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Chapter 2 of Statistical Physics by David Tong. Sections 2.1–2.2 give ideal classical partition functions; section 2.5 gives the density virial expansion and Mayer f-function. No separately contributed chapter author is established.

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

A hard-sphere gas is a classical model of identical impenetrable spheres, each of diameter $\sigma$ and radius $\sigma /2$. Its pair potential is $u(r)=+\infty$ for centre separation $r<\sigma$ and $u(r)=0$ for $r\ge \sigma$; the convention at exact contact has no effect on the integral. It isolates the effect of particle size without an attractive potential energy.

The upper-left panel is a cross-section of a three-dimensional forbidden region: the second centre cannot enter a sphere of radius $\sigma$ around the first centre. The particles themselves have radius $\sigma /2$. Consequently, with particle volume $v_{\mathrm{s}}$ and pair excluded volume $v_{\mathrm{ex}}$,The last equality follows directly from the Mayer f-function: $f=-1$ throughout the forbidden volume and zero outside it. Confusing particle radius with contact separation changes the answer by a factor of eight; omitting the unordered-pair factor changes it by a further factor of two.

Define the packing fraction $\eta =n{v}_{\mathrm{s}}$, where $n$ is number density. Then the virial expansion beginsFor example, at the toy value $\eta =0.01$, the leading correction predicts $Z\approx 1.04$; the omitted terms still matter if accuracy beyond this approximation is needed. The classical second virial coefficient is independent of temperature because the geometry of forbidden configurations does not change when the gas is warmed. The pressure itself still scales with temperature at fixed number density.

This model makes the positive correction particularly clear: eliminating overlapping configurations changes the dependence of the available configuration space on volume, even though the allowed configurations all have zero interaction energy. The figure's other panels add a toy attractive shell and show how its Boltzmann factor competes with that geometry. The hard-sphere gas is a useful reference model, while a low-order virial expansion remains a dilute-gas approximation rather than a dense sphere packing formula.

The Mayer f-function measures how a pair potential changes a pair's Boltzmann factor relative to the noninteracting value one:It vanishes where $u=0$, lies between $-1$ and $0$ for repulsion, and is positive for attraction. Even an infinite hard-core potential energy produces the finite value $f=-1$. This is why expanding in Mayer f-functions handles a hard-sphere gas more naturally than a Taylor series in $u$. ^[ref]

For pairwise-additive potential energy $U={\sum }_{i<j}u(r_{ij})$, the exact identityturns the configurational part of the canonical partition function into a sum of products associated with pairs. Drawing one edge for each chosen factor $f(r_{ij})$ gives the graphs used in the cluster expansion. The second virial coefficient uses just one pair's relative-position integral, $B_2=-\frac{1}{2}\int f{d}^3\mathbf{r}$.

For the square-well toy pair potential in second virial coefficient, $f=-1$ in the core, $f=e^{1/T^{\ast }}-1$ in the attractive shell, and $f=0$ beyond it. Its positive plateau need not be small: low number density, rather than small $f$ at every separation, controls the dilute-gas approximation. The integral weights each plateau by spherical-shell volume; equal-looking unweighted areas on a plot of $f$ versus $r$ need not cancel in $B_2$.

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 isFor each pair, translation of its center coordinate supplies $V$ and the other $N-2$ particles supply $V^{N-2}$. Neglecting boundary corrections,The Taylor expansion $\log (1+x)=x+O(x^2)$ and $N(N-1)\simeq N^2$ in the thermodynamic limit giveUsing $P=-(\partial F/\partial V{)}_{T,N}$ and the ideal gas free energy,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.

Use the classical dilute-gas conventionso the second virial coefficient has units of volume. The Mayer f-function formula for an isotropic pair potential isOnly separations smaller than $a$ contribute. Define $x=\beta \epsilon >0$ and substitute $t=r^3/a^3$, so $r^{2}dr=(a^3/3)dt$:This is the second virial coefficient of a bounded cubic repulsion. For every finite positive $T$, $1-e^{-x}<x$, so $B_2(T)>0$. There is no finite positive Boyle temperature. At high temperature, expanding the exponential givesThus the intended limiting statement isAt the other extreme the classical expression tends to $2\pi a^3/3$, the hard-exclusion value for separation $a$. This is a check on the integral, not a claim that a real molecular gas stays classical at arbitrarily low temperature. A molar virial expansion would instead use $B_{2,\mathrm{m}}=N_A{B}_2$.

The second virial coefficient is the leading interaction correction to the ideal gas law: at fixed temperature, $B_2(T)={\lim }_{n\to 0}(Z-1)/n$, where $Z=p/(n{k}_{B}T)$ and $n$ is number density. For identical classical particles in a homogeneous three-dimensional gas, with an isotropic pair potential $u(r)$ and $r$ the centre-to-centre separation,The function $f$ is the Mayer f-function. The factor $4\pi r^{2}dr$ counts the volume of a thin spherical shell; the factor $1/2$ counts each unordered pair once. The result has units of volume. Repulsive regions, $u>0$, contribute positively, while attractive regions, $u<0$, contribute negatively. ^[ref]

This is a striking bridge from a microscopic pair potential to a macroscopic pressure reading. At sufficiently small number density, $B_2>0$ raises pressure relative to an ideal gas at the same temperature and number density, while $B_2<0$ lowers it. A single value of $B_2$, however, cannot determine the entire pair potential: many different functions have the same weighted integral. Measurements over several temperatures supply more information.

The integral must exist. For example, a bounded short-range attraction outside a hard core has no difficulty. If instead $u(r)\sim A/r^s$ as $r\to \infty$, then $1-e^{-u/(k_{B}T)}\sim u/(k_{B}T)$ and the radial tail behaves like $\int r^{2-s}dr$: it converges only for $s>3$. A bare unscreened Coulomb potential therefore falls outside this formula's ordinary short-range setting. This condition on the tail does not by itself establish the stability or convergence of the full virial expansion.

For an exactly integrable toy attraction, choose the following square-well pair potential, with diameter $\sigma >0$, attraction depth $ϵ>0$, and range multiplier $a>1$:Set $T^{\ast }=k_{B}T/ϵ$ and $b_{\mathrm{HS}}=2\pi {\sigma }^3/3$, the second virial coefficient of a hard-sphere gas. Integrating the core and attractive shell separately gives the exact toy resultThe core contributes $b_{\mathrm{HS}}$; the shell contributes $-b_{\mathrm{HS}}(a^3-1)(e^{1/T^{\ast }}-1)$. Heating weakens the relative Boltzmann factor enhancement of attractive configurations, so $B_2$ approaches the positive hard-core value as $T^{\ast }\to \infty$. For the illustrative choice $a=3/2$, the zero occurs atAt this temperature, $Z=1+B_3{n}^2+\cdots$: cancellation removes the linear correction, not every correction. At low $T^{\ast }$, the negative shell contribution becomes large, requiring a correspondingly smaller number density for a useful truncated virial expansion. The square well is a deliberately chosen model, not a fit to the carbon dioxide experiment.

For a classical dilute gas with pair potential $u(r)=ϵ[1-(r/a{)}^3]$ for $r<a$ and zero otherwise, put $x=ϵ/(k_{B}T)>0$. Substitution $t=(r/a{)}^3$ in the Mayer f-function formula givesIt is positive for finite positive temperature, tends to $2\pi a^3/3$ as $T\to 0$ in the classical model, and behaves as $\pi a^3ϵ/(3k_{B}T)$ at high temperature. The low-temperature mathematical limit need not remain in the dilute classical regime for a real molecular gas.