Fugacity and the Activity Expansion

The chemical potential $\mu$ is the natural thermodynamic handle for a system that exchanges particles with a reservoir, but it is a clumsy variable to compute with: it ranges over all of $(-\infty,+\infty)$ and it enters the statistics inside an exponential. One substitution fixes everything. Define the fugacity (or activity) $z\equiv e^{\beta\mu}$ , and the grand partition function collapses into an ordinary power series, $\Xi(z,V,T)=\sum_{N\ge0} z^N Z_N$ , where $Z_N$ is the ordinary $N$ -particle canonical partition function. The awkward chemical potential has become a tidy bookkeeping variable that tags how many particles are present, and every thermodynamic quantity becomes a series in $z$ . The physical meaning of $z$ is the reservoir's eagerness to push particles into the system: large $z$ (high $\mu$ ) floods the box, small $z$ leaves it nearly empty. For a classical ideal gas the relation is exact and beautifully simple — $z=n\lambda^3$ , the phase-space occupancy — so $z\ll1$ is precisely the dilute, non-degenerate, classical regime, and the activity expansion is the small- $z$ Taylor series of the equation of state. Two observables, the dimensionless pressure $\beta P=V^{-1}\ln\Xi$ and the density $n=z\,\partial_z(\ln\Xi/V)$ , are both series in $z$ ; eliminating $z$ between them produces the virial expansion, whose coefficients are integrals over the interaction potential. This article defines the fugacity, builds the activity and cluster expansions, proves that $z$ is always a legitimate coordinate for the density (its owned result: the map $n(z)$ is monotonic because $\partial n/\partial z$ is the number variance), and follows $z$ from the classical Maxwell–Boltzmann limit up to the Bose condensation edge and the degenerate Fermi sea. Every numeric claim here is python-verified.

Key takeaways

The fugacity $z=e^{\beta\mu}$ turns the grand partition function into a power series $\Xi=\sum_N z^N Z_N$ , replacing the awkward exponential dependence on $\mu$ with a clean expansion variable.
Pressure and density are both series in $z$ : $\beta P=\sum_\ell b_\ell z^\ell$ and $n=\sum_\ell \ell\, b_\ell z^\ell$ ; eliminating $z$ gives the virial expansion $\beta P=n+B_2 n^2+\cdots$ .
The second virial coefficient is a Mayer integral over the pair potential, $B_2=-\tfrac12\int d^3r\,(e^{-\beta u}-1)$ , linking microscopic forces to the measured equation of state.
For the classical ideal gas $z=n\lambda^3$ exactly (no bound $z\le1$ ); for quantum gases $n\lambda^3=g_{3/2}(z)$ (Bose) or $f_{3/2}(z)$ (Fermi), and $z\to1$ marks the onset of Bose degeneracy.
Even the ideal quantum gas carries a statistical exchange virial $B_2=\mp\lambda^3/2^{5/2}$ — attractive for bosons ( $-$ ), repulsive for fermions ( $+$ ) — with no interaction at all.
Key idea: replace $\mu$ by the activity $z=e^{\beta\mu}$ ; then every thermodynamic function is a power series in $z$ whose coefficients are interaction integrals, and $z$ is a rigorous density coordinate because $\partial n/\partial z$ equals the particle-number variance.

The core insight

Replace the chemical potential by the fugacity $z=e^{\beta\mu}$ ; then $\Xi=\sum_N z^N Z_N$ is a power series and every thermodynamic function becomes an expansion in $z$ . Pressure and density are both series in $z$ , and eliminating $z$ gives the virial expansion whose coefficients are interaction integrals. The fugacity is always a valid density coordinate because $\partial n/\partial z=\langle\Delta N^2\rangle/(Vz)>0$ .

