---
title: The Classical Limit and Quantum Concentration
module: Quantum Statistics
moduleNumber: 7
lessonNumber: 3
order: 703
summary: >
  When every single-particle level is nearly empty, both quantum distributions
  collapse to the Maxwell-Boltzmann form, and the fugacity equals the ratio of
  the number density to the quantum concentration $n_Q = 1/\lambda^3$. The gas is
  classical when $n \ll n_Q$, degenerate when $n \gtrsim n_Q$. The chemical
  potential is large and negative in the classical regime and rises through zero
  as the gas degenerates. The leading quantum correction to the ideal-gas law is
  a second virial term that lowers the pressure for bosons and raises it for
  fermions — a statistical attraction and repulsion with no interaction behind it.
topics: [Quantum Statistics]
sources:
  - book: Schroeder
    ref: "Ch. 7 — Quantum Statistics; §7.2–7.3"
  - book: Reif
    ref: "Ch. 9 — Quantum Statistics of Ideal Gases; §9.9–9.10"
  - book: Kardar (Statistical Physics of Particles)
    ref: "Ch. 7 — Quantum Statistical Mechanics; §7.2"
  - book: Pathria & Beale
    ref: "Ch. 5 §5.5 and Ch. 6 §6.1"
draft: false
---

The occupation formulas of the previous lesson reduce to the Boltzmann law when
$e^{\beta(\varepsilon-\mu)}\gg 1$. This lesson makes that limit quantitative. The
single control parameter is the fugacity $z = e^{\beta\mu}$, which equals the
number density measured in units of a natural quantum scale, the
quantum concentration $n_Q = \lambda^{-3}$ built from the thermal de Broglie
wavelength. A gas is classical when $z\ll 1$ 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 $PV = Nk_BT$: 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
$\langle n_k\rangle \approx z\,e^{-\beta\varepsilon_k}$ with $z = e^{\beta\mu}$.
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

$$
N = \sum_k \langle n_k\rangle \approx z\sum_k e^{-\beta\varepsilon_k}
   = z\,\frac{V}{\lambda^3},
\qquad
\lambda = \frac{h}{\sqrt{2\pi m k_B T}},
$$

where $\lambda$ is the **thermal de Broglie wavelength**. Solving for the fugacity,

$$
z = \frac{N}{V}\,\lambda^3 = n\lambda^3 = \frac{n}{n_Q},
\qquad
n_Q \equiv \frac{1}{\lambda^3} = \left(\frac{2\pi m k_B T}{h^2}\right)^{3/2}.
$$

> **Definition (Quantum concentration).** The **quantum concentration** $n_Q =
> \lambda^{-3}$ is the number density at which one particle occupies a cube of
> side equal to the thermal de Broglie wavelength. The dimensionless
> **degeneracy parameter** is $n\lambda^3 = n/n_Q$, and it equals the fugacity in
> the classical limit.[^schroeder-nq]

The Maxwell-Boltzmann approximation $z\ll1$ is therefore the statement

$$
n\lambda^3 \ll 1 \quad\Longleftrightarrow\quad n \ll n_Q,
$$

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 $\lambda$, so the packets do not overlap and the
statistics is classical. The prefactor differs from the survey lesson's
$(3mk_BT)^{-3/2}$ because that estimate used the mean thermal speed rather than the
partition-function-consistent $2\pi$; the physical content is identical.

$$
% caption: In the classical regime every level carries a small fractional occupation; in the degenerate regime the low levels approach full occupation (fermions) or macroscopic occupation (bosons).
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
% classical ladder
\node[black,anchor=south,font=\scriptsize] at (1.3,3.3) {classical: n much less than nQ};
\foreach \y/\r in {0/1.0, 0.7/0.7, 1.4/0.45, 2.1/0.28, 2.8/0.16} {
  \draw[black,thick] (0.3,\y) -- (2.3,\y);
  \fill[acc] (1.3,\y+0.13) circle (\r pt);
}
\node[acc,font=\scriptsize,anchor=west] at (2.5,0.13) {each n much less than 1};
% degenerate ladder
\begin{scope}[xshift=6.0cm]
\node[black,anchor=south,font=\scriptsize] at (1.3,3.3) {degenerate: n near nQ};
\foreach \y in {0,0.7,1.4} {
  \draw[black,thick] (0.3,\y) -- (2.3,\y);
  \fill[black] (1.3,\y+0.13) circle (3.0pt);
}
\foreach \y in {2.1,2.8} {
  \draw[black,thick] (0.3,\y) -- (2.3,\y);
  \fill[black] (1.3,\y+0.13) circle (0.9pt);
}
\node[black,font=\scriptsize,anchor=west] at (2.5,0.7) {low levels filled};
\end{scope}
\end{tikzpicture}
$$

