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The European Physical Journal Plus

Thermostatistics of the Dunkl-deformed boson gas: algebra, Fock space, condensation, and Debye crystal model

Abdullah Algin, Won Sang Chung

Published Mar 16, 2026• 2 citations
Publisher

Abstract

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 $$\theta$$ θ in the range $$\theta > 0$$ 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 $$\theta$$ 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.

DOI
10.1140/epjp/s13360-026-07440-y

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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.
A q-deformed field theory at finite temperature is presented and its related thermo field dynamics is constructed. This enables us to introduce the quon field, whose Lagrangian leads to a q-deformed partition function. Accordingly, the general thermostatistical properties of a gas model of quon fields, such as the q-deformed statistical distribution function describing an intermediate-statistics behavior and the equation of state in two and three dimensions, are investigated. For low temperatures, the conditions under which Bose–Einstein-like condensation would occur in the quon gas model are discussed. It is shown that the critical temperature of such a gas is higher than that of the usual Bose gas for values of the model parameter q in the range [Formula: see text]. For high temperatures, possible anyonic behavior of the present quon gas model in two spatial dimensions is studied through an analysis of the effect of deformation on the second and third virial coefficients in the equation of state. The results obtained in this work reveal the ability of the model for analyzing a parastatistical behavior of systems with quasiparticles and put forward it as a possible candidate for effectively modeling the properties of some exotic quantum states, such as in the case of dark matter constituents.
Abstract Recently, deformed quantum systems have received lots of attention in the literature. Dunkl formalism differs from others by containing the difference-differential and reflection operator. It is one of the most interesting deformations since it let us discuss the solutions according to the even and odd solutions. In this work, we studied the ideal Bose gas and the blackbody radiation via the Dunkl formalism. To this end, we made a liaison between the coordinate and momentum operators with the creation and annihilation operators, which allowed us to obtain the expressions of the partition function, the condensation temperature, and the ground state population of the Bose gas. We found that Dunkl-condensation temperature increases with increasing θ value. In the blackbody radiation phenomena, we found how the Dunkl formalism modifies total radiated energy. Then, we examined the thermal quantities of the system. We found that the Dunkl deformation causes an increase in entropy and specific heat functions as well as in the total radiation energy. However, we observed a decrease in the Dunk-corrected Helmholtz free energy in this scenario. Finally, we found that the equation of state is invariant even in the considered formalism.
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.
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.
The thermostatistical properties of an ideal gas of q -deformed bosons trapped in a D -dimensional power-law potential are studied, based on the q -deformed Bose–Einstein distribution. The effects of q -deformation on the properties of the system are discussed. It is shown that q -deformed bosons ( q ≠ 1) possess many different characteristics from those of ordinary bosons, which include the condition that Bose–Einstein condensation (BEC) occurs, the critical temperature and the continuity of heat capacity.