## Maxwell’s Demon at work: Two types of Bose condensate fluctuations in power-law traps

Optics Express, Vol. 1, Issue 10, pp. 262-271 (1997)

http://dx.doi.org/10.1364/OE.1.000262

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### 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.

© Optical Society of America

*δN*

_{ν}of the occupation numbers

*N*

_{ν}of an ideal Bose gases’s

*ν*-th single-particle state are given by [1,2]

*δN*

_{0}of the ground state occupation number

*N*

_{0}of

*isolated*Bose gases: if the temperature approaches zero, all

*N*particles of an isolated Bose gas occupy the ground state, so that the actual fluctuations vanish, whereas Eq. (1) predicts fluctuations

*δN*

_{0}of the order

*N*. This seems to be one of the most important examples where the different statistical ensembles can not be regarded as equivalent. When computing low-temperature fluctuations of the ground state occupation number for isolated Bose gases, one therefore has to give up the convenient grand canonical point of view, and to resort to a microcanonical treatment.

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]

*δN*

_{0}have been developed only recently [5

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]

*δN*

_{0}is provided by

*N*ideal Bosons trapped in a one-dimensional harmonic potential. Since quasi one-dimensional harmonic trapping potentials can be realized as limiting cases of strongly anisotropic three-dimensional traps [9

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]

*E*equals the number of partitions of the integer

*n*=

*E*/(

*ħω*) into no more than

*N*summands, with

*Μ*being the oscillator frequency. Using the appropriate asymptotic formulae from partition theory, one finds that the microcanonical fluctuations

*δN*

_{0}for this model system vanish linearly with temperature

*T*[11

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]

*k*

_{B}is the Boltzmann constant, and

*δN*

_{0}/

*N*to the corresponding grand canonical fluctuations and to the approximation (2), for

*N*= 10

^{6}particles, this approximation is quite good indeed. The very same result (2) has also been obtained by Wilkens [13] within a

*canonical*approach, that is, for a trap in contact with a heat bath.

*N*

_{ex}|

*E*) even for

*N*

_{ex}>

*N*, which appears to be unphysical: after all, the excitation energy

*E*can not be distributed over more than the

*N*particles. However,

*provided*the microcanonical distributions for finding

*N*

_{ex}out of

*N*particles in an excited trap state,

*N̅*

_{ex}≪

*N*, which should be the case for temperatures well below the onset of Bose-Einstein condensation, we will have Φ(

*N*

_{ex}\

*E*)/Ω(

*E*\

*N*) ≈ 0 for

*N*

_{ex}>

*N*. In that case the generating function (3) would be quite useful, since one could obtain the microcanonical expectation value 〈

*N*

_{0}〉 for the ground state occupation number, and its fluctuation, from

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]

*E*is actually spread out over all

*N*particles carry only negligible statistical weight, so that the overwhelming majority of all microstates leaves a fraction of the particles in the ground state, forming the Bose condensate. Then the restriction on the number of microstates caused by the fact that there is only a finite number

*N*of particles becomes meaningless, so that, loosely speaking, “the system has no chance to know how many particles the condensate consists of”. But if this is the case, i.e., if the system’s properties become insensitive to the actual number of particles contained in the condensate, then one can act as if the condensate particles constituted an

*infinite*reservoir. Thus, the generating function (3) may be regarded as the partition function of a rather unusual ensemble, consisting of the excited-states subsystems of Bose gases that exchange particles with the ground state “reservoirs” without exchanging energy. Since such an exchange process, if performed by hand, requires a genius who is able to separate the hot, excited particles from the cold ones in the ground state, this new ensemble has been called the “Maxwell’s Demon ensemble” [14

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]

*independent of the total particle number N*, as they should be if the system really has no knowledge of the number of condensate particles, and thus of

*N*. The awe-inspiring agreement with the actual microcanonical fluctuations depicted in Fig. 1 leaves no doubt that this approximation is reliable. To further substantiate the new ensemble, we also consider the microcanonical fluctuations

*δN*

_{0}for an ideal Bose gas trapped by a three-dimensional isotropic harmonic oscillator potential [15

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]

