On Dunkl-Bose-Einstein Condensation in Harmonic Traps
DOI:
https://doi.org/10.31349/RevMexFis.70.051701Keywords:
Bose Einstein condensate, Dunkl derivative; harmonic potential traps; thermal quantitiesAbstract
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 and
finally 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.
References
V. Bagnato, D. E. Pritchard, and D. Kleppner, Bose-Einstein condensation in an external potential, Phys. Rev. A 35, (1987) 4354, https://doi.org/10.1103/PhysRevA.35.4354
H. A. Gersch, Bose-Einstein Gas in a Gravitational Field, J. Chem. Phys. 27 (1957) 928, https://doi.org/10.1063/1.1743881
A. Widom, Superfluid Phase Transitions in One and Two Dimensions, Phys. Rev. 176 (1968) 254, https://doi.org/10.1103/PhysRev.176.254
D. B. Baranov, and V. S. Yarunin, Bose-Einstein condensation in atomic trap: gravity shift of critical temperature, Phys. Lett. A 285 (2001) 34, https://doi.org/10.1016/S0375-9601(01)00325-5
J. I. Rivas, and A. Camacho, Bose-Einstein condensates in a homogeneous gravitational field, Mod. Phys. Lett. A 26 (2011) 481, https://doi.org/10.1142/S0217732311035109
T. G. Liu et al., Bose-Einstein condensation of bouncing balls, Physica A 388 (2009) 2383, https://doi.org/10.1016/j.physa.2009.02.035
C. F. Du, H. Li, Z. Q. Lin, and X. M. Kong, The condensation of ideal Bose gas in a gravitational field, Physica B 407 (2012) 4375, https://doi.org/10.1016/j.physb.2012.07.037
K. Kirsten, and D. J. Toms, Bose-Einstein condensation of atomic gases in a general harmonic-oscillator confining potential trap, Phys. Rev. A 54 (1996) 4188, https://doi.org/10.1103/PhysRevA.54.4188
W. Ketterle, and N. J. van Druten, Bose-Einstein condensation of a finite number of particles trapped in one or three dimensions, Phys. Rev. A 54 (1996) 656, https://doi.org/10.1103/PhysRevA.54.656
W. J. Mullin, Bose-Einstein condensation in a harmonic potential, J. Low Temp. Phys. 106 (1997) 615, https://doi.org/10.1007/BF02395928
Q. -J. Zeng, Z. Cheng, and J. -H. Yuan, Bose-Einstein condensation of a q-deformed boson system in a harmonic potential trap, Physica A 391 (2012) 563, https://doi.org/10.1016/j.physa.2011.09.011
Q. -J. Zeng, Y. -S. Luo, Y. -G. Xu, and H. Luo, Bose-Einstein condensation of a two-dimensional harmonically trapped qdeformed boson system, Physica A 398 (2014) 116, https://doi.org/10.1016/j.physa.2013.12.021
W. J. Mullin, and A. R. Sakhel, Generalized Bose-Einstein Condensation, J. Low Temp. Phys. 166 (2012) 125, https://doi.org/10.1007/s10909-011-0412-7
E. P. Wigner, Do the Equations of Motion Determine the Quantum Mechanical Commutation Relations?, Phys. Rev. 77 (1950) 711, https://doi.org/10.1103/PhysRev.77.711
L. M. Yang, A Note on the Quantum Rule of the Harmonic Oscillator, Phys. Rev. 84 (1951) 788, https://doi.org/10.1103/PhysRev.84.788
W. S. Chung, and H. Hassanabadi, New deformed Heisenberg algebra with reflection operator, Eur. Phys. J. Plus 136 (2021) 239, https://doi.org/10.1140/epjp/s13360-021-01186-5
C. F. Dunkl, Differential-difference operators associated to reflection groups, T. Am. Math. Soc. 311 (1989) 167, https://doi.org/10.1090/S0002-9947-1989-0951883-8
R. D. Mota, D. Ojeda-Guillén, M. Salazar-Ramírez, and V. D. Granados, Exact solutions of the 2D Dunkl-Klein-Gordon equation: The Coulomb potential and the Klein-Gordon oscillator, Mod. Phys. Lett. A 36 (2021) 2150171, https://doi.org/10.1142/S0217732321501716
