Microcanonical fluctuations of the condensate in weakly interacting Bose gases

0.33 MB·English

by  Zbigniew Idziaszek

Microcanonical fluctuations of the condensate in weakly interacting Bose gases
Zbigniew Idziaszek1,2
1Istituto Nazionale per la Fisica della Materia, BEC-INFM Trento, I-38050 Povo, Italy
2Centrum Fizyki Teoretycznej, Polska Akademia Nauk, 02-668 Warsaw, Poland
5
WestudyfluctuationsofthenumberofBosecondensedatomsinweaklyinteractinghomogeneous
0
andtrappedgases. Forahomogeneoussystemweapplytheparticle-number-conservingformulation
0
of the Bogoliubov theory and calculate the condensate fluctuations within the canonical and the
2
microcanonicalensembles. Wedemonstratethat,atleastinthelow-temperatureregime,predictions

n of the particle-number-conserving and traditional, nonconserving theory are identical, and lead to
a theanomalous scaling offluctuations. Furthermore,themicrocanonical fluctuationsdifferfrom the
J
canonical ones by a quantity which scales normally in the number of particles, thus predictions of

5 both ensembles are equivalent in the thermodynamic limit. We observe a similar behavior for a
2 weakly interacting gas in a harmonic trap. This is in contrast to the trapped, ideal gas, where
microcanonical and canonical fluctuations are different in thethermodynamiclimit.
]
h
PACSnumbers: 03.75.Hh,05.30.Jp
c
e
m
I. INTRODUCTION extensivequantity,andpredictionsofbothensemblesbe-
-
t come equivalent in the thermodynamic limit. For a ho-
a
mogeneoussystemthis featureisalreadyobservedforan
t Experimental realizationof atomic Bose-Einsteincon-
s ideal gas. Nevertheless, in a harmonic trap, the micro-
densationanddegeneratedFermigases,hasstimulateda
.
at largeinterestinthephysicsofultracoldgases[1]. Among canonical fluctuations of noninteracting condensate re-
main different from the canonical ones even in the ther-
m others, the issue of fluctuations in the number of con-
modynamic limit. We show here, that inclusion of weak
densed atoms has been a subject of intensive theoretical
-
d studies [2]-[18]. It is well-known that the standard text- interactions described in terms of Bogoliubov theory, re-
n book approach based on the grand-canonical ensemble stores the equivalence of microcanonical and canonical
o description, and gives rise to the anomalous scaling of
predicts unphysically large fluctuations. This pathologi-
c fluctuations in both considered ensembles. Noteworthy,
cal behavior is avoided, however, when one employs the
[
canonical or the microcanonical ensemble, where the to- forahomogeneoussystemweobservethatapplicationof

3 tal number of particles is fixed. From the experimental the particle-number-conservingmethodleadsto thesim-
v ilarresultsasobtainedfromthestandard,nonconserving
point of view, trapped atoms are under conditions of al-
2 approach. Therefore, for a trapped gas we carry out our
most complete isolation. This favors the use of micro-
5
derivations only within the nonconserving theory, which
canonical ensemble, at least in the situations when pre-
0
we find more convenient in this case. We discuss this is-
6 dictions of different statistical ensembles are not equiv-
sueinthelastsectionexplainingwhyatlowtemperatures
0 alent. While in ideal gases the microcanonical fluctua-
4 tions have been thoroughly investigated [3]-[8], the cor- the fluctuations are insensitive to the actual number of
0 responding problem for interacting gases has been only atoms.
