Abstract

We discuss the exact particle number counting statistics of degenerate ideal Bose gases in the microcanonical, canonical, and grand-canonical ensemble, respectively, for various trapping potentials. We then invoke the Maxwell’s Demon ensemble [Navez et el, Phys. Rev. Lett. (1997)] and show that for large total number of particles the root-mean-square fluctuation of the condensate occupation scales δn 0 α [T/Tc ] rNs with scaling exponents r = 3/2, s = 1/2 for the 3D harmonic oscillator trapping potential, and r = 1, s = 2/3 for the 3D box. We derive an explicit expression for r and s in terms of spatial dimension D and spectral index σ of the single-particle energy spectrum. Our predictions also apply to systems where Bose-Einstein condensation does not occur. We point out that the condensate fluctuations in the microcanonical and canonical ensemble respect the principle of thermodynamic equivalence.

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References

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  1. M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.M. Kurn, D.S. Durfee, and W. Ketterle, "Bose-Einstein condensation in tightly confining dc magnetic trap", Phys. Rev. Lett. 77, 416-420 (1996).
    [CrossRef] [PubMed]
  2. J.R. Ensher, D.S. Jin, M.R. Matthews, C.E. Wieman, and E.A. Cornell, "Bose-Einstein condensation in a dilute gas: Measurement of energy and ground state occupation", Phys. Rev. Lett. 77 4984 (1996).
    [CrossRef] [PubMed]
  3. E. Schroedinger, Statistical Thermodynamics (Dover Publ. New York, 1989).
  4. 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]
  5. M. Wilkens and C. Weiss, "Particle number uctuations in an ideal Bose gas", J. Mod. Opt., in press (1997).
    [CrossRef]
  6. R.M. Ziff, G.E. Uhlenbeck, and M. Kac, "The ideal Bose-Einstein gas, revisited", Phys. Rep. 32, 169-248 (1977).
    [CrossRef]
  7. M. Wilkens, "From Chinese Wok to Mexican Hat: Bose-Einstein Condensation in an isolated Bose gas", (pending, 1996); see also: Konstanz Annual Report (1996).
  8. P. T. Landsberg, Thermodynamics - with quantum statistical illustrations, Interscience Publishers, New York 1961.
  9. P. Borrmann and G. Franke, "Recursion formulas for quantum statistical partition functions", J. Chem. Phys. 98, 2484 (1993).
    [CrossRef]
  10. F. Brosens, J.T. Devreese, and L.F. Lemmens, "Canonical Bose-Einstein condensation in a parabolic well", Solid State Commun. 100, 123-127 (1996).
    [CrossRef]
  11. M. Wilkens and C. Weiss, "Universality classes and particle number uctuations of trapped ideal Bose gases", (submitted, 1997).
  12. M. Gajda and K. Rzazewski, "Fluctuations of Bose-Einstein Condensate", Phys. Rev. Lett. 78, 2686 (1997).
    [CrossRef]
  13. S. Grossmann and M. Holthaus, "Microcanonical fluctuations of a Bose system's ground state occupation number", Phys. Rev. E 54, 3495-3498 (1996).
    [CrossRef]
  14. S. Grossmann and M. Holthaus, "Fluctuations of the particle number in a trapped Bose condensate", (Preprint, Marburg, 1997).
  15. S. Grossmann and M. Holthaus, "From number theory to statistical mechanics: Bose-Einstein condensation in isolated traps", (Preprint, Marburg 1997).
  16. S. Grossmann and M. Holthaus, "Maxwell's Demon at work: Two types of Bose condensate fluctuations in power law traps", (Preprint, Marburg, 1997).
  17. H. D. Politzer, "Condensate fluctuations of a trapped, ideal Bose gas", Phys. Rev. A 54, 5048-5054 (1996).
    [CrossRef] [PubMed]
  18. P. Navez, D. Bitouk, M. Gajda, Z. Idziaszek, and K. Rzazewski, "The fourth statistical ensemble for the Bose-Einstein condensate", Phys. Rev. Lett. (1997).
    [CrossRef]
  19. S.R. de Groot, G.J. Hooyman, and C.A. ten Seldam, "On the Bose-Einstein condensation", Proc. R. Soc. London A 203, 266-286 (1950).
    [CrossRef]
  20. W.J. Mullin, "Bose-Einstein Condensation in a Harmonic Trap", J. Low Temp. Phys. 106, 615 (1997).
    [CrossRef]
  21. K.C. Chase, A.Z. Mekjian and L. Zamick, "Canonical and Microcanonical Ensemble Approaches to Bose-Einstein Condensation: The Thermodynamics of Particles in Harmonic Traps" [cond-mat/9708070].

