The virial expansion of ideal quantum gases reveals some interesting and amusing properties when considered as a function of dimensionality $d$ . In particular, the convergence radius $ρ_c(d)$ of the expansion is particulary large at {\em exactly\/} $d=3$ dimensions, $ρ_c(3) = 7.1068\ldots \times \lim_{d\to3} ρ_c(d)$ . The same phenomenon occurs in a few other special (non-integer) dimensions. We explain the origin of these facts, and discuss more generally the structure of singularities governing the asymptotic behavior of the ideal gas virial expansion.The virial expansion of ideal quantum gases reveals some interesting and amusing properties when considered as a function of dimensionality $d$. In particular, the convergence radius $ρ_c(d)$ of the expansion is particulary large at {\em exactly\/} $d=3$ dimensions, $ρ_c(3) = 7.1068\ldots \times \lim_{d\to3} ρ_c(d)$. The same phenomenon occurs in a few other special (non-integer) dimensions. We explain the origin of these facts, and discuss more generally the structure of singularities governing the asymptotic behavior of the ideal gas virial expansion.

arXiv:1502.01360Feb 4, 20150 citations

Scaling in the ideal Bose gas

Carol K. Hall

Journal of Statistical PhysicsAug 1, 197517 citations

Leading- and next-to-leading-order semiclassical approximation to the first seven virial coefficients of spin-1/2 fermions across spatial dimensions

Y. Hou, A. J. Czejdo, Jared De Chant +2 more

Following up on recent calculations, we investigate the leading- and next-to-leading-order semiclassical approximation to the virial coefficients of a two-species fermion system with a contact interaction. Using the analytic result for the second-order virial coefficient as a renormalization condition, we derive expressions for up to the seventh-order virial coefficient $\mathrm{\ensuremath{\Delta}}{b}_{7}$. Our results at leading order, though approximate, furnish simple analytic formulas that relate $\mathrm{\ensuremath{\Delta}}{b}_{n}$ to $\mathrm{\ensuremath{\Delta}}{b}_{2}$ for arbitrary dimension, providing a glimpse into the behavior of the virial expansion across dimensions and coupling strengths. As an application, we calculate the pressure and Tan's contact of the two-dimensional attractive Fermi gas and examine the radius of convergence of the virial expansion as a function of the coupling strength.Following up on recent calculations, we investigate the leading- and next-to-leading-order semiclassical approximation to the virial coefficients of a two-species fermion system with a contact interaction. Using the analytic result for the second-order virial coefficient as a renormalization condition, we derive expressions for up to the seventh-order virial coefficient $\mathrm{\ensuremath{\Delta}}{b}_{7}$. Our results at leading order, though approximate, furnish simple analytic formulas that relate $\mathrm{\ensuremath{\Delta}}{b}_{n}$ to $\mathrm{\ensuremath{\Delta}}{b}_{2}$ for arbitrary dimension, providing a glimpse into the behavior of the virial expansion across dimensions and coupling strengths. As an application, we calculate the pressure and Tan's contact of the two-dimensional attractive Fermi gas and examine the radius of convergence of the virial expansion as a function of the coupling strength.

arXiv:1907.10120Physical Review ADec 12, 20199 citations

Fisher zeros of a unitary Bose gas

W. van Dijk, Calvin Lobo, Allison MacDonald +1 more

For real inverse temperature β, the canonical partition function is always positive, being a sum of positive terms. There are zeros, however, on the complex β plane that are called Fisher zeros. In the thermodynamic limit, the Fisher zeros coalesce into continuous curves. In case there is a phase transition, the zeros tend to pinch the real-β axis. For an ideal trapped Bose gas in an isotropic three-dimensional harmonic oscillator, this tendency is clearly seen, signalling Bose–Einstein condensation (BEC). The calculation can be formulated exactly in terms of the virial expansion with temperature-dependent virial coefficients. When the second virial coefficient of a strongly interacting attractive unitary gas is included in the calculation, BEC seems to survive, with the condensation temperature shifted to a lower value for the unitary gas. This shift is consistent with a direct calculation of the heat capacity from the canonical partition function of the ideal and the unitary gas.For real inverse temperature β, the canonical partition function is always positive, being a sum of positive terms. There are zeros, however, on the complex β plane that are called Fisher zeros. In the thermodynamic limit, the Fisher zeros coalesce into continuous curves. In case there is a phase transition, the zeros tend to pinch the real-β axis. For an ideal trapped Bose gas in an isotropic three-dimensional harmonic oscillator, this tendency is clearly seen, signalling Bose–Einstein condensation (BEC). The calculation can be formulated exactly in terms of the virial expansion with temperature-dependent virial coefficients. When the second virial coefficient of a strongly interacting attractive unitary gas is included in the calculation, BEC seems to survive, with the condensation temperature shifted to a lower value for the unitary gas. This shift is consistent with a direct calculation of the heat capacity from the canonical partition function of the ideal and the unitary gas.

arXiv:1412.5112Canadian Journal of PhysicsDec 19, 201421 citations

The virial expansion of ideal quantum gases reveals some interesting and amusing properties when considered as a function of dimensionality

d

. In particular, the convergence radius

ρ_c(d)

of the expansion is particulary large at {\em exactly\/}

d=3

dimensions,

ρ_c(3) = 7.1068\ldots \times \lim_{d\to3} ρ_c(d)

. The same phenomenon occurs in a few other special (non-integer) dimensions. We explain the origin of these facts, and discuss more generally the structure of singularities governing the asymptotic behavior of the ideal gas virial expansion.