What is meant by the chemical potential $\mu$ of a thermodynamic system? Derive the Gibbs distribution for a system at temperature $T$ and chemical potential $\mu$ (and fixed volume) with variable particle number $N$.

Consider a non-interacting, two-dimensional gas of $N$ fermionic particles in a region of fixed area, at temperature $T$ and chemical potential $\mu$. Using the Gibbs distribution, find the mean occupation number $n_{F}(\varepsilon)$ of a one-particle quantum state of energy $\varepsilon$. Show that the density of states $g(\varepsilon)$ is independent of $\varepsilon$ and deduce that the mean number of particles between energies $\varepsilon$ and $\varepsilon+d \varepsilon$ is very well approximated for $T \ll \varepsilon_{F}$ by

$$\frac{N}{\varepsilon_{F}} \frac{d \varepsilon}{e^{\left(\varepsilon-\varepsilon_{F}\right) / T+1}}$$

where $\varepsilon_{F}$ is the Fermi energy. Show that, for $T$ small, the heat capacity of the gas has a power-law dependence on $T$, and find the power.