Phonon heat capacity of a dilute Bose gas ($C_V=2{\pi }^{2}V{k}_B(k_{B}T{)}^3/(15{\hslash }^3{c}_s^3)$)

One linear Bogoliubov branch $E_k=\hslash c_{s}k$ in three dimensions gives thermal energy ${\pi }^{2}V(k_{B}T{)}^4/(30{\hslash }^3{c}_s^3)$ and the displayed cubic heat capacity. There is one branch, rather than the three polarizations in a simple crystal's Debye model. The regime requires $\hslash c_s/L\ll k_{B}T\ll \mu$ for a box: the lower inequality permits a continuum integral, and the upper one selects phonons. Fixed finite boxes ultimately have discrete-level behaviour.