For the <partition function> $Z=\sum_{\{S\}}e^{-\beta E}$, differentiation with respect to the external field gives
$$
\frac{\partial\log Z}{\partial B}
=\frac1Z\sum_{\{S\}}\beta\!\left(\sum_iS_i\right)e^{-\beta E}
=\beta\sum_i\langle S_i\rangle.
$$
Therefore the <magnetization> per site is
$$
\boxed{m=\frac1{N\beta}\frac{\partial\log Z}{\partial B}}.
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-303/1/a/i/solution