Abstract

After discussing the key idea underlying the Maxwell’s Demon ensemble, we employ this idea for calculating fluctuations of ideal Bose gas condensates in traps with power-law single-particle energy spectra. Two essentially different cases have to be distinguished. If the heat capacity remains continuous at the condensation point in the large-N-limit, the fluctuations of the number of condensate particles vanish linearly with temperature, independent of the trap characteristics. If the heat capacity becomes discontinuous, the fluctuations vanish algebraically with temperature, with an exponent determined by the trap. Our results are based on an integral representation that yields the solution to both the canonical and the microcanonical fluctuation problem in a singularly transparent manner.

© 1997 Optical Society of America

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References

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  1. L.D. Landau and E.M. Lifshitz, Statistical Physics (Pergamon, London, 1959).
  2. R.K. Pathria, Statistical Mechanics (Pergamon, Oxford, 1985).
  3. I. Fujiwara, D. ter Haar, and H. Wergeland, “Fluctuations in the population of the ground state of Bose systems”, J. Stat. Phys. 2, 329–346 (1970).
    [Crossref]
  4. R.M. Ziff, G.E. Uhlenbeck, and M. Kac, “The ideal Bose-Einstein gas, revisited”, Phys. Rep. 32, 169–248 (1977).
    [Crossref]
  5. M. Gajda and K. Rzążewski, “Fluctuations of Bose-Einstein condensate”, Phys. Rev. Lett. 78, 2686–2689 (1997).
    [Crossref]
  6. M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor”, Science 269, 198–201 (1995).
    [Crossref] [PubMed]
  7. K.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn, and W. Ketterle, “Bose-Einstein condensation in a gas of sodium atoms”, Phys. Rev. Lett. 75, 3969–3973 (1995).
    [Crossref] [PubMed]
  8. C.C. Bradley, C.A. Sackett, and R.G. Hulet, “Bose-Einstein condensation of lithium: observation of limited condensate number”, Phys. Rev. Lett. 78, 985–989 (1997).
    [Crossref]
  9. 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, 656–660 (1996).
    [Crossref] [PubMed]
  10. N.J. van Druten and W. Ketterle, “Two-step condensation of the ideal Bose gas in highly anisotropic traps”, Phys. Rev. Lett. 79, 549–552 (1997).
    [Crossref]
  11. S. Grossmann and M. Holthaus, “Microcanonical fluctuations of a Bose system’s ground state occupation number”, Phys. Rev. E 54, 3495–3498 (1996).
    [Crossref]
  12. S. Grossmann and M. Holthaus, “From number theory to statistical mechanics: Bose-Einstein condensation in isolated traps”, to appear in Chaos, Solitons & Fractals (Proceedings of the 178th Heraeus-Seminar Pattern formation in nonlinear optical systems, Bad Honnef, June 23-25, 1997).
  13. M. Wilkens, “From Chinese wok to Mexican hat: Bose-Einstein condensation in an isolated Bose gas” (Preprint, Konstanz, 1996).
  14. P. Navez, D. Bitouk, M. Gajda, Z. Idziaszek, and K. Rzążewski, “The fourth statistical ensemble for the Bose-Einstein condensate”, Phys. Rev. Lett. 79, 1789–1792 (1997).
    [Crossref]
  15. S. Grossmann and M. Holthaus, “Fluctuations of the particle number in a trapped Bose-Einstein condensate”, Phys. Rev. Lett. 79, 3557–3560 (1997).
    [Crossref]
  16. P. Borrmann and G. Franke, “Recursion formulas for quantum statistical partition functions”, J. Chem. Phys. 98, 2484–2485 (1993).
    [Crossref]
  17. B. Eckhardt, “Eigenvalue statistics in quantum ideal gases”. In: Emerging applications of number theory, edited by D. Hejhal, F. Chung, J. Friedman, M. Gutzwiller, and A. Odlyzko (Springer, New York, to appear 1997).
  18. M. Wilkens and C. Weiss, “Universality classes and particle number fluctuations of trapped ideal Bose gases” (Preprint, Potsdam, 1997).
  19. S.R. de Groot, G.J. Hooyman, and C.A. ten Seldam, “On the Bose-Einstein condensation”, Proc. Roy. Soc. London A 203, 266–286 (1950).
    [Crossref]
  20. V.S. Nanda, “Bose-Einstein condensation and the partition theory of numbers”, Proc. Nat. Inst. Sci. (India)  19, 681–690 (1953).
  21. H.D. Politzer, “Condensate fluctuations of a trapped, ideal Bose gas”, Phys. Rev. A 54, 5048–5054 (1996).
    [Crossref] [PubMed]
  22. F.Y. Wu, G. Rollet, H.Y. Huang, J.M. Maillard, C.-K. Hu, and C.-N. Chen, “Directed compact lattice animals, restricted partitions of an integer, and the infinite-states Potts model”, Phys. Rev. Lett. 76, 173–176 (1996).
    [Crossref] [PubMed]

