The Classical Limit and Quantum Concentration

The occupation formulas of the previous lesson reduce to the Boltzmann law when $e^{β (ε - μ)} ≫ 1$ . This lesson makes that limit quantitative. The single control parameter is the fugacity $z = e^{β μ}$ , which equals the number density measured in units of a natural quantum scale, the quantum concentration $n_{Q} = λ^{- 3}$ built from the thermal de Broglie wavelength. A gas is classical when $z ≪ 1$ and degenerate when $z$ approaches or exceeds one. The same parameter controls the chemical potential, which sweeps from large negative values in the classical regime up through zero, and it sets the size of the first quantum correction to $P V = N k_{B} T$ : a second virial coefficient of purely statistical origin, opposite in sign for bosons and fermions.

The fugacity and the quantum concentration

In the Maxwell-Boltzmann limit the mean occupation of every level is $⟨ n_{k} ⟩ \approx z e^{- β ε_{k}}$ with $z = e^{β μ}$ . Summing over levels fixes $z$ from the total number. For free particles in a volume $V$ the level sum becomes the single-particle partition function, and

N = k \sum ⟨ n_{k} ⟩ \approx z k \sum e^{- β ε_{k}} = z \frac{V}{λ ^{3}}, λ = \frac{h}{2 π m k _{B} T},

where $λ$ is the thermal de Broglie wavelength. Solving for the fugacity,

z = \frac{N}{V} λ^{3} = n λ^{3} = \frac{n}{n _{Q}}, n_{Q} \equiv \frac{1}{λ ^{3}} = (\frac{2 π m k _{B} T}{h ^{2}})^{3/2} .

The Maxwell-Boltzmann approximation $z ≪ 1$ is therefore the statement

n λ^{3} ≪ 1 ⟺ n ≪ n_{Q},

the same criterion the survey lesson derived from wave-packet overlap, now read as a comparison of the actual density to the quantum concentration. The interparticle spacing $n^{- 1/3}$ exceeds $λ$ , so the packets do not overlap and the statistics is classical. The prefactor differs from the survey lesson's $(3 m k_{B} T)^{- 3/2}$ because that estimate used the mean thermal speed rather than the partition-function-consistent $2 π$ ; the physical content is identical.

In the classical regime every level carries a small fractional occupation; in the degenerate regime the low levels approach full occupation (fermions) or macroscopic occupation (bosons).

The classical-quantum boundary for real systems

The boundary $n = n_{Q}$ is a curve in the density-temperature plane. Because $n_{Q} \propto T^{3/2}$ , holding the density fixed and lowering the temperature always drives a gas across it eventually; holding the temperature fixed and compressing does the same. Whether a real system is quantum is decided by where it sits relative to this line, and the mass enters through $n_{Q} \propto m^{3/2}$ : light particles have a small quantum concentration and degenerate readily.

Air at room temperature has $n \approx 2.5 \times 1 0^{25} m^{- 3}$ and $n_{Q} \sim 1 0^{30} m^{- 3}$ , so $n λ^{3} \sim 1 0^{- 5}$ : thoroughly classical.
Liquid $^{4}$ He at $4 K$ has $n λ^{3} \sim 1$ : quantum, and it condenses.
Conduction electrons in a metal have $n \sim 1 0^{29} m^{- 3}$ and, because $m_{e}$ is small, $n λ^{3} \sim 1 0^{3}$ – $1 0^{4}$ at room temperature: a degenerate Fermi gas whose degeneracy temperature is tens of thousands of kelvin.
Electrons in a white dwarf reach $n λ^{3} ≫ 1$ and are relativistically degenerate.

The line

n = n_{Q} \propto T^{3/2}

divides the density-temperature plane; systems above and to the left are quantum-degenerate, those below and to the right are classical.

The chemical potential across the crossover

Inverting $z = n λ^{3}$ gives the chemical potential of the classical ideal gas,

μ = k_{B} T ln (n λ^{3}) = k_{B} T ln \frac{n}{n _{Q}} .

