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arXiv:2308.10891 · Revista Mexicana de Física

On Dunkl-Bose-Einstein Condensation in Harmonic Traps

A. Hocine, B. Hamil, F. Merabtine, B. C. Lütfüoğlu, Mohamed Benarous

Published Sep 1, 2024• 12 citations
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Abstract

The use of the Dunkl derivative, defined by a combination of the difference-differential and reflection operators, allows the classification of the solutions according to even and odd solutions. Recently, we considered the Dunkl formalism to investigate the Bose-Einstein condensation of an ideal Bose gas confined in a gravitational field. In this work, we address another essential problem and examine an ideal Bose gas trapped by a three-dimensional harmonic oscillator within the Dunkl formalism. To this end, we derive an analytic expression for the critical temperature of the N particle system, discuss its value at large-N limit andfinally derive and compare the ground state population with the usual case result. In addition, we explore two thermal quantities, namely the Dunkl-internal energy and the Dunkl-heat capacity functions.

DOI
10.31349/revmexfis.70.051701

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The use of the Dunkl derivative, defined by a combination of the difference-differential and reflection operators, allows the classification of the solutions according to even and odd solutions. Recently, we considered the Dunkl formalism to investigate the Bose-Einstein condensation of an ideal Bose gas confined in a gravitational field. In this work, we address another essential problem and examine an ideal Bose gas trapped by a three-dimensional harmonic oscillator within the Dunkl formalism. To this end, we derive an analytic expression for the critical temperature of the N particle system, discuss its value at large-N limit andfinally derive and compare the ground state population with the usual case result. In addition, we explore two thermal quantities, namely the Dunkl-internal energy and the Dunkl-heat capacity functions.
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.
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Bose–Einstein condensation is the anomalous accumulation of particles in the ground state of a system of bosons, when compared with the population in a system of particles obeying classical statistics. I use undergraduate-level statistical mechanics and a symbolic algebra computer program to study the occupation numbers of the energy levels of a finite number of noninteracting particles confined in a three-dimensional harmonic oscillator potential. I also calculate the heat capacity of the gas. The harmonic oscillator potential simplifies the calculations and approximates the conditions of the recent experiments achieving Bose–Einstein condensation in laser-cooled alkali vapors.
The critical temperature for Bose-Einstein condensation, ground-state fraction, and heat capacity of an ideal gas of Bose particles that are confined by an n-dimensional generic power-law potential trap are derived. The conditions for Bose-Einstein condensation and the discontinuous conditions of the heat capacity at critical temperature are obtained. All these quantities and conditions are found to vary markedly with the dimensionality of space, characteristics of particles, and shape of the external potential. The results obtained here are universal.
Abstract We have proposed an analytical solution to the problem of Bose–Einstein condensation (BEC) of harmonically trapped, two-dimensional, and ideal atoms. It is found that the number of atoms in vapor is characterized by an analytical function, which involves a series of q -digamma functions in mathematics. We employ the analytical solution to calculate the internal energy E , the entropy S , the Helmholtz free energy F , and the heat capacity at constant number C N of ideal Bose atoms in a two-dimensional isotropic harmonic trap. The first main finding in this paper is that the variation with temperature of the internal energy of a finite number of ideal Bose atoms has an inflection point, which occurs at the transition temperature T c . The second main finding is that the internal energy of a finite number of ideal Bose atoms has a classic limit E = 2 N k B T as T ≫ T c . The third main finding is that the variation with temperature of the heat capacity of a finite number of ideal Bose atoms has a maximal value, which occurs at T c . The fourth main finding is that the heat capacity of a finite number of ideal Bose atoms has a classic limit C N / k B = 2 N as T ≫ T c . The fifth main finding is that in the thermodynamic limit, at critical temperature T c the heat capacity at constant number shows a cusp singularity, which is analogous to the λ -transition of liquid helium. Consequently, the transition between normal and condensed states is a second-order phase transition.