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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2024

Paper 1

With $r=|x|$, set
$$
r=V^{1/3}(k_BT)^{1/(2n)}u.
$$
Then
$$
\int e^{-\beta U(x)}d^3x
=4\pi V(k_BT)^{3/(2n)}I_n.
$$
Part (b) gives
$$
Z_1=\frac{4}{\pi(\hbar c)^3}
V(k_BT)^{3+3/(2n)}I_n.
$$
For $N$ identical classical non-interacting particles,
$$
Z_N=\frac{Z_1^N}{N!}.
$$
Using $\log N!=N\log N-N+o(N)$,
$$
F=-Nk_BT\left[\log V+A\log(k_BT)+\log I_n+B\right]+o(N),
$$
where
$$
\boxed{
A=3+\frac{3}{2n},
\qquad
B=1+\log\!\left(\frac{4}{\pi N(\hbar c)^3}\right).}
$$
This is the <canonical partition function for an ultrarelativistic gas in a power-law trap>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 36B / c / i / Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 36B / c / i / Solution

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