/
t studied in the first-orderperturbation theory [11], or for The anomalous scaling of the fluctuations predicted
a
relatively small systems by means of numerical methods by the Bogoliubov theory is closely related with the
m
[12]. On the other hand, the Bogoliubov theory [19], existence of phonon excitations at low energies. On
- which has provedto be immensely successful to describe the contrary, application of the first-order perturbation
d
n weakly interacting gases, has been applied to study the theory results in the normal scaling of the fluctuations
o fluctuations in the canonical ensemble [10, 14, 15], but [11,12,15]. Whiletheperturbativetreatmentismoreap-
c to ourbest knowledge,it was not usedto investigatethe propriateforfinitesizesystemswithsufficientlysmallin-
v: microcanonical fluctuations. teractions,inthethermodynamiclimitthesystemshould
i The purposeofthis paper isto addressthe problemof be rather described within the Bogoliubov theory, since
X
microcanonical fluctuations in the framework of Bogoli- the presence of arbitrary small interactions leads to the
r ubov theory. For a homogeneous system we perform our phononexcitationsatlowenergiesinthethermodynamic
a
analysis within the particle-number-conserving formula- limit. We note that there is some controversy on using
tion of the Bogoliubov theory, developed by Girardeau theBogoliubovmethodtocalculatethequantitieshigher
andArnowitt[21][35]. In this waywe avoidthe possible than the second order in the field operators [17]. An ex-
errors resulting from the nonconservation of the number cellent agreement of the Bogoliubov theory with the ex-
of particles in the Bogoliubov method. We obtain that act statistics calculated for the one dimensional trapped
the microcanonical fluctuations, similarly to the canoni- gas [18] provides, however, strong arguments in favor of
cal ones [10, 14, 15], exhibit anomalous scaling with the the Bogoliubov approach. The ultimate test of the the-
total number of particles. Moreover, fluctuations in the oretical predictions should be provided by experiments,
microcanonical and the canonical ensemble differ by an either throughthe measurementofthe second-ordercor-
2
relation functions, or by detecting the statistics of scat- case of three dimensional box, where resulting integral
tered photons [20]. for the condensate fluctuations is infrared divergent. To
The paper is organized as follows. In section II, for avoid these difficulties, in our study the thermodynamic
the sake of completeness, we analyze the fluctuations quantities are evaluated from the integral representa-
of ideal Bose gases confined in a three dimensional box tion involving spectral Zeta function [7, 8]. Employ-
and a harmonic trap. The behavior of fluctuations in ing integral representation of the exponential function
an interacting gas confined in a three dimensional box is e−t = (2πi)−1 c+i∞dz t−zΓ(z), one can express N
studied in section III. In particular, in section IIIA we and δ2N as cco−ni∞tour integrals of Mellin-Barnes htypeei
brieflydescribetheGirardeau-Arnowittparticle-number- e R
[8]
conserving formalism. Section IIIB is devoted to the (cid:10) (cid:11)
analysis of fluctuations in the canonical ensemble, while c+i∞ dz
the microcanonical fluctuations are calculated in section hNeiCN = 2πiZ(β,z)ζ(z)Γ(z), (3)
IIIC. Section IV presents the results for a weakly inter- Zc−i∞
