---
title: Quantum Gases with Interactions and Statistical Exchange
module: Interacting Gases
moduleNumber: 10
lessonNumber: 3
order: 1003
summary: >
  A quantum gas has a nonzero second virial coefficient even with no forces
  between the particles: symmetrization alone produces an effective statistical
  interaction, attractive for bosons and repulsive for fermions, with range the
  thermal wavelength $\lambda$. This lesson derives that exchange contribution
  $B_2=\mp\lambda^3/2^{5/2}g$, writes it as a statistical potential
  $v_s(r)=-k_BT\ln(1\pm e^{-2\pi r^2/\lambda^2})$, and shows how real interactions
  add on top through the Beth-Uhlenbeck phase-shift formula, reducing at low
  temperature to a single scattering length.
topics: [Interacting Gases]
sources:
  - book: Pathria & Beale
    ref: "Ch. 10 — Statistical Mechanics of Interacting Systems: The Method of Cluster Expansions; §10.3–10.5"
  - book: Kardar (Statistical Physics of Particles)
    ref: "Ch. 7 — Quantum Statistical Mechanics; §7.2, and Ch. 5"
  - book: Huang
    ref: "Ch. 10 — Approach to Equilibrium; Quantum Cluster Expansion and the Beth-Uhlenbeck formula"
draft: false
---

The virial expansion of the two preceding lessons was classical: the pair
potential entered through the Boltzmann factor $e^{-\beta u}$, and the particles
were distinguishable points. A quantum gas departs from ideality for a second
reason that has nothing to do with forces. Identical bosons and fermions obey
symmetrized statistics, and symmetrization correlates their positions even when
the Hamiltonian contains no interaction term at all. The result is a nonzero
second virial coefficient for the ideal quantum gas — a statistical "exchange"
interaction that is attractive for bosons, repulsive for fermions, and has range
equal to the thermal de Broglie wavelength. Real interactions add to this
statistical piece, and the full quantum second virial coefficient separates
cleanly into the two contributions.

## The second virial coefficient of the ideal quantum gas

The ideal Bose and Fermi gases already carry a density expansion. From the
[grand-canonical treatment](/statistical-mechanics/quantum-statistics/deriving-the-quantum-distributions),
the pressure and density are set by the fugacity $z=e^{\beta\mu}$ through the
Bose-Einstein and Fermi-Dirac functions,

$$
\frac{P\lambda^3}{g\,k_BT} = g_{5/2}(z) = z \pm \frac{z^2}{2^{5/2}} + \cdots,
\qquad
\frac{n\lambda^3}{g} = g_{3/2}(z) = z \pm \frac{z^2}{2^{3/2}} + \cdots,
$$

where $g$ is the spin degeneracy, $\lambda = h/\sqrt{2\pi m k_BT}$ is the thermal
wavelength, and the upper sign is for bosons, the lower for fermions.[^funcs]
Eliminating $z$ between the two series gives the pressure as a function of density.
Writing $x = n\lambda^3/g$ and inverting the density series, $z = x \mp
x^2/2^{3/2} + \cdots$, then substituting into the pressure series,

$$
\frac{P}{n k_BT} = 1 \mp \frac{n\lambda^3}{2^{5/2}\,g} + \cdots,
$$

so the quantum gas has a second virial coefficient

$$
B_2^{\text{stat}}(T) = \mp\frac{\lambda^3}{2^{5/2}\,g},
$$

negative for bosons and positive for fermions. No potential was assumed anywhere
in the derivation — this coefficient is a pure consequence of quantum statistics.
Its sign matches the qualitative picture from the occupation functions: bosons
prefer to share states and cluster together, lowering the pressure like an
attraction, while the Pauli principle keeps fermions apart, raising the pressure
like a repulsion.

[^funcs]: The polylogarithm functions
$g_\nu(z)=\sum_{\ell\ge1}z^\ell/\ell^\nu$ (bosons) and
$f_\nu(z)=\sum_{\ell\ge1}(-1)^{\ell-1}z^\ell/\ell^\nu$ (fermions) and the ideal
quantum-gas thermodynamics are developed in **Pathria & Beale**, _Statistical
Mechanics_ (4th ed.), §6.1–6.2, and the [ideal quantum-gas framework
lesson](/statistical-mechanics/quantum-statistics/ideal-quantum-gases-general-framework).

