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Physical review. B, Condensed matter

Fractional exclusion statistics of two-species gases

Wung-Hong Huang

Published Dec 15, 1999• 4 citations
Publisher

Abstract

The thermodynamics of two-species gases described by the Haldane exclusion statistics is investigated. I use the high-temperature perturbation to find the third Virial coefficient for the gas in an arbitrary dimension. The perturbative result indicates that the two-dimensional thermodynamic potential becomes the linear function of the element of the Haldane statistics matrix, if the matrix is symmetric and particle mass is species independent. I also present a nonperturbative method to prove this factorizable property and an exact expression of the thermodynamic potential is suggested.

DOI
10.1103/physrevb.60.15742

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We propose that the notion of Haldane's dimension and the corresponding statistics be treated in a probabilistic spirit. Motivated by the experience of dimensional regularization, the dimension of a space is defined as a trace of a diagonal ``unit operator'' where the diagonal matrix elements are not in unity in general but are probabilities to find a system in a given state. These probabilities are then uniquely defined by the rules of Haldane's statistics. The partition function for an ideal gas of the particles, a state-counting procedure, the entropy and a distribution function for this probabilistic definition are investigated and compared with previous works in the context of g particles. The corresponding creation and annihilation operators are introduced and Hamiltonians for interacting g particles are constructed. The stability of the thermodynamical properties of a gas of particles with exclusion statistics under the influence of hopping (or scattering) between states is investigated. \textcopyright{} 1996 The American Physical Society.

Physical review. B, Condensed matter•Feb 1, 1996•6 citations
The thermodynamics of two-species gases described by the Haldane exclusion statistics is investigated. I use the high-temperature perturbation to find the third Virial coefficient for the gas in an arbitrary dimension. The perturbative result indicates that the two-dimensional thermodynamic potential becomes the linear function of the element of the Haldane statistics matrix, if the matrix is symmetric and particle mass is species independent. I also present a nonperturbative method to prove this factorizable property and an exact expression of the thermodynamic potential is suggested.
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.
The quantum statistical mechanics of an ideal gas with fractional exclusion (i.e., Haldane-Wu) statistics in arbitrary dimensions is discussed. The general formulation for pressure and density of the system is obtained in a closed form in terms of the DDD-dimensional momentum representation, which can be regarded as a natural generalization of the classic results for Fermi and Bose gases. Using this, it is shown that ideal gases with fractional exclusion statistics can be regarded as composites of fermions and bosons, and that no condensation occurs at low temperature except for the pure boson case.
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 propose that the notion of Haldane's dimension and the corresponding statistics be treated in a probabilistic spirit. Motivated by the experience of dimensional regularization, the dimension of a space is defined as a trace of a diagonal ``unit operator'' where the diagonal matrix elements are not in unity in general but are probabilities to find a system in a given state. These probabilities are then uniquely defined by the rules of Haldane's statistics. The partition function for an ideal gas of the particles, a state-counting procedure, the entropy and a distribution function for this probabilistic definition are investigated and compared with previous works in the context of g particles. The corresponding creation and annihilation operators are introduced and Hamiltonians for interacting g particles are constructed. The stability of the thermodynamical properties of a gas of particles with exclusion statistics under the influence of hopping (or scattering) between states is investigated. \textcopyright{} 1996 The American Physical Society.