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Molecular Physics

Intermediate statistics

A. M. Guénault, D. K. C. MacDonald

Published Sep 1, 1962• 9 citations
Publisher

Abstract

The properties of a gas of particles obeying a hypothetical ‘intermediate statistics’ (first proposed by Gentile) are discussed. Although of no known direct application to a physical system, the theoretical results can help in understanding the physical properties of Fermi-Dirac and Bose-Einstein statistics.
DOI
10.1080/00268976200100591

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We overwiev the properties of a quantum gas of particles with the intermediate statistics defined by Haldane. Although this statistics has no direct connection to the symmetry of the multiparticle wave function, the statistical distribution function interpolates continuously between the Fermi-Dirac and the Bose-Einstein limits. We present an explicit solution of the transcendental equation for the didtribution function in a general case, as well as determine the thermodynamic properties in both low- and high-temperature limits.
It is shown that intermediate statistics do not correspond to any physical process. The stationary probability distributions of intermediate statistics are not compatible with any mechanism which allows a variation between Fermi–Dirac and Bose–Einstein statistics. The binomial and negative binomial distributions, characterizing Fermi–Dirac and Bose–Einstein statistics, respectively, transform into the Poisson distribution, descriptive of classical statistics, as the number of energy cells increases without limit. These distributions are shown to be the laws of error leading to the average value as the most probable value.
Beginning with the assumption that the quantum state of a many-particle system is a functional on the internal space of the particles, a unified quantum statistics, which may interpolate smoothly between Bose-Einstein and Fermi-Dirac statistics, is proposed. The commutation relations for such particle creation and annihilation operators are derived, and statistical partition function and thermodynamical properties of an ideal gas of the particles are investigated. Besides the Bose-Einstein condensation, another kind of particle condensation may appear.
Analytic expressions for the distribution function of an ideal gas of particles (exclusons) which have statistics intermediate between Fermi-Dirac and Bose-Einstein are obtained for all values of the Haldane statistics parameter \ensuremath{\alpha}\ensuremath{\in}[0,1]. The analytic structure of the distribution function is investigated and found to have no singularities in the physical region when the parameter \ensuremath{\alpha} lies in the range 0\ensuremath{\le}1. High- and low-temperature series are also derived for the internal energy E and heat capacity CV{\mathit{C}}_{\mathit{V}}CV​ of the D-dimensional excluson gas. The low-temperature series for the thermodynamic properties illustrate the pseudofermion nature of exclusons. \textcopyright{} 1996 The American Physical Society.
Synopsis: ƒe connection between a particle’s spin and the statistics it obeys is given. Bose-Einstein and Fermi-Dirac statistics are explained and illustrated with products of wave functions. A new type of particle basis state is described and two possibilities for the commutation properties of the associated raising (“creation”) and lowering (“annihilation”) ladder operators are examined, giving rise to the Bose-Einstein and Fermi-Dirac cases. ƒe relation between particle occupation number and energy is inferred from detailed balance, and cubical enclosures of particles are studied. Two physics mysteries encountered in the earlier chapters (“zweideutigkeit” and the vanishing electronic component of specific heat) are explained. ƒe Hartree-Fock theoretical framework for the approximate treatment of many-particle atomic systems is constructed.