## The statistical potential

The exchange effect can be repackaged as an effective classical interaction. Two
identical free particles have a symmetrized two-body density matrix, and forming
the relative-coordinate probability density shows that their positions are
correlated exactly as if they interacted through the **statistical potential**

$$
v_s(r) = -k_BT\,\ln\!\Bigl(1 \pm e^{-2\pi r^2/\lambda^2}\Bigr),
$$

with the upper sign for bosons and the lower for fermions.[^stat] For bosons the
argument of the logarithm exceeds one, so $v_s<0$: an attractive well of depth
$k_BT\ln 2$ at contact. For fermions the argument is less than one, $v_s>0$, and
it diverges as $r\to 0$ — the Pauli exclusion appears as an infinite repulsive
core in the statistical potential. Both fall off on the scale of the thermal
wavelength, so $\lambda$ is the **exchange length**: identity effects are felt
only when two particles approach within a de Broglie wavelength of each other, and
vanish in the classical limit $\lambda\to 0$.

[^stat]: The Uhlenbeck-Gropper statistical potential and its derivation from the
free two-body density matrix appear in **Huang**, _Statistical Mechanics_
(2nd ed.), Ch. 10, and **Pathria & Beale**, _Statistical Mechanics_ (4th ed.),
§10.3. The Gaussian factor is $e^{-2\pi r^2/\lambda^2}=e^{-\pi r^2 m k_BT/\hbar^2}$.

$$
% caption: The statistical potential $v_s(r)$ in units of $k_BT$ versus $r/\lambda$. Bosons feel an attractive well of depth $k_BT\ln 2$ at contact; fermions feel a repulsion that diverges as $r\to 0$ (the exchange hole). Both decay on the scale of the thermal wavelength $\lambda$.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(6.6,0) node[right,black]{$r$};
\draw[->,black] (0.3,-1.6)--(0.3,3.2) node[above,black]{$v_s$};
\draw[black,dashed] (0.3,0)--(6.4,0);
% fermion: repulsive, diverging near 0
\draw[black,very thick,densely dashed] plot[domain=0.34:1.5,samples=120]
  ({0.3+\x*3.6},{-2.0*ln(1-exp(-2*3.14159*\x*\x))});
\node[black] at (2.6,2.4) {fermions: repulsive};
% boson: attractive well
\draw[acc,very thick] plot[domain=0.06:1.5,samples=120]
  ({0.3+\x*3.6},{-2.0*ln(1+exp(-2*3.14159*\x*\x))});
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\draw[black,dashed] (0.3+3.6,-1.5)--(0.3+3.6,3.0);
\node[black,above] at (0.3+3.6,3.0) {range};
\end{tikzpicture}
$$

## The pair correlation function

The same symmetrization shows up directly in the **pair correlation function**
$g(r)$, the probability of finding a second particle a distance $r$ from a given
one, normalized to unity for uncorrelated positions. For the ideal quantum gas,

$$
g(r) = 1 \pm \frac{1}{g}\,e^{-2\pi r^2/\lambda^2}.
$$

At short range bosons show $g(0)=1+1/g>1$, a **bunching** enhancement — identical
bosons are more likely to be found close together, the effect measured in the
Hanbury Brown–Twiss intensity correlations of thermal light. Fermions show
$g(0)=1-1/g<1$, an **exchange hole** (antibunching) surrounding each particle, of
radius $\sim\lambda$, inside which the density of like-spin partners is depleted.
The correction integrates to exactly the exchange virial coefficient: $B_2^{\text{stat}}
= -\tfrac12\int[g(r)-1]\,\d^3 r$ reproduces $\mp\lambda^3/2^{5/2}g$.

$$
% caption: The ideal-gas pair correlation $g(r)$ versus $r/\lambda$ (spinless case $g=1$). Bosons bunch, $g(0)=2$; fermions carry an exchange hole, $g(0)=0$; classical particles are uncorrelated, $g=1$. Both approach the classical value beyond a thermal wavelength.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(6.6,0) node[right,black]{$r$};
\draw[->,black] (0,0)--(0,4.2) node[above,black]{$g(r)$};
\draw[black,dashed] (0,1.8)--(6.4,1.8) node[right,black]{$g=1$};
\draw[acc,very thick] plot[domain=0:1.9,samples=120]
  ({\x*3.2},{1.8*(1+exp(-2*3.14159*\x*\x))});
\draw[black,very thick,densely dashed] plot[domain=0:1.9,samples=120]
  ({\x*3.2},{1.8*(1-exp(-2*3.14159*\x*\x))});
\node[acc] at (2.3,3.35) {bosons: bunching};
\node[black] at (2.7,0.7) {fermions: exchange hole};
\end{tikzpicture}
$$

