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arXiv:1502.01360

Properties of the Virial Expansion and Equation of State of Ideal Quantum Gases in Arbitrary Dimensions

Kåre Olaussen, Asle Sudbø

Published Feb 4, 2015• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.1502.01360

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<ノート>多体問題とグリーン関数との関係の研究--グリーン関数と多体問題(11)量子統計力学(3)

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ln this paper, next subject is discussed, §32. The equation of state of an ideal gas. Here we consider the equation of state of a spinless ideal Fermi gas. ln high temperatures and low density, we obtain the form of a virial expansion of the equation of state. The corrections to the classical ideal gas law are due to quantum effects. The second virial coefficient is deduced. In low temperature and high densities, we obtain the expansion for the chemical potential in the expansion parameter /<ε_F>. Next we obtain the asynptotic expansion of the internal energyU in the expansion parameter /<ε_F>. We also obtain the specific heat at constant volume C_V.~

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The virial expansion of ideal quantum gases reveals some interesting and amusing properties when considered as a function of dimensionality ddd. In particular, the convergence radius ρc(d)ρ_c(d)ρc​(d) of the expansion is particulary large at {\em exactly/} d=3d=3d=3 dimensions, ρc(3)=7.1068…×lim⁡d→3ρc(d)ρ_c(3) = 7.1068\ldots \times \lim_{d\to3} ρ_c(d)ρc​(3)=7.1068…×limd→3​ρc​(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.
Many-body quantum systems provide an interesting playground for experimentalists and theorists alike. In particular, ultracold atomic gases provide not only a realm of study for which experiments have a high degree of control but also provides theorists with intriguing, yet challenging systems that may be accessed computationally. It is for these reasons that it is here where the majority of our work will focus. We discuss the necessary physics, thermodynamics, and computational methods for carrying out this work, and present several methods for computing coefficients of the quantum virial expansion across dimensions for Fermi gases with zero-range interactions. The techniques discussed are non-perturbative and involve stochastic methods as well as a semi-classical lattice approximation. Through these algorithms, we have computed the interaction dependence (both attractive and repulsive where applicable) of the virial coefficients to a relatively high order as compared to current literature in one, two, and three spatial dimensions. Through one of said methods, we are able to provide a prediction for the radius of convergence of the virial expansion in one spatial dimension.
Abstract In this paper, we show how to recover the low-temperature and high-density information of ideal quantum gases from the high-temperature and low-density approximation by the Padé approximant. The virial expansion is a high-temperature and low-density expansion and in practice, often, only the first several virial coefficients can be obtained. For Bose gases, we determine the BEC phase transition from a truncated virial expansion. For Fermi gases, we recover the low-temperature and high-density result from the virial expansion.
In the framework of the Gross-Pitaevskii equation, we consider the problem of the expansion of quantum gases into a vacuum. For them, the chemical potential μ has a power-law dependence on the density n with the exponent ν = 2/D, whereD is the space dimension. For gas condensates of Bose atoms as the temperature T → 0, s scattering gives the main contribution to the interaction of atoms in the leading order in the gas parameter. Therefore, the exponent ν = 1 for an arbitrary D. In the three-dimensional case, ν = 2/3 is realized for condensates of Fermi atoms in the so-called unitary limit. For ν = 2/D, the Gross-Pitaevskii equation has an additional symmetry under Talanov transformations of the conformal type, which were first found for the stationary self-focusing of light. A consequence of this symmetry is the virial theorem relating the average size R of an expanding gas cloud to its Hamiltonian. The quantity R asymptotically increases linearly with time as t →∞. In the semiclassical limit, the equations of motion coincide with those of the hydrodynamics of an ideal gas with the adiabatic exponent γ = 1+2/D. In this approximations, self-similar solutions describe angular deformations of the gas cloud against the background of the expanding gas in the framework of equations of the Ermakov-Ray-Reid type.
ln this paper, next subject is discussed, §32. The equation of state of an ideal gas. Here we consider the equation of state of a spinless ideal Fermi gas. ln high temperatures and low density, we obtain the form of a virial expansion of the equation of state. The corrections to the classical ideal gas law are due to quantum effects. The second virial coefficient is deduced. In low temperature and high densities, we obtain the expansion for the chemical potential in the expansion parameter /<ε_F>. Next we obtain the asynptotic expansion of the internal energyU in the expansion parameter /<ε_F>. We also obtain the specific heat at constant volume C_V.~