Because $n_{Q}$ grows as $T^{3/2}$ , at high temperature $n ≪ n_{Q}$ and $μ$ is large and negative: adding a particle to a dilute hot gas lowers the free energy, since the entropy gained outweighs the energy cost. As the temperature falls at fixed density, $n_{Q}$ shrinks toward $n$ , the logarithm rises toward zero, and the classical formula predicts $μ \to 0$ near $n λ^{3} \sim 1$ . That is exactly where the classical formula stops being valid, and the two statistics part ways:

For fermions, $μ$ continues to rise past zero and approaches the positive Fermi energy $ε_{F}$ as $T \to 0$ . The chemical potential of a metal is set by the filled Fermi sea, not by the Boltzmann logarithm.
For bosons, $μ$ approaches zero from below and is pinned at (just under) the ground-state energy once condensation begins; it cannot cross into the spectrum.

The chemical potential rises as the gas cools at fixed density. The classical logarithm is accurate only while it is large and negative; near the crossover the fermion branch climbs to a positive Fermi energy while the boson branch is pinned just below the ground state.

The first quantum correction to the ideal-gas law

Keeping the next term beyond the Maxwell-Boltzmann leading order gives the first quantum correction to the equation of state. The number density and pressure of an ideal quantum gas are exact power series in the fugacity,

n λ^{3} = z \pm \frac{z ^{2}}{2 ^{3/2}} + \dots, \frac{P λ ^{3}}{k _{B} T} = z \pm \frac{z ^{2}}{2 ^{5/2}} + \dots,

with the upper sign for bosons and the lower for fermions in the convention that the boson series has $+ z^{2} / 2^{3/2}$ .² Eliminating $z$ between the two series to second order gives a virial expansion in the density,

\frac{P V}{N k _{B} T} = 1 \mp \frac{n λ ^{3}}{2 ^{5/2}} + \dots = 1 + B_{2} (T) n + \dots, B_{2} (T) = \mp \frac{λ ^{3}}{2 ^{5/2}},

the upper sign now giving the boson result $- n λ^{3} / 2^{5/2}$ and the lower the fermion result $+ n λ^{3} / 2^{5/2}$ . The correction is of order the degeneracy parameter, so it is small exactly when the gas is classical and grows as degeneracy sets in.

The sign is the occupation asymmetry of the earlier lessons in thermodynamic dress: bosons cluster, so they sit closer than random and the momentum flux on the walls drops; fermions exclude, so they hold apart and the flux rises. The factor $2^{- 5/2} = 0.1768$ fixes the size. For helium vapor at $5 K$ and atmospheric pressure $n λ^{3} \sim 1 0^{- 2}$ , so the statistical correction is a fraction of a percent — measurable, and of the sign predicted by Bose statistics.

The compressibility factor against the degeneracy parameter to leading order. Bose statistics bends it below unity and Fermi statistics above; the classical gas sits on the line at one.

Summary

In the classical limit the mean occupation is $⟨ n_{k} ⟩ \approx z e^{- β ε_{k}}$ , and the fugacity equals the degeneracy parameter $z = n λ^{3} = n / n_{Q}$ .
The quantum concentration is $n_{Q} = λ^{- 3} = (2 π m k_{B} T / h^{2})^{3/2}$ ; a gas is classical for $n ≪ n_{Q}$ and degenerate for $n ≳ n_{Q}$ , with the boundary $n = n_{Q} \propto T^{3/2}$ crossed by cooling or compressing. Light particles degenerate first.
The chemical potential $μ = k_{B} T ln (n / n_{Q})$ is large and negative when classical and rises toward zero at the crossover; below it the fermion branch climbs to $+ ε_{F}$ and the boson branch is pinned just under the ground state.
The leading quantum correction is $P V / N k_{B} T = 1 \mp n λ^{3} / 2^{5/2}$ , a statistical attraction for bosons and repulsion for fermions of range $λ$ and no dynamical origin.

Schroeder, An Introduction to Thermal Physics, §7.2–7.3 — the thermal de Broglie wavelength, the quantum concentration $n_{Q}$ , and the degeneracy criterion $n λ^{3} ≲ 1$ separating the classical and quantum regimes. ↩
Kardar, Statistical Physics of Particles, §7.2 — the fugacity expansions of $n λ^{3}$ and $P λ^{3} / k_{B} T$ and the resulting statistical second virial coefficient $B_{2} = \mp λ^{3} / 2^{5/2}$ . Consistent with Pathria & Beale, §6.1. ↩