Few-Body Data in the Virial Expansion

The virial expansion is a controlled many-body expansion in fugacity whose coefficients are fixed by few-body clusters. At second order, bound states and scattering phase shifts determine the interaction correction exactly; at third order, genuine three-body spectra and scattering enter. Its control parameter is $z=e^{\beta\mu}$, not the interaction strength.

Required background. Effective Range, Shallow Poles, and Universality Windows supplies two-body spectral data, and Thermal Density Operators and the KMS Condition fixes the grand-canonical ensemble.

Helpful background. Efimov Physics and the Three-Body Parameter supplies the three-body input needed for bosonic $b_3$.

Fugacity expansion

For a balanced two-component gas of equal mass with common fugacity $z$ and thermal wavelength

$$\lambda_T=\sqrt{\frac{2\pi}{mT}},$$

write

$$\frac{P}{T} =\frac{2}{\lambda_T^3} \left(z+b_2z^2+b_3z^3+\cdots\right).$$

This convention gives the ideal Fermi value $b_2^{(0)}=-2^{-5/2}$. Other authors absorb spin degeneracy or factorials into $b_n$; compare the displayed pressure rather than coefficient symbols alone.

The density follows from $n=z\,\partial(P/T)/\partial z$,

$$n=\frac{2}{\lambda_T^3} \left(z+2b_2z^2+3b_3z^3+\cdots\right).$$

A truncated series is credible only on a fugacity interval where the last retained contribution decreases and the result is stable when the next coefficient or a resummation consistent with known coefficients is included.

Beth–Uhlenbeck formula

Let $\Delta Z_{\rm rel}$ be the interaction-induced change in the relative two-body partition function in an $s$-wave channel. Then

$$\Delta Z_{\rm rel} =\sum_b e^{\beta|E_b|} +\frac1\pi\int_0^\infty\!\mathrm dk\, e^{-\beta k^2/m}\frac{\mathrm d\delta_0}{\mathrm dk},$$

with the threshold contribution treated consistently with Levinson’s theorem. In the balanced two-component convention above,

$$\Delta b_2=\sqrt2\,\Delta Z_{\rm rel}.$$

The formula shows explicitly how a shallow dimer and continuum phase shifts share spectral weight. Counting the dimer while using a phase-shift integral whose Levinson contribution already includes it double counts the threshold rearrangement. The original spectral relation is due to Beth and Uhlenbeck 1937.

At zero-range unitarity the properly regulated threshold limit gives $\Delta b_2=1/\sqrt2$. Effective range produces corrections organized by $r_e/\lambda_T$ and microscopic range by $R/\lambda_T$.

Third and higher coefficients

$b_3$ contains connected three-body information after subtracting products of lower clusters. For resonant identical bosons it depends on the Efimov spectrum, $\kappa_*\lambda_T$, and inelastic or stability assumptions. For two-component equal-mass fermions there is no identical-boson Efimov parameter in the leading $s$-wave sector, but three-body scattering still determines the coefficient.

At order $z^n$, only clusters with at most $n$ particles enter. That hierarchy is exact, but computing the cluster partition functions can be difficult. Bound-state degeneracies, center-of-mass factors, quantum statistics, and trap-to-homogeneous conversions must be kept consistent.

Contact and derivatives

The grand-canonical adiabatic relation gives

$$\frac{C}{V}=4\pi m\frac{\partial P}{\partial a^{-1}} \bigg|_{T,\mu}.$$

Inserting the virial series expresses the thermal contact order by order through $\partial b_n/\partial a^{-1}$. This provides a strong cross-check: differentiating the phase-shift representation of $b_2$ must agree with a direct two-body contact calculation in the same convention.

Common pitfalls

Using density rather than fugacity as the control parameter. They agree only at leading dilute order; interactions and statistics modify their relation.

Mixing coefficient conventions. Spin factors and thermal wavelengths can move between the prefactor and $b_n$.

Adding bound and continuum terms inconsistently. Levinson’s theorem fixes how spectral weight moves through threshold.

Exercises

Derive density through second order

Differentiate the pressure series and retain terms through $z^2$.

Solution

Since $n=\partial P/\partial\mu=z\partial(P/T)/\partial z$, differentiating gives $n=2\lambda_T^{-3}(z+2b_2z^2)+O(z^3)$. The factor two multiplying $b_2$ counts the particles in a two-body cluster.

Estimate range control

If $|r_e|/\lambda_T=0.08$ and $R/\lambda_T=0.02$, which correction limits a zero-range $b_2$ prediction?

Solution

The effective-range ratio is larger, so it supplies the leading expected correction, about eight percent absent an anomalously small coefficient. The smaller microscopic-range ratio does not justify ignoring the measured $r_e$.

Continue

Resonant Bose Matter and Metastable Branches explains why equilibrium cluster thermodynamics may be preempted by loss. From Few-Body Inputs to Many-Body Predictions combines fugacity truncation with range and parameter errors. Universal Relations and Tan Contact supplies the adiabatic derivative.

References

• Beth, E., and G. E. Uhlenbeck. “The Quantum Theory of the Non-Ideal Gas. II. Behaviour at Low Temperatures.” Physica 4 (1937): 915–924. DOI.

Further reading

• Ho, Tin-Lun, and Erich J. Mueller. “High Temperature Expansion Applied to Fermions near Feshbach Resonance.” Physical Review Letters 92 (2004): 160404. DOI.
• Liu, Xia-Ji. “Virial Expansion for a Strongly Correlated Fermi System and Its Application to Ultracold Atomic Fermi Gases.” Physics Reports 524 (2013): 37–83. DOI.