Other

M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.M. Kurn, D.S. Durfee, and W. Ketterle, "Bose-Einstein condensation in tightly confining dc magnetic trap", Phys. Rev. Lett. 77, 416-420 (1996).
[CrossRef] [PubMed]

J.R. Ensher, D.S. Jin, M.R. Matthews, C.E. Wieman, and E.A. Cornell, "Bose-Einstein condensation in a dilute gas: Measurement of energy and ground state occupation", Phys. Rev. Lett. 77 4984 (1996).
[CrossRef] [PubMed]

E. Schroedinger, Statistical Thermodynamics (Dover Publ. New York, 1989).

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]

M. Wilkens and C. Weiss, "Particle number uctuations in an ideal Bose gas", J. Mod. Opt., in press (1997).
[CrossRef]

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

M. Wilkens, "From Chinese Wok to Mexican Hat: Bose-Einstein Condensation in an isolated Bose gas", (pending, 1996); see also: Konstanz Annual Report (1996).

P. T. Landsberg, Thermodynamics - with quantum statistical illustrations, Interscience Publishers, New York 1961.

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

F. Brosens, J.T. Devreese, and L.F. Lemmens, "Canonical Bose-Einstein condensation in a parabolic well", Solid State Commun. 100, 123-127 (1996).
[CrossRef]

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

M. Gajda and K. Rzazewski, "Fluctuations of Bose-Einstein Condensate", Phys. Rev. Lett. 78, 2686 (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, "Fluctuations of the particle number in a trapped Bose condensate", (Preprint, Marburg, 1997).

S. Grossmann and M. Holthaus, "From number theory to statistical mechanics: Bose-Einstein condensation in isolated traps", (Preprint, Marburg 1997).

S. Grossmann and M. Holthaus, "Maxwell's Demon at work: Two types of Bose condensate fluctuations in power law traps", (Preprint, Marburg, 1997).

H. D. Politzer, "Condensate fluctuations of a trapped, ideal Bose gas", Phys. Rev. A 54, 5048-5054 (1996).
[CrossRef] [PubMed]

P. Navez, D. Bitouk, M. Gajda, Z. Idziaszek, and K. Rzazewski, "The fourth statistical ensemble for the Bose-Einstein condensate", Phys. Rev. Lett. (1997).
[CrossRef]

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

W.J. Mullin, "Bose-Einstein Condensation in a Harmonic Trap", J. Low Temp. Phys. 106, 615 (1997).
[CrossRef]

K.C. Chase, A.Z. Mekjian and L. Zamick, "Canonical and Microcanonical Ensemble Approaches to Bose-Einstein Condensation: The Thermodynamics of Particles in Harmonic Traps" [cond-mat/9708070].

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Figures (4)

Figure 1.
Figure 1.

Particle number counting statistics of the ground state occupation of an ideal Bose gas in a three-dimensional isotropic harmonic oscillator trapping potential for a total number of particles N = 200 in the microcanonical ensemble (dashed), canonical ensemble (solid), and grand-canonical ensemble (dotted). Temperatures are from left to right T/Tc = 1.252, 0.905, 0.557 (no grand-canonical curve for this case).

Figure 2.
Figure 2.

Exact data for the condensate root-mean-square fluctuations in a three-dimensional isotropic harmonic oscillator trapping potential for N = 10, 42 and 100 particles in the canonical ensemble (full) and microcanonical ensemble (dashed), respectively. Temperature is measured in units of the energy gap ∆ between the trap ground state and trap first excited state.

Figure 3.
Figure 3.

Exact data for a Bose gas of N = 400 particles in a one-dimensional box in the microcanonical ensemble (dashed), canonical ensemble (full) grand-canonical ensemble (dotted). Left: Ground state occupation counting statistics for T/Tc = 2.0, 0.3. Right: Root-mean-square fluctuations of ground state occupation with asymptotics (dotted-dashed) according to Eq. (28). Inset: Ground state mean occupation.

Figure 4.
Figure 4.

Exact data (full line) for the condensate root-mean-square fluctuations in a three-dimensional isotropic harmonic oscillator trapping potential (left) and three-dimensional box (right) for N = 104 particles (lower curves) and N = 105 particles (upper curves) in the canonical ensemble. Dotted-dashed line: predictions of the asymptotic formulas. Dashed line: asymptotic formula with finite-N corrections.

