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Journal of Physics A Mathematical and General

Planck distribution for a q-boson gas

Miguel Angel Martin-Delgado

Published Nov 7, 1991• 73 citations
Publisher

Abstract

The energy-density distributions for bosons obeying the q-deformation of the harmonic oscillator algebra have been studied in order to obtain some physical insight into the parameter q. The action of this parameter on the energy distributions resembles the action of the temperature parameter in the classical Planck law (q=1).
DOI
10.1088/0305-4470/24/21/006

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Just as for the ordinary quantum harmonic oscillators, we expect the zero-point energy to play a crucial role in the correct high temperature behavior. We accordingly reformulate the theory of the statistical distribution function for the q-deformed boson oscillators and develop an approximate theory incorporating the zero-point energy. We are then able to demonstrate that for small deformations, ln q≪1, the theory reproduces the correct limits both for very high temperatures and for very low temperatures. The deformed theory thus reduces to the undeformed theory in these extreme cases.
In the framework of the q -deformed oscillator algebra we investigate the nature of the interaction among the particles of a free q -boson gas with a view to interpreting the interaction in terms of the general properties of the q -deformation. It is shown that many thermodynamic quantities characterizing the q -boson system can be expressed as power series in the number of particles operator or as a series in the deformation parameter q , implying that the q -deformation can originate from the interaction among the particles of the ensemble. As an example, we derive the virial expansion of the q -boson gas, showing that the lowest order coefficient reduces to the standard virial coefficient of an undeformed boson gas, while the higher order coefficients contain effects arising from interaction.
Complex-q boson gas as an application of complex-q oscillator algebra is reinvestigated by using an algebra that was introduced recently by us 5 as an improvement over the earlier studies of Gupta et al.2 and Angelopoulou et al.3 The improvements are in terms of the use of Hermitian operators (real eigenvalues) and constant boson number. The formalism is therefore exact and results in the modifications of some of the approximate results of Gupta et al.2 A possible connection of the q-deformed, effective Planck constant to the (minimum) q-uncertainty relation is also discussed.