acting trapped gas. The canonical fluctuations are con- c+i∞ dz
δ2N = Z(β,z)ζ(z 1)Γ(z), (4)
sidered in section IVA, whereas section IVB describes e CN 2πi −
Zc−i∞
the results for the microcanonical ensemble. We end in (cid:10) (cid:11)
section V presenting some conclusions. Finally, four ap- whereζ(z)denotesRiemannZetafunction,andZ(β,z)is
pendices give some technical details. thespectralZetafunction: Z(β,z)= (βε )−z. The
ν6=0 ν
contour of integration lies to the right of all the poles of
P
the integrand. Now, the problem of calculating fluctua-
II. IDEAL GAS tions has been reduced to determination of the poles of
Z(β,z). For the considered case of a three dimensional
In this section we analyze fluctuations of the num- boxwithperiodicboundaryconditions,thespectralZeta
ber of condensed atoms in an ideal gas. First, we con- function reads
siderhomogeneoussystemwithN noninteractingbosons
ζ (z)
confined in a box of the size L with periodic boundary Z(β,z)= E (5)
(β∆)z
conditions. Below the critical temperature of the Bose-
Einstein condensation, the fluctuations of the number of
where ζ (z) denotes the Zeta function of Epstein [24]
condensed atoms can be calculated with the help of the E
Maxwell’s Demon ensemble (MDE) [4]. In this approach
1 3
the condensate serves as an infinite reservoir of particles ζ (z)= , Rez > , (6)
E (n2 +n2+n2)z 2
for the subsystem of excited states. Such treatment is ~n6=0 x y z
X
justified as long as the probability of states with all the
atoms excited is negligible, hence it is not valid in the and ∆ = 2π2~2/mL2 is the energy of the first excited
vicinity of the critical temperature. The mean number state. Function ζE(z) has merely single pole at z = 3/2
of excited atoms N and its fluctuations δ2N in the equal to 2π. The straightforwardcalculation of the con-
e e
h i
canonical ensemble, can be calculated from the grand tour integrals (3) and (4) with Zeta function (5), leads
canonical partition function Ξ (z,β) of the(cid:10)excit(cid:11)ed sub- to
e
system [4]
N = π3/2ζ(3/2)t3/2+ζ (1)t, (7)
∂ e−βεν h eiCN E
hNeiCN =z∂z lnΞe(z,β) = 1 e−βεν, (1) δ2Ne CN = ζE(2)t2+π3/2ζ(1/2)t3/2, (8)
(cid:12)z=1 ν6=0 −
(cid:12) X (cid:10) (cid:11)
∂ ∂ (cid:12) where t=kBT/∆. In derivation of Eqs. (7) and (8), we
δ2N =z z lnΞ (z(cid:12),β) have included leading and next to the leading rightmost
e CN ∂z ∂z e
(cid:12)z=1 pole of the integrand. Thus, the result accounts also for
(cid:10) (cid:11) e−βεν (cid:12)(cid:12) correction due to the finite size of the system. As was
= . (cid:12) (2)
(1 e−βεν)2 first pointed out in [25], the canonical fluctuations in a
ν6=0 −
X box are anomalous. This can be observed by rewriting
In this formula ε denote the single–particle energy of Eq. (8) in the form containing explicit dependence on
ν
the level ν, β = 1/k T is the inverse temperature and the number of particles
B
z is the fugacity. In the canonical ensemble, the to-
tal number of particles N is conserved, and obviously δ2N = ζE(2) N4/3T˜2+ ζ(12)NT˜3/2, (9)
hN0iCN = N − hNeiCN, and δ2N0 CN = δ2Ne CN, e CN π2ζ(32)4/3 ζ(32)
where N and δ2N denote the mean number (cid:10) (cid:11)
of condhen0seiCdNatoms and i0tsCflNuc(cid:10)tuation(cid:11)s, respe(cid:10)ctively(cid:11). In whereT˜ =T/T ,andk T /∆=π−1(N/ζ(3))2/3 isthe
(cid:10) (cid:11) C B C 2
the thermodynamic limit, summation on the r.h.s. of critical temperature.