## Interactions on top of statistics: the Beth-Uhlenbeck formula

Real particles interact, and the full quantum second virial coefficient is the
sum of the statistical piece and a genuine interaction piece,

$$
B_2(T) = B_2^{\text{stat}}(T) + B_2^{\text{int}}(T).
$$

The interaction part is fixed entirely by the two-body scattering problem. Beth
and Uhlenbeck showed that $B_2^{\text{int}}$ is built from the energies of the
two-body bound states and the scattering phase shifts $\delta_\ell(k)$ of the
continuum states,

$$
B_2^{\text{int}}(T) \propto -\lambda^3\left[\sum_{\text{bound }b} e^{-\beta E_b}
+ \frac{1}{\pi}\int_0^\infty e^{-\beta\hbar^2 k^2/m}\,\frac{\d\delta(k)}{\d k}\,\d k\right],
$$

with $\delta(k)$ the appropriately weighted sum of partial-wave phase shifts and
the proportionality constant of order unity fixed by the reduced-mass
convention.[^bethuhl] Each two-body bound state contributes a negative term
weighted by its Boltzmann factor $e^{-\beta E_b}$ — bound pairs act as an
effective attraction — while the scattering states contribute through the rate at
which the phase shift changes with momentum. A repulsive potential produces
$\delta<0$ and a positive $B_2^{\text{int}}$; an attractive one produces
$\delta>0$ and a negative $B_2^{\text{int}}$. The formula is the quantum
counterpart of $B_2=-\tfrac12\int f\,\d^3r$: it replaces the classical Mayer
integral by the spectrum of the two-body Hamiltonian.

[^bethuhl]: The Beth-Uhlenbeck formula and the partial-wave sum, with the
statistics-dependent restriction to even or odd $\ell$ for identical particles in
a given spin state, are derived in **Huang**, _Statistical Mechanics_ (2nd ed.),
Ch. 10, and **Pathria & Beale**, _Statistical Mechanics_ (4th ed.), §10.3–10.5.
The two-body relative motion uses the reduced mass $m/2$, so its thermal
wavelength is $\sqrt2\,\lambda$.

$$
% caption: The quantum second virial coefficient splits into a statistical part from symmetrization (sign set by the statistics) and an interaction part from the two-body bound states and scattering phase shifts (the Beth-Uhlenbeck spectrum).
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\node[draw,black,thick,fill=black!6,align=center,minimum width=3.0cm,minimum height=1.2cm] (stat) at (0,1.5) {statistical exchange\\(symmetrization)};
\node[draw,black,thick,fill=black!6,align=center,minimum width=3.0cm,minimum height=1.2cm] (int) at (0,-0.6) {real interaction\\(bound states $+$ phase shifts)};
\node[draw,acc,thick,fill=acc!10,align=center,minimum width=3.0cm,minimum height=1.2cm] (tot) at (5.4,0.45) {quantum $B_2$};
\draw[->,black,thick] (stat.east)--(tot.west);
\draw[->,black,thick] (int.east)--(tot.west);
\node[black] at (3.4,1.55) {sign from statistics};
\node[black] at (3.4,-0.7) {sign from phase shift};
\end{tikzpicture}
$$

## The dilute quantum gas and the scattering length

At low temperature only the lowest partial wave survives. For a short-range
potential the s-wave phase shift behaves as $\delta_0(k)\approx -a_s k$ at small
$k$, defining the **scattering length** $a_s$, and the Beth-Uhlenbeck integral
collapses to a single term linear in $a_s$. The entire interaction reduces to one
length, and the low-energy physics of a dilute gas is governed by the effective
contact coupling

$$
g_{\text{int}} = \frac{4\pi\hbar^2 a_s}{m},
$$

positive for a repulsive effective interaction ($a_s>0$) and negative for an
attractive one. The diluteness condition is $n|a_s|^3 \ll 1$: the mean spacing far
exceeds the scattering length, so the gas is nearly ideal and interactions enter
perturbatively. For a dilute Bose gas the leading correction to the ground-state
energy per particle is $E/N = (2\pi\hbar^2 a_s/m)\,n\,[1 + \tfrac{128}{15\sqrt\pi}
(n a_s^3)^{1/2} + \cdots]$, the Lee-Huang-Yang expansion, whose first term is the
mean-field shift and whose square-root correction is the first beyond-mean-field
effect.[^lhy] This is the regime realized in the ultracold-atom condensates that
made [Bose-Einstein condensation](/statistical-mechanics/bose-systems/bose-einstein-condensation-derived)
directly observable.

