Blundell Thermal Physics · Problem 30.4

Quantum gases and condensates

Physics — Statistical Mechanics · bose-gas-heat-capacity

Problem

Show that the heat capacity of a system of non-interacting bosons is given by

\begin{aligned} C_V &= \frac{15}{4}\frac{\zeta(\frac{5}{2})}{\zeta(\frac{3}{2})}Nk_{\rm B}\left(\frac{T}{T_{\rm c}}\right)^{3/2}, \quad T < T_{\rm c}, \nonumber\\ C_V &= \frac{3}{2}Nk_{\rm B}\left(\frac{5}{2}\frac{{\rm Li}_{5/2}(z)}{{\rm Li}_{3/2}(z)} - \frac{3}{2}\frac{{\rm Li}_{3/2}(z)}{{\rm Li}_{1/2}(z)}\right), \quad T > T_{\rm c}. \tag{30.65} \end{aligned}