Eqs. (1) and (2), can be replaced by integration. How- Now we turn to the microcanonical ensemble. In or-
ever, this is not always possible, in particular, in the der to calculate fluctuations of the number of condensed
3
atoms, we employ the following relation [4, 8]

[ δN δE ]2 numerical
δ2Ne MC = δ2Ne CN− h δe2E iCN , (10) 60 analytic
h iCN (a)
(cid:10) (cid:11) (cid:10) (cid:11)
>
which links microcanonicalfluctuations with the quanti- e
N
tiescalculatedinthecanonicalensemble: particle-energy 40
correlation δN δE = (N N )(E E ) ,and < canonical
e CN e e CN
h i h −h i −h i i
fluctuations of the system’s energy δ2E = (E
CN
E )2 . The formercanbe determihned firomthehmea−n microcanonical
CN
h i i 20
number of excited particles N
e
h i
∂N
e
δN δE = . (11)
h e iCN − ∂β N = 1000
(cid:18) (cid:19)z(cid:12)z=1 0
(cid:12) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
(cid:12)
Inasimilarmanner,onecancalculate(cid:12)thefluctuationsof 40
energy from its expectation value numerical
analytic
δ2E = ∂E , (12) > 30 (b)
CN − ∂β e
(cid:18) (cid:19)z(cid:12)z=1 N
(cid:10) (cid:11) (cid:12)
whilethemeanenergy E canbeexp(cid:12)(cid:12)ressedasacontour < 20 canonical
h i
integral containing spectral Zeta function Z(β,z)
1 c+i∞ dz microcanonical
E = Z(β,z 1)ζ(z)Γ(z). (13) 10
h iCN β 2πi −
Zc−i∞
After straightforward calculations, we arrive at the fol- N = 1000
0
lowingresultforthemicrocanonicalfluctuationsinabox 0.0 0.2 0.4 0.6 0.8 1.0
T/T
C
ζ(1)2 3ζ(3) FIG. 1: Fluctuations of the number of excited particles in
δ2N =ζ (2)t2+π3/2 2 − 5 2 t3/2. (14) themicrocanonical and canonical ensemble for the system of
e MC E ζ(1)
2 N = 1000 noninteracting bosons confined in a box with pe-
(cid:10) (cid:11) riodic boundary conditions (upper panel) and in a harmonic
By comparison with the canonical fluctuations (8), we trap (lower panel). Analytical results (solid line) derived in
see that the leading-order term, which gives rise to the MDE approximation are compared to the exact numerical
anomalous scaling, is the same. The difference between data (dashed line). The temperature is scaled in the units
fluctuations in the considered ensembles appears in the of the critical temperatureTC.
finite size correction term, thus the microcanonical and
canonical fluctuations become equal in the thermody-
N ,ω 0andNω3 =const,the leadingorderterm
namic limit. →∞ →
in Eq. (16) is proportionalto N and the fluctuations ex-
Let us turn now to the case of a trapped gas. For
hibit normal behavior. In the microcanonical ensemble
simplicity we assume that the atoms are confined in a
weemploythe identity (10),whichleadstothe following
spherically symmetric harmonic potential, however gen-
result for the fluctuations
eralizationofthe resultsfor anisotropictrapsis straight-
forward. The spectral Zeta function for the harmonic 3ζ(3)2
δ2N =τ3 ζ(2) +τ2 3lnτ + ,
trap of frequency ω is given by [8] e MC − 4 ζ(4) 2 C
(cid:18) (cid:19)
(cid:10) (cid:11) (cid:0) (cid:1)(17)
Z(β,z)=τz 1ζ(z 2)+ 3ζ(z 1)+ζ(z) , (15)
2 − 2 − with
with τ = k T/(cid:0)(~ω). Substituting this func(cid:1)tion into 3ζ(2)ζ(3) 9 ζ(3)2
B =ζ(2)+ 5 + 3γ + . (18)
Eq. (4) and taking into account leading and next to the C 4 2 − 2 ζ(4) 16ζ(4)2
leading rightmost poles, we obtain the following result
We observethat the leading order term, which gives rise
for the fluctuations in the canonical ensemble
to the normal scaling of fluctuation in the thermody-
δ2N =ζ(2)τ3+τ2 3lnτ +ζ(2)+ 5 + 3γ , (16) namic limit, is different than in the canonical ensemble.
e CN 2 4 2
Hence the microcanonicaland the canonicalfluctuations
w(cid:10)here γ(cid:11) 0.577 denotes E(cid:0)uler’s constant. The lo(cid:1)garith- in the harmonic trap remain different even in the ther-
≃
mic correction in (16) originates from a double pole of modynamic limit, in contrast to their behavior in a ho-
the integrand at z = 2. In the thermodynamic limit: mogeneous system.
