Fugacity and Chemical Potential in Quantum Gases

Every ideal quantum gas — the electrons in a metal, the photons in a furnace, the ultracold rubidium atoms in a magnetic trap — is governed by one dimensionless dial. Fix the temperature $T$ and the density $n$ , and the physics is decided by a single number, the fugacity $z=e^{\beta\mu}$ , where $\mu$ is the chemical potential and $\beta=1/k_BT$ . The fugacity is the reservoir's "price of admission": each particle you add multiplies the grand-canonical weight by $z$ , exactly as each unit of energy multiplies it by $e^{-\beta\varepsilon}$ . Because $z$ enters every single-particle level only through the combination $z\,e^{-\beta\varepsilon}$ , it is the natural variable in which the whole theory of quantum gases is written. The power of the fugacity is that it turns three separate stories — the classical Maxwell-Boltzmann gas, the Bose gas that condenses, the Fermi gas that resists compression — into one continuous story parametrized by $z$ . When $z\ll1$ occupations are tiny, statistics are irrelevant, and $z\to n\lambda_T^3$ : the fugacity is the degeneracy parameter (the thermal de Broglie wavelength $\lambda_T$ is owned by aq-sm-thermal-de-broglie-wavelength-sm4, and the crossover number $n\lambda_T^3$ by aq-sm-quantum-concentration-classicality-criterion). As $z$ climbs toward one, the gas becomes degenerate and Bose and Fermi statistics part ways decisively. Statistics constrain $z$ in a way that encodes the two great low-temperature phenomena. For fermions any $z\in(0,\infty)$ is legal; a deeply degenerate gas has $\mu\approx\varepsilon_F>0$ and $z\to\infty$ . For bosons the ground-state occupation $z/(1-z)$ must stay finite, forcing $z\le1$ — and the boundary $z\to1$ is precisely where Bose-Einstein condensation sets in. This article owns the fugacity as the classical-to-quantum control knob, and the density–fugacity relation $n\lambda_T^3=g_{3/2}(z)$ that inverts to give $\mu(n,T)$ across every regime. It takes the grand-canonical machinery and the individual distributions from their owners and shows how one variable ties them together.

Key takeaways

The fugacity $z=e^{\beta\mu}$ is the dimensionless control variable conjugate to particle number: the grand-canonical weight of a level enters only through $z\,e^{-\beta\varepsilon}$ , so $z$ tunes the occupation of every mode at once.
The classical limit is $z\ll1$ , where $z\to n\lambda_T^3$ — the fugacity literally equals the degeneracy parameter, and $\mu=k_BT\ln(n\lambda_T^3)<0$ .
Statistics fix the range: $0<z<\infty$ for fermions ( $\mu\to\varepsilon_F>0$ , $z\to\infty$ degenerate), but $0<z\le1$ for bosons because $\mu\le0$ ; reaching $z=1$ triggers Bose-Einstein condensation.
Number and pressure are encoded by the Bose/Fermi functions $g_\nu(z)=\sum_{\ell\ge1}z^\ell/\ell^\nu$ and $f_\nu(z)=\sum_{\ell\ge1}(-1)^{\ell-1}z^\ell/\ell^\nu$ ; the master relation is $n\lambda_T^3=g_{3/2}(z)$ (Bose) or $f_{3/2}(z)$ (Fermi).
The Bose ceiling $g_{3/2}(1)=\zeta(3/2)\approx2.612$ is the BEC threshold; the identity $z\,\partial_z g_\nu=g_{\nu-1}$ generates all thermodynamics.
Key idea: the fugacity $z=e^{\beta\mu}$ enters the grand-canonical sum only through $z\,e^{-\beta\varepsilon}$ per mode, so inverting the single relation $n\lambda_T^3=g_{3/2}(z)$ or $f_{3/2}(z)$ converts the degeneracy parameter $n\lambda_T^3$ into the chemical potential $\mu(n,T)$ across the classical, condensed, and degenerate regimes alike.

