Quantum statistics effects near the critical point in systems with different inter-particle interactions

Equation of state with quantum statistics corrections is derived for systems of the Fermi and Bose particles by using their van der Waals (vdW) and effective density-dependent Skyrme mean-field interactions. First few orders of these corrections over the small quantum statistics parameter, \(\varepsilon \approx \hbar^3 n(mT)^{-3/2}g^{-1}\), where \(n\) and \(T\) are the particle number density and temperature, \(m\) and \(g\) the mass and degeneracy factor of particles, are analytically obtained. For interacting system of nucleon and \(\alpha\) - particles, a small impurity of \(\alpha\) - particles to a nucleon system at leading first order in both \(\alpha \)-particle and nucleon small parameters \(\varepsilon\) does not change much the basic results for the symmetric nuclear matter in the quantum vdW consideration. Our approximate analytical results for the quantum vdW and Skyrme mean-field approaches are in a good agreement with accurate numerical calculations.