4
Fig.1presentsfluctuationsofthenumberofcondensed the variational calculus leads, however, to the excita-
atomsinthecanonicalandthemicrocanonicalensembles, tion spectrum which possess a gap [21], which is not
calculated for the system of N =1000 atoms confined in appropriate for studying the system in the thermody-
athree-dimensionalbox(upperpanel)andathreedimen- namic limit. Gapless Bogoliubov spectrum is recovered,
sional harmonic trap (lower panel). It compares the an- whenoneminimizestheexpectationvalueoftheparticle-
alyticalpredictionsofEqs.(8), (14)(box)andEqs.(16), number-conserving Bogoliubov Hamiltonian
(17)(harmonictrap)withtheexactnumericalresultscal-
culatedwiththehelpoftherecurrencealgorithmsforthe Hˆ = g Nˆ(Nˆ 1)+ ~2k2 + gNˆ0 aˆ†aˆ +
canonicalandmicrocanonicalpartitionfunctions. Wesee B 2V − 2m V ! k k
k=6 0
that analytic results agree very well with the numerical X
g
data,apartfromthe regioncloseto the criticaltempera- + 2V aˆ†kaˆ†−kaˆ0aˆ0+H.c., (21)
ture,andaboveit,wheretheMDEapproximationceases kX=6 0(cid:16) (cid:17)
to be valid.
where g = 4π~2a/m, V = L3, and Nˆ is the opera-
tor of the total number of particles. This approach as-
sumes omitting the Hartree-Fock and pair-pair interac-
III. WEAKLY INTERACTING
tion terms in the part of the total Hamiltonian, which
HOMOGENEOUS GAS
gives nonzero contribution in the variational ground
state. Theseterms,however,arenegligibleintheconsid-
A. Girardeau-Arnowitt particle-number-conserving
eredrangeoftemperatures. Theminimizationprocedure
formalism
yields [21]
In this section we present a brief description of the ε +gn εB
particle-number-conserving version of the Bogoliubov tanhψk = k gn− k , (22)
theory,formulatedbyGirardeauandArnowitt(GA)[21].
where ε = ~2k2/2m, n = N/V is the atomic density,
Inthe following,werestrictouranalysisto the regimeof k
temperatures much lower than the critical temperature. and εBk = ε2k+2gnεk are the energy levels of the Bo-
Thisallowsustomakethefollowingassumptions: (i)the goliubov spectrum.
p
meannumber ofcondensedatomsismuchlargerthatits The excited states in GA theory are constructed in
fluctuations N δN , thus we can we can exclude basically similar manner to the variationalgroundstate
0 0
h i ≫ h i
the possibility of exciting the state where the conden- Ψ =Uˆ Φ , (23)
sate is totally depleted, (ii) we neglect the influence of a {nk} {nk}
finite temperature on the excitation spectrum, through where (cid:12) (cid:11) (cid:12) (cid:11)
(cid:12) (cid:12)
the thermal depletion of the condensate, which is pre- (aˆ†)N0 (aˆ†)nk
dicted in the Popov theory [26]. In addition we assume Φ = 0 k 0 , (24)
dilute-gas regime na3 1, where n is the density of {nk} (N0!)1/2 (nk!)1/2 | i
k=6 0
atoms and a is the s-w≪ave scattering length, which al- (cid:12)(cid:12) (cid:11) Y
and N =N n . Here, n is the set of integer
lowstoneglectthe influence ofthe quantumdepletionof 0 − k=6 0 k { k}
numbers, where n represents the number of elementary
the condensate on the Bogoliubov excitation spectrum. k
Let us introduce unitary operators βˆ =(Nˆ +1)−1/2aˆ , excitations withPenergy εBk.
0 0 0
βˆ† =aˆ†(Nˆ +1)−1/2,βˆ†βˆ =βˆ βˆ† =1,whereaˆ ,aˆ† are,
0 0 0 0 0 0 0 k k
respectively, the annihilation and creation operators of B. Fluctuations in the canonical ensemble
the particles with momentum ~k, and Nˆ = aˆ†aˆ . The
0 0 0
particle-number-conservinggroundstateofGAtheoryis
In an interacting gas, canonical fluctuations of the
number of condensed atoms can be calculated with the
(aˆ†)N help of the following generating function.