The core insight

The chemical potential is an awkward variable to carry through every formula; the fugacity $z=e^{\beta\mu}$ is its natural exponential form. Because a level of energy $\varepsilon$ feels $\mu$ only through $z\,e^{-\beta\varepsilon}$ , the entire equation of state collapses onto functions of the single number $z$ . Fixing $n$ and $T$ fixes $z$ through $n\lambda_T^3=g_{3/2}(z)$ (bosons) or $f_{3/2}(z)$ (fermions), and $\mu=k_BT\ln z$ follows — one dial for classical, condensed, and degenerate gases.

Where this lives today

The fugacity is the daily working variable of ultracold-atom physics. Experimenters measure the density and temperature of a trapped gas, compute $n\lambda_T^3$ , and invert $g_{3/2}(z)$ or $f_{3/2}(z)$ to extract $z$ — and hence the chemical potential, the equation of state, and the distance to condensation or degeneracy. The pinning of $z$ at one and the growth of the condensate fraction $1-(T/T_c)^{3/2}$ are the textbook signatures used to confirm Bose-Einstein condensation, first achieved in dilute rubidium and sodium in 1995. In condensed matter the same variable organizes semiconductor carrier statistics: electron and hole densities are Fermi functions $f_{1/2}(z)$ of the local fugacity, reducing to the Boltzmann law $n\propto e^{\mu/k_BT}$ (that is, $z\ll1$ ) in lightly doped material and requiring the full large- $z$ treatment in degenerate, heavily doped semiconductors and metals. Wherever a gas of identical quantum particles reaches equilibrium, the fugacity is the knob that says how classical or how quantum it has become.

The physical picture

Picture the reservoir as a shopkeeper quoting a price $\mu$ to add one particle. The grand-canonical weight of a configuration with $N$ particles and energy $E$ is $z^N e^{-\beta E}$ , so $z=e^{\beta\mu}$ is the "discount factor" per particle, the exact analog of the Boltzmann factor $e^{-\beta\varepsilon}$ per unit of energy. A level at energy $\varepsilon$ therefore sees only the product $z\,e^{-\beta\varepsilon}$ : raise the price (larger $z$ ) or lower the level (smaller $\varepsilon$ ) and the level fills up. That single product is the occupation variable of quantum statistics. The first figure is the hero: the fugacity dial. As $z$ sweeps from $0$ to $1$ (and beyond, for fermions), the gas passes from the classical Maxwell-Boltzmann regime, through the onset of degeneracy, to the two divergent fates — a Bose gas that hits the condensation wall at $z=1$ , and a Fermi gas that sails on to $z\to\infty$ .

Figure

The fugacity dial from classical to quantum

The punchline: bosons can never pass $z=1$ — they run into the condensation wall — while fermions ignore it and keep going. The same dial, read two ways, contains both of the deep low-temperature phenomena.

Because $z$ multiplies the Boltzmann factor of every level uniformly, the whole crossover is governed by one number. Raising the temperature at fixed density shrinks $\lambda_T$ , drives $n\lambda_T^3$ down, and pushes $z\to0$ ; cooling inflates $\lambda_T^3$ , drives the gas toward degeneracy, and pushes $z$ toward $1$ (bosons) or toward large values (fermions). The second figure shows the quantitative version of the dial: the density–fugacity curve $n\lambda_T^3=g_{3/2}(z)$ for bosons, which saturates at the finite value $\zeta(3/2)$ as $z\to1$ , versus the Fermi curve $f_{3/2}(z)$ , which keeps rising without bound.

Figure

Density versus fugacity: the Bose ceiling and the unbounded Fermi curve

Near the origin both curves start as $n\lambda_T^3\approx z$ : the classical limit, where fugacity and degeneracy parameter coincide. They separate only once $z$ approaches one.

The core equations

The fugacity is defined as the exponential of the reduced chemical potential. It is dimensionless, and inverting it recovers $\mu$ .

