A system consists of $N$ weakly interacting non-relativistic fermions, each of mass $m$, in a three-dimensional volume, $V$. Derive the Fermi-Dirac distribution

$$n(\epsilon)=K V g \frac{\epsilon^{1 / 2}}{\exp ((\epsilon-\mu) / k T)+1}$$

where $n(\epsilon) d \epsilon$ is the number of particles with energy in $(\epsilon, \epsilon+d \epsilon)$ and $K=2 \pi(2 m)^{3 / 2} / h^{3}$. Explain the physical significance of $g$.

Explain how the constant $\mu$ is determined by the number of particles $N$ and write down expressions for $N$ and the internal energy $E$ in terms of $n(\epsilon)$.

Show that, in the limit $\kappa \equiv e^{-\mu / k T} \gg 1$,

$$N=\frac{V}{A \kappa}\left(1-\frac{1}{2 \sqrt{2} \kappa}+O\left(\frac{1}{\kappa^{2}}\right)\right)$$

where $A=h^{3} / g(2 \pi m k T)^{3 / 2}$.

Show also that in this limit

$$E=\frac{3}{2} N k T\left(1+\frac{1}{4 \sqrt{2} \kappa}+O\left(\frac{1}{\kappa^{2}}\right)\right) \text {. }$$

Deduce that the condition $\kappa \gg 1$ implies that $A N / V \ll 1$. Discuss briefly whether or not this latter condition is satisfied in a white dwarf star and in a dilute electron gas at room temperature.

$\left[\right.$ Note that $\left.\int_{0}^{\infty} d u e^{-u^{2} a}=\frac{1}{2} \sqrt{\frac{\pi}{a}}\right]$.