leads to the product of delta-functions \(\prod_{a=2}^{N}\delta^{(3)}(\mathbf{\xi}_{a})\), so the integrals over \(\mathbf{\xi}_{a}\) (\(a=2,\dots,N\)) disappear. The density matrix with off-diagonal elements for the particles with numbers \(b\
eq 1\) replaced with zeros is known as the \"single-momentum density matrix for particle 1\" [34]:  
\[\rho_{SM}(q,\mathbf{\xi}_{1})=\langle q-\xi/2|\hat{\rho}|q+\xi/2\rangle_{\xi_{ 2}=\dots=\xi_{N}=0}. \tag{11}\]  
Integrating it over coordinates \(q\), one obtains the \"single-particle \(\xi\)-distribution function\":  
\[f(\mathbf{\xi})=\int\mathrm{d}q\,\rho_{SM}(q,\mathbf{\xi}_{1})\Big{|}_{\xi= \mathbf{\xi}_{1}}. \tag{12}\]  
The Fourier transform of \(\rho_{SM}\) results in the single-particle MDF:  
\[F(\mathbf{p})=\int\mathrm{d}\mathbf{\xi}\,\mathrm{e}^{\frac{\mathrm{i}}{\hbar }p\xi}\,f(\xi). \tag{13}\]  
As the macroscopic system is isotropic, the single-particle \(\xi\)-distribution function and MDF depend only on the modules of the vectors \(\mathbf{\xi}\) and \(\mathbf{p}\), and the 3-dimensional Fourier transform can be reduced to the 1-dimensional sine transform:  
\[F(p)=\int_{0}^{\infty}\mathrm{d}\xi\,4\pi\xi^{2}f(\xi)\frac{\sin(p\xi/\hbar)} {p\xi/\hbar}. \tag{14}\]The expressions for PCFs contains multiple integrals over \(\mathfrak{p}_{1},\ldots,\mathfrak{p}_{N}\) and can be transformed in a similar way. Due to the product of the delta-functions, \(\prod_{n=1}^{N}\delta^{(3)}(\xi_{a})\), only the diagonal elements of the density matrix remain. The density matrix with non-diagonal elements replaced with zeros is known as the \"diagonal density matrix\" [34]:  
\[|q\rangle =\frac{1}{N!}\sum_{P_{a},P_{d}}(-1)^{P_{a}+P_{d}}\,|\mathbf{q}_{p _{1}}\rangle|\mathbf{q}_{p_{2}}\rangle\dots|\mathbf{q}_{p_{N}}\rangle,\]0  
Further integration over \(N-2\) coordinates gives the corresponding PCF.