Abstract
We report exact numerical calculations of the chemical potential, condensate fraction and specific heat of N non-interacting bosons confined in an isotropic harmonic oscillator trap in one, two and three dimensions, as also for interacting bosons in a 3D trap. Quasi phase transitions (QPT) are observed in all these cases, including in one-dimension, as shown by a rapid change of several thermodynamic quantities at the transition point. The change becomes more rapid as N increases in 2D and 3D cases. However with increase in N, the sudden change in the nature of specific heat, gets gradually wiped out in 1D, while it becomes more drastic in 2D and 3D. But the sudden changes in the condensate fraction and chemical potential become more drastic as N increases, even in 1D. This shows that a QPT is possible in 1D also. We define the transition exponent, which characterizes the nature of a thermodynamic quantity at the transition point of a quasi phase transition, and evaluate them by careful numerical calculations, very near the transition temperature. These exponents are found to be independent of the size of the system. They are also the same for interacting and non-interacting bosons. These demonstrate the universality property of the transition exponents.
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References
K. Huang, Statistical Mechanics, 2nd edn. (Wiley, New York, 1987)
F. Dalfovo, S. Giorgini, L.P. Pitaevskii, S. Stringari, Rev. Mod. Phys. 71, 463 (1999)
V. Bagnato, D. Kleppner, Phys. Rev. A 44, 7439 (1991)
V.I. Yukalov, Phys. Rev. A 72, 033608 (2005)
W. Ketterle, N.J. van Druten, Phys. Rev. A 54, 656 (1996)
R.K. Pathria, Statistical Mechanics, 2nd edn. (Butterworth-Heinemann, Oxford, 1996)
C.J. Pethick, H. Smith, Bose-Einstein Condensation in Dilute Gases (Cambridge University Press, Cambridge, 2002)
V. Bagnato, D.E. Pritchard, D. Kleppner, Phys. Rev. A 35, 4354 (1987)
J.L. Roberts et al., Phys. Rev. Lett. 86, 4211 (2001)
S. Stringari, Phys. Rev. Lett. 77, 2360 (1996)
R. Napolitano, J. De Luca, V.S. Bagnato, G.C. Marques, Phys. Rev. A 55, 3954 (1997)
A. Biswas, J. Phys. B 42, 215302 (2009)
J.M. Yeomans, Statistical Mechanics of Phase Transitions (Clarendon Press, Oxford, New York, 1992)
T.K. Das, S. Canuto, A. Kundu, B. Chakrabarti, Phys. Rev. A 75, 042705 (2007)
D. Blume, C.H. Greene, Phys. Rev. A 63, 063601 (2001)
A. Kundu, B. Chakrabarti, T.K. Das, S. Canuto, J. Phys. B 40, 2225 (2007)
T.K. Das, B. Chakrabarti, Phys. Rev. A 70, 063601 (2004)
B. Chakrabarti, A. Kundu, T.K. Das, J. Phys. B 38, 2457 (2005)
A. Biswas, T.K. Das, J. Phys. B 41, 231001 (2008)
B. Chakrabarti, T.K. Das, P.K. Debnath, Phys. Rev. A 79, 053629 (2009)
T.K. Das, A. Kundu, S. Canuto, B. Chakrabarti, Phys. Lett. A 373, 258 (2009)
J.L. Ballot, M. Fabre de la Ripelle, Ann. Phys. 127, 62 (1980)
M. Fabre de la Ripelle Ann. Phys. 147, 281 (1983)
M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972)
T.K. Das, H.T. Coelho, M. Fabre de la Ripelle, Phys. Rev. C 26, 2281 (1982)
B. Chakrabarti, T.K. Das, Phys. Rev. A 78, 063608 (2008)
B. Chakrabarti, T.K. Das, Phys. Rev. A 81, 015601 (2010)
Acknowledgements
We would like to thank Dr. Parongama Sen for drawing our attention to the critical exponent and universality, as well as for useful discussions. S.G. acknowledges CSIR (India) for a Senior Research Fellowship (Sanction No.: 09/028(0762)/2010-EMR-I). TKD acknowledges DST (India) for financial assistance through the USERS program.
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Goswami, S., Das, T.K. & Biswas, A. Thermodynamic Properties of Ultracold Bose Gas: Transition Exponents and Universality. J Low Temp Phys 172, 184–201 (2013). https://doi.org/10.1007/s10909-013-0860-3
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DOI: https://doi.org/10.1007/s10909-013-0860-3