1997 (5)

M. Gajda and K. Rzążewski, “Fluctuations of Bose-Einstein condensate”, Phys. Rev. Lett. 78, 2686–2689 (1997).
[Crossref]

C.C. Bradley, C.A. Sackett, and R.G. Hulet, “Bose-Einstein condensation of lithium: observation of limited condensate number”, Phys. Rev. Lett. 78, 985–989 (1997).
[Crossref]

N.J. van Druten and W. Ketterle, “Two-step condensation of the ideal Bose gas in highly anisotropic traps”, Phys. Rev. Lett. 79, 549–552 (1997).
[Crossref]

P. Navez, D. Bitouk, M. Gajda, Z. Idziaszek, and K. Rzążewski, “The fourth statistical ensemble for the Bose-Einstein condensate”, Phys. Rev. Lett. 79, 1789–1792 (1997).
[Crossref]

S. Grossmann and M. Holthaus, “Fluctuations of the particle number in a trapped Bose-Einstein condensate”, Phys. Rev. Lett. 79, 3557–3560 (1997).
[Crossref]

1996 (4)

H.D. Politzer, “Condensate fluctuations of a trapped, ideal Bose gas”, Phys. Rev. A 54, 5048–5054 (1996).
[Crossref] [PubMed]

F.Y. Wu, G. Rollet, H.Y. Huang, J.M. Maillard, C.-K. Hu, and C.-N. Chen, “Directed compact lattice animals, restricted partitions of an integer, and the infinite-states Potts model”, Phys. Rev. Lett. 76, 173–176 (1996).
[Crossref] [PubMed]

S. Grossmann and M. Holthaus, “Microcanonical fluctuations of a Bose system’s ground state occupation number”, Phys. Rev. E 54, 3495–3498 (1996).
[Crossref]

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, 656–660 (1996).
[Crossref] [PubMed]

1995 (2)

M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor”, Science 269, 198–201 (1995).
[Crossref] [PubMed]

K.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn, and W. Ketterle, “Bose-Einstein condensation in a gas of sodium atoms”, Phys. Rev. Lett. 75, 3969–3973 (1995).
[Crossref] [PubMed]

1993 (1)

P. Borrmann and G. Franke, “Recursion formulas for quantum statistical partition functions”, J. Chem. Phys. 98, 2484–2485 (1993).
[Crossref]

1977 (1)

R.M. Ziff, G.E. Uhlenbeck, and M. Kac, “The ideal Bose-Einstein gas, revisited”, Phys. Rep. 32, 169–248 (1977).
[Crossref]

1970 (1)

I. Fujiwara, D. ter Haar, and H. Wergeland, “Fluctuations in the population of the ground state of Bose systems”, J. Stat. Phys. 2, 329–346 (1970).
[Crossref]

1953 (1)

V.S. Nanda, “Bose-Einstein condensation and the partition theory of numbers”, Proc. Nat. Inst. Sci. (India)  19, 681–690 (1953).

