What is meant by the chemical potential of a thermodynamic system? Derive the Gibbs distribution with variable particle number $N$, for a system at temperature $T$ and chemical potential $\mu$. (You may assume that the volume does not vary.)

Consider a non-interacting gas of fermions in a box of fixed volume, at temperature $T$ and chemical potential $\mu$. Use the Gibbs distribution to find the mean occupation number of a one-particle quantum state of energy $\varepsilon$. Assuming that the density of states is $C \varepsilon^{1 / 2}$, for some constant $C$, deduce that the mean number of particles with energies between $\varepsilon$ and $\varepsilon+d \varepsilon$ is

$$\frac{C \varepsilon^{\frac{1}{2}} d \varepsilon}{e^{(\varepsilon-\mu) / T}+1} .$$

Why can $\mu$ be identified with the Fermi energy $\varepsilon_{F}$ when $T=0$ ? Estimate the number of particles with energies greater than $\varepsilon_{F}$ when $T$ is small but non-zero.