*E*|

*N*) of microstates for some given excitation energy

*E*=

*nħω*can then be obtained from the canonical

*N*-particle partition function

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

*β*= 1/(

*k*

_{B}

*T*) denotes the inverse temperature. By means of numerical saddle-point inversions of Eq. (5), we compute the desired numbers Ω(

*E*\

*N*

_{ex}) for

*N*

_{ex}ranging from 1 to

*N*[15

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]

*N*= 1000 and several “low” temperatures. What we find is exactly what is needed for Maxwell’s Demon: the distributions are well peaked for temperatures below the onset of condensation, and remarkably close to Gaussians [12]. It is then no surprise that the corresponding microcanonical low-temperature fluctuations

*δN*

_{0}, obtained from the widths of these distributions, are — as long as a condensate exists! — once again independent of

*N*, as exemplified in Fig. 3 for

*N*= 200, 500, and 1000. As discussed above, it is precisely this

*N*-independence, expressed mathematically by the appearance of the upper summation bound “∞” rather than “

*N*” in Eq. (3), that lies at the bottom of the Maxwell’s Demon ensemble. But whereas this

*N*-independence is, by construction,

*put into*this ensemble, it has

*come out*here as the result of a truly microcanonical calculation [12,15

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]

*N*, not with ∞.

*δN*

_{0}. To this end, we consider ideal Bose gases in

*d*-dimensional traps with arbitrary single-particle energies

*ε*

_{ν}we stipulate

*ε*

_{0}= 0. Denoting the grand canonical partition function by Ξ(

*z*,

*β*), we base our analysis on its “excited” part Ξ

_{ex}(

*z*,

*β*) ≡ (1 -

*z*)Ξ(

*z*,

*β*). Since, by virtue of Eq. (7),

*canonical*moments

*M*

_{k}(

*β*), and generates the

*microcanonical*moments

*N*

_{ex}ranging from 0 to ∞) approximate the true moments of the physical set {Φ(

*N*

_{ex}|

*E*)} (where the number

*N*

_{ex}of excited particles can not exceed the total particle number

*N*); in this approximation one has the identity

*μ*

_{0}(

*E*) = Ω(

*E*|

*N*) [15

**79**, 3557–3560 (1997). [CrossRef]

*N*

_{ex}〉 =

*N*- 〈

*N*

_{0}〉 of the number of excited particles is given by

*δN*

_{0}

*δN*

_{ex}

*t*) and ζ(

*t*) denote the Gamma function and Riemann’s Zeta function, respectively. All the information about the specific trap under consideration is embodied in its spectral Zeta function

*ε*

_{0}= 0. The real number

*τ*in Eqs. (13) and (14) has to be chosen such that the path of integration up the complex

*t*-plane sees all poles to its left.

*d*-dimensional traps with power-law single-particle spectra

*c*

_{i}characterize the trap’s anisotropy, normalized such that the lowest

*c*

_{i}is unity; the characteristic energy ∆ measures the gap between the ground state and the first excited state, and the exponent

*σ*is determined by the potential’s shape. Such systems have been studied first by de Groot

*et al*. [19

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]

*N*-asymptotically large systems and disregard finite-

*N*-effects, that is, if we focus on gases consisting of at least some 10

^{5}particles, say, then a good approximation to the density of states is provided by

*c*

_{i}are not too different from each other [10

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]

*d*/

*σ*> 1 there is a sharp onset of Bose-Einstein condensation at the temperature

*T*

_{0}given by [19

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]

*k*

_{B}

*T*, the behavior of either integral is determined by the pole of its integrand farthest to the right in the complex plane. Keeping in mind that ζ(

*z*) has merely one single pole at

*z*= 1, with residue 1, while the poles of Γ(

*z*) are located at

*z*= 0, - 1, -2, …, the decisive pole is provided

*either*by the system’s Zeta function ζ(

*t*+ 1 -

*d*/

*σ*),

*or*by the other Zeta function that is determined by the order of the cumulant one is asking for: by ζ(

*t*), if one asks for the first cumulant 〈

*N*

_{ex}〉, or by ζ(

*t*- 1), if one asks for the second cumulant (

*δN*

_{0})

^{2}. To see what the argument boils down to, let us first consider the evaluation of Eq. (20), where the system’s pole at