1950 (1)

S.R. de Groot, G.J. Hooyman, and C.A. ten Seldam, “On the Bose-Einstein condensation”, Proc. Roy. Soc. London A 203, 266–286 (1950).
[Crossref]

Anderson, M.H.

M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor”, Science 269, 198–201 (1995).
[Crossref] [PubMed]

Andrews, M.R.

K.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn, and W. Ketterle, “Bose-Einstein condensation in a gas of sodium atoms”, Phys. Rev. Lett. 75, 3969–3973 (1995).
[Crossref] [PubMed]

Bitouk, D.

P. Navez, D. Bitouk, M. Gajda, Z. Idziaszek, and K. Rzążewski, “The fourth statistical ensemble for the Bose-Einstein condensate”, Phys. Rev. Lett. 79, 1789–1792 (1997).
[Crossref]

Borrmann, P.

P. Borrmann and G. Franke, “Recursion formulas for quantum statistical partition functions”, J. Chem. Phys. 98, 2484–2485 (1993).
[Crossref]

Bradley, C.C.

C.C. Bradley, C.A. Sackett, and R.G. Hulet, “Bose-Einstein condensation of lithium: observation of limited condensate number”, Phys. Rev. Lett. 78, 985–989 (1997).
[Crossref]

Chen, C.-N.

F.Y. Wu, G. Rollet, H.Y. Huang, J.M. Maillard, C.-K. Hu, and C.-N. Chen, “Directed compact lattice animals, restricted partitions of an integer, and the infinite-states Potts model”, Phys. Rev. Lett. 76, 173–176 (1996).
[Crossref] [PubMed]

Cornell, E.A.

M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor”, Science 269, 198–201 (1995).
[Crossref] [PubMed]

Davis, K.B.

K.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn, and W. Ketterle, “Bose-Einstein condensation in a gas of sodium atoms”, Phys. Rev. Lett. 75, 3969–3973 (1995).
[Crossref] [PubMed]

de Groot, S.R.

S.R. de Groot, G.J. Hooyman, and C.A. ten Seldam, “On the Bose-Einstein condensation”, Proc. Roy. Soc. London A 203, 266–286 (1950).
[Crossref]

Druten, N.J. van

N.J. van Druten and W. Ketterle, “Two-step condensation of the ideal Bose gas in highly anisotropic traps”, Phys. Rev. Lett. 79, 549–552 (1997).
[Crossref]

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, 656–660 (1996).
[Crossref] [PubMed]

K.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn, and W. Ketterle, “Bose-Einstein condensation in a gas of sodium atoms”, Phys. Rev. Lett. 75, 3969–3973 (1995).
[Crossref] [PubMed]

Durfee, D.S.

K.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn, and W. Ketterle, “Bose-Einstein condensation in a gas of sodium atoms”, Phys. Rev. Lett. 75, 3969–3973 (1995).
[Crossref] [PubMed]

Eckhardt, B.

B. Eckhardt, “Eigenvalue statistics in quantum ideal gases”. In: Emerging applications of number theory, edited by D. Hejhal, F. Chung, J. Friedman, M. Gutzwiller, and A. Odlyzko (Springer, New York, to appear 1997).

Ensher, J.R.

M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor”, Science 269, 198–201 (1995).
[Crossref] [PubMed]

Franke, G.

P. Borrmann and G. Franke, “Recursion formulas for quantum statistical partition functions”, J. Chem. Phys. 98, 2484–2485 (1993).
[Crossref]

Fujiwara, I.

I. Fujiwara, D. ter Haar, and H. Wergeland, “Fluctuations in the population of the ground state of Bose systems”, J. Stat. Phys. 2, 329–346 (1970).
[Crossref]

Gajda, M.

