Low-temperature heat capacity of a three-dimensional Fermi gas ($C_V=({\pi }^2/2)N{k}_B(T/T_F)$)

For a three-dimensional ideal gas with quadratic dispersion and fixed particle number, the total density of states at the Fermi energy is $g(E_F)=3N/(2E_F)$. The Sommerfeld coefficient then gives $C_V=({\pi }^2/2)N{k}_B(T/T_F)$ to leading order, where $T_F=E_F/k_B$. The fixed-number energy correction is $\Delta U=({\pi }^2/4)N(k_{B}T{)}^2/E_F$. Only states near the Fermi surface change occupation, explaining the small active fraction.