## 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

According to grand canonical statistics, the root-mean-square fluctuations
*δN*_{ν}
of the occupation numbers
*N*_{ν}
of an ideal Bose gases’s
*ν*-th single-particle state are given by [1,2]

This expression follows without any approximation from the grand canonical approach, but
it faces a severe problem when applied to the fluctuations
*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.

Although this problem had been well recognized and discussed some time ago [3,4], tools for computing the microcanonical fluctuations
*δN*
_{0} have been developed only recently [5], spurred by the progress in preparing Bose-Einstein condensates
of alkali atoms in magnetic traps [6,7,8]. A particularly instructive model system for illustrating the
microcanonical approach to fluctuations *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 [9,10], this system is not merely of academic interest. The value of
the model lies in the fact that it allows one to map the problem of evaluating the
microcanonical statistics to problems also arising in analytic number theory, because
the number of microstates accessible at some excitation energy *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* [11,12]:

where *k*_{B}
is the Boltzmann constant, and
${T}_{0}^{\left(1\right)}$ denotes the temperature below which the
ground state occupation becomes significant. As illustrated in Fig. 1, which compares the relative microcanonical fluctuations
*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.

How can one generalize this finding to other trap types? A rather interesting suggestion
has been made by Navez *et al*. [14]. Denoting, for an ideal *N*-particle Bose gas in
some arbitrary trap, the number of microstates with *exactly
N*_{ex}
excited particles as
Φ(*N*
_{ex}|*E*), so that the total
number of microstates accessible at the given energy *E* reads
Ω(*E*\*N*) = Σ^{N}
_{Nex=0}
Φ(*N*
_{ex}|*E*), these authors
consider the generating function

This function involves Φ(*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,

are strongly peaked around some value *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

respectively [14]. The proviso can be formulated in more intuitive terms: the
required well-peakedness of the distributions (4) means that those microstates where the
energy *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].

But can we rely on Maxwell’s Demon, that is, does the proviso hold? This
question needs to be answered first. A strong argument in favour of the
Maxwell’s Demon ensemble has already been provided by the approximation (2)
to the low-temperature fluctuations for a Bose gas in a one-dimensional oscillator
potential: these fluctuations are *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]. The numbers
Ω(*E*|*N*) of microstates for some given
excitation energy *E* = *nħω* can
then be obtained from the canonical *N*-particle partition function

which, in turn, can be calculated numerically with the help of the recursion relation [16,17,18]

As usual, *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], and get the differences

that determine the microcanonical distributions (4). Some of these distributions are
displayed in Fig. 2, for *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] that works with the actual *N*, not with
∞.

Having thus gained confidence in the abilities of Maxwell’s Demon, we
now¿set it to work in order to compute condensate fluctuations
*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),

this function has the decisive property

i.e., it yields directly the non-normalized *canonical* moments
*M*_{k}
(*β*), and generates the
*microcanonical* moments

We then employ the Maxwell’s Demon approximation: as long as there is a
condensate, these moments (with *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]. Now the calculations within the canonical ensemble become
remarkably simple. The canonical expectation value
〈*N*
_{ex}〉 = *N* -
〈*N*
_{0}〉 of the number of excited
particles is given by

the canonical condensate fluctuations (*δN*
_{0}${)}_{\mathit{\text{cn}}}^{2}$ = (*δN*
_{ex}${)}_{\mathit{\text{cn}}}^{2}$ follow from

Without any further approximation, these expressions can be rewritten as complex integrals:

and

where Γ(*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

where the sum runs over the trap spectrum, excluding the ground state energy
*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.

So far, the analysis is quite general. We now specialize the further deliberations to
ideal Bose gases in *d*-dimensional traps with power-law single-particle
spectra

where the dimensionless coefficients *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]; we have adopted here the notation also employed by Wilkens and
Weiss [18].