M. Gajda and K. Rzążewski, “Fluctuations of Bose-Einstein condensate”, Phys. Rev. Lett. 78, 2686–2689 (1997).
[Crossref]

P. Navez, D. Bitouk, M. Gajda, Z. Idziaszek, and K. Rzążewski, “The fourth statistical ensemble for the Bose-Einstein condensate”, Phys. Rev. Lett. 79, 1789–1792 (1997).
[Crossref]

Grossmann, S.

S. Grossmann and M. Holthaus, “Fluctuations of the particle number in a trapped Bose-Einstein condensate”, Phys. Rev. Lett. 79, 3557–3560 (1997).
[Crossref]

S. Grossmann and M. Holthaus, “Microcanonical fluctuations of a Bose system’s ground state occupation number”, Phys. Rev. E 54, 3495–3498 (1996).
[Crossref]

S. Grossmann and M. Holthaus, “From number theory to statistical mechanics: Bose-Einstein condensation in isolated traps”, to appear in Chaos, Solitons & Fractals (Proceedings of the 178th Heraeus-Seminar Pattern formation in nonlinear optical systems, Bad Honnef, June 23-25, 1997).

Holthaus, M.

S. Grossmann and M. Holthaus, “Fluctuations of the particle number in a trapped Bose-Einstein condensate”, Phys. Rev. Lett. 79, 3557–3560 (1997).
[Crossref]

S. Grossmann and M. Holthaus, “Microcanonical fluctuations of a Bose system’s ground state occupation number”, Phys. Rev. E 54, 3495–3498 (1996).
[Crossref]

S. Grossmann and M. Holthaus, “From number theory to statistical mechanics: Bose-Einstein condensation in isolated traps”, to appear in Chaos, Solitons & Fractals (Proceedings of the 178th Heraeus-Seminar Pattern formation in nonlinear optical systems, Bad Honnef, June 23-25, 1997).

Hooyman, G.J.

S.R. de Groot, G.J. Hooyman, and C.A. ten Seldam, “On the Bose-Einstein condensation”, Proc. Roy. Soc. London A 203, 266–286 (1950).
[Crossref]

Hu, C.-K.

F.Y. Wu, G. Rollet, H.Y. Huang, J.M. Maillard, C.-K. Hu, and C.-N. Chen, “Directed compact lattice animals, restricted partitions of an integer, and the infinite-states Potts model”, Phys. Rev. Lett. 76, 173–176 (1996).
[Crossref] [PubMed]

Huang, H.Y.

F.Y. Wu, G. Rollet, H.Y. Huang, J.M. Maillard, C.-K. Hu, and C.-N. Chen, “Directed compact lattice animals, restricted partitions of an integer, and the infinite-states Potts model”, Phys. Rev. Lett. 76, 173–176 (1996).
[Crossref] [PubMed]

Hulet, R.G.

C.C. Bradley, C.A. Sackett, and R.G. Hulet, “Bose-Einstein condensation of lithium: observation of limited condensate number”, Phys. Rev. Lett. 78, 985–989 (1997).
[Crossref]

Idziaszek, Z.

P. Navez, D. Bitouk, M. Gajda, Z. Idziaszek, and K. Rzążewski, “The fourth statistical ensemble for the Bose-Einstein condensate”, Phys. Rev. Lett. 79, 1789–1792 (1997).
[Crossref]

Kac, M.

R.M. Ziff, G.E. Uhlenbeck, and M. Kac, “The ideal Bose-Einstein gas, revisited”, Phys. Rep. 32, 169–248 (1977).
[Crossref]

Ketterle, W.

N.J. van Druten and W. Ketterle, “Two-step condensation of the ideal Bose gas in highly anisotropic traps”, Phys. Rev. Lett. 79, 549–552 (1997).
[Crossref]

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, 656–660 (1996).
[Crossref] [PubMed]

K.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn, and W. Ketterle, “Bose-Einstein condensation in a gas of sodium atoms”, Phys. Rev. Lett. 75, 3969–3973 (1995).
[Crossref] [PubMed]

Kurn, D.M.