Where this lives today

The fugacity is the working variable at both ends of modern many-body physics. In ultracold-atom laboratories, the ideal-quantum-gas relations $n\lambda^3=g_{3/2}(z)$ and $f_{3/2}(z)$ are quantitatively tested: experiments measure the equation of state of trapped bosonic and fermionic gases as a function of $z$ , map the Bose saturation at $\zeta(3/2)$ as the condensation threshold, and trace the crossover to Fermi degeneracy — the polylogarithm curves of the punchline figure are directly reconstructed from absorption images. The interaction virial coefficients $B_2(T)$ , $B_3(T)$ remain the standard bridge from measured $P$ – $V$ – $T$ data to intermolecular potentials, tabulated for gases from argon to methane and used to calibrate the Lennard-Jones and more elaborate potentials that feed molecular simulations. Beyond the ideal gas, the activity expansion launches the whole theory of interacting fluids: the Mayer cluster expansion and its resummations (hypernetted-chain, Percus–Yevick) underpin the statistical mechanics of liquids and colloids. In chemistry the activity $a=z/z^{\ominus}$ generalizes concentration in non-ideal mixtures, and equilibrium constants are products of activities — the microscopic foundation of electrolyte solutions, Debye–Hückel theory, and reactor design. Even adsorption reduces to exact activity expressions: the Langmuir coverage $\theta=z/(1+z)$ is a one-site grand partition function, and adsorbate interactions turn the higher cluster terms into cooperative isotherms.

The physical picture

Picture the grand partition function $\Xi=\sum_N z^N Z_N$ as a generating function in which $z$ is a tag counting particles: the $N$ -particle sector is weighted by $z^N$ . Differentiating $\ln\Xi$ with respect to $\ln z$ pulls down a factor $N$ , so $z\,\partial_z\ln\Xi=\langle N\rangle$ — the fugacity is the dial that reads off the mean occupancy. Turn $z$ up and particles pour in; turn it down and the box empties. The dilute, classical regime is $z\ll1$ , where only the vacuum and single-particle sectors matter and the leading term of every series dominates.

Figure

The fugacity z is the reservoir's dial for particle number. Small z (low μ) leaves the box nearly empty; large z (high μ) floods it. ⟨N⟩ = z ∂(ln Ξ)/∂z reads off the occupancy.

Because $\ln\Xi$ is extensive, the dimensionless pressure $\beta P=\ln\Xi/V$ and the density $n$ are both clean power series in $z$ . The entire art of equation-of-state theory is to compute the cluster integrals $b_\ell$ that are these series' coefficients — each $b_\ell$ an integral of connected diagrams of Mayer functions $f(r)=e^{-\beta u(r)}-1$ — and then invert the density series to express the pressure in powers of $n$ . That inversion is the virial expansion, and its first coefficient $B_2(T)$ is a direct experimental window onto the pair potential $u(r)$ . The same single variable $z$ also sweeps the entire phase diagram of an ideal quantum gas, and this is the punchline. For bosons and fermions the mode sums are geometric/alternating series in $z\,e^{-\beta\varepsilon}$ , so the density becomes a polylogarithm: $n\lambda^3=g_{3/2}(z)$ for bosons, $f_{3/2}(z)$ for fermions, and $z$ for the classical line. The three curves start together at small $z$ (where all reduce to $n\lambda^3=z$ ) and then split dramatically: the Bose curve saturates at $g_{3/2}(1)=\zeta(3/2)\approx2.612$ — the excited states can hold no more, and any extra particles condense into the ground state — while the Fermi curve climbs without bound as the sea fills.

Figure

The degeneracy parameter nλ³ vs fugacity z. Bose (blue) saturates at ζ(3/2) ≈ 2.612 — the BEC onset knee. Fermi (red) climbs without bound. Classical (dashed) is the line nλ³ = z. All three merge as z → 0.

The core equations

The fugacity linearizes the dependence of the statistics on the chemical potential. It is defined as a single exponential:

z \equiv e^{\beta\mu},\qquad \beta=\frac{1}{k_BT} \qquad (1)

where: $z$ — fugacity (activity), dimensionless; $\mu$ — chemical potential; $\beta$ — inverse temperature. Because $\mu$ enters the Gibbs factor as $e^{\beta\mu N}=z^N$ , the fugacity is the object that actually appears in the sums; the chemical potential is recovered by $\mu=k_BT\ln z$ . With $z^N$ pulled out of each particle-number sector, the grand partition function is an ordinary power series whose coefficients are the canonical partition functions:

\Xi(z,V,T)=\sum_{N=0}^{\infty} z^{N}\,Z_N(V,T),\qquad Z_0\equiv 1 \qquad (2)

where: $\Xi$ — grand partition function; $Z_N$ — $N$ -particle canonical partition function; $V$ — volume. This is the master identity: an intractable sum over states of all particle numbers has become a tidy power series in $z$ , with each term weighted by the corresponding closed-system partition function. Taking the logarithm makes the extensive quantities appear. The pressure and the density are both power series in $z$ with the same cluster integrals $b_\ell$ :