G. J. Heckman, A Remark on the Dunkl Differential-Difference Operators, Prog. Math. 101 (1991) 181, https://doi.org/10.1007/978-1-4612-0455-8_8
C. F. Dunkl, and Y. Xu, Orthogonal Polynomials of Several Variables, Encyclopedia of Mathematics and Its Applications 81 (Cambridge University Press, Cambridge, 2001)
H. de Bie, B. Orsted, P. Somberg, and V. Soucek, Dunkl operators and a family of realizations of osp(1|2), Trans. Am. Math. Soc. 364 (2012) 3875, https://doi.org/10.1090/S0002-9947-2012-05608-X
M. S. Plyushchay, Deformed Heisenberg Algebra, Fractional Spin Fields, and Supersymmetry without Fermions, Ann. Phys. 245 (1996) 339, https://doi.org/10.1006/aphy.1996.0012
S. Kakei, Common algebraic structure for the Calogero - Sutherland models, J. Phys. A 29 (1996) L619, https://doi.org/10.1088/0305-4470/29/24/002
L. Lapointe, and L. Vinet, Exact Operator Solution of the Calogero-Sutherland Model, Commun. Math. Phys. 178 (1996) 425, https://doi.org/10.1007/BF02099456
M. S. Plyushchay, Deformed Heisenberg algebra with reflection, Nucl. Phys. B 491 (1997) 619, https://doi.org/10.1016/S0550-3213(97)00065-5
M. S. Plyushchay, Hidden Nonlinear Supersymmetries in pure Parabosonic systems, Int. J. Mod. Phys. A 15 (2000) 3679, https://doi.org/10.1142/S0217751X00001981
V. Genest, M. Ismail, L. Vinet, and A. Zhedanov, The Dunkl oscillator in the plane: I. Superintegrability, separated wavefunctions and overlap coefficients, J. Phys. A 46 (2013) 145201, https://doi.org/10.1088/1751-8113/46/14/145201
V. Genest, L. Vinet, and A. Zhedanov, The singular and the 2:1 anisotropic Dunkl oscillators in the plane, J. Phys. A 46 (2013) 325201, https://doi.org/10.1088/1751-8113/46/32/325201
V. Genest, M. Ismail, L. Vinet, and A. Zhedanov, The Dunkl Oscillator in the Plane II: Representations of the Symmetry Algebra, Commun. Math. Phys. 329 (2014) 999, https://doi.org/10.1007/s00220-014-1915-2
V. Genest, L. Vinet, and A. Zhedanov, The Dunkl oscillator in three dimensions, J. Phys. Conf. Ser. 512 (2014) 012010, https://doi.org/10.1088/1742-6596/512/1/012010
W. S. Chung, and H. Hassanabadi, One-dimensional quantum mechanics with Dunkl derivative, Mod. Phys. Lett. A 34 (2019) 1950190, https://doi.org/10.1142/S0217732319501906
H. Hassanabadi, M. de Montigny, W. S. Chung, and P. Sedaghatnia, Superstatistics of the Dunkl oscillator, Physica A 580 (2021) 126154, https://doi.org/10.1016/j.physa.2021.126154
R. D. Mota, D. Ojeda-Guillén, M. Salazar-Ramírez, and V. D. Granados, Exact solutions of the 2D Dunkl-Klein-Gordon equation: The Coulomb potential and the Klein-Gordon oscillator, Mod. Phys. Lett. A 36 (2021) 2150171, https://doi.org/10.1142/S0217732321501716
W. S. Chung, and H. Hassanabadi, The Wigner-DunklNewton mechanics with time-reversal symmetry, Rev. Mex. Fis. 66 (2020) 308, https://doi.org/10.31349/RevMexFis.66.308
M. R. Ubriaco, Thermodynamics of boson systems related to Dunkl differential-difference operators, Physica A 414 (2014) 128, https://doi.org/10.1016/j.physa.2014.06.087
Y. Kim, W. S. Chung, and H. Hassanabadi, Deviation of inverse square law based on Dunkl derivative: deformed Coulomb’s law, Rev. Mex. Fis. 66 (2020) 411, https://doi.org/10.31349/RevMexFis.66.411
H. Hassanabadi, M. de Montigny, and W. S. Chung, Superstatistics of the Dunkl oscillator, Physica A 580 (2021) 126154, https://doi.org/10.1016/j.physa.2021.126154