K.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn, and W. Ketterle, “Bose-Einstein condensation in a gas of sodium atoms”, Phys. Rev. Lett. 75, 3969–3973 (1995).
[Crossref] [PubMed]

Landau, L.D.

L.D. Landau and E.M. Lifshitz, Statistical Physics (Pergamon, London, 1959).

Lifshitz, E.M.

L.D. Landau and E.M. Lifshitz, Statistical Physics (Pergamon, London, 1959).

Maillard, J.M.

F.Y. Wu, G. Rollet, H.Y. Huang, J.M. Maillard, C.-K. Hu, and C.-N. Chen, “Directed compact lattice animals, restricted partitions of an integer, and the infinite-states Potts model”, Phys. Rev. Lett. 76, 173–176 (1996).
[Crossref] [PubMed]

Matthews, M.R.

M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor”, Science 269, 198–201 (1995).
[Crossref] [PubMed]

Mewes, M.-O.

K.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn, and W. Ketterle, “Bose-Einstein condensation in a gas of sodium atoms”, Phys. Rev. Lett. 75, 3969–3973 (1995).
[Crossref] [PubMed]

Nanda, V.S.

V.S. Nanda, “Bose-Einstein condensation and the partition theory of numbers”, Proc. Nat. Inst. Sci. (India)  19, 681–690 (1953).

Navez, P.

P. Navez, D. Bitouk, M. Gajda, Z. Idziaszek, and K. Rzążewski, “The fourth statistical ensemble for the Bose-Einstein condensate”, Phys. Rev. Lett. 79, 1789–1792 (1997).
[Crossref]

Pathria, R.K.

R.K. Pathria, Statistical Mechanics (Pergamon, Oxford, 1985).

Politzer, H.D.

H.D. Politzer, “Condensate fluctuations of a trapped, ideal Bose gas”, Phys. Rev. A 54, 5048–5054 (1996).
[Crossref] [PubMed]

Rollet, G.

F.Y. Wu, G. Rollet, H.Y. Huang, J.M. Maillard, C.-K. Hu, and C.-N. Chen, “Directed compact lattice animals, restricted partitions of an integer, and the infinite-states Potts model”, Phys. Rev. Lett. 76, 173–176 (1996).
[Crossref] [PubMed]

Rzazewski, K.

P. Navez, D. Bitouk, M. Gajda, Z. Idziaszek, and K. Rzążewski, “The fourth statistical ensemble for the Bose-Einstein condensate”, Phys. Rev. Lett. 79, 1789–1792 (1997).
[Crossref]

M. Gajda and K. Rzążewski, “Fluctuations of Bose-Einstein condensate”, Phys. Rev. Lett. 78, 2686–2689 (1997).
[Crossref]

Sackett, C.A.

C.C. Bradley, C.A. Sackett, and R.G. Hulet, “Bose-Einstein condensation of lithium: observation of limited condensate number”, Phys. Rev. Lett. 78, 985–989 (1997).
[Crossref]

ten Seldam, C.A.

S.R. de Groot, G.J. Hooyman, and C.A. ten Seldam, “On the Bose-Einstein condensation”, Proc. Roy. Soc. London A 203, 266–286 (1950).
[Crossref]

ter Haar, D.

I. Fujiwara, D. ter Haar, and H. Wergeland, “Fluctuations in the population of the ground state of Bose systems”, J. Stat. Phys. 2, 329–346 (1970).
[Crossref]

Uhlenbeck, G.E.

R.M. Ziff, G.E. Uhlenbeck, and M. Kac, “The ideal Bose-Einstein gas, revisited”, Phys. Rep. 32, 169–248 (1977).
[Crossref]

Weiss, C.