\frac{\ln\Xi}{V}=\beta P=\sum_{\ell=1}^{\infty} b_\ell\,z^{\ell},\qquad n=\frac{\langle N\rangle}{V}=z\,\frac{\partial}{\partial z}\!\left(\frac{\ln\Xi}{V}\right)=\sum_{\ell=1}^{\infty}\ell\, b_\ell\,z^{\ell} \qquad (3)

where: $P$ — pressure; $n$ — number density; $b_\ell$ — cluster integral (coefficient of $z^\ell$ in $\beta P$ ). The density carries an extra factor $\ell$ in each term because $z\,\partial_z$ acting on $z^\ell$ returns $\ell z^\ell$ . For the ideal gas only $b_1=1/\lambda^3$ survives, giving $\beta P=z/\lambda^3=n$ , the ideal-gas law, and the exact relation $z=n\lambda^3$ . Eliminating $z$ between the two series of $(3)$ turns the activity expansion into the virial expansion — pressure as a series in density instead of fugacity:

\beta P=n+B_2(T)\,n^2+B_3(T)\,n^3+\cdots,\qquad B_2(T)=-2\pi\int_0^{\infty}\big(e^{-\beta u(r)}-1\big)\,r^2\,\mathrm dr \qquad (4)

where: $B_2(T)$ — second virial coefficient; $u(r)$ — pair potential; $r$ — separation. The virial coefficients are the measurable equation-of-state corrections, and $B_2$ is a direct integral over the pair potential: repulsion ( $u>0$ ) makes $e^{-\beta u}-1<0$ and pushes $B_2>0$ ; attraction pulls $B_2$ down, vanishing at the Boyle temperature. For a quantum ideal gas the density is a polylogarithm of the fugacity, the single equation that interpolates all statistics:

n\lambda^3=\begin{cases} z & \text{Maxwell–Boltzmann}\\[2pt] g_{3/2}(z)=\displaystyle\sum_{\ell\ge1}\frac{z^\ell}{\ell^{3/2}} & \text{Bose}\\[2pt] f_{3/2}(z)=\displaystyle\sum_{\ell\ge1}\frac{(-1)^{\ell+1}z^\ell}{\ell^{3/2}} & \text{Fermi}\end{cases} \qquad (5)

where: $\lambda=h/\sqrt{2\pi m k_BT}$ — thermal de Broglie wavelength; $g_\nu,f_\nu$ — Bose/Fermi polylogarithms. All three agree at small $z$ (where each is just $z$ ); the Bose branch saturates at $g_{3/2}(1)=\zeta(3/2)\approx2.612$ , fixing the condensation threshold, while the Fermi branch grows without bound into the degenerate sea.

Derivation AThe fugacity is always a valid density coordinate — monotonicity from the number variance

This is the article's owned structural result. The activity expansion is only useful if we can invert the density series $n(z)$ to eliminate $z$ in favour of $n$ — that inversion is exactly what produces the virial expansion. The goal is to prove that the map $z\mapsto n$ is strictly monotonic (hence invertible) for any stable system, by showing that its slope $\partial n/\partial z$ equals the particle-number variance divided by $Vz$ , which is manifestly positive. The strategy is to relate $\partial n/\partial z$ to derivatives of $\ln\Xi$ and recognize the variance.

Step 1. Write the mean and the variance as fugacity derivatives. From $(3)$ , $\langle N\rangle=z\,\partial_z\ln\Xi$ . Differentiate once more with the same operator: $\langle\Delta N^2\rangle=\langle N^2\rangle-\langle N\rangle^2=z\,\partial_z\langle N\rangle=z\,\partial_z(z\,\partial_z\ln\Xi)$ . Why: the operator $z\partial_z$ acting on $\ln\Xi$ generates the cumulants of the particle-number distribution — the first application gives the mean, the second gives the variance, exactly as $\partial_\beta$ on $\ln Z$ generates the energy cumulants.

Step 2. Differentiate the density with respect to $z$ . Since $n=\langle N\rangle/V=(z\,\partial_z\ln\Xi)/V$ , we have $\dfrac{\partial n}{\partial z}=\dfrac{1}{V}\,\partial_z\big(z\,\partial_z\ln\Xi\big)$ . Why: $n$ is just $\langle N\rangle$ per unit volume, so its $z$ -derivative is the $z$ -derivative of $\langle N\rangle$ scaled by $1/V$ .

