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.

© 1997 Optical Society of America

After decades of plenteous investigation devoted to the ideal Bose gas, it has only recently transpired that the counting statistics in these systems poses an interesting problem of its own. Indeed, while the textbook grand-canonical prediction of the condensate mean occupation agrees fairly well with the Bose-Einstein condensation of trapped atomic gases [1,2], this cannot even be approximately true with respect to the counting statistics

PνG(n)=11+nν¯(nν¯1+nν¯)n,

which gives the probability to find n particles in a given single-particle state ν where the mean occupation is nν ̅. Below the Bose-Einstein condensation temperature, where the ground state mean-occupation is macroscopic, n 0̅ ~ , the distribution P0G(n) becomes extremely broad with the mean-square variance δ2n0n02¯n0¯2 given by

δ2n0=(1[TTc]η)2N¯2,

where η is a trap-dependent exponent. Besides the unusual scaling δ 2 n 0 2, which defies all thermodynamics [3], the fluctuations actually grow with decreasing temperature. This latter prediction is surely at odds with the isolated Bose gas, where for sufficiently small energy all particles are expected to occupy the ground state with no fluctuations left [4–7]. One may then argue that the unusual behavior predicted in Eq. (2) is in fact due to the “ideality” of the gas and its being in diffusive contact with an infinite particle reservoir, but this is wrong. A simple exercise in combinatorics reveals that a physical particle reservoir implies Poissonian number fluctuations, δ 2 n 0 ~ , which is in strong contrast to the grand-canonical prediction (2) [5]. As was first pointed out by Ziff et al. [6], the anomalous behavior (2) is just a mathematical artefact of the standard grand-canonical ensemble which becomes unphysical below the condensation point.

 

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).

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It is interesting to compare the counting statistics (1) with the predictions of other statistical ensembles. Two ensembles which are frequently used are the canonical ensemble, where the system exchanges energy with a heat bath of a given temperature, and the microcanonical ensemble, where the system is completely isolated from its environment. The counting statistics P 0(n) of these ensembles are illustrated in Fig. 1 for the ideal Bose gas in a three-dimensional harmonic oscillator trapping potential for a total of N = 200 particles at three different temperatures. While for the high temperature all three ensemble predict the same counting statistics, this is certainly not the case for the low temperatures. Here the broad distribution of the grand-canonical statistics differs most dramatically from the single-peaked distribution of the canonical and microcanonical statistics, respectively. In fact, it is this single-peaked statistics which one would naively expect for a genuine Bose condensate, may it be isolated or in contact with a heat bath.

The distributions depicted in Fig. 1 are simple plots of the exact counting statistics in the respective ensembles. The corresponding analytical expressions are derived in the appendix; here we only summarize the results. In the canonical ensemble the probability to find n particles occupying the single-particle state ν is given by

PνC(n)=enβενZNnZNe(n+1)βενZNn1ZN.

Here ZN is the canonical partition function for N particles, εν is the single-particle energy of the level ν, β = (kBT)-1, and we set the ground state energy ∊0 = 0. Except for some exactly solvable models [7], the formula (3) seems to be of little use because it involves the canonical partition function which is difficult to calculate. Fortunately, for ideal Bose gases quite a powerful recurrence relation exists1

ZN(β)=1Nn=1NZ1(nβ)ZNn(β),Z0=1,

which enables one to numerically compute the entire counting statistics (3) for up to N = 105 particles, say.

Incidentally, the same line of reasoning which leads to Eq. (3) can also be employed for a derivation of the counting statistics in the microcanonical ensemble - see the appendix for details. The result reads

PνM(n)=ΩNn(Enεν)ΩN(E)ΩNn1(E(n+1)εν)ΩN(E),

where the microcanonical partition function obeys the recurrence relation

ΩN(E)=1Nn=1NνΩNn(Enεν).
 

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.

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For the particular case of a isotropic harmonic oscillator trapping potential in D spatial dimensions, where the single-particle energies assume integer values in suitably scaled units, this recurrence relation simplifies

ΩN(E)=1Nn=1Nν=0f(D)(ν)ΩNn(Enħων),

where f(D)(ν)1(D1)!k=1D1(ν+Dk).