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

where: $z$ — fugacity (dimensionless); $\mu$ — chemical potential; $\beta$ — inverse temperature. Physical reading: $z$ is the multiplicative weight per particle in the grand sum, the particle-number analog of the Boltzmann factor $e^{-\beta\varepsilon}$ for energy. When $\mu<0$ (dilute gas) $z<1$ ; when $\mu>0$ (degenerate fermions) $z>1$ . The grand partition function factorizes over independent modes — a result owned by aq-sm-grand-canonical-ideal-quantum-gases — and each mode's factor depends on $\mu$ only through $z\,e^{-\beta\varepsilon}$ .

\mathcal{Z}=\prod_{\varepsilon}\big(1\mp z\,e^{-\beta\varepsilon}\big)^{\mp1}, \qquad \ln\mathcal{Z}=\mp\sum_\varepsilon\ln\big(1\mp z\,e^{-\beta\varepsilon}\big) \qquad (2)

where: the upper sign is Bose, the lower sign Fermi; the product runs over single-particle levels $\varepsilon$ . Physical reading: the whole thermodynamics is packed into $\ln\mathcal{Z}$ , and $z$ appears everywhere in the pair $z\,e^{-\beta\varepsilon}$ — this is why a single variable suffices. For bosons each factor requires $z\,e^{-\beta\varepsilon}<1$ , the seed of the $z\le1$ constraint. Differentiating $\ln\mathcal{Z}$ with respect to $\ln z$ gives the mean occupation of a level. This is the single expression from which aq-sm-bose-einstein-distribution and aq-sm-fermi-dirac-distribution are read off; here it shows the fugacity's role explicitly.

\langle n_\varepsilon\rangle=\frac{z\,e^{-\beta\varepsilon}}{1\mp z\,e^{-\beta\varepsilon}}=\frac{1}{z^{-1}e^{\beta\varepsilon}\mp1} \qquad (3)

where: $\langle n_\varepsilon\rangle$ — mean occupation; upper sign Bose, lower Fermi. Physical reading: with $z=e^{\beta\mu}$ the factor $z^{-1}e^{\beta\varepsilon}=e^{\beta(\varepsilon-\mu)}$ , recovering the familiar $1/(e^{\beta(\varepsilon-\mu)}\mp1)$ . The Bose $-1$ can drive $\langle n\rangle\to\infty$ ; the Fermi $+1$ caps it at $1$ . Summing the occupation over all levels, converted to a continuum integral with the free-particle density of states, produces the master relation of this article: the density expressed as a Bose or Fermi function of $z$ .

\frac{N}{V}\lambda_T^3=g_{3/2}(z)\ \text{(Bose)},\qquad \frac{N}{V}\lambda_T^3=f_{3/2}(z)\ \text{(Fermi)} \qquad (4)

where: $\lambda_T=h/\sqrt{2\pi m k_BT}$ — thermal de Broglie wavelength; $g_\nu,f_\nu$ — the Bose/Fermi functions below. Physical reading: the left side is exactly the degeneracy parameter $n\lambda_T^3$ . Invert this one implicit equation and you have $z(n,T)$ , hence $\mu(n,T)$ — the whole point of the fugacity formalism. The Bose and Fermi functions are the special functions that organize every quantum-gas thermodynamic quantity. They are polylogarithms in disguise.

g_\nu(z)=\sum_{\ell=1}^{\infty}\frac{z^\ell}{\ell^\nu}=\mathrm{Li}_\nu(z),\qquad f_\nu(z)=\sum_{\ell=1}^{\infty}\frac{(-1)^{\ell-1}z^\ell}{\ell^\nu}=-\mathrm{Li}_\nu(-z) \qquad (5)

where: $\nu$ — order ( $3/2$ for number, $5/2$ for pressure in 3D); $\mathrm{Li}_\nu$ — polylogarithm. Physical reading: both start as $\approx z$ for small $z$ (the classical term), and the recursion $z\,\partial_z g_\nu(z)=g_{\nu-1}(z)$ (same for $f$ ) lets you climb between orders — it is the fugacity-derivative that turns $N$ into fluctuations and thermodynamic derivatives. The pressure has the same form one order higher, and the Bose function saturates at a finite value that fixes the condensation threshold.

