 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-3/35b/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

A grand canonical ensemble describes a system exchanging both energy and particles with a reservoir at fixed

T

μ

. For a noninteracting fermion level

r

, occupation is

0

1

, so

Z_{r} = 1 + e^{- β (E_{r} - μ)} .

Factorization over levels gives

Z = \prod_{r} Z_{r}

β μ

⟨ n_{r} ⟩ = \frac{1}{e ^{β (E_{r} - μ)} + 1} .

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