## 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 $n\approx 2.5\times10^{25}\ \mathrm{m^{-3}}$ and
  $n_Q\sim10^{30}\ \mathrm{m^{-3}}$, so $n\lambda^3\sim10^{-5}$: thoroughly
  classical.
- **Liquid $^4$He** at $4\ \mathrm{K}$ has $n\lambda^3\sim1$: quantum, and it
  condenses.
- **Conduction electrons in a metal** have $n\sim10^{29}\ \mathrm{m^{-3}}$ and,
  because $m_e$ is small, $n\lambda^3\sim10^{3}$–$10^{4}$ at room temperature:
  a degenerate Fermi gas whose degeneracy temperature is tens of thousands of
  kelvin.
- **Electrons in a white dwarf** reach $n\lambda^3\gg1$ and are relativistically
  degenerate.

$$
% caption: The line $n=n_Q\propto T^{3/2}$ divides the density-temperature plane; systems above and to the left are quantum-degenerate, those below and to the right are classical.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0) -- (7.6,0) node[right,black,font=\scriptsize] {temperature};
\draw[->,black] (0,0) -- (0,4.6) node[above,black,font=\scriptsize] {number density};
% boundary n ~ T^{3/2}
\draw[acc,very thick] plot coordinates {(0.6,0.5)(1.5,1.0)(2.6,1.7)(3.8,2.5)(5.2,3.4)(6.6,4.3)};
\node[text=acc,font=\scriptsize,anchor=west,rotate=32] at (3.9,2.9) {n equals nQ};
\node[black,font=\scriptsize] at (1.7,3.6) {degenerate};
\node[black,font=\scriptsize] at (5.3,1.9) {classical};
% systems
\fill[black] (5.6,0.9) circle (2.2pt); \node[black,font=\scriptsize,anchor=west] at (5.7,0.9) {air};
\fill[black] (1.4,2.2) circle (2.2pt); \node[black,font=\scriptsize,anchor=west] at (1.5,2.2) {liquid helium};
\fill[black] (3.0,3.9) circle (2.2pt); \node[black,font=\scriptsize,anchor=west] at (3.1,3.9) {metal electrons};
\fill[black] (0.9,4.2) circle (2.2pt); \node[black,font=\scriptsize,anchor=west] at (1.0,4.2) {white dwarf};
\end{tikzpicture}
$$

## The chemical potential across the crossover

Inverting $z = n\lambda^3$ gives the chemical potential of the classical ideal
gas,

$$
\mu = k_B T\ln\!\big(n\lambda^3\big) = k_B T\ln\!\frac{n}{n_Q}.
$$

Because $n_Q$ grows as $T^{3/2}$, at high temperature $n\ll n_Q$ and $\mu$ 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 $\mu\to 0$ near $n\lambda^3\sim1$. That is exactly where
the classical formula stops being valid, and the two statistics part ways:

- For **fermions**, $\mu$ continues to rise past zero and approaches the positive
  Fermi energy $\varepsilon_F$ as $T\to0$. The chemical potential of a metal is
  set by the filled Fermi sea, not by the Boltzmann logarithm.
- For **bosons**, $\mu$ approaches zero from below and is pinned at (just under)
  the ground-state energy once condensation begins; it cannot cross into the
  spectrum.

$$
% caption: The chemical potential rises as the gas cools at fixed density. The classical logarithm is accurate only while it is large and negative; near the crossover the fermion branch climbs to a positive Fermi energy while the boson branch is pinned just below the ground state.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,1.8) -- (7.4,1.8) node[right,black,font=\scriptsize] {temperature};
\draw[->,black] (0,0) -- (0,4.2) node[above,black,font=\scriptsize] {chemical potential};
\node[black,font=\scriptsize,anchor=east] at (0,1.8) {0};
% classical branch (valid high T, mu negative, large T at right): dashed where invalid
\draw[black,thick] plot coordinates {(7.0,1.2)(6.0,1.0)(5.0,0.75)(4.2,0.5)};
\draw[black,thick,densely dashed] plot coordinates {(4.2,0.5)(3.6,0.7)(3.2,1.1)(3.0,1.6)};
\node[text=black,font=\scriptsize,anchor=west] at (5.6,0.75) {classical log};
% fermion branch: rises to positive E_F
\draw[black!70,very thick,densely dotted] plot coordinates {(4.2,0.5)(3.6,0.9)(3.0,1.6)(2.2,2.4)(1.2,2.95)(0.3,3.1)};
\node[text=black!70,font=\scriptsize,anchor=west] at (0.4,3.35) {fermion, toward Fermi energy};
% boson branch: pinned just below zero
\draw[acc,very thick] plot coordinates {(4.2,0.5)(3.6,0.85)(3.0,1.35)(2.2,1.62)(1.0,1.72)(0.3,1.74)};
\node[text=acc,font=\scriptsize,anchor=west] at (0.4,1.2) {boson branch};
\end{tikzpicture}
$$