Ψ =Uˆ 0 0 , (19)
| 0i (N!)1/2 | i X(z,β)=Tr zNˆee−βHˆB , (25)
n o
with the unitary operator Uˆ where Nˆ = aˆ†aˆ is the operator of the number
e k=6 0 k k
of noncondensedparticles, and the trace has to be taken
P
1 over all eigenstates of the system with the total number
Uˆ =exp2 ψk (βˆ0†)2aˆ−kaˆk−aˆ†kaˆ†−k(βˆ0)2 . (20) of particles equal to N. For z = 1, generating function
kX=6 0 h i X(z,β) is equal, by definition, to the canonical parti-
  tionfunction. Thefluctuationsofthenumberofnoncon-
Coefficients ψk are real, even function of k. In the GA densed atoms δ2N are given by
e
theory they are determined by minimizing the expecta-
tion value of the full Hamiltonian in variational ground (cid:10) (cid:11) ∂ ∂
δ2N = z z lnX(z,β) , (26)
state (19). The inclusion of the total Hamiltonian in e CN ∂z ∂z
(cid:12)z=1
(cid:10) (cid:11) (cid:12)
(cid:12)
(cid:12)
5
Obviously,thetotalnumberofparticlesisconserved,and Incomparisonto the idealgas result(2), thermalfluctu-
thefluctuationsinthenumberofcondensedandnoncon- ations of the number of elementary excitations, are mul-
densedatomsareequal: δ2N = δ2N . Wecal- tiplied by the factor (u2 +v2)2 +4u2v2. Second term
0 CN e CN k k k k
culate X(z,β) by performing trace in the basis of eigen- onr.h.s. ofEq. (31), describes the quantumfluctuations
states Ψ{nk} of GA the(cid:10)ory (cid:11) (cid:10) (cid:11) at zero temperature δ2Nq ≡2 k=6 0u2kvk2. We note that
a similar expression can be derived in the framework of
X(z,(cid:12)(cid:12)β)= (cid:11) Φ Uˆ−1zNˆee−βHˆBUˆ Φ thestandard,particle-number-Pnonconservingtheory[10].
{nk} {nk}
We comment on this issue in the last section of the pa-
{Xnk}(cid:10) (cid:12) (cid:12) (cid:11) per. Now,werewrite δ2N intermsofcontourintegral
= Φ{nk}(cid:12)zUˆ−1NˆeUˆe−βUˆ−1Hˆ(cid:12)BUˆ Φ{nk} , involving spectral Zeta funection. Calculations similar to
(cid:10) (cid:11)
{Xnk}(cid:10) (cid:12) (cid:12) (cid:11) the derivation of (3) and (4), yield the following integral
(cid:12) (cid:12) (27) representations
where summation runs over sets n of the occupation
{ k} c+i∞ dz
numbers of elementary excitations. In principle, accord- δ2N = Z(t,z,α)ζ(z 1)Γ(z)
ing to the definition of Ψ{nk} , the number of elemen- e CN Zc−i∞ 2πi −
taryexcitation k=6 0nk(cid:12)canno(cid:11)texceedthe totalnumber (cid:10) (cid:11) +δ2Nq, (32)
ofparticlesN. However,(cid:12)wellbelowthecriticaltempera-
P
ture,theprobabilityofexcitingsuchastateisnegligible,
with spectral Zeta function
and we can consider the summation in Eq. (27) as un-
constrained. Similar approximationis assumedin MDE,
where we treat the condensate as a reservoir of particles n4+2αn2+2α2
Z(t,z,α)=tz , (33)
for excited subsystem, and as a consequence we consider √n4+2αn2z+2
thepartitionsofexcitedatomsinthecanonicalensemble nX6=0
without the particle number constrain. The transformed
operatorsUˆ−1NˆeUˆ andUˆ−1HˆBUˆ canbe found with the where α = gn/∆ = 2Na/πL, and n = (nx,ny,nz) is
help of the following identity [21] a vector composed of integer numbers. Spectral Zeta
function Z (t,z,α) can be expressed as a sum of two
1
Uˆ−1aˆ Uˆ =u aˆ +v (βˆ )2aˆ†, (28) simpler functions
k k k k 0 k
whereu =coshψ andv = sinhψ aretheusualam-
k k k − k Z(t,z,α)=tz θ(z,α)+2α2θ(z+2,α) , (34)
