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Physical review. B, Condensed matter

Generalized Lagrange theorem and thermodynamics of a multispecies quasiparticle gas with mutual fractional exclusion statistics

Kazumoto Iguchi

Published Sep 15, 1998• 8 citations
Publisher

Abstract

We discuss the relationship between the classical Lagrange theorem in mathematics and the quantum statistical mechanics and thermodynamics of an ideal gas of multispecies quasiparticles with mutual fractional exclusion statistics. First, we show that the thermodynamic potential and the density of the system are analytically expressed in terms of the language of generalized cluster expansions, where the cluster coefficients are determined from Wu's functional relations for describing the distribution functions of mutual fractional exclusion statistics. Second, we generalize the classical Lagrange theorem for inverting the one complex variable functions to that for the multicomplex variable functions. Third, we explicitly obtain all the exact cluster coefficients by applying the generalized Lagrange theorem.

DOI
10.1103/physrevb.58.6892

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Physical review. B, Condensed matter•Dec 15, 1999•4 citations
We discuss the relationship between the classical Lagrange theorem in mathematics and the quantum statistical mechanics and thermodynamics of an ideal gas of multispecies quasiparticles with mutual fractional exclusion statistics. First, we show that the thermodynamic potential and the density of the system are analytically expressed in terms of the language of generalized cluster expansions, where the cluster coefficients are determined from Wu's functional relations for describing the distribution functions of mutual fractional exclusion statistics. Second, we generalize the classical Lagrange theorem for inverting the one complex variable functions to that for the multicomplex variable functions. Third, we explicitly obtain all the exact cluster coefficients by applying the generalized Lagrange theorem.
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.
The Fermi liquid theory may provide a good description of the thermodynamic properties of an interacting particle system when the interaction between the particles contributes to the total energy of the system with a quantity which may depend on the total particle number, but does not depend on the temperature. In such a situation, the ideal part of the Hamiltonian, i.e. the energy of the system without the interaction energy, also provides a good description of the system's thermodynamics. If the total interaction energy of the system, being a complicated function of the particle populations, is temperature dependent, then the Landau's quasiparticle gas cannot describe accurately the thermodynamics of the system. A general solution to this problem is presented in this paper, in which the quasiparticle energies are redefined in such a way that the total energy of the system is identical to the sum of the energies of the quasiparticles. This implies also that the thermodynamic properties of the system and those of the quasiparticle gas are identical. By choosing a perspective in which the quasiparticle energies are fixed while the density of states along the quasiparticle axis vary, we transform our quasiparticle system into an ideal gas which obey fractional exclusion statistics.
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 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.