Step 3. Compare the two expressions. The variance from Step 1 is $\langle\Delta N^2\rangle=z\,\partial_z\big(z\,\partial_z\ln\Xi\big)$ , which is exactly $z$ times what appears in Step 2. Therefore $\dfrac{\partial n}{\partial z}=\dfrac{1}{V}\cdot\dfrac{\langle\Delta N^2\rangle}{z}=\dfrac{\langle\Delta N^2\rangle}{Vz}$ . Why: the only difference between the variance and $V\,\partial_z n$ is a single factor of $z$ , which the algebra makes explicit.

Step 4. Read off the conclusion. A variance is non-negative, and $V,z>0$ , so $\partial n/\partial z\ge0$ , with strict positivity whenever the number fluctuates at all. Hence $n(z)$ is a strictly increasing function and can always be inverted to give $z(n)$ — the fugacity is a legitimate coordinate for the density, and the activity series can always be re-summed into a density (virial) series. Why: strict monotonicity is precisely the condition for a smooth, single-valued inverse; the number variance being positive (thermodynamic stability, $\kappa_T>0$ ) is what guarantees it. *Verified: $\partial n/\partial z=\langle\Delta N^2\rangle/(Vz)$ confirmed symbolically ( $\langle\Delta N^2\rangle=z\partial_z\langle N\rangle$ , $\langle N\rangle=z\partial_z\ln\Xi$ ) in verify_aq-sm-fugacity-activity-expansion.py.*

Go deeper · Derivation B · 1 minFrom the grand sum to the cluster series, and the second virial coefficient

Derivation BFrom the grand sum to the cluster series, and the second virial coefficient

The goal is to show that $\ln\Xi$ is a genuine power series in $z$ whose coefficients $b_\ell$ are intensive cluster integrals, and to compute the first non-trivial one, $b_2$ , obtaining the second virial coefficient as a Mayer integral over the pair potential. The strategy is to expand the logarithm of the grand sum, identify the connected-cluster subtraction, and evaluate the two-particle cluster.

Step 1. Expand the logarithm. With $\Xi=1+zZ_1+z^2Z_2+\cdots$ , the series for the logarithm is $\ln\Xi=zZ_1+z^2\big(Z_2-\tfrac12 Z_1^2\big)+\cdots$ . Why: $\ln(1+x)=x-\tfrac12 x^2+\cdots$ with $x=zZ_1+z^2Z_2$ ; the $-\tfrac12 Z_1^2$ term subtracts the disconnected (reducible) part — this is the origin of "connected clusters."

Step 2. Define the cluster integrals by extensivity. Write $Vb_1=Z_1$ and $Vb_2=Z_2-\tfrac12 Z_1^2$ . Extensivity guarantees each bracket is proportional to $V$ , so the $b_\ell$ are intensive. For the ideal gas $Z_1=V/\lambda^3$ gives $b_1=1/\lambda^3$ . Why: only connected clusters scale as a single power of $V$ ; the subtraction removes the $V^2$ piece of $Z_1^2$ , leaving $Vb_2$ .

Step 3. Evaluate the two-particle cluster. With $Z_2=\tfrac{1}{2\lambda^6}\int d^3r_1\,d^3r_2\,e^{-\beta u(r_{12})}$ and $\tfrac12 Z_1^2=\tfrac{V^2}{2\lambda^6}$ , the difference is $Vb_2=\tfrac{1}{2\lambda^6}\int d^3r_1\,d^3r_2\,(e^{-\beta u}-1)$ . Integrating out the centre of mass gives a factor $V$ , so $b_2=\tfrac{1}{2\lambda^6}\int d^3r\,f(r)$ with the Mayer function $f(r)=e^{-\beta u(r)}-1$ . Why: subtracting the "1" replaces the bare Boltzmann factor by the Mayer function, which vanishes wherever the particles do not interact, making the integral converge.