## 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,

$$
n\lambda^3 = z \pm \frac{z^2}{2^{3/2}} + \cdots,
\qquad
\frac{P\lambda^3}{k_B T} = z \pm \frac{z^2}{2^{5/2}} + \cdots,
$$

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

$$
\frac{PV}{Nk_B T} = 1 \mp \frac{n\lambda^3}{2^{5/2}} + \cdots
= 1 + B_2(T)\,n + \cdots,
\qquad
B_2(T) = \mp\,\frac{\lambda^3}{2^{5/2}},
$$

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

> **Result.** Quantum statistics shifts the ideal-gas pressure at fixed density
> and temperature even with no interaction present:
> $$
> \frac{PV}{Nk_B T} = 1 - \frac{n\lambda^3}{2^{5/2}}\ \text{(bosons)},
> \qquad
> \frac{PV}{Nk_B T} = 1 + \frac{n\lambda^3}{2^{5/2}}\ \text{(fermions)}.
> $$
> Bose statistics lowers the pressure, as if the particles attracted one another;
> Fermi statistics raises it, as if they repelled. The effective range of this
> **statistical interaction** is the thermal wavelength $\lambda$, and its origin
> is the exchange symmetry of the wave function, not any term in the Hamiltonian.

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 $5\ \mathrm{K}$ and
atmospheric pressure $n\lambda^3\sim10^{-2}$, so the statistical correction is a
fraction of a percent — measurable, and of the sign predicted by Bose statistics.

$$
% caption: The compressibility factor against the degeneracy parameter to leading order. Bose statistics bends it below unity and Fermi statistics above; the classical gas sits on the line at one.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,1.8) -- (6.6,1.8) node[right,black,font=\scriptsize] {degeneracy parameter};
\draw[->,black] (0,0) -- (0,3.6) node[above,black,font=\scriptsize] {P V over N kT};
\node[black,font=\scriptsize,anchor=east] at (0,1.8) {1};
\draw[black,thick,densely dashed] (0,1.8) -- (6.2,1.8);
\node[text=black,font=\scriptsize,anchor=south east] at (6.2,1.8) {classical};
% fermion: 1 + x/2^{5/2}, x=0..2 -> slope; tikz y = 1.8 + (x/5.657)*scale
\draw[black!70,very thick,densely dotted] plot coordinates {(0,1.8)(1.5,2.05)(3.0,2.35)(4.5,2.7)(6.0,3.1)};
\node[text=black!70,font=\scriptsize,anchor=south] at (4.6,2.75) {fermion};
% boson: 1 - x/2^{5/2}
\draw[acc,very thick] plot coordinates {(0,1.8)(1.5,1.55)(3.0,1.25)(4.5,0.9)(6.0,0.5)};
\node[text=acc,font=\scriptsize,anchor=north] at (4.6,0.85) {boson};
\end{tikzpicture}
$$

## Summary

- In the classical limit the mean occupation is $\langle n_k\rangle\approx z\,
  e^{-\beta\varepsilon_k}$, and the fugacity equals the degeneracy parameter
  $z = n\lambda^3 = n/n_Q$.
- The quantum concentration is $n_Q=\lambda^{-3}=(2\pi m k_BT/h^2)^{3/2}$; a gas
  is classical for $n\ll n_Q$ and degenerate for $n\gtrsim n_Q$, with the boundary
  $n=n_Q\propto T^{3/2}$ crossed by cooling or compressing. Light particles
  degenerate first.
- The chemical potential $\mu = k_BT\ln(n/n_Q)$ is large and negative when
  classical and rises toward zero at the crossover; below it the fermion branch
  climbs to $+\varepsilon_F$ and the boson branch is pinned just under the ground
  state.
- The leading quantum correction is $PV/Nk_BT = 1\mp n\lambda^3/2^{5/2}$, a
  statistical attraction for bosons and repulsion for fermions of range $\lambda$
  and no dynamical origin.

[^schroeder-nq]: **Schroeder**, _An Introduction to Thermal Physics_, §7.2–7.3 — the thermal de Broglie wavelength, the quantum concentration $n_Q$, and the degeneracy criterion $n\lambda^3\lesssim1$ separating the classical and quantum regimes.
[^kardar-corr]: **Kardar**, _Statistical Physics of Particles_, §7.2 — the fugacity expansions of $n\lambda^3$ and $P\lambda^3/k_BT$ and the resulting statistical second virial coefficient $B_2=\mp\lambda^3/2^{5/2}$. Consistent with **Pathria & Beale**, §6.1.