[^lhy]: The scattering-length description of the dilute Bose gas and the
Lee-Huang-Yang correction are in **Pathria & Beale**, _Statistical Mechanics_
(4th ed.), §10.4–10.5, and **Kardar**, _Statistical Physics of Particles_, §5.4;
MIT OCW 8.333, <https://ocw.mit.edu/courses/8-333-statistical-mechanics-i-statistical-mechanics-of-particles-fall-2013/>.

## When statistics and interactions compete

Two length scales decide which effects matter: the range $r_0$ (or scattering
length) of the real potential, and the thermal wavelength $\lambda$ that sets the
range of the statistical potential. Their ratio is fixed by temperature, since
$\lambda\propto T^{-1/2}$ grows without bound as the gas is cooled.

- **$\lambda \ll r_0$ (high $T$).** The thermal wavelength is smaller than the
  interaction range; exchange is negligible and the classical virial expansion of
  the earlier lessons applies, with $B_2$ from the Mayer integral.
- **$\lambda \gg r_0$ (low $T$).** Wave packets overlap on scales far larger than
  the potential range; statistics dominate, and the gas is a nearly ideal quantum
  gas with the exchange $B_2^{\text{stat}}$ as its leading correction.
- **$\lambda \sim r_0$.** Both contributions are comparable and the full quantum
  $B_2 = B_2^{\text{stat}} + B_2^{\text{int}}$ is required. This crossover, where
  degeneracy and interactions set in together, is the operating regime of
  ultracold atomic gases.

$$
% caption: The two competing scales versus temperature. The thermal wavelength $\lambda\propto T^{-1/2}$ falls with temperature while the interaction range $r_0$ is fixed; they cross at a temperature below which exchange (statistics) dominates and above which the classical interaction picture holds.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(6.8,0) node[right,black]{temperature};
\draw[->,black] (0,0)--(0,4.2) node[above,black]{length};
\draw[acc,very thick] plot[domain=0.42:6.2,samples=120] ({\x},{3.0/sqrt(\x)});
\node[acc] at (1.5,3.2) {thermal wavelength};
\draw[black,very thick] (0,1.8)--(6.4,1.8);
\node[black,right] at (5.0,2.05) {interaction range};
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\node[black,above right] at (2.78,1.85) {crossover};
\node[black] at (1.5,0.6) {statistics dominate};
\node[black] at (5.2,0.6) {classical};
\end{tikzpicture}
$$

## Summary

- Quantum statistics gives the ideal gas a second virial coefficient
  $B_2^{\text{stat}} = \mp\lambda^3/2^{5/2}g$ with no forces present: negative
  (attractive) for bosons, positive (repulsive) for fermions.
- The exchange effect is equivalent to a statistical potential
  $v_s(r) = -k_BT\ln(1\pm e^{-2\pi r^2/\lambda^2})$, an attractive well for bosons
  and a diverging repulsion for fermions, of range the thermal wavelength.
- The pair correlation $g(r)=1\pm g^{-1}e^{-2\pi r^2/\lambda^2}$ shows boson
  bunching ($g(0)>1$) and the fermion exchange hole ($g(0)<1$); its integral
  returns $B_2^{\text{stat}}$.
- Real interactions add $B_2^{\text{int}}$, given by the Beth-Uhlenbeck formula
  from the two-body bound-state energies and scattering phase shifts — the quantum
  replacement for the classical Mayer integral.
- At low temperature the interaction reduces to the scattering length $a_s$ and
  the contact coupling $g_{\text{int}}=4\pi\hbar^2 a_s/m$; the dilute regime
  $n|a_s|^3\ll1$ with $\lambda\sim r_0$ is that of ultracold atomic gases.