M. Wilkens and C. Weiss, “Universality classes and particle number fluctuations of trapped ideal Bose gases” (Preprint, Potsdam, 1997).

Wergeland, H.

I. Fujiwara, D. ter Haar, and H. Wergeland, “Fluctuations in the population of the ground state of Bose systems”, J. Stat. Phys. 2, 329–346 (1970).
[Crossref]

Wieman, C.E.

M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor”, Science 269, 198–201 (1995).
[Crossref] [PubMed]

Wilkens, M.

M. Wilkens, “From Chinese wok to Mexican hat: Bose-Einstein condensation in an isolated Bose gas” (Preprint, Konstanz, 1996).

M. Wilkens and C. Weiss, “Universality classes and particle number fluctuations of trapped ideal Bose gases” (Preprint, Potsdam, 1997).

Wu, F.Y.

F.Y. Wu, G. Rollet, H.Y. Huang, J.M. Maillard, C.-K. Hu, and C.-N. Chen, “Directed compact lattice animals, restricted partitions of an integer, and the infinite-states Potts model”, Phys. Rev. Lett. 76, 173–176 (1996).
[Crossref] [PubMed]

Ziff, R.M.

R.M. Ziff, G.E. Uhlenbeck, and M. Kac, “The ideal Bose-Einstein gas, revisited”, Phys. Rep. 32, 169–248 (1977).
[Crossref]

J. Chem. Phys. (1)

P. Borrmann and G. Franke, “Recursion formulas for quantum statistical partition functions”, J. Chem. Phys. 98, 2484–2485 (1993).
[Crossref]

J. Stat. Phys. (1)

I. Fujiwara, D. ter Haar, and H. Wergeland, “Fluctuations in the population of the ground state of Bose systems”, J. Stat. Phys. 2, 329–346 (1970).
[Crossref]

Phys. Rep. (1)

R.M. Ziff, G.E. Uhlenbeck, and M. Kac, “The ideal Bose-Einstein gas, revisited”, Phys. Rep. 32, 169–248 (1977).
[Crossref]

Phys. Rev. A (2)

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, 656–660 (1996).
[Crossref] [PubMed]

H.D. Politzer, “Condensate fluctuations of a trapped, ideal Bose gas”, Phys. Rev. A 54, 5048–5054 (1996).
[Crossref] [PubMed]

Phys. Rev. E (1)

S. Grossmann and M. Holthaus, “Microcanonical fluctuations of a Bose system’s ground state occupation number”, Phys. Rev. E 54, 3495–3498 (1996).
[Crossref]

Phys. Rev. Lett. (7)

N.J. van Druten and W. Ketterle, “Two-step condensation of the ideal Bose gas in highly anisotropic traps”, Phys. Rev. Lett. 79, 549–552 (1997).
[Crossref]

P. Navez, D. Bitouk, M. Gajda, Z. Idziaszek, and K. Rzążewski, “The fourth statistical ensemble for the Bose-Einstein condensate”, Phys. Rev. Lett. 79, 1789–1792 (1997).
[Crossref]

S. Grossmann and M. Holthaus, “Fluctuations of the particle number in a trapped Bose-Einstein condensate”, Phys. Rev. Lett. 79, 3557–3560 (1997).
[Crossref]

M. Gajda and K. Rzążewski, “Fluctuations of Bose-Einstein condensate”, Phys. Rev. Lett. 78, 2686–2689 (1997).
[Crossref]

K.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn, and W. Ketterle, “Bose-Einstein condensation in a gas of sodium atoms”, Phys. Rev. Lett. 75, 3969–3973 (1995).
[Crossref] [PubMed]

C.C. Bradley, C.A. Sackett, and R.G. Hulet, “Bose-Einstein condensation of lithium: observation of limited condensate number”, Phys. Rev. Lett. 78, 985–989 (1997).
[Crossref]

F.Y. Wu, G. Rollet, H.Y. Huang, J.M. Maillard, C.-K. Hu, and C.-N. Chen, “Directed compact lattice animals, restricted partitions of an integer, and the infinite-states Potts model”, Phys. Rev. Lett. 76, 173–176 (1996).
[Crossref] [PubMed]

Proc. Nat. Inst. Sci. (1)

V.S. Nanda, “Bose-Einstein condensation and the partition theory of numbers”, Proc. Nat. Inst. Sci. (India)  19, 681–690 (1953).