Step 4. Convert to the virial coefficient. Inverting $n(z)$ to first order and substituting into $\beta P$ gives $B_2=-b_2\lambda^6=-\tfrac12\int d^3r\,f(r)=-2\pi\int_0^\infty(e^{-\beta u}-1)r^2\,dr$ . For hard spheres of diameter $\sigma$ ( $u=\infty$ for $r<\sigma$ , else $0$ ), $f=-1$ inside and $B_2=\tfrac23\pi\sigma^3$ — four times the molecular volume. Why: the elimination of $z$ carries the sign flip $B_2=-b_2\lambda^6$ ; the hard-sphere case is the cleanest check because the Mayer function is a step and the integral is elementary. *Verified: $\ln\Xi$ coefficients $b_1=1/\lambda^3$ and $Vb_2=Z_2-\tfrac12 Z_1^2$ , plus hard-sphere $B_2=\tfrac23\pi\sigma^3$ , confirmed symbolically in the verify script.*

Limiting cases

The fugacity $z$ smoothly tracks the physical regime of the gas:

Case	Result	Meaning
Classical dilute gas	$z=n\lambda^3\ll1$	Maxwell–Boltzmann; the leading activity term dominates, corrections are small interaction effects. No bound on $z$ .
Ideal Bose gas at condensation	$z\to1^-$ , $n\lambda^3\to\zeta(3/2)\approx2.612$	The fugacity saturates at $1$ ; excited states hold no more and the excess condenses into the ground state.
Degenerate Fermi gas	$z\gg1$ , $\ln z\approx\beta E_F$	Large fugacity signals a filled Fermi sea; the Sommerfeld expansion in $1/\ln z$ replaces the activity series.
Empty system	$z\to0$ , $\Xi\to1$ , $\langle N\rangle\to zZ_1$	Only vacuum and single-particle sectors matter; the expansion is exact to first order.
Quantum exchange (weak degeneracy)	$B_2=\mp\lambda^3/2^{5/2}$	Ideal gas already has a virial correction: attractive for bosons, repulsive for fermions, purely statistical.

Common trapDo not assume the fugacity is always less than one, or confuse z with the partition function Z

Two traps. First, $z\le1$ is not a general law — it is special to the ideal Bose gas, where it enforces non-negative occupation of every mode ( $\langle n_k\rangle=z e^{-\beta\varepsilon_k}/(1-z e^{-\beta\varepsilon_k})\ge0$ requires $z<1$ ). For a classical ideal gas $z=n\lambda^3$ climbs freely above $1$ at high density or low temperature with no pathology whatsoever, and for a Fermi gas $z=e^{\beta E_F/k_BT}$ becomes enormous in the degenerate limit. Assuming $z\le1$ everywhere is a common and serious error. Second, the fugacity $z$ (lowercase, the activity $e^{\beta\mu}$ ) is a completely different object from the partition function $Z$ (uppercase); they are related only through $\Xi=\sum_N z^N Z_N$ . Third, the activity expansion is an asymptotic low-density series: truncating it near condensation, near close-packing, or in the degenerate regime gives nonsense. Use it where $z$ is small, and switch to the polylogarithm or Sommerfeld machinery where it is not.

Worked examples

1 · The statistical exchange virial of a quantum ideal gas (a real correction from statistics alone)

An ideal Bose or Fermi gas has no interparticle forces, yet quantum statistics already produces a second virial coefficient. Compute $B_2$ for each, and evaluate its size for helium-4 at $T=1\,\text{K}$ .

Step 1. Get the two leading terms of the density and pressure polylogarithms. From $(5)$ , $n\lambda^3=z\pm\dfrac{z^2}{2^{3/2}}+\cdots$ and $\beta P\lambda^3=z\pm\dfrac{z^2}{2^{5/2}}+\cdots$ , where the upper sign is Bose, the lower is Fermi (each $\ell$ -th term carries $(\pm1)^{\ell+1}/\ell^{3/2}$ or $/\ell^{5/2}$ ).

Step 2. Invert the density series for $z(n)$ . To the needed order, $z=n\lambda^3\mp\dfrac{(n\lambda^3)^2}{2^{3/2}}+\cdots$ .

Step 3. Substitute into $\beta P$ . Collecting the $n^2$ term gives $\beta P=n+B_2 n^2+\cdots$ with $\mathbf{B_2=\mp\dfrac{\lambda^3}{2^{5/2}}}$ — negative (attractive) for bosons, positive (repulsive) for fermions. The magnitude is $\lambda^3/2^{5/2}=0.1768\,\lambda^3$ .

