Semi-classical expansion of distribution function using modified Hermite polynomials for quantum gas

The author proposes the semi-classical expansion of the distribution function using modified Hermite polynomials to calculate moment equations for quantum gas. The completeness of the semi-classical expansion of the distribution function is not satisfied, whereas we can conjecture that moment equations obtained using the semi-classical expansion coincides with those obtained using Uehling–Uhlenbeck equation. Actually, Grad’s 13 moment equations, which are calculated using correct Grad’s 13 moment equation, coincide with those, which are calculated using the semi-classical expansion of the distribution function, when the collisional term of the Uehling–Uhlenbeck equation is replaced with the quantum Bhatnagar–Gross–Krook model.