Bose–Einstein condensation of nonideal gas

The canonical distribution with a constraint is applied to the Bogoliubov model of a nonideal Bose-gas. Two approximate solutions of variational equations for the Bose-condensate density and a constraint parameter are found. They are treated as “nontrivial” and “trivial” parts of the process of Bose-condensation as far as they contribute to the thermodynamic properties at a temperature $T$ below and above the critical $T_c$ respectively. The corresponding branches of the spectrum are investigated. The heat capacity $C_V$ is considered with the help of the derived solutions for all the temperatures. Low-temperature behaviour $C_V \sim T^{3/2}$ and $C_V \sim T^3$ due to the contributions of free Boson and phonon excitations is proved. The asymptotics of $C_V$ for a temperature close to the critical temperature $T_c$ below and above it are calculated for numerical values of parameters of $\operatorname{He}^4$ in a qualitative agreement with the experiment.