The low-temperature counting statistics in Fig. 1 also display some differences in the microcanonical and canonical ensemble, respectively. While both are narrow-peaked around the condensate mean occupation, and the width of both peaks decreases with decreasing temperatures, the width of the microcanonical peak is slightly smaller than the width of the canonical peak. This difference is also seen Fig. 2 which shows the root-mean-square fluctuations δn 0 = [〈n02〉 - 〈n 02]1/2 in the canonical and microcanonical ensemble, respectively, as a function of the temperature for total number of particles N = 10, 42,100. Although the particle numbers in these plots are quite small, general trends can be identified. Save for extremely low temperatures, where the internal energy is of the order of the energy gap ∆, fluctuations display the same temperature dependence in both the ensembles, although they are slightly smaller in the microcanonical ensemble than in the canonical ensemble. Second, for sufficiently low temperatures, in both the ensembles the fluctuations are independent of N. As we shall see subsequently, the temperature dependence is in fact α T 3/2, and the N-independence is in fact a manifestation of the ideality of the gas.

In the reminder of this paper we analyse δn 0 for temperatures below the condensation point in the large-N asymptotic limit [11]. We shall concentrate here on the canonical ensemble; the case of the microcanonical ensemble is covered in a work by Gajda and Rzazewski [12] and a series of papers by Grossmann and Holthaus [13,14,15,16]. Our analysis is guided by an idea of Politzer [17] where, instead of considering the statistics of the no particles in the trap ground state, one considers the counting statistics of the remaining N ex = N - n 0 particles in the trap excited states. This detour, which recently was rigorously formulated by Navez et al [18], proves to be quite useful, as the many excited state configurations of the trap allow one to apply grand-canonical techniques in the end.

Working in the canonical ensemble, the numbers n 0 and N ex = Σν≠0 nν are stochastic quantities while the total number of particles N = n 0 + N ex is fixed and non-fluctuating. Since the alternatives of “being in the trap ground state” and “being in a trap excited state” are exclusive and complete, the probability to find M particles in the trap excited states, P ex(M), is just given by the probability to find N - M particles in the trap ground state, P ex(M) = P0C(N - M), where P0C(n) is given in Eq. (3). Consider now the N ex-moment generating function

ϒ(z)ZNM=0NzMP0C(NM).

From this we obtain [18]

Nn0=zzlnϒ(z)|z=1,
δ2n0=(zz)2lnϒ(z)|z=1,

where we have used δ 2 n 0δ 2 N ex, and the angular brackets denote expectation values with respect to P0C(n), that is 〈n0k〉 ≡ Σn=0N nk P0C(n).

We now assume Bose degeneracy for the trap ground state

n0~O(N)andδn0n0,

where the second condition assures P 0(n) ≈ 0 for small n. If this condition holds - which may be well expected for sufficiently low temperatures, see Fig. 1 - the upper limit of the M-sum in Eq. (8) may be taken to infinity. Inserting P0C (Eq. (3)) and shifting the summation variable one obtains

ϒ(z)(1z)M=0zMZM(β).

Here the right-hand side is easily identified with the the grand-canonical partition function for the occupation of the trap excited states. More explicitly

lnϒν0ln(1zeβεν),

where ν ≠ 0 means that the single-particle ground state is excluded from the sum.

The validity of the assumption (11) must be proven selfconsistently in the approximation (12). If it applies, the N ex counting statistics does not depend on the total number of particles N: according to Eq. (9), the adding of N′ particles to the system just means adding these N′ particles to the condensate without affecting the statistics. Due to this insensitivity to the number of particles in the condensate, which in this form can only be expected for non-interacting particles, one may regard the condensate as an infinite particle reservoir for the above-condensate part, as was first pointed out by Politzer [17]. However, since the particle reservoir has zero temperature, the exchange of particles requires the ability to distinguish between cold and hot atoms. Quite appropriately, Navez et al. call the somewhat unusual ensemble which emerges from these considerations the “Maxwell’s Demon ensemble” [18].

 

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.