Equations (46)

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P ν G ( n ) = 1 1 + n ν ¯ ( n ν ¯ 1 + n ν ¯ ) n ,
δ 2 n 0 = ( 1 [ T T c ] η ) 2 N ¯ 2 ,
P ν C ( n ) = e n β ε ν Z N n Z N e ( n + 1 ) β ε ν Z N n 1 Z N .
Z N ( β ) = 1 N n = 1 N Z 1 ( n β ) Z N n ( β ) , Z 0 = 1 ,
P ν M ( n ) = Ω N n ( E n ε ν ) Ω N ( E ) Ω N n 1 ( E ( n + 1 ) ε ν ) Ω N ( E ) ,
Ω N ( E ) = 1 N n = 1 N ν Ω N n ( E n ε ν ) .
Ω N ( E ) = 1 N n = 1 N ν = 0 f ( D ) ( ν ) Ω N n ( E n ħ ων ) ,
ϒ ( z ) Z N M = 0 N z M P 0 C ( N M ) .
N n 0 = z z ln ϒ ( z ) | z = 1 ,
δ 2 n 0 = ( z z ) 2 ln ϒ ( z ) | z = 1 ,
n 0 ~ O ( N ) and δ n 0 n 0 ,
ϒ ( z ) ( 1 z ) M = 0 z M Z M ( β ) .
ln ϒ ν 0 ln ( 1 z e β ε ν ) ,
ε ν = Δ i = 1 D c i ν i , σ 0 < σ 2 ,
k B T c Δ ~ [ i = 1 D c i ] 1 D [ Γ ( 1 σ + 1 ) ] σ [ N ζ ( D σ ) ] σ D ,
n 0 ~ ( 1 [ T T c ] D σ ) N .
k B T c ħ ω ~ N ln N .
δ 2 n 0 ν 0 1 4 sinh 2 ( β ε ν 2 ) ,
δ n 0 2 k = 1 k [ Z ( k β ) 1 ] ,
S ( β ) ν = 1 e β ν σ ,
δ n 0 2 d = 1 D D d k = 1 k S ( k β ) d .
S ( k β ) ~ Γ ( 1 + 1 σ ) e k β [ k β ] 1 σ .
δ 2 n 0 ~ d = 1 D D d Γ ( 1 + 1 σ ) d g d σ 1 ( e d β ) β d σ
β η g η 1 ( e β ) ~ { ζ ( η 1 ) β η for η > 2 β 2 ln ( β 1 ) for η = 2 Γ ( 2 η ) β 2 for η < 2
δ 2 n 0 ~ C [ k B T Δ ] D σ ,
C = Γ ( 1 + 1 σ ) D [ i = 1 D c i ] 1 σ ζ ( D σ 1 ) .
δ 2 n 0 ~ C [ k B T Δ ] 2 ln ( k B T Δ ) ,
δ n 0 2 ~ C [ k B T Δ ] 2 ,
C = ν 0 1 [ ν 1 σ + ν 2 σ + + ν D σ ] 2 .
δ n 0 ~ A [ T T c ] r N s ,
r = { D 2 σ if D > 2 σ 1 if D < 2 σ , s = { 1 2 if D > 2 σ σ D if D < 2 σ .
T T c < 1 A N 1 s .
H = ν = 0 ε ν n ν ,
Ω N ( E ) = n 0 = 0 n 1 = 0 n = 0 δ H , E δ Σ n ν , N ,
P ( { n } ) = 1 Z N exp { β ν = 0 ε ν n ν } δ Σ n ν , N ,
Z N ( β ) = n 0 = 0 n ν = 0 n = 0 e β H δ Σ n ν , N .
P ν M ( n ) = Ω N n ( E n ε ν ) Ω N ( E ) Ω N n 1 ( E ( n + 1 ) ε ν ) Ω N ( E ) ,
P ν C ( n ) = e n β ε ν Z N n Z N e ( n + 1 ) β ε ν Z N n 1 Z N ,
n ν = 1 Z N n = 1 N e n β ε ν Z N n .
N = 1 Z N n = 1 N Z N n ν e n β ε ν ,
Z N ( β ) = 1 N n = 1 N Z 1 ( n β ) Z N n ( β ) , Z 0 = 1
n ν n = 1 N n P ν ( n | E , N ) .
n ν = n = 1 N Ω N n ( E n ε ν ) Ω N ( E ) .
N ν n ν
Ω N ( E ) = 1 N n = 1 N ν Ω N n ( E n ε ν ) .
Ω N ( E ) = 1 N n = 1 N ν = 0 f ( d ) ( ν ) Ω N n ( E n ħ ων ) .

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