\frac{PV}{k_BT}=\ln\mathcal{Z}=\frac{gV}{\lambda_T^3}\begin{cases}g_{5/2}(z)\\[2pt] f_{5/2}(z)\end{cases},\qquad g_{3/2}(1)=\zeta\!\left(\tfrac32\right)\approx2.612 \qquad (6)

where: $g$ — spin/polarization degeneracy; $\zeta$ — Riemann zeta. Physical reading: because $g_{3/2}$ is bounded on $(0,1]$ (it converges since $3/2>1$ ), a Bose gas cannot accommodate more than $\zeta(3/2)\,V/\lambda_T^3$ particles in its excited states — the surplus condenses. The Fermi $f_{3/2}(z)$ has no such ceiling.

Derivations

Derivation AThe density is a Bose or Fermi function of the fugacity

The goal is to derive the master relation $n\lambda_T^3=g_{3/2}(z)$ (bosons) or $f_{3/2}(z)$ (fermions) starting from the mode sum $N=\sum_\varepsilon\langle n_\varepsilon\rangle$ , converting it to a continuum integral over the free-particle density of states, and expanding the occupation in powers of the fugacity. The strategy is to trade the awkward denominator $z^{-1}e^{\beta\varepsilon}\mp1$ for a geometric series in $z\,e^{-\beta\varepsilon}$ , integrate term by term against $\sqrt\varepsilon$ , and recognize the resulting sum as the defining series of $g_\nu$ or $f_\nu$ . Watch the convergence condition $z\,e^{-\beta\varepsilon}<1$ , which for the ground state $\varepsilon\to0$ becomes the constraint $z<1$ for bosons.

Step 1. Convert the sum to an integral. Using the 3D free-particle density of states $g(\varepsilon)=\frac{gV}{4\pi^2}\left(\frac{2m}{\hbar^2}\right)^{3/2}\sqrt\varepsilon$ (owned by aq-sm-density-of-states-free-particles), write $N=\int_0^\infty g(\varepsilon)\langle n_\varepsilon\rangle\,\mathrm{d}\varepsilon$ . Substituting $x=\beta\varepsilon$ collapses the prefactors into $\lambda_T^3$ : $N=\frac{gV}{\lambda_T^3}\frac{2}{\sqrt\pi}\int_0^\infty\frac{x^{1/2}\,\mathrm{d}x}{z^{-1}e^{x}\mp1}$ . Why: the temperature dependence in $g(\varepsilon)$ and in $\beta^{3/2}$ combine exactly into the thermal de Broglie wavelength cubed, leaving a pure dimensionless integral.

Step 2. Expand the occupation as a geometric series. Since $z\,e^{-x}<1$ , $\dfrac{1}{z^{-1}e^{x}\mp1}=\dfrac{z\,e^{-x}}{1\mp z\,e^{-x}}=\sum_{\ell\ge1}(\pm1)^{\ell-1}z^\ell e^{-\ell x}$ . Why: the sign in the denominator threads through the geometric expansion — bosons ( $-$ ) give all-positive terms, fermions ( $+$ ) give alternating terms, and this single sign becomes the difference between $g_\nu$ and $f_\nu$ .

Step 3. Integrate term by term. Using $\frac{2}{\sqrt\pi}\int_0^\infty x^{1/2}e^{-\ell x}\,\mathrm{d}x=\ell^{-3/2}$ (from $\Gamma(3/2)=\sqrt\pi/2$ ), the integral becomes $\sum_{\ell\ge1}(\pm1)^{\ell-1}z^\ell\ell^{-3/2}$ . Why: each exponential integrates to a clean power of $\ell$ , and the sum over $\ell$ is exactly the defining series of the Bose ( $g_{3/2}$ ) or Fermi ( $f_{3/2}$ ) function.

