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arXiv:1211.0926 · Journal of Physics A Mathematical and Theoretical

Finite temperature analytical results for a harmonically confined gas obeying exclusion statistics ind-dimensions

Zachary MacDonald, Brandon P. van Zyl

Published Jan 3, 2013• 5 citations
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Abstract

Closed form, analytical results for the finite-temperature one-body density matrix, and Wigner function of a d -dimensional, harmonically trapped gas of particles obeying exclusion statistics are presented. As an application of our general expressions, we consider the intermediate particle statistics arising from the Gentile statistics, and compare its thermodynamic properties to the Haldane fractional exclusion statistics. At low temperatures, the thermodynamic quantities derived from both distributions are shown to be in excellent agreement. As the temperature is increased, the Gentile distribution continues to provide a good description of the system, with deviations only arising well outside of the degenerate regime. Our results illustrate that the exceedingly simple functional form of the Gentile distribution is an excellent alternative to the generally only implicit form of the Haldane distribution at low temperatures.

DOI
10.1088/1751-8113/46/4/045001

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Closed form, analytical results for the finite-temperature one-body density matrix, and Wigner function of a d -dimensional, harmonically trapped gas of particles obeying exclusion statistics are presented. As an application of our general expressions, we consider the intermediate particle statistics arising from the Gentile statistics, and compare its thermodynamic properties to the Haldane fractional exclusion statistics. At low temperatures, the thermodynamic quantities derived from both distributions are shown to be in excellent agreement. As the temperature is increased, the Gentile distribution continues to provide a good description of the system, with deviations only arising well outside of the degenerate regime. Our results illustrate that the exceedingly simple functional form of the Gentile distribution is an excellent alternative to the generally only implicit form of the Haldane distribution at low temperatures.
We present a closed-form expression for the finite temperature first-order density matrix of an isotropic harmonically trapped ideal Fermi gas in any dimension. This constitutes a much sought after generalization of the recent results in the literature, where analytical expressions have been limited to quantities derived from the diagonal first-order density matrix. We compare our results with the Thomas-Fermi approximation (TFA) and demonstrate numerically that the TFA provides an excellent description of the first-order density matrix in the large-NNN limit. As an interesting application, we derive a closed-form expression for the finite temperature Hartree-Fock exchange energy of a two-dimensional (2D) parabolically confined quantum dot. We numerically test this result against the 2D TF exchange functional, and comment on the applicability of the local-density approximation to the exchange energy of an inhomogeneous 2D Fermi gas.
We present a new, simple closed form expression for the exact Wigner function of an ideal gas of harmonically trapped fermions or bosons at arbitrary temperature and dimensionality. One of the primary advantages of our expression is that it provides a clean separation of the spatial and momentum variables, which can significantly simplify analytical calculations involving the Wigner function. Our analysis is based on the explicit Fourier transform of the exact first-order density matrix, which affords an exceedingly simple derivation of the Wigner function. We demonstrate the utility of our approach by presenting several analytical results which may be trivially derived from the Wigner function, but otherwise require more involved calculations. We also examine the temperature and dimensionality dependence of the Fermi Wigner function in detail, thereby pointing out some subtle issues regarding the applicability of the Thomas–Fermi approximation to the Wigner function in the limit of a 'macroscopic' number of particles, and finite-temperatures.
PACS. 05.30.-d – Quantum statistical mechanics. Abstract. – We consider a gas of neutral fermionic atoms at ultra-low temperatures, with the attractive interaction tuned to Feshbach resonance. We calculate, the variation of the chemical potential and the energy per particle as a function of temperature by assuming the system to be an ideal gas obeying the Haldane-Wu fractional exclusion statistics. Our results for the untrapped gas compare favourably with the recently published Monte Carlo calculations of two groups. For a harmonically trapped gas, the results agree with experiment, and also with other published work. Consider a dilute gas of neutral fermionic atoms at a low temperature. In general, the low-energy properties of the gas are determined by the scattering length a, the number density n, and the temperature T of the gas ( the effective range r0 is small, so that r0/|a | → 0 as a becomes large). When the attractive interaction between the atoms is increased continuously by magnetic tuning from weak to strong, the scattering length a goes from a small negative to a small positive value. In between, there is a zero-energy two-body bound state, and |a|
We use the Thomas-Fermi method to examine the thermodynamics of particles obeying Haldane exclusion statistics. Specifically, we study Calogero-Sutherland particles placed in a given external potential in one dimension. For the case of a simple harmonic potential (constant density of states), we obtain the exact one-particle spatial density and a closed form for the equation of state at finite temperature, which are both new results. We then solve the problem of particles in a x2/3{x}^{2/3}x2/3 potential (linear density of states) and show that Bose-Einstein condensation does not occur for any statistics other than bosons.
We consider a gas of neutral fermionic atoms at ultra-low temperatures, with the attractive interaction tuned to Feshbach resonance. We calculate the variation of the chemical potential and the energy per particle as a function of temperature by assuming the system to be an ideal gas obeying the Haldane–Wu fractional exclusion statistics. Our results for the untrapped gas compare favourably with the recently published Monte Carlo calculations of two groups. For a harmonically trapped gas, the results agree with the experiment, and also with other published works.
We analytically calculate some thermodynamic quantities of an ideal ggg-on gas obeying generalized exclusion statistics. We show that the specific heat of a ggg-on gas (g≠0g \neq 0g=0) vanishes linearly in any dimension as T→0T \to 0T→0 when the particle number is conserved and exhibits an interesting dual symmetry that relates the particle-statistics at ggg to the hole-statistics at 1/g1/g1/g at low temperatures. We derive the complete solution for the cluster coefficients bl(g)b_l(g)bl​(g) as a function of Haldane's statistical interaction ggg in DDD dimensions. We also find that the cluster coefficients bl(g)b_l(g)bl​(g) and the virial coefficients al(g)a_l(g)al​(g) are exactly mirror symmetric (lll=odd) or antisymmetric (lll=even) about g=1/2g=1/2g=1/2. In two dimensions, we completely determine the closed forms about the cluster and the virial coefficients of the generalized exclusion statistics, which exactly agree with the virial coefficients of an anyon gas of linear energies. We show that the ggg-on gas with zero chemical potential shows thermodynamic properties similar to the photon statistics. We discuss some physical implications of our results.