Quantum Gases with Interactions and Statistical Exchange

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, the pressure and density are set by the fugacity $z=e^{\beta \mu }$ through the Bose-Einstein and Fermi-Dirac functions,

where $g$ is the spin degeneracy, $\lambda =h/\sqrt{2\pi m{k}_{B}T}$ is the thermal wavelength, and the upper sign is for bosons, the lower for fermions.¹ 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,

so the quantum gas has a second virial coefficient

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.

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

with the upper sign for bosons and the lower for fermions.² For bosons the argument of the logarithm exceeds one, so $v_s<0$: an attractive well of depth $k_{B}T\ln 2$ 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 $\lambda \to 0$.

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,

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: $B_2^{\text{stat}}=-\frac{1}{2}\int [g(r)-1]{\mathbf{d}}^{3}r$ reproduces $\mp {\lambda }^3/2^{5/2}g$.

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,

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,

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.³ 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 $B_2=-\frac{1}{2}\int f{\mathbf{d}}^{3}r$: it replaces the classical Mayer integral by the spectrum of the two-body Hamiltonian.

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

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 $E/N=(2\pi {\hslash }^2{a}_s/m)n[1+\frac{128}{15\sqrt{\pi }}(n{a}_s^3{)}^{1/2}+\cdots ]$, 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.⁴ This is the regime realized in the ultracold-atom condensates that made Bose-Einstein condensation directly observable.

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.

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 $v_s(r)=-k_{B}T\ln (1\pm e^{-2\pi r^2/{\lambda }^2})$, 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 $g_{\text{int}}=4\pi {\hslash }^2{a}_s/m$; the dilute regime $n|a_s{|}^3\ll 1$ with $\lambda \sim r_0$ is that of ultracold atomic gases.