Step 4. Read off the master relation. Collecting, $\dfrac{N}{V}\lambda_T^3=g_{3/2}(z)$ for bosons and $f_{3/2}(z)$ for fermions (with $g=1$ ). Why: the left side is the degeneracy parameter $n\lambda_T^3$ ; inverting this equation numerically gives $z(n,T)$ and then $\mu=k_BT\ln z$ — the fugacity has done its job of packaging the thermodynamics into one special function of one variable. Verified: the geometric expansion, the Gamma-function integral $\frac{2}{\sqrt\pi}\int x^{1/2}e^{-\ell x}dx=\ell^{-3/2}$ , and the numerical equality of the full Bose/Fermi density integrals with $g_{3/2}(z)$ / $f_{3/2}(z)$ at $z=0.2,0.5,0.9$ are checked in verify_fugacity-chemical-potential-quantum-gases.py.

Go deeper · Derivation B · 1 minThe classical limit: the fugacity equals the degeneracy parameter

Derivation BThe classical limit: the fugacity equals the degeneracy parameter

The goal is to prove that in the dilute, high-temperature regime the fugacity reduces to the degeneracy parameter, $z\to n\lambda_T^3$ , and hence $\mu=k_BT\ln(n\lambda_T^3)$ — the classical Maxwell-Boltzmann chemical potential, strongly negative when the gas is dilute. This is the owned structural result that anchors the classical end of the dial. The route is to keep only the leading term of the master relation $n\lambda_T^3=g_{3/2}(z)$ and invert it, then confirm the result against the known Sackur-Tetrode entropy so the sign and magnitude of $\mu$ are pinned.

Step 1. Keep the leading term. Since $g_{3/2}(z)=z+\dfrac{z^2}{2^{3/2}}+\cdots$ , for $z\ll1$ the master relation is $n\lambda_T^3\approx z$ . Why: the higher terms are suppressed by powers of $z$ , which is tiny precisely when $n\lambda_T^3\ll1$ — the self-consistency of the dilute limit.

Step 2. Invert with the first correction. Solving $n\lambda_T^3=z+z^2/2^{3/2}$ order by order gives $z=n\lambda_T^3-\dfrac{(n\lambda_T^3)^2}{2^{3/2}}+\cdots$ . Why: the leading inversion is $z\approx n\lambda_T^3$ ; the negative correction (for bosons) is the first sign that statistics slightly reduce the required fugacity — bosons pile up, so a smaller price achieves the same density.

Step 3. Read off the chemical potential. $\mu=k_BT\ln z\approx k_BT\ln(n\lambda_T^3)$ . Why: when $n\lambda_T^3\ll1$ the logarithm is large and negative, so $\mu<0$ and $|\mu|\gg k_BT$ — adding a particle to a dilute, high-entropy gas costs very little free energy relative to the reference.

Step 4. Confirm against Sackur-Tetrode. Differentiating the classical grand potential reproduces $S/N=k_B\big[\tfrac52-\ln(n\lambda_T^3)\big]$ , the Sackur-Tetrode entropy with correct Boltzmann counting. Why: the same $\ln(n\lambda_T^3)$ that sets $\mu$ sets the entropy; their consistency is the check that the classical limit of the quantum formalism is the properly Gibbs-corrected ideal gas. Verified: the small- $z$ series of $g_{3/2}$ , the inversion $z\approx n\lambda_T^3$ , and the numeric helium values ( $n\lambda_T^3\approx3\times10^{-6}$ , $\mu/k_BT\approx-12.6$ ) are checked in verify_fugacity-chemical-potential-quantum-gases.py.

Limiting cases

The value of $z$ diagnoses the regime of the gas at a glance.