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Even though we are working in the canonical ensemble, the Maxwell’s Demon approximation (13) has turned the calculation of condensate fluctuations into an exercise in grand-canonical statistics. To complete this little exercise we shall consider traps which are parametrized by a single-particle energy spectrum of the form [19]

εν=Δi=1Dciνi,σ0<σ2,

where D is the spatial dimension of the system, ∆ is the energy gap between the trap ground state and trap first excited state, ci are geometric coefficients of order unity with the convention mini ci = 1, σ is the single-particle energy spectral index, and νi are integer quantum number for the ith cartesian direction. For spatial dimensions D > σ the temperature for the onset of Bose-Einstein condensation is given by [19]

kBTcΔ~[i=1Dci]1D[Γ(1σ+1)]σ[Nζ(Dσ)]σD,

where the right-hand side should be divided by 2D in the case of the box with periodic boundary conditions. ζ(x) denotes the Riemannian zeta function and Γ(x) the gamma function. Below Tc , the condensate mean-occupation is given by

n0~(1[TTc]Dσ)N.

At the lower critical dimension D = σ - the one-dimensional harmonic oscillator trapping potential, say - quasi-condensation occurs for a temperature [7,13]2

kBTcħω~NlnN.

For spatial dimensions D < σ - the one-dimensional box, say - Bose-Einstein condensation does not occur. But there still exists a temperature below which the counting statistics P0C(n) displays a single peak, see Fig. 3. Since the Maxwell’s Demon approximation (12) was solely based on the condition (11), it may still be employed to calculate the low-temperature fluctuations of the ground state occupation even for this exotic range of spatial dimensions.

We now calculate the fluctuations (10) in the Maxwell’s Demon approximation. Using (13) in Eq.(10), differentiating twice, and setting z = 1 leads to

δ2n0ν014sinh2(βεν2),

where the summation extends over all single-particle states save for the single-particle ground state. For the subsequent asymptotic analysis, this representation turns out to be useful only for low-dimensional traps D < 2σ. For high-dimensional traps D > 2σ a representation in terms of Bose functions proves to be more useful. Expanding ln ϒ(z) in a power-series in z before derivatives are taken leads to

δn02k=1k[Z(kβ)1],

where Z is the single-particle canonical partition function.

To simplify the subsequent derivation we now assume isotropic traps, ci = 1, and we set β∆ → β. Since εν is additive in the cartesian directions, Z factorizes, Z = [S + 1]D, where we have defined the one-dimensional partition function

S(β)ν=1eβνσ,

which excludes the contribution from ν = 0. Expanding the product [1 + S]D and subtracting unity, Eq. (19) becomes

δn02d=1DDdk=1kS(kβ)d.

Evidently, the fluctuations are given by a sum of D terms of increasing spatial dimensionality. In previous investigations [11,16] only that term was taken into account where the dimensionality is largest. Since it is well known that fluctuations are the larger the smaller the spatial dimension, we are well advised to keep all contributions before the analysis is finally complete.

We are interested here in the temperature range T < Tc but T ≫ ∆/kB which in scaled units amount to the small-β limit. In order to extract the series expansion in β -1 we apply the Euler summation formula for the evaluation of the one-dimensional partition functions S. In leading order

S(kβ)~Γ(1+1σ)ekβ[kβ]1σ.

The exponential is also found in the large-β asymptotic limit of S; we keep it here since it guarantees convergence of the k-sum for arbitrary spectral index σ. Using (22) in (21), we finally obtain

δ2n0~d=1DDdΓ(1+1σ)dgdσ1(edβ)βdσ

where gμ(z)k=1zkkμ is a Bose function. For the remaining steps we recall the small-β Robinson expansion

βηgη1(eβ)~{ζ(η1)βηforη>2β2ln(β1)forη=2Γ(2η)β2forη<2

Depending of the type of trap σ and spatial Dimension D we find:

 

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.

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  • For D > 2σ, the d = D term in Eq. (23) provides the leading order in the limit β → 0. In gDσ1(eDβ) the argument may safely be replaced by unity and one obtains

    δ2n0~C[kBTΔ]Dσ,

    where

    C=Γ(1+1σ)D[i=1Dci]1σζ(Dσ1).

    For the three-dimensional harmonic oscillator C =ζ(2), and we recover the formula previously derived by Politzer [17].

  • For D = 2σ the d = D term still dominates in the small-β limit, but the Bose-function g 1 develops a logarithmic singularity g 1(e -) ~ - ln[β]. In this case

    δ2n0~C[kBTΔ]2ln(kBTΔ),

    with C given in Eq. (26).

