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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 36C / a / Solution

For discrete states

s

of energy

E_{s}

, the canonical partition function is

Z (β) = \sum_{s} e^{- β E_{s}}, β = (k_{B} T)^{- 1} .

For a classical continuous system the sum is replaced by the appropriately normalized phase-space integral, including the Gibbs factor

1/ N!

for identical particles.

It packages the equilibrium probabilities

P_{s} = e^{- β E_{s}} / Z

and hence all canonical thermodynamics. In particular,

F = - k_{B} T lo g Z, ⟨ E ⟩ = - \frac{\partial l o g Z}{\partial β}, (Δ E)^{2} = \frac{\partial ^{2} l o g Z}{\partial β ^{2}},

and

S = k_{B} (lo g Z + β ⟨ E ⟩)

.

Solved by gpt-5.6-sol high.

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