Step 4. Put in numbers for helium-4 at $1\,\text{K}$ . With $m=6.646\times10^{-27}\,\text{kg}$ , the thermal wavelength is $\lambda=h/\sqrt{2\pi m k_BT}\approx0.87\,\text{nm}$ , so $\lambda^3\approx6.6\times10^{-28}\,\text{m}^3$ and $\mathbf{B_2^{\text{Bose}}=-\lambda^3/2^{5/2}\approx-1.2\times10^{-28}\,\text{m}^3}$ . Bosons behave as if weakly attracted (they "bunch"), fermions as if weakly repelled (they "anti-bunch"), purely from the exchange symmetry of the wavefunction — a measurable correction to the ideal-gas law with no potential in the Hamiltonian.

2 · Hard-sphere gas and the Boyle temperature (exam level)

Compute the second virial coefficient of a hard-sphere gas of diameter $\sigma$ , and explain why a real gas with attraction has a temperature where $B_2=0$ .

Step 1. For hard spheres, $u(r)=\infty$ for $r<\sigma$ and $0$ for $r>\sigma$ , so the Mayer function is $f(r)=e^{-\beta u}-1=-1$ for $r<\sigma$ and $0$ beyond.

Step 2. Evaluate the integral: $B_2=-2\pi\int_0^\infty f(r)\,r^2\,dr=-2\pi\int_0^\sigma(-1)\,r^2\,dr=2\pi\cdot\dfrac{\sigma^3}{3}=\dfrac{2\pi\sigma^3}{3}$ . This is $\mathbf{B_2=\tfrac23\pi\sigma^3}$ , exactly four times the volume of one sphere $\tfrac16\pi\sigma^3$ — the excluded-volume repulsion raises the pressure.

Step 3. Add attraction. A real potential (a hard core plus a $-\varepsilon$ attractive well, like square-well or Lennard-Jones) contributes a positive piece to $\int f\,d^3r$ from the region where $u<0$ (there $e^{-\beta u}-1>0$ ), which subtracts from $B_2$ . At low $T$ the attraction dominates and $B_2<0$ ; at high $T$ the hard core wins and $B_2>0$ .

Step 4. The crossover is the Boyle temperature $T_B$ , where $B_2(T_B)=0$ and the gas mimics ideal behaviour over an extended pressure range. Measuring $B_2(T)$ and locating $T_B$ is a direct experimental probe of the pair potential $u(r)$ — the whole point of the activity-to-virial machinery is that these coefficients are computable integrals over the microscopic forces.

Origins · 1 minfugacity and how this was discovered

fugacity

physical-chemistry concept, Lewis 1901 · escaping tendency → z = e^(βμ)

The fugacity was introduced by the American chemist Gilbert Newton Lewis in 1901, who was troubled that the chemical potential of a real gas has no finite reference value and diverges logarithmically as pressure goes to zero. He defined the fugacity as an "escaping tendency" — an effective pressure, engineered so that a real gas obeys the same thermodynamic relations as an ideal one when written in terms of $z$ instead of $P$ . In the dilute limit the fugacity equals the pressure; departures measure non-ideality, quantified by the fugacity coefficient $\phi=z/P$ still used throughout chemical engineering. Statistical mechanics then revealed the deeper identity: the fugacity is $z=e^{\beta\mu}$ , the natural expansion variable of Gibbs's grand ensemble, in which $\Xi=\sum_N z^N Z_N$ is a power series. What Lewis introduced as a practical accounting device turned out to be the exact control parameter for the equation of state — organizing the Mayer cluster expansion and, through the polylogarithms $g_\nu(z)$ and $f_\nu(z)$ , the entire crossover from Maxwell–Boltzmann gases to Bose condensation and Fermi degeneracy. The same activity reappears in solution thermodynamics and reaction equilibria, replacing concentrations in the law of mass action.

History

1901

— Lewis introduces the fugacity as an "escaping tendency" for real-gas thermodynamics.

1901

— Kamerlingh Onnes writes the empirical virial equation of state as a power series in density.

1907

— Lewis introduces the closely related concept of activity for solutions and mixtures.

1927

— Ursell expresses the cluster expansion of the classical partition function in connected graphs.

1937

— Mayer develops the systematic cluster/virial theory with the Mayer f-function.

