## 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*/*T*_{c}
]
^{r}*N*^{s}
with scaling exponents *r* = 3/2, *s* = 1/2 for the
3*D* harmonic oscillator trapping potential, and *r* = 1,
*s* = 2/3 for the 3*D* 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.

© 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

which gives the probability to find *n* particles in a given single-particle
state *ν* where the mean occupation is *n*_{ν}*
̅*.
Below the Bose-Einstein condensation temperature, where the ground state mean-occupation is
macroscopic, *n*
_{0}̅ ~
*N̅*, the distribution
${P}_{0}^{\mathrm{G}}$(*n*) becomes extremely broad with the
mean-square variance
${\delta}^{2}{n}_{0}\equiv \overline{{n}_{0}^{2}}-{\overline{{n}_{0}}}^{2}$ given by

where *η* is a trap-dependent exponent. Besides the unusual scaling
*δ*
^{2}
*n*
_{0}
*∝*
*N̅*^{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}
~ *N̅*, 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.

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

Here *Z*_{N}
is the canonical partition function for *N*
particles, *ε*_{ν}
is the single-particle energy of the level
*ν*, *β* =
(*k*_{B}*T*)^{-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 exists^{1}

which enables one to numerically compute the entire counting statistics (3) for up to
*N* = 10^{5} 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

where the microcanonical partition function obeys the recurrence relation

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

where ${f}^{\left(D\right)}\left(\nu \right)\equiv \frac{1}{\left(D-1\right)!}{\prod}_{k=1}^{D-1}\left(\nu +D-k\right).$

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} =
[〈${n}_{0}^{2}$〉 -
〈*n*
_{0}〉^{2}]^{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} = Σ_{ν≠0}
*n*_{ν}
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*) =
${P}_{0}^{\mathrm{C}}$(*N* - *M*), where
${P}_{0}^{\mathrm{C}}$(*n*) is given in Eq. (3). Consider now the *N*
_{ex}-moment generating function

From this we obtain [18]

where we have used *δ*
^{2}
*n*
_{0}
≡ *δ*
^{2}
*N*
_{ex}, and the
angular brackets denote expectation values with respect to
${P}_{0}^{\mathrm{C}}$(*n*), that is
〈${n}_{0}^{k}$〉 ≡ ${\mathrm{\Sigma}}_{n=0}^{N}$
*n*^{k}
${P}_{0}^{\mathrm{C}}$(*n*).

We now assume Bose degeneracy for the trap ground state

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 ${P}_{0}^{C}$ (Eq. (3)) and shifting the summation variable one obtains

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

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].

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]

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

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

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

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
${P}_{0}^{\mathrm{C}}$(*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

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* < 2*σ*. For
high-dimensional traps *D* > 2*σ* 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

where *Z* is the single-particle canonical partition function.

To simplify the subsequent derivation we now assume isotropic traps,
*c*_{i}
= 1, and we set *β*∆
→ *β*. Since *ε*_{ν}
is
additive in the cartesian directions, *Z* factorizes, *Z* =
[*S* + 1]^{D}, where we have defined the one-dimensional partition function

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

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* <
*T*_{c}
but *T* ≫
∆/*k*_{B}
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

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

where ${g}_{\mu}\left(z\right)\equiv \sum _{k=1}^{\infty}\frac{{z}^{k}}{{k}^{\mu}}$ is a Bose function. For the remaining steps we recall the
small-*β* Robinson expansion

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

- For
*D*> 2*σ*, the*d*=*D*term in Eq. (23) provides the leading order in the limit*β*→ 0. In ${g}_{\frac{D}{\sigma}-1}\left({e}^{-D\beta}\right)$ the argument may safely be replaced by unity and one obtainswhere

$$C=\frac{\Gamma {\left(1+\frac{1}{\sigma}\right)}^{D}}{{\left[{\prod}_{i=1}^{D}{c}_{i}\right]}^{\frac{1}{\sigma}}}\zeta \left(\frac{D}{\sigma}-1\right).$$For the three-dimensional harmonic oscillator

*C*=ζ(2), and we recover the formula previously derived by Politzer [17]. - For
*D*= 2*σ*the*d*=*D*term still dominates in the small-*β*limit, but the Bose-function*g*_{1}develops a logarithmic singularity*g*_{1}(*e*^{-Dβ}) ~ - ln[*β*]. In this case$${\delta}^{2}{n}_{0}~C{\left[\frac{{k}_{\text{B}}T}{\Delta}\right]}^{2}\mathrm{ln}\left(\frac{{k}_{\text{B}}T}{\Delta}\right),$$with C given in Eq. (26).