*t*=

*d*/

*σ*competes with the cumulant-order pole at

*t*= 1:

- If
*d*/*σ*> 1, the low-temperature behavior of 〈*N*_{ex}〉 is governed by the pole of ζ(*t*+ 1 -*d*/*σ*) at*t*=*d*/*σ*. Hence the residue theorem yields - If
*d*/*σ*= 1, both Zeta functions in Eq. (20) coincide. We then encounter a double pole at*t*= 1, and findwhere γ = 0.5772… is Euler’s constant. This corresponds to a result obtained already in 1950 by Nanda [20] with the help of the Euler-Maclaurin summation formula. - If 0 <
*d*/σ < 1, the pole of ζ(*t*) at*t*= 1 takes over:so that for sufficiently low temperatures 〈*N*_{ex}〉 now depends linearly on*T*, regardless of the value of*d*/*σ*that characterizes the trap.A mere glance at Eq. (21) then suffices to reveal that the*very same scenario*— a first pole at*t*=*d*/*σ*that endows the temperature dependence with a trap-specific exponent as long as it lies to the right of a second one, which yields universal behavior when it becomes dominant — also governs the canonical condensate fluctuations, with the only difference that the second pole now is located at*t*= 2: - If
*d*/*σ*> 2, the pole of ζ(*t*+ 1 -*d*/*σ*) at*t*=*d*/*σ*wins, givingIf*d*/*σ*= 2, we find at*t*= 2 the already familiar double pole, resulting in - If 0 <
*d*/*σ*< 2, the pole of ζ(*t*- 1) at*t*= 2 lies to the right of its rival, yielding

*d*= 1,

*σ*= 1,

*A*= 1 and ∆ =

*ħω*, so that we recover our previous microcanonical result (2) within the canonical ensemble, recalling that ζ(2) =

*π*

^{2}/6.

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

*microcanonical*quantities now requires saddle-point inversions of Eq. (9) in order to obtain the microcanonical moments

*μ*

_{k}(

*E*) from the canonical moments

*M*

_{k}(

*β*). Performing these inversions, and reexpressing energy in terms of temperature, we find that the integral (13) — and, hence, the results (22), (23), and (24) for the number of excited particles — remains valid within the microcanonical ensemble. The fluctuations require more care: Whereas canonical and microcanonical fluctuations coincide in the large-

*N*-limit for

*d*/

*σ*< 2, the micro-canonical mean-square fluctuations (

*δN*

_{0}

*d*/

*σ*> 2:

*d*/

*σ*> 1, and temperatures below the condensation temperature

*T*

_{0}, the heat capacity per particle is given by

*T*

_{0}by

*z*approaches unity from below when

*T*approaches

*T*

_{0}from above, so that the Bose function

*g*

_{α}(

*z*) approaches

*ζ*(

*α*), we see that the heat capacity remains continuous at

*T*

_{0}for 0 <

*d*/

*σ*≤ 2, but exhibits a jump of size

*d*/

*σ*> 2.

*d*and trap exponent

*σ*> 0, the fluctuation of the number of condensate particles is independent of the total particle number

*N*. For isolated traps, this insensitivity of the system with respect to

*N*reflects the well-peakedness of the microcanonical distributions (4), see Fig. 2: if there is a condensate, the behavior of the ideal Bose gas does not depend on how many particles the condensate consists of. If

*d*/

*σ*< 2, so that the heat capacity remains continuous in the large-

*N*-limit, canonical and microcanonical fluctuations

*δN*

_{0}vanish linearly with temperature, see Eq. (27). If

*d*/

*σ*= 2, there appears a logarithmic correction to the linear

*T*-dependence, as quantified by Eq. (26). But if

*d*/

*σ*> 2, so that the heat capacity becomes discontinuous, then the fluctuations

*δN*

_{0}vanish proportionally to

*T*

^{d/2σ}, so that now the properties of the trap determine the way the fluctuations depend on temperature. In addition, in this case the microcanonical fluctuations are markedly lower than the fluctuations in a trap that exchanges energy with a heat bath.

