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 provide a simple approach to the analytical evaluation of inverse integral transforms that does not require any knowledge of complex analysis. The central idea behind our method is to reduce the inverse transform to the solution of an ordinary differential equation. We illustrate the utility of our approach by providing examples of the evaluation of transforms without the use of tables. We also demonstrate how the method may be used to obtain a general representation of a function in the form of a series involving the Dirac delta distribution and its derivatives, which has applications in quantum mechanics, semiclassical, and nuclear physics. PACS Nos: 02.30.Uu, 02.30.Zz, 01.40.gbRésumé : Nous proposons une approche simple pour évaluer analytiquement l'inverse d'une transformation intégrale, qui ne requiert pas de connaître l'analyse complexe. L'idée centrale de la méthode est de remplacer la transformation inverse par la solution d'une équation différentielle ordinaire. Nous illustrons l'utilité de l'approche à l'aide d'exemples d'évaluation de transformations, sans utiliser de tables. Nous démontrons aussi comment la méthode peut être utilisée pour obtenir une représentation générale d'une fonction sous la forme d'une série impliquant la distribution delta de Dirac et de ses dérivées, avec applications en mécanique quantique et semi-classique et en physique nucléaire.[Traduit par la Rédaction]
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