S. H. Dong, W. H. Huang, W. S. Chung, P. Sedaghatnia, and H. Hassanabadi, Exact solutions to generalized Dunkl oscillator and its thermodynamic properties, EPL 135 (2021) 30006, https://doi.org/10.1209/0295-5075/ac2453
S. Sargolzaeipor, H. Hassanabadi, and W. S. Chung, Effect of the Wigner-Dunkl algebra on the Dirac equation and Dirac harmonic oscillator, Mod. Phys. Lett. A 33 (2018) 1850146, https://doi.org/10.1142/S0217732318501468
S. Ghazouani, I. Sboui, M. A. Amdouni, and M. B. El Hadj Rhouma, The Dunkl-Coulomb problem in three-dimensions: energy spectrum, wave functions and h-spherical harmonics, J. Phys. A: Math. Theor. 52 (2019) 225202, https://doi.org/10.1088/1751-8121/ab0d98
R. D. Mota, D. Ojeda-Guillén, M. Salazar-Ramírez, and V. D. Granados, Exact solution of the relativistic Dunkl oscillator in (2 + 1) dimensions, Ann. Phys. 411 (2019) 167964, https://doi.org/10.1016/j.aop.2019.167964
B. Hamil, and B. C. Lütfüoğlu, Dunkl graphene in constant magnetic field, Eur. Phys. J. Plus 137 (2022) 1241, https://doi.org/10.1140/epjp/s13360-022-03463-3
R. D. Mota, D. Ojeda-Guillén, M. Salazar-Ramírez, and V. D. Granados, Landau levels for the (2 + 1) Dunkl-Klein-Gordon oscillator, Mod. Phys. Lett. A 36 (2021) 2150066, https://doi.org/10.1142/S0217732321500668
A. Merad, and M. Merad, The Dunkl-Duffin-Kemmer-Petiau Oscillator, Few-Body Syst. 62 (2021) 98, https://doi.org/10.1007/s00601-021-01683-4
B. Hamil, and B. C. Lütfüoğlu, Dunkl-Klein-Gordon Equation in Three-Dimensions: The Klein-Gordon Oscillator and Coulomb Potential, Few-Body Syst. 63 (2022) 74, https://doi.org/10.1007/s00601-022-01776-8
B. Hamil, and B. C. Lütfüoğlu, Thermal properties of relativistic Dunkl oscillators, Eur. Phys. J. Plus 137 (2022) 812, https://doi.org/10.1140/epjp/s13360-022-03055-1
B. Hamil, and B. C. Lütfüoğlu, The condensation of ideal Bose gas in a gravitational field in the framework of Dunkl-statistic, Physica A 623 (2023) 128841, https://doi.org/10.1016/j.physa.2023.128841
F. Merabtine, B. Hamil, B. C. Lütfüoğlu, A. Hocine, and M. Benarous, Ideal Bose gas and blackbody radiation in the Dunkl formalism, J. Stat. Mech. 2023 (2023) 053102, https://doi.org/10.1088/1742-5468/acd106
B. Hamil, and B. C. Lütfüoğlu, GUP to all Orders in the Planck Length: Some Applications, Int. J. Theor. Phys. 61 (2022) 202, https://doi.org/10.1007/s10773-022-05188-6
L. N. Chang, D. Minic, N. Okamura, and T. Takeuchi, Exact solution of the harmonic oscillator in arbitrary dimensions with minimal length uncertainty relations, Phys. Rev. D 65 (2002) 125027, https://doi.org/10.1103/PhysRevD.65. 125027
P. Bosso, Rigorous Hamiltonian and Lagrangian analysis of classical and quantum theories with minimal length, Phys. Rev. D 97 (2018) 126010, https://doi.org/10.1103/PhysRevD.97.126010
S. Grossmann, and M. Holthaus, On Bose-Einstein condensation in harmonic traps, Phys. Lett. A 208 (1995) 188, https://doi.org/10.1016/0375-9601(95)00766-V
L. F. Matin, and S. Miraboutalebi, Statistical aspects of harmonic oscillator under minimal length supposition, Physica A. 425 (2015) 10, https://doi.org/10.1016/j.physa.2015.01.041
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2024 A. Hocine, B. Hamil, F. Merabtine, Bekir Can Lütfüoğlu, M. Benarous
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Authors retain copyright and grant the Revista Mexicana de Física right of first publication with the work simultaneously licensed under a CC BY-NC-ND 4.0 that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