- For
*D*< 2*σ*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

where the amplitude is now given by

For the particular case of the one-dimensional harmonic oscillator $C\prime =\frac{{\pi}^{2}}{6}$; 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* =
10^{4} and 10^{5} 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* = 10^{4} 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
*T*_{c}
-dependent expression (16) for
〈*n*
_{0}〉 we use Eq. (15) to eliminate ∆ in Eqs.(25)–(28) and obtain [11]

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

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* > 2*σ* the
root-mean-square fluctuations display the proper thermodynamic scaling
*δ*
*n*
_{0} ~
√*N*, while in low-dimensional traps, *σ*
< *D* < 2*σ*, the condensate
counting statistics is super-fluctuant,
*δ*
*n*
_{0} ~
*N*^{s}
with 1/2 < *s* < 1, and
*δ*
*n*
_{0} ~
*N*/ln(*N*) at the lower critical dimension *D* =
*σ*. In any case, for *D* ≥
*σ* 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

from which one infers the validity of the Maxwell’s Demon approximation below
*T*_{c}
.

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* < 2*σ* even the
amplitude *A* turns out to be the same in both the canonical and microcanonical ensemble.^{3} For high-dimensional traps *D* >
2*σ*, 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* >
2*σ*, but not for *D* <
2*σ*, 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

where *ν* stands for a set of quantum numbers which label a given
single-particle trap state, *ε*_{ν}
is the associated energy
with ground-state energy *ε*
_{0} = 0 by convention, and the
occupation numbers assume values *n*_{ν}
= 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},*n*_{i}
,…}, the
microcanonical partition function Ω ≡ Ω_{N}(*E*) is given by

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/(*k*_{B}*T*), the probability to find a
particular system microstate {*n*} =
{*n*
_{0},*n*_{i}
,…} is given by the
Boltzmann distribution

where *Z*_{N}
≡
*Z*_{N}
(*β*) is the canonical partition
function,

The particle counting statistics for the *ν*th trap level are denoted
*P*_{ν}
(*n*) ≡
〈*δn*_{ν}
,*n*〉 which
gives the probability to find *n* particles occupying trap level
*ν*. This quantity is most conveniently represented in the form ${P}_{\nu}\left(n\right)=P\underset{\nu}{\ge}\left(n\right)-P\underset{\nu}{\ge}\left(n+1\right),$ where $P\underset{\nu}{\ge}\left(n\right)\equiv \u3008{\theta}_{{n}_{\nu}-n}\u3009$ 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 $P\underset{\nu}{\ge}\left(n\right)$ is elementary. In the microcanonical ensemble, we just have to count the
number of microstates for which *N* - *n* particles share an
energy *E* - *nε*_{ν}
. This number being just
Ω_{N-n}(*E* -
*nε*_{ν}
), we have $P\underset{\nu}{\ge}\left(n\right)={\Omega}_{N-n}\left(E-n{\epsilon}_{\nu}\right)/{\Omega}_{N}\left(E\right)$ and therefore

where the super-script M specifies the microcanonical ensemble.

In the canonical ensemble, the computation of $P\underset{\nu}{\ge}\left(n\right)$ involves summations as in Eq. (36), the only difference being that the sum over
*n*_{ν}
starts at *n*_{ν}
= *n*
and not at *n*_{ν}
= 0. Upon choosing the summation variable *n*′_{ν} = *n*_{ν}
- *n* one extracts a factor
*e*
^{-βnεν} which
leads to $P\underset{\nu}{\ge}\left(n\right)={e}^{-n\beta {\epsilon}_{\nu}}\frac{{Z}_{N-n}}{{Z}_{N}}$ and therefore

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
〈*n*_{ν}
〉 ≡ ${\mathrm{\Sigma}}_{n=0}^{N}$
${\mathit{\text{nP}}}_{\nu}^{C}$
(*n*).
Using Eqs (38) one finds after some simple reshuffling

Summing over *ν*, and recalling that the total number of particles is given
by *N* ≡ Σ_{ν}〈*n*_{ν}
〉, we find

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

which is the desired recurrence relation.

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

Eq. (37) can be used to derive:

The total number of particles is fixed:

together with Eq. (43) this leads to:

with boundary conditions Ω_{N≥0}(0)
≡ 1 and Ω_{0}(*E* > 0) ≡ 0
(those boundary conditions follow from the fact that *ε*
_{0}
≡ 0 and *Z*
_{0} ≡ 1 with
*Z*_{N}
≡ Σ_{E}
*e*
^{-Eβ}Ω_{N}(*E*)). For finite *E* the sum over *ν* is
finite because of Ω_{N}(*E* < 0) ≡ 0. For the
*d*-dimensional harmonic oscillator Eq. (45) simplifies to:

where ${f}^{\left(d\right)}\left(\nu \right)\equiv \frac{1}{\left(d-1\right)!}{\prod}_{k=1}^{d-1}\left(\nu +d-k\right).$

## Footnotes

^{1} | See 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. |

^{2} | We 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. |

^{3} | The amplitude found in Ref. [16] is incorrect; see the discussion of Eq. (28). |

^{4} | Additional 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. |

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