  • For D < 2σ the series representation (23) ceases to be useful since all the terms including their corrections contribute in order T 2. However, the leading term of the small-β expansion of the denominator of (18) provides a convergent series such that

    δn02~C[kBTΔ]2,

where the amplitude is now given by

C=ν01[ν1σ+ν2σ++νDσ]2.

For the particular case of the one-dimensional harmonic oscillator C=π26; we here recover the results previously obtained by Grossmann and Holthaus in the microcanonical ensemble [13], and in [7] for the canonical ensemble. In passing we note that for the standard box C′ = 15.5787&, 6.008684&, 2ζ(4) in spatial dimensions D = 3, 2, 1.

These predictions are well confirmed in Fig. 4 where we compare exact canonical data for N = 104 and 105 particles with the corresponding asymptotic formulas for the case of a three-dimensional harmonic oscillator potential and three-dimensional equal-sided box. Even for N = 104 the agreement between the exact data and the asymptotic formulas is already quite good. Further improvement is obtained by taking finite-N corrections into account, see also [12]. Note that the formula (28) also applies for spatial dimensions D < σ which are too small to support Bose-Einstein condensation. For the particular case of a one-dimensional box the asymptotic prediction is in excellent agreement with the exact data, see Fig. (3).

To complete our analysis we must now check whether the condition δn 0 ≪ 〈n 0〉 underlying the Maxwell’s Demon approximation is satisfied. The examples indicate that this is the case. By virtue of the Tc -dependent expression (16) for 〈n 0〉 we use Eq. (15) to eliminate ∆ in Eqs.(25)–(28) and obtain [11]

δn0~A[TTc]rNs,

where the scaling exponents r and s and amplitude A only depend on the trap spatial dimension D and spectral index σ

r={D2σifD>2σ1ifD<2σ,s={12ifD>2σσDifD<2σ.

Although s is not a genuine scaling exponent, it enables us to explain the N-dependence of the rms-fluctuations at the condensation temperature. For high dimensional traps D > 2σ the root-mean-square fluctuations display the proper thermodynamic scaling δ n 0 ~ √N, while in low-dimensional traps, σ < D < 2σ, the condensate counting statistics is super-fluctuant, δ n 0 ~ Ns with 1/2 < s < 1, and δ n 0 ~ N/ln(N) at the lower critical dimension D = σ. In any case, for Dσ the normalized rms fluctuations δ n 0/N vanish in the limit N → ∞ for all temperatures. Comparing 〈n 0〉, Eq. (16), with δ n 0, Eq. (30) one finds that the condition δ n 0 ≤ 〈n 0〉 is satisfied for temperatures

TTc<1AN1s.

from which one infers the validity of the Maxwell’s Demon approximation below Tc .

In a recent investigation of the fluctuations of trapped Bose gases, Grossmann and Holthaus [16] have found the same scaling exponents (31) in microcanonical ensemble. For low dimensional traps D < 2σ even the amplitude A turns out to be the same in both the canonical and microcanonical ensemble.3 For high-dimensional traps D > 2σ, however, the amplitude is O(1) larger in the canonical ensemble.

Quite remarkably, these findings fully respect the principle of thermodynamic equivalence. This principle states that, save for criticality, mean-square fluctuations of extensive quantities in the various ensembles may differ at most by an extensive quantity. In our case this means that different amplitudes may be expected - and indeed are found - for D > 2σ, but not for D < 2σ, since in this latter case the mean-square fluctuations are “more than extensive”.

Note added: After submission of this work we learned that Eqs. (5)–(7) where derived indepentently by Chase, Mekjian and Zamick [21].

The material of this paper was presented at the International Conference Quantum Optics IV which took place in Jaszowiec, Poland, June 17-24, 1997. MW would like to thank the participants of this workshop, and in particular M. Gajda, K. Rzazewski and P. Navez for truly helpful comments. CW acknowledges support by the Studienstiftung des Deutschen Volkes. MW acknowledges support by the Forschergruppe “Quan-tengase” of the Deutsche Forschungsgemeinschaft and long-lasting hospitality in the group of Jürgen Mlynek.

A APPENDIX

We here derive the counting statistics and recurrence relations for the ideal Bose gas in the canonical and microcanonical ensemble, respectively.