Proc. Roy. Soc. London A (1)

S.R. de Groot, G.J. Hooyman, and C.A. ten Seldam, “On the Bose-Einstein condensation”, Proc. Roy. Soc. London A 203, 266–286 (1950).
[Crossref]

Science (1)

M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor”, Science 269, 198–201 (1995).
[Crossref] [PubMed]

Other (6)

L.D. Landau and E.M. Lifshitz, Statistical Physics (Pergamon, London, 1959).

R.K. Pathria, Statistical Mechanics (Pergamon, Oxford, 1985).

B. Eckhardt, “Eigenvalue statistics in quantum ideal gases”. In: Emerging applications of number theory, edited by D. Hejhal, F. Chung, J. Friedman, M. Gutzwiller, and A. Odlyzko (Springer, New York, to appear 1997).

M. Wilkens and C. Weiss, “Universality classes and particle number fluctuations of trapped ideal Bose gases” (Preprint, Potsdam, 1997).

S. Grossmann and M. Holthaus, “From number theory to statistical mechanics: Bose-Einstein condensation in isolated traps”, to appear in Chaos, Solitons & Fractals (Proceedings of the 178th Heraeus-Seminar Pattern formation in nonlinear optical systems, Bad Honnef, June 23-25, 1997).

M. Wilkens, “From Chinese wok to Mexican hat: Bose-Einstein condensation in an isolated Bose gas” (Preprint, Konstanz, 1996).

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Figures (3)

Fig. 1.
Fig. 1.

Full red line: microcanonical fluctuations δN 0/N for a system of N = 106 ideal Bose particles trapped by a one-dimensional harmonic potential [11]. The temperature T 0 = T0(1) denotes the characteristic temperature below which the ground state occupation becomes significant, see Eq. (2). Black short-dashed line: grand canonical fluctuations for the same system. Blue dashed line: low-temperature approximation provided by Eq. (2).

Fig. 2.
Fig. 2.

Microcanonical probability distributions P ex(N ex\n) for finding N ex out of N = 1000 ideal Bose particles, trapped by a three-dimensional isotropic harmonic potential, excited when the total excitation energy E is nħω, with ω denoting the oscillator frequency. The number n determines the temperature T. The normalized temperatures T/T 0 corresponding to the blue, Gaussian-like distributions range from 0.3 to 0.9 (left to right, in steps of 0.1); T 0 = (ħω/kB )(N/ζ(3))1/3. Due to finite-N-effects, the condensation temperature is lowered from T 0 to about 0.93 T 0. The temperature corresponding to the rightmost, red distribution is T = 0.95T 0, lying slightly above the condensation point.

Fig. 3.
Fig. 3.

Microcanonical fluctuations δN 0 for N = 200, 500, and 1000 ideal Bose particles trapped by a three-dimensional, isotropic harmonic potential. The fluctuations are maximal close to the respective condensation points. These maximal fluctuations scale approximately as √N, cf. Eqs. (18) and (25). Note that the low-temperature fluctuations for all three systems agree perfectly, thus demonstrating the N-independence of δN 0 below the condensation point.

