Annals of Physics

Volume 270, Issue 1, 20 November 1998, Pages 198-230

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Regular Article

Condensate Fluctuations in Trapped Bose Gases: Canonical vs. Microcanonical Ensemble ☆ ☆☆

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Martin Holthaus, Eva Kalinowski, Klaus Kirsten
Fachbereich Physik, Philipps-Universität, Renthof 6, D-35032, Marburg, Germanyf1E-mail: holthaus@stat.physik.uni.marburg.def1Institut für Theoretische Physik, Universität Leipzig, Augustusplatz 10, D-04109, Leipzig, Germany

Received 22 April 1998, Available online 15 April 2002

doi:10.1006/aphy.1998.5852

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Abstract

We study the fluctuation of the number of particles in ideal Bose–Einstein condensates, both within the canonical and the microcanonical ensemble. Employing the Mellin–Barnes transformation, we derive simple expressions that link the canonical number of condensate particles, its fluctuation, and the difference between canonical and microcanonical fluctuations to the poles of a Zeta function that is determined by the excited single-particle levels of the trapping potential. For the particular examples of one- and three-dimensional harmonic traps we explore the microcanonical statistics in detail, with the help of the saddle-point method. Emphasizing the close connection between the partition theory of integer numbers and the statistical mechanics of ideal Bosons in one-dimensional harmonic traps, and utilizing thermodynamical arguments, we also derive an accurate formula for the fluctuation of the number of summands that occur when a large integer is partitioned.
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References

    REFERENCES

  • 1 

    E. Schrödinger

    Statistical Thermodynamics, Cambridge Univ. Press, Cambridge (1946)

  • 2 

    M. Fierz

    Helv. Phys. Acta, 29 (1956), p. 47

  • 3 

    I. Fujiwara, D. ter Haar, H. Wergeland

    J. Stat. Phys., 2 (1970), p. 329

  • 4 

    R.M. Ziff, G.E. Uhlenbeck, M. Kac

    Phys. Rep., 32 (1977), p. 169

    Article PDF (5004K)

  • 5 

    M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, E.A. Cornell

    Science, 269 (1995), p. 198

  • 6 

    K.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn, W. Ketterle

    Phys. Rev. Lett., 75 (1995), p. 3969

  • 7 

    C.C. Bradley, C.A. Sackett, R.G. Hulet

    Phys. Rev. Lett., 78 (1997), p. 985

  • 8 

    U. Ernst, A. Marte, F. Schreck, J. Schuster, G. Rempe

    Europhys. Lett., 41 (1998), p. 1

  • 9 

    H.-J. Miesner, W. Ketterle, Solid State Commun.

  • 10 

    S. Grossmann, M. Holthaus

    Phys. Rev. E, 54 (1996), p. 3495

  • 11 

    H.D. Politzer

    Phys. Rev. A, 54 (1996), p. 5048

  • 12 

    M. Gajda, K. Rzażewski

    Phys. Rev. Lett., 78 (1997), p. 2686

  • 13 

    M. Wilkens, C. Weiss

    J. Mod. Opt., 44 (1997), p. 1801

  • 14 

    C. Weiss, M. Wilkens

    Opt. Express, 1 (1997), p. 272

  • 15 

    P. Navez, D. Bitouk, M. Gajda, Z. Idziaszek, K. Rzażewski

    Phys. Rev. Lett., 79 (1997), p. 1789

  • 16 

    S. Grossmann, M. Holthaus, Opt. Express, 1, 1997, 262

  • 17 

    K. Kirsten, D.J. Toms

    Phys. Rev. A, 54 (1996), p. 4188

  • 18 

    S. Inouye, M.R. Andrews, J. Stenger, H.-J. Miesner, D.M. Stamper-Kurn, W. Ketterle

    Nature, 392 (1998), p. 151

  • 19 

    L.B. Richmond

    Acta Arith., 26 (1975), p. 411

  • 20 

    S. Grossmann, M. Holthaus

    Phys. Rev. Lett., 79 (1997), p. 3557

  • 21 

    E.T. Whittaker, G.N. Watson

    A Course of Modern Analysis, Cambridge Univ. Press, Cambridge (1962)

  • 22 

    S.R. de Groot, G.J. Hooyman, C.A. ten Seldam

    Proc. R. Soc. London A, 203 (1950), p. 266

  • 23 

    S. Grossmann, M. Holthaus

    Z. Naturforsch. A., 50 (1995), p. 921

  • 24 

    W. Ketterle, N.J. van Druten

    Phys. Rev. A, 54 (1996), p. 656

  • 25 

    K. Kirsten, D.J. Toms

    Phys. Lett. A, 222 (1996), p. 148

    Article PDF (314K)

  • 26 

    H. Haugerud, T. Haugset, F. Ravndal

    Phys. Lett. A, 225 (1997), p. 18

    Article PDF (435K)

  • 27 

    E.W. Barnes

    Trans. Cambridge Philos. Soc., 19 (1903), p. 426

  • 28 

    J.S. Dowker

    J. Math. Phys., 35 (1994), p. 4989

  • 29 

    N.J. van Druten, W. Ketterle

    Phys. Rev. Lett., 79 (1997), p. 549

  • 30 

    H.A. Bethe

    Rev. Mod. Phys., 9 (1937), p. 69

  • 31 

    G.H. Hardy, S. Ramanujan

    Proc. London Math. Soc., 17 (1918), p. 75

  • 32 

    S. Grossmann, M. Holthaus, Chaos Solitons Fractals

  • 33 

    G.E. Andrews

    Theory of Partitions, Addison–WesleyEncyclopedia of Mathematics and Its Applications, Reading (1976)

  • 34 

    P. Erdös, J. Lehner

    Duke Math. J., 8 (1941), p. 335

  • 35 

    V.S. Nanda

    Proc. Cambridge Philos. Soc., 47 (1951), p. 591

  • 36 

    K. Kirsten, D.J. Toms

    Phys. Lett. A, 243 (1998), p. 137

    Article PDF (403K)

  • 37 

    I.S. Gradshteyn, I.M. Ryzhik

    Tables of Integrals, Series, and Products, Academic Press, San Diego (1980)

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☆
Dedicated to Professor Dr. Siegfried Großmann on the occasion of his 68th birthday.
☆☆
G.-C. Rota
f1
E-mail: holthaus@stat.physik.uni.marburg.de

Copyright © 1998 Academic Press. All rights reserved.