If we consider *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

Using this density, and assuming that the anisotropy coefficients
*c*_{i}
are not too different from each other [10], the usual line of reasoning shows that for
*d*/*σ* > 1 there is a sharp
onset of Bose-Einstein condensation at the temperature *T*
_{0}
given by [19]

Moreover, the spectral Zeta functions can now be well approximated by

so Eqs. (13) and (14) adopt the transparent forms

and

For ∆ ≪ *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$$\u3008{N}_{\mathrm{ex}}\u3009\approx A\phantom{\rule{.2em}{0ex}}\zeta \left(\frac{d}{\sigma}\right){\left(\frac{{k}_{B}T}{\Delta}\right)}^{\frac{d}{\sigma}}.$$This canonical result, valid for

*T*<*T*_{0}, coincides precisely with the result of the customary grand canonical analysis [19]. For example, in the case of the three-dimensional isotropic harmonic oscillator potential (i.e., for*d*= 3,*σ*= 1,*A*= 1 and ∆ =*ħω*) Eq. (22) yields the familiar formula - If
*d*/*σ*= 1, both Zeta functions in Eq. (20) coincide. We then encounter a double pole at*t*= 1, and find$$\u3008{N}_{\mathrm{ex}}\u3009\approx A\frac{{k}_{B}T}{\Delta}\left[\mathrm{ln}\left(\frac{{k}_{B}T}{\Delta}\right)+\gamma \right],$$where 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:$$\u3008{N}_{\mathrm{ex}}\u3009\approx \frac{A}{\Gamma \left(\frac{d}{\sigma}\right)}\zeta \left(2-\frac{d}{\sigma}\right)\frac{{k}_{B}T}{\Delta},$$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, giving$${\left(\delta {N}_{0}\right)}_{\mathit{cn}}^{2}\approx \mathrm{A\zeta}\left(\frac{d}{\sigma}-1\right){\left(\frac{{k}_{B}T}{\Delta}\right)}^{\frac{d}{\sigma}}.$$If

*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

In particular, for the one-dimensional harmonic oscillator we have *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.

The above canonical fluctuations, derived from the integral representation (14), reduce to the expression obtained by Politzer [21] in the case of the three-dimensional isotropic trap, and match the results obtained by Wilkens and Weiss [18].

The calculation of the corresponding *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}${)}_{\mathit{\text{mc}}}^{2}$ are distinctly lower than their canonical counterparts for
*d*/*σ* > 2:

Thus, the exponent of *T* is the same for both
(*δN*
_{0}${)}_{\mathit{\text{cn}}}^{2}$ and (*δN*
_{0}${)}_{\mathit{\text{mc}}}^{2}$ , but the prefactors can differ substantially. This Eq. (28) contains as a special case the result obtained for the
three-dimensional isotropic trap by Navez *et al* [14].

Before summarizing these findings, it is useful to also consider the heat capacities for
trapped ideal Bose gases with the single-particle spectra (16): for
*d*/*σ* > 1, and temperatures
below the condensation temperature *T*
_{0}, the heat capacity per
particle is given by

above *T*
_{0} by

Since the fugacity *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

for *d*/*σ* > 2.

We thus arrive at the following picture: For any dimension *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.

Intuitively, one might have expected some sort of square root law for the fluctuations.
Because of the *N*-independence of the condensate fluctuations, there is,
of course, no “√*N*-dependence” of
*δN*
_{0}. The square root is hidden elsewhere:
Since *δN*
_{0} =
*δN*
_{ex} for ensembles with fixed particle number
*N*, we find

*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,

with proportionality constants that are independent of temperature, both canonically and
microcanonically. The first of these relations is just what one might have guessed, but
the crossover from normal fluctuations for
*d*/*σ* > 2 to much stronger
fluctuations for 0 < *d*/
< 1 appears noteworthy.

It remains to be seen how much of this ideal structure survives in the case of weakly
interacting Bose gases. It should also be recognized that Maxwell’s Demon,
though it has provided the microcanonical low-temperature fluctuations, can not solve
all problems of the ideal gas. When considering *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]), but the description of the Bose-Einstein transition itself is
beyond the capabilities of Maxwell’s Demon. Namely, that description requires
the computation of the numbers
Ω(*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].

Nonetheless, the results obtained with the help of the Maxwell’s Demon
approximation have some interesting number-theoretical implications. Going once more
back to Eq. (2) for the one-dimensional oscillator, and inserting the
energy-temperature relation *n* =
*E*/(*ħω*) ≈
ζ(2)(*k*_{B}*T*/*ħω*)^{2},
we find the truly remarkable formula

This has a twofold interpretation. The physicist, puzzeled by the loss of the square root
fluctuation law at the level of 〈*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*

- certainly one of the most amazing examples for the occurrence of square root fluctuations! The ease with which the solution to a seemingly difficult number-theoretical question has been obtained here is even aesthetically appealing. It is pleasing to conclude that ongoing developments in statistical mechanics, themselves being motivated by recent experimental achievements [6,7,8], have a high potential for further fertilization across subfield boundaries.

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