= Phonon heat capacity of a dilute Bose gas
{title2=$C_V=2\pi^2Vk_B(k_BT)^3/(15\hbar^3c_s^3)$}
One linear Bogoliubov branch $E_k=\hbar c_sk$ in three dimensions gives thermal <energy> $\pi^2V(k_BT)^4/(30\hbar^3c_s^3)$ 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 $\hbar c_s/L\ll k_BT\ll\mu$ for a box: the lower inequality permits a continuum integral, and the upper one selects <phonons>. Fixed finite boxes ultimately have discrete-level behaviour.
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