We consider a set of N ideal Bosons in a trap. The many-particle energy is given by the Hamiltonian

H=ν=0ενnν,

where ν stands for a set of quantum numbers which label a given single-particle trap state, εν is the associated energy with ground-state energy ε 0 = 0 by convention, and the occupation numbers assume values nν = 0,1, 2,….

In the microcanonical ensemble the system is assumed to be found with equal probability 1/Ω in any microstate which is compatible with the total energy E and total number of particles N.4 The microstates being nothing but the configurations of occupation numbers, {n} = {n 0,ni ,…}, the microcanonical partition function Ω ≡ ΩN(E) is given by

ΩN(E)=n0=0n1=0n=0δH,EδΣnν,N,

where the Kronecker-Deltas assure that the system contains exactly N particles which share a total amount of energy E.

Systems which exchange energy (but no particles) with a heat bath of a given temperature T are described by the canonical ensemble. Denoting β = 1/(kBT), the probability to find a particular system microstate {n} = {n 0,ni ,…} is given by the Boltzmann distribution

P({n})=1ZNexp{βν=0ενnν}δΣnν,N,

where ZNZN (β) is the canonical partition function,

ZN(β)=n0=0nν=0n=0eβHδΣnν,N.

The particle counting statistics for the νth trap level are denoted Pν (n) ≡ 〈δnν ,n〉 which gives the probability to find n particles occupying trap level ν. This quantity is most conveniently represented in the form Pν(n)=Pν(n)Pν(n+1), where Pν(n)θnνn denotes the probability to find at least n particles in trap state ν and θ nν-n is the discrete Heavyside function.

The computation of Pν(n) is elementary. In the microcanonical ensemble, we just have to count the number of microstates for which N - n particles share an energy E - ν . This number being just ΩN-n(E - ν ), we have Pν(n)=ΩNn(Enεν)/ΩN(E) and therefore

PνM(n)=ΩNn(Enεν)ΩN(E)ΩNn1(E(n+1)εν)ΩN(E),

where the super-script M specifies the microcanonical ensemble.

In the canonical ensemble, the computation of Pν(n) involves summations as in Eq. (36), the only difference being that the sum over nν starts at nν = n and not at nν = 0. Upon choosing the summation variable nν = nν - n one extracts a factor e -βnεν which leads to Pν(n)=enβενZNnZN and therefore

PνC(n)=enβενZNnZNe(n+1)βενZNn1ZN,

where the super-script C specifies the canonical ensemble.

The derivation of the recurrence relation for the canonical partition function proceeds by considering the mean occupation of a given trap state ν, that is 〈nν 〉 ≡ Σn=0N nPνC (n). Using Eqs (38) one finds after some simple reshuffling

nν=1ZNn=1NenβενZNn.

Summing over ν, and recalling that the total number of particles is given by N ≡ Σνnν 〉, we find

N=1ZNn=1NZNnνenβεν,

where the last sum over states is nothing but the single-particle canonical partition function Z 1(). Since in the canonical ensemble N is not fluctuating,

ZN(β)=1Nn=1NZ1(nβ)ZNn(β),Z0=1

which is the desired recurrence relation.

For the microcanonical partition function one invokes similar lines of reasoning. The average number of particles in the νth state is given by:

nνn=1NnPν(n|E,N).

Eq. (37) can be used to derive:

nν=n=1NΩNn(Enεν)ΩN(E).

The total number of particles is fixed:

Nνnν

together with Eq. (43) this leads to:

ΩN(E)=1Nn=1NνΩNn(Enεν).

with boundary conditions ΩN≥0(0) ≡ 1 and Ω0(E > 0) ≡ 0 (those boundary conditions follow from the fact that ε 0 ≡ 0 and Z 0 ≡ 1 with ZN ≡ ΣE e -ΩN(E)). For finite E the sum over ν is finite because of ΩN(E < 0) ≡ 0. For the d-dimensional harmonic oscillator Eq. (45) simplifies to:

ΩN(E)=1Nn=1Nν=0f(d)(ν)ΩNn(Enħων).

where f(d)(ν)1(d1)!k=1d1(ν+dk).