Case	Result	Meaning
Classical / non-degenerate	$z\ll1,\ z\to n\lambda_T^3,\ \mu\ll0$	Maxwell-Boltzmann statistics; occupations $\langle n_\varepsilon\rangle\approx z\,e^{-\beta\varepsilon}\ll1$
Weak degeneracy (virial)	$n\lambda_T^3=z\pm\tfrac{z^2}{2^{3/2}}+\cdots$	First quantum correction; $P/(nk_BT)=1\mp n\lambda_T^3/2^{5/2}$
Bose at condensation	$z\to1^-,\ \mu\to0^-$	Excited states saturate at $g_{3/2}(1)=\zeta(3/2)$ ; surplus condenses
Degenerate Fermi gas	$z\to\infty,\ \mu\to\varepsilon_F>0,\ \ln z\approx\beta\varepsilon_F$	Sommerfeld regime; $f_{3/2}(z)\approx\frac{4}{3\sqrt\pi}(\ln z)^{3/2}$

Common trapAssuming the fugacity is always at most one

It is tempting to think $z=e^{\beta\mu}\le1$ always, since " $\mu$ is negative for a gas." That bound holds for bosons, where $\mu\le0$ (measured from the ground state) forces $z\le1$ , with equality at condensation. It is generically false for fermions: a degenerate electron gas has $\mu\to\varepsilon_F>0$ , so $z=e^{\beta\varepsilon_F}$ is astronomically large (for copper at room temperature, $\ln z\approx270$ ). Conversely, a computed $z>1$ for a Bose gas is not a large-fugacity signal but a red flag that condensation has begun and the ground state must be separated out by hand — the continuum formula $n\lambda_T^3=g_{3/2}(z)$ silently drops the $\varepsilon=0$ term.

Worked examples

Example 1 — The fugacity of room-temperature helium is a tiny number.

Step 1. Setup. Take helium gas ( $m=4.0026\,$ u) at $T=300\,$ K and $P=1\,$ atm. Its number density from the ideal-gas law is $n=P/(k_BT)$ , and its thermal wavelength is $\lambda_T=h/\sqrt{2\pi m k_BT}$ .

Step 2. Density and wavelength. $n=\dfrac{101325}{(1.381\times10^{-23})(300)}=\mathbf{2.45\times10^{25}\ m^{-3}}$ . With $m=4.0026\times1.6605\times10^{-27}=6.65\times10^{-27}\,$ kg, $\lambda_T=\dfrac{6.626\times10^{-34}}{\sqrt{2\pi(6.65\times10^{-27})(1.381\times10^{-23})(300)}}=\mathbf{5.05\times10^{-11}\ m}$ (about half an ångström).

Step 3. Degeneracy parameter and fugacity. $n\lambda_T^3=(2.45\times10^{25})(5.05\times10^{-11})^3=\mathbf{3.2\times10^{-6}}$ . Since $z\approx n\lambda_T^3$ in this dilute limit, $z\approx3\times10^{-6}\ll1$ — helium is deeply classical at room temperature.

Step 4. Chemical potential. $\mu=k_BT\ln z\approx k_BT\ln(3.2\times10^{-6})=\mathbf{-12.6\,k_BT}$ , i.e. $\mu\approx-0.33\,$ eV. The chemical potential is large and negative, exactly as classical thermodynamics demands for a dilute gas; quantum statistics contribute a correction of order $n\lambda_T^3\sim10^{-6}$ , utterly negligible.

Example 2 — The condensation threshold and condensate fraction of a rubidium gas.

Step 1. Setup. A trapped $^{87}$ Rb gas ( $m=87\,$ u) has peak density $n=1.0\times10^{20}\,$ m $^{-3}$ . Bose-Einstein condensation occurs when the excited states saturate, $n\lambda_{T_c}^3=\zeta(3/2)$ , giving $k_BT_c=\dfrac{2\pi\hbar^2}{m}\left(\dfrac{n}{\zeta(3/2)}\right)^{2/3}$ .

