In the canonical ensemble, temperature $T$ is:

Canonical ensemble: fixed $T$, $V$, $N$ — system exchanges heat with reservoir.

Correct answer: Fixed (system in contact with a heat reservoir)

The partition function $Z = \sum_i e^{-\beta E_i}$ for a two-level system ($E = 0$ and $E = \varepsilon$) is:

$Z = e^0 + e^{-\beta\varepsilon} = 1 + e^{-\beta\varepsilon}$.

Correct answer: $1 + e^{-\beta\varepsilon}$

Write the Fermi-Dirac distribution and explain what it describes.

At $T=0$, all states below $\mu$ (Fermi energy) are filled, all above are empty.

Correct answer: $f(E) = \frac{1}{e^{(E-\mu)/k_BT}+1}$. It gives the average occupation number of a quantum state with energy $E$ for fermions, where $\mu$ is the chemical potential.