For an ideal Fermi gas in equilibrium at temperature $T$ and chemical potential $\mu$, the average occupation number of the $k$ th energy state, with energy $E_{k}$, is

$$\bar{n}_{k}=\frac{1}{e^{\left(E_{k}-\mu\right) / k_{B} T}+1} .$$

Discuss the limit $T \rightarrow 0$. What is the Fermi energy $\epsilon_{F} ?$ How is it related to the Fermi momentum $p_{F}$ ? Explain why the density of states with momentum between $p$ and $p+d p$ is proportional to $p^{2} d p$ and use this fact to deduce that the fermion number density at zero temperature takes the form

$$n \propto p_{F}^{3} .$$

Consider an ideal Fermi gas that, at zero temperature, is either (i) non-relativistic or (ii) ultra-relativistic. In each case show that the fermion energy density $\epsilon$ takes the form

$$\epsilon \propto n^{\gamma}$$

for some constant $\gamma$ which you should compute.