Equations (34)

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( δ N ν ) 2 = N ν ( N ν + 1 ) .
δ N 0 π 6 k B T ħ ω for T T 0 ( 1 ) ħ ω k B N ln N ,
ϒ ( z , E ) = N ex = 0 z N ex Φ ( N ex E ) .
p ex ( N ex E ) = Φ ( N ex E ) Ω ( E N ) , N ex = 1,2 , , N ,
N N 0 = z z ln ϒ ( z , E ) | z = 1 and ( δ N 0 ) 2 = ( z z ) 2 ln ϒ ( z , E ) z = 1 ,
Z N ( β ) = n = 0 e nβħω Ω ( n ħ ω N ) ,
Z N ( β ) = 1 N k = 1 N Z 1 ( k β ) Z N k ( β ) .
Φ ( N ex | E ) = Ω ( E | N ex ) Ω ( E | N ex 1 )
Ξ ex ( z , β ) = N ex = 0 z N ex E Φ ( N ex E ) e β E ,
( z z ) k Ξ ex ( z , β ) | z = 1 = E ( N ex = 0 N ex k Φ ( N ex E ) ) e β E M k ( β ) ,
μ k ( E ) N ex = 0 N ex k Φ ( N ex E ) with k = 0,1,2 , .
N ex = M 1 ( β ) M 0 ( β ) ;
( δ N 0 ) cn 2 = M 2 ( β ) M 0 ( β ) ( M 1 ( β ) M 0 ( β ) ) 2 .
N ex = 1 2 πi τ i τ + i d t Γ ( t ) Z ( β , t ) ζ ( t )
( δ N 0 ) cn 2 = 1 2 πi τ i τ + i d t Γ ( t ) Z ( β , t ) ζ ( t 1 ) ,
Z ( β , t ) = ν = 1 1 ( β ε ν ) t
ε { ν i } = Δ i = 1 d c i ν i σ , ν i = 0,1,2 , , σ > 0 ,
ρ ( E ) = A Γ ( d σ ) ( E Δ ) d σ 1 1 Δ with A Γ ( 1 σ + 1 ) d ( i = 1 d c i ) 1 σ .
k B T 0 Δ = 1 A σ d ( N ζ ( d σ ) ) σ d .
Z ( β , t ) A Γ ( d σ ) ( β Δ ) t ζ ( t + 1 d σ ) ,
N ex A Γ ( d σ ) 1 2 πi τ i τ + i d t ( β Δ ) t Γ ( t ) ζ ( t + 1 d σ ) ζ ( t )
( δ N 0 ) cn 2 A Γ ( d σ ) 1 2 πi τ i τ + i d t ( β Δ ) t Γ ( t ) ζ ( t + 1 d σ ) ζ ( t 1 ) .
N ex A ζ ( d σ ) ( k B T Δ ) d σ .
N 0 = N N ex = N [ 1 ( T T 0 ) 3 ] for T < T 0 = ħ ω k B ( N ζ ( 3 ) ) 1 3 .
N ex A k B T Δ [ ln ( k B T Δ ) + γ ] ,
N ex A Γ ( d σ ) ζ ( 2 d σ ) k B T Δ ,
( δ N 0 ) cn 2 ( d σ 1 ) ( k B T Δ ) d σ .
( δ N 0 ) cn 2 A ( k B T Δ ) 2 [ ln ( k B T Δ ) + γ + 1 ] .
( δ N 0 ) cn 2 A Γ ( d σ ) ζ ( 3 d σ ) ( k B T Δ ) 2 .
( δ N 0 ) cn 2 ( δ N 0 ) mc 2 A d d + σ ζ 2 ( d σ ) ζ ( d σ + 1 ) ( k B T Δ ) d σ for d σ > 2 and T < T 0 .
C < N k B = d σ ( d σ + 1 ) ζ ( d σ + 1 ) ζ ( d σ ) ( T T 0 ) d σ ,
C < N k B = d σ ( d σ + 1 ) g d σ + 1 ( z ) g d σ ( z ) d 2 σ 2 g d σ ( z ) g d σ 1 ( z ) .
C < C > N k B | T 0 = d 2 σ 2 ζ ( d σ ) ζ ( d σ 1 )
δ N 0 n .

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