PX284 - M3 - degeneracy pressure

\begin{aligned} ⟨ E ⟩ & = \frac{U}{N} = \frac{\int_{0}^{\infty} E g (E) f_{F D} (E) d E}{\int_{0}^{\infty} g (E) f_{F D} (E) d E} \\ = \frac{\int_{0}^{E_{F}} E g (E) d E}{\int_{0}^{E_{F}} g (E) d E} \\ \approx \frac{\int_{0}^{E_{F}} E E^{1 / 2} d E}{\int_{0}^{E_{F}} E^{1 / 2} d E} \\ \approx \frac{E_{F}^{5 / 2} / (5 / 2)}{E_{F}^{3 / 2} / (3 / 2)} \\ ∴ ⟨ E ⟩ & \approx \frac{3}{5} E_{F} \end{aligned}

where, $g (E) \sim E^{1 / 2}$

U = \frac{3 N}{5} E_{F}

\begin{matrix} U \propto V^{- 2 / 3} \\ \ln U = - \frac{2}{3} \ln V + constant \\ \frac{d U}{U} = - \frac{2}{3} \frac{d V}{V} \end{matrix}

p = - {(\frac{\partial U}{\partial V})}_{S} = \frac{2}{3} \frac{U}{V} = \frac{2}{5} n E_{F}