Going further

References. Pathria & Beale, Statistical Mechanics, 3rd/4th ed., Ch. 4 (grand ensemble) and Ch. 9–10 (cluster & virial expansions) · Kardar, Statistical Physics of Particles, Ch. 5 (interacting particles, cumulant/cluster expansion) · Reif, Fundamentals of Statistical and Thermal Physics, Ch. 9 · Sethna, Statistical Mechanics, Ch. 6 · Landau & Lifshitz, Statistical Physics Part 1, §§35–37, 74–75 · Mayer & Mayer, Statistical Mechanics (original cluster expansion).

Provenance. The activity/cluster expansion follows Pathria & Beale (Ch. 4, 10) and Kardar (Ch. 5). A verifier should reproduce: $\ln\Xi=zZ_1+z^2(Z_2-\tfrac12 Z_1^2)+\cdots$ from the logarithm of the series; the owned monotonicity $\partial n/\partial z=\langle\Delta N^2\rangle/(Vz)$ ; $B_2=-\tfrac12\int d^3r\,(e^{-\beta u}-1)$ and the hard-sphere value $\tfrac23\pi\sigma^3$ ; the ideal-gas relation $z=n\lambda^3$ ; the polylogarithm density $n\lambda^3=g_{3/2}(z)$ (Bose) / $f_{3/2}(z)$ (Fermi) with $g_{3/2}(1)=\zeta(3/2)\approx2.612$ and $f_{3/2}(1)=\eta(3/2)\approx0.7651$ ; and the quantum exchange virial $B_2=\mp\lambda^3/2^{5/2}=\mp0.1768\,\lambda^3$ (helium-4 at $1\,\text{K}$ : $\lambda\approx0.87\,\text{nm}$ , $B_2^{\text{Bose}}\approx-1.2\times10^{-28}\,\text{m}^3$ ). Numbers recomputed in verify_aq-sm-fugacity-activity-expansion.py. $h$ , $k_B$ (CODATA).

Builds on: Grand Canonical Ensemble · Grand Partition Function · Gibbs Factor and Chemical Potential · Canonical Partition Function · Boltzmann Factor. Leads to: Virial Expansion · Mayer Cluster Expansion · Ideal Bose and Fermi Gases · Thermodynamic Quantities from the Partition Function.

What is the difference between the fugacity z and the partition function Z?

They are entirely different objects that happen to share a letter. The fugacity $z=e^{\beta\mu}$ (lowercase) is a single number, the exponentiated chemical potential — the reservoir's "escaping tendency" that dials particle number. The partition function $Z$ (uppercase) is a sum over states that encodes the thermodynamics of a closed system at fixed particle number. They are connected only through the grand partition function $\Xi=\sum_N z^N Z_N$ , where $z$ is the expansion variable and each $Z_N$ is a coefficient. So $z$ tells you how strongly the reservoir pushes particles in, and $Z_N$ tells you the internal statistics once $N$ particles are present. Confusing the two — for instance treating $z$ as if it summed over states, or expecting $Z$ to be dimensionless like $z$ — is a classic slip. A good mnemonic: little $z$ is the little knob (the activity), big $Z$ is the big sum.

Why is z the "right" expansion variable rather than the chemical potential μ itself?

Because $\mu$ enters the statistics inside an exponential and ranges over the whole real line, so an expansion in $\mu$ is neither natural nor convergent. The Gibbs factor weights the $N$ -particle sector by $e^{\beta\mu N}=z^N$ , so with $z=e^{\beta\mu}$ the grand partition function becomes an ordinary power series $\Xi=\sum_N z^N Z_N$ — polynomial structure that you can expand, differentiate, and invert. In the dilute limit $z$ is small (for the ideal gas $z=n\lambda^3$ ), so the series converges rapidly and truncating it gives systematic corrections. Working directly with $\mu$ would mean expanding $e^{\beta\mu N}$ itself, reintroducing the exponentials you were trying to organize. The fugacity is the change of variable that linearizes the particle-number dependence, which is exactly what makes the cluster and virial expansions possible.

How can an ideal gas have a non-zero second virial coefficient if there are no forces?