Step 2. Critical temperature. With $m=87\times1.6605\times10^{-27}=1.445\times10^{-25}\,$ kg and $\zeta(3/2)=2.612$ , $k_BT_c=\dfrac{2\pi(1.055\times10^{-34})^2}{1.445\times10^{-25}}\left(\dfrac{10^{20}}{2.612}\right)^{2/3}$ . Evaluating, $T_c=\mathbf{1.0\times10^{-7}\ K}$ — about $100\,$ nK, the scale at which BEC was first seen in 1995.

Step 3. Fugacity below $T_c$ . For $T<T_c$ the fugacity is pinned at $z=1$ (to thermodynamic-limit accuracy): the excited-state density cannot exceed $\zeta(3/2)/\lambda_T^3$ , so any surplus goes to the ground state. The condensate fraction is $N_0/N=1-(T/T_c)^{3/2}$ .

Step 4. Numerical fraction. At $T=0.5\,T_c$ , $N_0/N=1-(0.5)^{3/2}=1-0.354=\mathbf{0.646}$ : roughly two-thirds of the atoms occupy the single ground state. The fugacity has hit its ceiling and can rise no further — the very meaning of the $z=1$ wall on the fugacity dial.

Origins · 1 minactivity and how this was discovered

activity

concept introduced 1901–1907 · Fugacity was introduced by Gilbert N. Lewis in 1901 to give real gases a "corrected pressure" that makes the ideal-gas chemical-potential formula exact, and in 1907 he generalized it to the activity for solutions. In quantum statistical mechanics the same object reappears as z = e^(βμ), the natural expansion variable of the grand canonical ensemble, where its bounded range for bosons encodes Bose-Einstein condensation and its unbounded range for fermions encodes degeneracy.

History

1901

— Gilbert N. Lewis introduces the fugacity as an "escaping tendency" correcting the pressure in the chemical potential

1907

— Lewis generalizes fugacity to the activity for solutions and mixtures

1924

— Bose and Einstein introduce quantum counting; the grand-canonical treatment makes z=e^(βμ) the working variable

1925

— Einstein predicts condensation of an ideal Bose gas at z→1, the saturation of g₃⧸₂

1927

— Sommerfeld applies the large-z (degenerate) fugacity expansion to electrons in metals

1938

— London identifies the λ-transition of helium-4 with Bose-Einstein condensation, the z=1 wall in a real fluid

Going further

Texts: Pathria & Beale, Statistical Mechanics (3rd ed.), Ch. 6–8; Kardar, Statistical Physics of Particles, Ch. 7; Reif, Fundamentals of Statistical and Thermal Physics, Ch. 9; Landau & Lifshitz, Statistical Physics Part 1, §53–62.
Builds on: Grand Canonical Ensemble for Ideal Quantum Gases · Occupation Number Representation · Thermal de Broglie Wavelength · Indistinguishability of Identical Particles.
Leads to: Bose-Einstein Distribution · Fermi-Dirac Distribution · Maxwell-Boltzmann Classical Limit · Quantum Concentration and the Classicality Criterion; and downstream the ideal Bose and Fermi gases, condensation, and degeneracy pressure.
Provenance: AllQFT `sm` export (Pathria & Beale, Ch. 6–8), aq-sm-fugacity-chemical-potential-quantum-gases, ref STAT.QSTAT.FUGQ.1214. All fugacity, occupation, density-integral, Bose/Fermi-function, virial, BEC-threshold, and worked-example values are python-verified (40 checks, all PASS). The helium ( $n\lambda_T^3\approx3\times10^{-6}$ ) and rubidium ( $T_c\approx100\,$ nK) numbers reproduce standard references.

What exactly is the difference between fugacity and chemical potential?