*δN*_{ex}∝ 〈*N*_{ex}〉^{1/2}for 2 <*b*/*σ*;*δN*_{ex}∝ 〈*N*_{ex}〉^{σ/d}for 1 <*b*/*σ*< 2;*δN*_{ex}∝ 〈*N*_{ex}〉 for 0 <*b*/*σ*< 1,

*d*/

*σ*> 2 to much stronger fluctuations for 0 <

*d*/ < 1 appears noteworthy.

*d*-dimensional isotropic harmonic traps, the Maxwell’s Demon approximation (i.e., the replacement of the true upper summation bound “

*N*” in Eq. (10) by “∞”) is

*exact*below the “restriction temperature” (i.e., that temperature where the number

*n*=

*E*/(

*ħω*) of energy quanta equals the number

*N*of particles [15

**79**, 3557–3560 (1997). [CrossRef]

*nħω*|

*N*) of microstates also under conditions where the restriction due to the finite N becomes decisive. Incidentially, one meets the task of computing such restricted partitions of integers also in other problems of statistical mechanics, for example in the theory of the so-called compact lattice animals, or of the infinite-state Potts model [22

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]

*n*=

*E*/(

*ħω*) ≈ ζ(2)(

*k*

_{B}

*T*/

*ħω*)

^{2}, we find the truly remarkable formula

*N*

_{ex}〉, finds a substitute:

*For ideal Bose particles trapped at low temperatures by a one-dimensional harmonic potential, the root-mean-square fluctuation of the number of ground state particles is given by the square root of the number of energy quanta*.The mathematician, who approaches Eq. (32) from the viewpoint of partition theory, sees the solution to another problem:*If one considers all unrestricted partitions of the integer n into positive, integer summands, and asks for the root-mean-square fluctuation of the number of summands, then the answer is (asymptotically) just*√*n*

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]

## References

1. | L.D. Landau and E.M. Lifshitz, |

2. | R.K. Pathria, |

3. | I. Fujiwara, D. ter Haar, and H. Wergeland, “Fluctuations in the population of the ground state of Bose systems”, J. Stat. Phys. |

4. | R.M. Ziff, G.E. Uhlenbeck, and M. Kac, “The ideal Bose-Einstein gas, revisited”, Phys. Rep. |

5. | M. Gajda and K. Rzążewski, “Fluctuations of Bose-Einstein condensate”, Phys. Rev. Lett. |

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 |

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. |

8. | C.C. Bradley, C.A. Sackett, and R.G. Hulet, “Bose-Einstein condensation of lithium: observation of limited condensate number”, Phys. Rev. Lett. |

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 |

10. | N.J. van Druten and W. Ketterle, “Two-step condensation of the ideal Bose gas in highly anisotropic traps”, Phys. Rev. Lett. |

11. | S. Grossmann and M. Holthaus, “Microcanonical fluctuations of a Bose system’s ground state occupation number”, Phys. Rev. E |

12. | S. Grossmann and M. Holthaus, “From number theory to statistical mechanics: Bose-Einstein condensation in isolated traps”, to appear |

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. |

15. | S. Grossmann and M. Holthaus, “Fluctuations of the particle number in a trapped Bose-Einstein condensate”, Phys. Rev. Lett. |

16. | P. Borrmann and G. Franke, “Recursion formulas for quantum statistical partition functions”, J. Chem. Phys. |

17. | B. Eckhardt, “Eigenvalue statistics in quantum ideal gases”. In: |

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 |

20. | V.S. Nanda, “Bose-Einstein condensation and the partition theory of numbers”, Proc. Nat. Inst. Sci. (India) |

21. | H.D. Politzer, “Condensate fluctuations of a trapped, ideal Bose gas”, Phys. Rev. A |

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. |

**OCIS Codes**

(000.6590) General : Statistical mechanics

(020.7010) Atomic and molecular physics : Laser trapping

**ToC Category:**

Focus Issue: Fluctuations and oscillations of Bose-Einstein

**History**

Original Manuscript: September 10, 1997

Published: November 10, 1997

**Citation**

Siegfried Grossmann and Martin Holthaus, "Maxwell's Demon at work: Two types of Bose condensate fluctuations in power-law traps," Opt. Express **1**, 262-271 (1997)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-1-10-262

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### References

- 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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