plitudes of the Bogoliubov transformation. After some
straightforwardalgebra we obtain (cid:0) (cid:1)
with
Uˆ−1Nˆ Uˆ = u2 +v2 aˆ†aˆ + v2
e k k k k k
k=6 0 k=6 0 1
X(cid:0) (cid:1) X θ(z,α)= , (35)
+ (βˆ0)−2aˆkaˆ−k+H.c. , (29) nX6=0(n4+2αn2)z/2
kX=6 0(cid:16) (cid:17)
Uˆ−1Hˆ Uˆ E + εBaˆ†aˆ , (30) In the thermodynamic limit: N , V , n =
B ≃ 0 k k k const, and α n1/3N2/3 . →Th∞us, for→suffi∞ciently
kX=6 0 large system w∼e can consid→er t∞he regime α 1, and in
≫
where E is the ground state energy of the system. The the subsequent derivation we will approximate function
0
transformation Uˆ−1 Uˆ does not diagonalize the Hamil- θ(z,α), by its asymptotic expansion for large parameter
tonianHˆ exactly, a·ndin the derivationof (30) we have α. The analytic properties of the function θ(z,α) are
B
neglected terms describing interactions between excited studiedin appendix A, while its asymptoticexpansionis
atoms. This approximation is consistent with neglection developedinappendix B. In the formeritis shown,that
ofthe Hartree-Fockandpair-pairscatteringtermsinthe thepolesofθ(z,α)arelocatedatz = 3 m,m=0,1,...,
2−
total Hamiltonian, and is well justified in the considered with residues equal to 2π(( 2α)m/m!)Γ(3 + m)/Γ(3
range of temperatures. m). Hence the residues of Z−(t,z,α) are p4ropo2rtiona4l t−o
2
Employing Eqs. (26), (27), (29) and (30) we obtain (α/t)m = (gn/k T)m, and inclusion in the calculation
B
the following result for the fluctuations in the canonical only the leading rightmost poles, would yield the result
ensemble whichisvalidonlyintheregimeoftemperaturesk T
B
≫
gn In order to obtain result, which holds also for low
δ2Ne CN = u2k+vk2 2+4u2kvk2 (1 e−eβ−εβBkεBk)2 tinetmegprearlat(u32re)s,:wkeBhTav<e tgons,uminotvheercaalllctuhlaetpioonlesofwchoinchtoluier
(cid:10) (cid:11) kX=6 0(cid:16)(cid:0) (cid:1) (cid:17) − tothe leftofthecontourofintegration. Thissummation
+2 u2kvk2. (31) canbedoneanalytically,the detailsofthe derivationare
k=6 0 presented in appendix C. The result for the canonical
X
6
fluctuations reads
δ2Ne CN =21 ζE(2)+ √π22α t2+ π62(2α)32 25 30
(cid:20) (cid:21) 25
(cid:10) (cid:11) 1 α α2 α 20
+ ζH 2,1+ 2πt + 2π2t2s1 2πt 20
(cid:20) (cid:18) (cid:19) (cid:16) (cid:17) >
α 3 α 3 Ne 15
+ 2πtζH(cid:18)2,1+ 2πt(cid:19)(cid:21)(πt)2, (36) < 15 100 2 4 6 = 108

where
10
∞
3k+2x
fluctuations
s (x)= (37)
1 k2(k+x)3/2 low T expansion
Xk=1 5 = 2.4 high T expansion
and ζH(s,a) is Zeta function of Hurwitz: ζH(s,a) = 0 2 4 6 8
∞k=0(k+a)−s, defined by the series for Re s > 1. The kBT/
leading order term: 1ζ (2)t2 is proportional to N4/3 in
P 2 E FIG. 2: Fluctuations of the number of noncondensed atoms
the thermodynamic limit, and the canonicalfluctuations
in the canonical ensemble for interacting Bose gas confined
in the interacting gas are anomalous, similarly to the
in abox with periodic boundaryconditions. Plots are shown
ideal gas. The prefactor of the anomalous term is ex-
for α = 2.4 and α = 10 (inset), where α = 2Na/πL. The
actly twice smaller than for the ideal gas, which can be
analyticalresultofEq. (36)iscomparedwithitslowandhigh
attributed to the coupling of k and k modes in the temperature expansions: Eqs. (39) and (38), respectively.