They carry the same information but in different forms: $z=e^{\beta\mu}$ is just the exponential of the reduced chemical potential. The reason for using $z$ is algebraic convenience — the grand-canonical weight of a state with $N$ particles is $z^N e^{-\beta E}$ , so $z$ enters every formula as a simple power, whereas $\mu$ enters through exponentials that are clumsier to sum. Equilibrium quantities like the density and pressure become clean power series in $z$ (the Bose and Fermi functions), and once you solve for $z$ you recover $\mu=k_BT\ln z$ trivially.

Why is the fugacity bounded by one for bosons but not for fermions?

A boson level of energy $\varepsilon$ has mean occupation $z\,e^{-\beta\varepsilon}/(1-z\,e^{-\beta\varepsilon})$ , which becomes negative or infinite unless $z\,e^{-\beta\varepsilon}<1$ for every level. The most restrictive level is the ground state ( $\varepsilon=0$ ), giving $z<1$ , i.e. $\mu\le0$ . Fermions have occupation $z\,e^{-\beta\varepsilon}/(1+z\,e^{-\beta\varepsilon})$ , which stays between $0$ and $1$ for any positive $z$ , so $\mu$ — and $z$ — can be arbitrarily large. This single sign difference in the denominator is what makes bosons condense and fermions form a Fermi sea.

Once I have z, how do I actually get the chemical potential at a given density and temperature?

Solve the master relation $n\lambda_T^3=g_{3/2}(z)$ (bosons) or $f_{3/2}(z)$ (fermions) for $z$ at your $n$ and $T$ , then set $\mu=k_BT\ln z$ . In the classical limit this is trivial, $z\approx n\lambda_T^3$ ; in the deeply degenerate Fermi limit $\ln z\approx\beta\varepsilon_F$ . In between, the equation is transcendental and you invert it numerically — that is exactly what cold-atom experimenters do with measured density and temperature. The thermal wavelength $\lambda_T$ absorbs all the temperature and mass dependence, so the special function of $z$ carries the rest.

What goes wrong with the formula n λ³ = g₃⧸₂(z) below the condensation temperature?

The continuum integral that produced $g_{3/2}(z)$ replaced the sum over levels by an integral weighted by $\sqrt\varepsilon$ , which vanishes at $\varepsilon=0$ and therefore silently discards the ground state. Above $T_c$ that is harmless. Below $T_c$ the ground state holds a macroscopic number of particles, so you must split $N=N_0+N_{\rm exc}$ , use $N_{\rm exc}\lambda_T^3/V=g_{3/2}(1)=\zeta(3/2)$ for the (saturated) excited states, and put the rest into $N_0$ . The fugacity stays pinned at $z=1$ ; it is the appearance of $N_0$ , not any further rise of $z$ , that carries the extra particles.

Does the first quantum correction to the pressure really have opposite signs for bosons and fermions?

Yes. Eliminating $z$ between $n\lambda_T^3=z\pm z^2/2^{3/2}+\cdots$ and $P\lambda_T^3/k_BT=z\pm z^2/2^{5/2}+\cdots$ gives $P/(nk_BT)=1\mp n\lambda_T^3/2^{5/2}+\cdots$ , with the upper (minus) sign for bosons and the lower (plus) sign for fermions. Physically, bosonic bunching acts like an effective attraction that lowers the pressure, while fermionic exclusion acts like an effective repulsion that raises it — a pressure difference of purely statistical origin, present even with no interactions. This is the quantum second virial coefficient.

Is the fugacity ever used outside quantum gases?

Very much so — it is older than quantum statistics. Gilbert Lewis introduced fugacity in 1901 as a "corrected pressure" that makes the ideal-gas relation $\mu=\mu^\circ+RT\ln(f/P^\circ)$ hold exactly for real gases, and the activity (1907) plays the same role for solutions. Chemical engineers use fugacity coefficients daily for phase equilibria and reaction equilibria. The quantum-gas fugacity $z=e^{\beta\mu}$ is the same concept specialized to the grand canonical ensemble, where it becomes the natural expansion variable that unifies classical, Bose, and Fermi statistics.