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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2020

Paper 4

Set $k_B=1$. After the <Gaussian integral> over momenta, the <canonical partition function> factors as
$$
Z=Z_{\rm kin}\frac1{N!}\int_{V^N}e^{-U(\mathbf x_1,\ldots,\mathbf x_N)/T}\,d^{3N}x,
$$
whereas the <ideal gas> has $Z_{\rm ideal}=Z_{\rm kin}V^N/N!$. Hence
$$
\frac Z{Z_{\rm ideal}}
=\frac1{V^N}\int_{V^N}e^{-U/T}\,d^{3N}x
=1+\frac1{V^N}\int_{V^N}(e^{-U/T}-1)\,d^{3N}x.
$$
Using the <Helmholtz free energy> $F=-T\log Z$ therefore gives
$$
\boxed{F=F_{\rm ideal}-T\log\left\{1+\frac1{V^N}\int_{V^N}(e^{-U/T}-1)\,d^{3N}x\right\}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 35A / i / Solution

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 35A / i / Solution

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Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 35A / i / Solution

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