Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 303 / 1 / a / i

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- Solution i

Solution

For the partition function

Z = \sum_{{S}} e^{- βE}

, differentiation with respect to the external field gives

\frac{\partial l o g Z}{\partial B} = \frac{1}{Z} \sum_{{S}} β (\sum_{i} S_{i}) e^{- βE} = β \sum_{i} ⟨ S_{i} ⟩ .

Therefore the magnetization per site is

m = \frac{1}{Nβ} \frac{\partial lo g Z}{\partial B} .

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