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International Journal of Modern Physics B

INTEGRAL REPRESENTATION FOR THE GR AND PARTITION FUNCTION IN QUANTUM STATISTICAL MECHANICS OF EXCLUSION STATISTICS

Kazumoto Iguchi, Kazuhiko Aomoto

Published Feb 20, 2000• 4 citations
Publisher

Abstract

We derive an exact integral representation for the gr and partition function for an ideal gas with exclusion statistics. Using this we show how the Wu's equation for the exclusion statistics appears in the problem. This can be an alternative proof for the Wu's equation. We also discuss that singularities are related to the existence of a phase transition of the system.

DOI
10.1142/s0217979200000455

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We derive an exact integral representation for the gr and partition function for an ideal gas with exclusion statistics. Using this we show how the Wu's equation for the exclusion statistics appears in the problem. This can be an alternative proof for the Wu's equation. We also discuss that singularities are related to the existence of a phase transition of the system.
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.
Abstract Chapter 11 presents a statistical mechanical treatment of the quantum ideal gases, i.e., the ideal Boltzmann, fermion, and boson gases. The discussion begins with the microscopic description and the solution of the eigenvalue problem for a quantum ideal gas. It is argued that calculating the partition function is most readily accomplished in the grand canonical ensemble using a second-quantized formulation. When the particles are distinguishable, the equation of state is identical to that of a classical ideal gas. For fermions and bosons, however, the problem of computing thermodynamic properties is significantly more complex and can only be solved exactly in certain limits. Away from these limits approximations are needed and are discussed in detail. The relevant distributions - the Fermi-Dirac and Bose-Einstein distributions are derived. The local density approximation of density functional theory is derived for the ideal electron gas. For the ideal boson gas, the phenomenon of Bose-Einstein condensation is discussed
Based on the density operator's coherent state representation [Phys. Lett.A252, 281 (1999)], we extend the well-known Bose distribution's partition function Ξ [Formula: see text] for ideal photon gas [Formula: see text] to that of an assembly of photon gas interacting with nonlinear medium described by the Hamiltonian [Formula: see text], where A=(a1,a2,…,an), Ω is a symmetric matrix, and we report that the corresponding partition function is [Formula: see text] As a result, the generalized Bose distribution and thermodynamic functions are derived.