Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 35B / a / Solution

A grand canonical ensemble describes a system exchanging both energy and particles with a reservoir at fixed $T$, volume, and chemical potential $\mu$. For a noninteracting fermion level $r$, occupation is $0$ or $1$, soFactorization over levels gives $\mathcal{Z}={\prod }_r{\mathcal{Z}}_r$, and differentiation with respect to $\beta \mu$ yields the Fermi-Dirac distribution