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Journal of Physics A Mathematical and General

Bose–Einstein condensation of a two-parameter deformed quantum group boson gas

Abdullah Algin, Bayram Deviren

Published Jun 15, 2005• 22 citations
Publisher

Abstract

We investigate the low temperature behaviour of a two-parameter generalized bosonic quantum gas with -symmetry, where q 1 and q 2 are real independent deformation parameters. We calculate, in the thermodynamical limit, several statistical and thermodynamical functions of the system through an -invariant bosonic Hamiltonian. In the low and high temperature limits, the specific heat of the system is obtained in terms of some functions of the deformation parameters q 1 and q 2 . At the critical temperature being higher than that of the free boson gas, the specific heat of the two-parameter generalized boson gas exhibits a λ-point transition behaviour. We also discuss the conditions under which the Bose–Einstein condensation would occur in the present two-parameter generalized boson model. However, the free boson gas results can be recovered in the limit q 1 = q 2 = 1.

DOI
10.1088/0305-4470/38/26/008

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We investigate the low temperature behaviour of a two-parameter generalized bosonic quantum gas with -symmetry, where q 1 and q 2 are real independent deformation parameters. We calculate, in the thermodynamical limit, several statistical and thermodynamical functions of the system through an -invariant bosonic Hamiltonian. In the low and high temperature limits, the specific heat of the system is obtained in terms of some functions of the deformation parameters q 1 and q 2 . At the critical temperature being higher than that of the free boson gas, the specific heat of the two-parameter generalized boson gas exhibits a λ-point transition behaviour. We also discuss the conditions under which the Bose–Einstein condensation would occur in the present two-parameter generalized boson model. However, the free boson gas results can be recovered in the limit q 1 = q 2 = 1.
We study the Bose-Einstein condensation of a gas with SUq(2){\mathrm{SU}}_{q}(2)SUq​(2) symmetry. We show, in the thermodynamic limit, that the boson interactions introduced by the quantum group symmetries enhance Bose-Einstein condensation giving a discontinuity in the heat capacity Cv{C}_{v}Cv​ at the critical temperature Tc.{T}_{c}.Tc​. The critical temperature and the gap in Cv{C}_{v}Cv​ increase with the value of the parameter qqq and become approximately constant for q>3q>3q>3.
The q -deformed Bose–Einstein distribution derived from the q -boson algebra is used to study the low-temperature behavior of an ideal q -deformed Bose gas with relativistic energy spectrum. The effects of q -deformation on the properties relative to Bose–Einstein condensation (BEC) are discussed. It is shown that q -deformation leads to some novel characteristics different from those of an original Bose gas, which include the criteria on the occurrence of BEC, critical temperature and jump of heat capacity at the critical point. The results obtained here provide a unified description for the properties of q -deformed Bose systems from the nonrelativistic case to the ultrarelativistic limit, so that some important conclusions in the literature are included in this paper.
Abstract Based on the framework established in a recent work (Ubriaco in Physica A 414:128, 2014), wherein a generalized bosonic model was formulated using the Dunkl operator formalism (DOF), we extend this approach to its different applications with introducing a statistical distribution—termed the Dunkl-deformed Bose–Einstein (DDBE) distribution—constructed by generalizing the usual Bose–Einstein distribution through the DOF. In this framework, we study further general thermodynamical and statistical properties of a gas model of the Dunkl-deformed bosons obeying the DDBE statistics. Emphasis is given to a careful analysis on the conditions under which the Bose–Einstein-like condensation would occur in the present Dunkl-deformed boson gas. It is shown that the critical temperature for such a system is higher than that of the usual (undeformed) Bose gas for values of the model deformation parameter θ\thetaθ θ in the range θ>0\theta > 0θ>0 θ > 0 . As another application of the DDBE statistics, we investigate the thermostatistics of the Dunkl-deformed Debye solid and the effects of the deformation parameter θ\thetaθ θ on the high- and low-temperature behavior of the model specific heat are discussed. Our results obtained in this study could provide much physical insight for approaching not only in the treatment of interacting bosons theories including collective excitations, but also in the condensation characteristics of quantum systems beyond the standard formalism.
Abstract Bose–Einstein condensation in a Bose gas is studied analytically, in any positive dimensionality ( d > 0) for identical bosons with any energy-momentum positive-exponent ( s > 0) plus an energy gap Δ between the ground state energy ε 0 and the first excited state, i.e. ε = ε 0 for k = 0 and ε = ε 0 + Δ + c s k s , for k > 0, where ℏ k is the particle momentum and c s a constant with dimensions of energy multiplied by a length to the power s > 0. Explicit formula with arbitrary d / s and Δ are obtained and discussed for the critical temperature and the condensed fraction, as well as for the equation of state from where we deduce a generalized Δ independent thermal de Broglie wavelength. Also the internal energy is calculated from where we obtain the isochoric specific heat and its jump at T c . When Δ > 0, for any d > 0 exists a Bose–Einstein critical temperature T c ≠ 0 where the internal energy shows a peak and the specific heat shows a jump. Both the critical temperature and the specific heat jump increase as functions of the gap but they decrease as functions of d / s . At sufficiently high temperatures Δ-independent classical results are recovered while for temperatures below the critical one the gap effects are predominant. For Δ = 0 we recover previous reported results.