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The cluster expansion organizes the <logarithm> of a dilute interacting system's <canonical partition function> as a sum over connected clusters built from <Mayer f-function>[Mayer bonds].

Cluster expansion

Let the <pair potential> be $u(r)$, so that in the dilute limit only one interacting pair need be retained at a time. The leading <Mayer f-function> expansion is
$$
e^{-U/T}-1\simeq\sum_{1\leq i<j\leq N}
\left(e^{-u(|\mathbf x_i-\mathbf x_j|)/T}-1\right).
$$
For each pair, translation of its center coordinate supplies $V$ and the other $N-2$ particles supply $V^{N-2}$. Neglecting boundary corrections,
$$
\frac1{V^N}\int(e^{-U/T}-1)\,d^{3N}x
\simeq \frac{N(N-1)}{2V}\int_{\mathbb R^3}(e^{-u(r)/T}-1)\,d^3r.
$$
The <Taylor expansion> $\log(1+x)=x+O(x^2)$ and $N(N-1)\simeq N^2$ in the <thermodynamic limit> give
$$
\boxed{F=F_{\rm ideal}+\frac{N^2T}{V}B(T)},
\qquad
\boxed{B(T)=\frac12\int_{\mathbb R^3}\left(1-e^{-u(r)/T}\right)d^3r
=2\pi\int_0^\infty\left(1-e^{-u(r)/T}\right)r^2\,dr}.
$$
Using $P=-(\partial F/\partial V)_{T,N}$ and the <ideal gas free energy>,
$$
\boxed{P=\frac{NT}{V}+\frac{N^2T}{V^2}B(T)}.
$$
With <number density> $n=N/V$, this is $P/(nT)=1+B(T)n+O(n^2)$, so $B(T)$ is precisely the <second virial coefficient>.

Solution

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

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{title2=$f(r)=e^{-u(r)/(k_BT)}-1$}

The Mayer f-function packages the departure of a pair's <Boltzmann factor> from one and is the basic bond in the dilute-gas <cluster expansion>.

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 Incoming links: Mayer f-function