Footnotes

1See p. 432ff in [8] and references cited therein. Recently, the canonical recurrence relation was used by several groups [9,10]. A derivation is included in the Appendix of the present paper.
2We do not indulge into a discussion of the “true” versus “marginal” Bose-Einstein condensation, “existence” versus “non-existence” of the thermodynamic limit, or “presence” versus “absence” of a phase transition. The interested reader may consult a recent work by Mullin where the shortcomings of these classificatory attempts are illuminated [20]. The convention here is that whenever (11) is realized the system is interesting.
3The amplitude found in Ref. [16] is incorrect; see the discussion of Eq. (28).
4Additional conserved quantities, like the total momentum or the total angular momentum, say, may be included; the corresponding ensemble becomes then useful for studies devoted to “super”-behavior, like superfluidity or superconductivity.

References

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. 774984 (1996). [CrossRef]   [PubMed]  

3. E. Schrödinger, 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 fluctuations 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 York1961.

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 fluctuations 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].

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. 774984 (1996).
    [Crossref] [PubMed]
  3. E. Schrödinger, 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 fluctuations 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 York1961.
  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 fluctuations 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].

1997 (3)

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

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]

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

1996 (5)

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]

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

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

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. 774984 (1996).
[Crossref] [PubMed]

1993 (1)

P. Borrmann and G. Franke, “Recursion formulas for quantum statistical partition functions”, J. Chem. Phys. 98, 2484 (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]

1950 (1)

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]

Andrews, M.R.

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]

Bitouk, D.

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]

Borrmann, P.

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

Brosens, F.

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]

Chase, K.C.

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].

Cornell, E.A.

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. 774984 (1996).
[Crossref] [PubMed]

de Groot, S.R.

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]

Devreese, J.T.

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]

Druten, N.J. van

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]

Durfee, D.S.

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]

Ensher, J.R.

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. 774984 (1996).
[Crossref] [PubMed]

Franke, G.

P. Borrmann and G. Franke, “Recursion formulas for quantum statistical partition functions”, J. Chem. Phys. 98, 2484 (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. Rzazewski, “Fluctuations of Bose-Einstein Condensate”, Phys. Rev. Lett. 78, 2686 (1997).
[Crossref]

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]

Grossmann, S.

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).

Holthaus, M.

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, “Maxwell’s Demon at work: Two types of Bose condensate fluctuations in power law traps”, (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, “Fluctuations of the particle number in a trapped Bose condensate”, (Preprint, Marburg, 1997).

Hooyman, G.J.

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]

Idziaszek, Z.

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]

Jin, D.S.

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. 774984 (1996).
[Crossref] [PubMed]

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.

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]

Kurn, D.M.

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]

Landsberg, P. T.

P. T. Landsberg, Thermodynamics - with quantum statistical illustrations, Interscience Publishers, New York1961.

Lemmens, L.F.

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]

Matthews, M.R.

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. 774984 (1996).
[Crossref] [PubMed]

Mekjian, A.Z.

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].

Mewes, M.-O.

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]

Mullin, W.J.

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

Navez, P.

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]

Politzer, H. D.

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

Rzazewski, K.

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]

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

Schrödinger, E.

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

ten Seldam, C.A.

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]

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”, (submitted, 1997).

M. Wilkens and C. Weiss, “Particle number fluctuations in an ideal Bose gas”, J. Mod. Opt., in press ( 1997).
[Crossref]

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.

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. 774984 (1996).
[Crossref] [PubMed]

Wilkens, M.

M. Wilkens and C. Weiss, “Particle number fluctuations in an ideal Bose gas”, J. Mod. Opt., in press ( 1997).
[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).

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

Zamick, L.

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].

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 (1993).
[Crossref]

J. Low Temp. Phys. (1)

W.J. Mullin, “Bose-Einstein Condensation in a Harmonic Trap”, J. Low Temp. Phys. 106, 615 (1997).
[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 (1)

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. (4)

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]

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. 774984 (1996).
[Crossref] [PubMed]

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

Proc. R. Soc. London A (1)

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]

Solid State Commun. (1)

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]

Other (9)

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

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 York1961.

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

M. Wilkens and C. Weiss, “Particle number fluctuations in an ideal Bose gas”, J. Mod. Opt., in press ( 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].

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).

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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)

Equations on this page are rendered with MathJax. Learn more.

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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