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.

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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. Zi, G.E. Uhlenbeck, and M. Kac, "The ideal Bose-Einstein gas, revisited", Phys. Rep. 32, 169-248 (1977).
    [CrossRef]
  5. 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]
  6. 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]
  7. 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] [PubMed]
  8. 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]
  9. 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] [PubMed]
  10. S. Grossmann and M. Holthaus, "Microcanonical uctuations of a Bose system's ground state occupation number", Phys. Rev. E 54, 3495-3498 (1996).
    [CrossRef]
  11. S. Grossmann and M. Holthaus, "From number theory to statistical mechanics: Bose-Einstein condensation in isolated traps" (Preprint, Marburg, 1997).
    [CrossRef]
  12. M. Wilkens, "From Chinese wok to Mexican hat: Bose-Einstein condensation in an isolated Bose gas" (Preprint, Konstanz, 1996).
  13. P. Navez, D. Bitouk, M. Gajda, Z. Idziaszek, and K. Rzazewski, "The fourth statistical ensemble
  14. S. Grossmann and M. Holthaus, "Fluctuations of the particle number in a trapped Bose condensate" (Preprint, Marburg, 1997).
    [CrossRef]
  15. P. Borrmann and G. Franke, "Recursion formulas for quantum statistical partition functions", J. Chem. Phys. 98, 2484-2485 (1993).
    [CrossRef]
  16. B. Eckhardt, "Eigenvalue statistics in quantum ideal gases" (Preprint, Oldenburg, 1997).
    [CrossRef]
  17. M. Wilkens and C. Weiss, "Universality classes and particle number uctuations of trapped ideal Bose gases" (Preprint, Potsdam, 1997).
  18. 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).
  19. M. Gajda and K. Rzazewski, "Fluctuations of Bose-Einstein condensate", Phys. Rev. Lett. 78, 2686-2689 (1997). See Eq. (13) therein.
    [CrossRef]
  20. J.E. Robinson, "Note on the Bose-Einstein integral functions", Phys. Rev. 83, 678-679 (1951). See also Ref. [2], Appendix D.
  21. 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 in finite-states Potts model", Phys. Rev. Lett. 76, 173-176 (1996).
    [CrossRef] [PubMed]

Other

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

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

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]

R.M. Zi, G.E. Uhlenbeck, and M. Kac, "The ideal Bose-Einstein gas, revisited", Phys. Rep. 32, 169-248 (1977).
[CrossRef]

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]

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] [PubMed]

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]

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] [PubMed]

S. Grossmann and M. Holthaus, "Microcanonical uctuations 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" (Preprint, Marburg, 1997).
[CrossRef]

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

P. Navez, D. Bitouk, M. Gajda, Z. Idziaszek, and K. Rzazewski, "The fourth statistical ensemble

S. Grossmann and M. Holthaus, "Fluctuations of the particle number in a trapped Bose condensate" (Preprint, Marburg, 1997).
[CrossRef]

P. Borrmann and G. Franke, "Recursion formulas for quantum statistical partition functions", J. Chem. Phys. 98, 2484-2485 (1993).
[CrossRef]

B. Eckhardt, "Eigenvalue statistics in quantum ideal gases" (Preprint, Oldenburg, 1997).
[CrossRef]

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

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).

M. Gajda and K. Rzazewski, "Fluctuations of Bose-Einstein condensate", Phys. Rev. Lett. 78, 2686-2689 (1997). See Eq. (13) therein.
[CrossRef]

J.E. Robinson, "Note on the Bose-Einstein integral functions", Phys. Rev. 83, 678-679 (1951). See also Ref. [2], Appendix D.

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 in finite-states Potts model", Phys. Rev. Lett. 76, 173-176 (1996).
[CrossRef] [PubMed]

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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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