This is the striking lesson of quantum statistics. Even with no interaction potential in the Hamiltonian, indistinguishable particles are correlated by the symmetry of the many-body wavefunction: bosons are more likely to be found close together ("bunching"), fermions less likely ("anti-bunching," from the Pauli principle). When you expand the ideal quantum gas equation of state $n\lambda^3=g_{3/2}(z)$ (Bose) or $f_{3/2}(z)$ (Fermi) and eliminate $z$ , the second term is a genuine virial correction, $B_2=\mp\lambda^3/2^{5/2}$ — negative (effectively attractive) for bosons and positive (effectively repulsive) for fermions. It has the size of the thermal volume $\lambda^3$ , so it matters when the gas approaches degeneracy ( $n\lambda^3\sim1$ ). This "statistical" or "exchange" virial is not a real force; it is the thermodynamic footprint of wavefunction symmetry, and it is the first hint of the dramatic differences (BEC, Fermi pressure) that appear when quantum statistics fully takes over.

Why does the Bose fugacity get stuck at z = 1 while the classical one does not?

For the ideal Bose gas the mean occupation of the ground state is $\langle n_0\rangle=z/(1-z)$ , which is only finite and non-negative for $z<1$ ; as $z\to1^-$ it diverges. The density carried by the excited states is $n\lambda^3=g_{3/2}(z)$ , which is a bounded, increasing function that reaches its maximum $g_{3/2}(1)=\zeta(3/2)\approx2.612$ exactly at $z=1$ . So if you keep raising the density (or lowering the temperature) past $n\lambda^3=2.612$ , the excited states physically cannot absorb the extra particles — the fugacity pins at $z=1$ and the surplus condenses into the ground state. That pinning is Bose–Einstein condensation. For a classical gas there is no such ceiling: $z=n\lambda^3$ is exact and unbounded, because Maxwell–Boltzmann statistics places no restriction like "non-negative mode occupation," so $z$ sails past 1 without anything happening. The bound $z\le1$ is a purely bosonic phenomenon.

What is the Boyle temperature, and how does it relate to B_2?

The Boyle temperature $T_B$ is the temperature at which the second virial coefficient vanishes, $B_2(T_B)=0$ . Since $\beta P=n+B_2 n^2+\cdots$ , when $B_2=0$ the first correction to the ideal-gas law disappears and the gas behaves ideally over an extended range of pressures — hence "Boyle," after the ideal-gas law $PV=\text{const}$ . Physically, $B_2$ is a competition: the hard-core repulsion contributes a positive (excluded-volume) piece, while the attractive part of the potential contributes a negative piece. At low temperature the attraction wins and $B_2<0$ ; at high temperature the fast-moving molecules barely feel the attraction and the repulsion wins, $B_2>0$ . At exactly $T_B$ they cancel. Measuring $T_B$ is therefore a clean experimental probe of the balance point of the pair potential, and it is one of the standard data points used to fit intermolecular potential models.

Is the activity expansion the same as the virial expansion?

They are two sides of the same computation in different variables. The activity expansion writes thermodynamic quantities as power series in $z=e^{\beta\mu}$ : $\beta P=\sum_\ell b_\ell z^\ell$ and $n=\sum_\ell \ell b_\ell z^\ell$ , with the cluster integrals $b_\ell$ coming straight out of $\ln\Xi$ . The virial expansion writes pressure as a series in density, $\beta P=n+B_2 n^2+\cdots$ . You get the virial form by inverting $n(z)$ to obtain $z(n)$ and substituting into $\beta P$ — always possible because $n(z)$ is monotonic (this article's owned result). The activity form is more fundamental (the $b_\ell$ are connected clusters); the virial form is what experiments measure, since density and pressure are observable and $\mu$ is not. Both encode the same physics.

Why does taking the logarithm of the grand sum matter for the cluster expansion?

Because $\ln\Xi$ , not $\Xi$ , is the extensive quantity ( $\ln\Xi=\beta PV$ ), and its expansion coefficients are the intensive cluster integrals. When you expand $\ln(1+zZ_1+z^2Z_2+\cdots)$ the logarithm automatically subtracts the "disconnected" pieces: the $z^2$ coefficient is $Z_2-\tfrac12 Z_1^2$ , not $Z_2$ , and that $-\tfrac12 Z_1^2$ removes the reducible part where two particles are independently placed. What survives is the connected two-particle cluster, which scales as a single power of the volume and is therefore intensive per unit volume. This connected-cluster structure is exactly what makes the virial coefficients finite and computable: each $b_\ell$ is an integral over connected diagrams of Mayer functions, and the Mayer functions vanish when particles are far apart, so the integrals converge. Working with $\Xi$ itself would keep the disconnected terms and give coefficients that grow with $V$ — useless. The logarithm is the operation that turns the grand sum into a well-behaved cluster series.