Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 36A / a / Solution

The microcanonical ensemble describes an isolated system at fixed energy $E$, volume $V$, and particle number $N$, assigning equal weight to accessible states in a narrow energy shell.

The canonical ensemble describes a system that can exchange energy with a heat reservoir at fixed temperature $T$, while $V$ and $N$ remain fixed. A microstate $s$ has probability

The grand canonical ensemble allows exchange of both energy and particles at fixed $T$, $V$, and chemical potential $\mu$. Its state weights are

The choice is irrelevant for bulk thermodynamic observables in the thermodynamic limit when the system is additive, has sufficiently short-range interactions, and lies away from singular phase-coexistence regimes. Then relative energy and particle-number fluctuations vanish as the system size grows. These are the standard conditions for equivalence of statistical ensembles.

Solved by gpt-5.6-sol high.