Bogoliubov Hamiltonian [14]. We not−e that our result The temperatureis scaled in theunitsof ∆=2π2~2/mL2.
agree in the leading order with the result of Ref. [10],
calculated in the framework of the standard, particle-
number-nonconserving theory.
Fig. 2 presents the fluctuations of the number of non-
It is interesting to investigate the behavior of fluctu-
condensed atoms in the canonical ensemble calculated
ations in the two limits: k T gn and k T gn.
B ≫ B ≪ for α = 2.4. The inset shows the same quantity for the
For the assumed range of temperatures, the former can
system with stronger interactions: α = 10. The high-
be realized in the system with sufficiently small interac-
est temperature presented in the plot, corresponds to
tions. The latter corresponds to the regime where the
T/T 0.5 for N = 1000. Fig. 2 compares the analyti-
thermal excitations occur mainly in the phonon part of C ≈
calresultof Eq. (36), with its lowand high temperature
theenergyspectrum. Inthelimitk T gn,weperform
B ≫ expansions: (39)and (38), respectively. We observethat
Taylor expansion of functions ζ (s,1+η) and s (η) for
H 1 the low temperature expansion correctly describes the
small η =α/2πt, and obtain
fluctuations in the whole regime of considered tempera-
δ2Ne CN ≃12 ζE(2)+ √π22α t2+ π62(2α)32 ttuortehs.e Tfluhcisturaetflioecntss, tchoemfeasctfrothmatththeepmhoaninoncopnatrrtiboufttiohne
(cid:20) (cid:21) spectrum. On the other hand, the high temperature ex-
(cid:10) (cid:11) +ζ 1 (πt)32 + αζ 3 (πt)12. (38) pansion is valid only for temperatures t&α.
2 4 2
(cid:0) (cid:1) (cid:0) (cid:1)
In the regime k T gn, we perform asymptotic ex-
B
≪
pansion of the functions ζ (s,1+η) and s (η) for large
H 1
η =α/2πt [27], and obtain the following result C. Fluctuations in the microcanonical ensemble
δ2Ne CN ≃ 12ζE(2)t2−π√2αt+ π42(2α)23. (39) We now turn to the the microcanonical ensemble.
Fluctuations of the number of noncondensed atoms can
(cid:10) (cid:11)
This resultcanbe alsoderived,by assumingphonondis- be calculated from the following generating function
persion relation for the energy spectrum: εB = ck, with
k
the sound velocity c = gn/m . In this case spectral
Y(z,E)=Tr zNˆeδ(E Hˆ ) , (41)
Zeta function takes form − B
p
n o
αtz z
Zph(t,z,α)= √2αzζE 2 +1 . (40) where delta operator δ(E HˆB) selects the states with
−
(cid:16) (cid:17) the total energy equal to E, and the trace is taken over
ThelastequationcanbeobtaineddirectlyfromEq. (33), allthe eigenstateswith a fixedtotalnumber ofparticles.
by approximating each term of the series by its asymp- Taking into account the definition of Y(z,E), the mean
totic behavior for α . number andfluctuations in the microcanonicalensemble
→∞
7
can be evaluated from noncondensedatoms. Ther.h.s. of(47)canbe rewritten
in the following way
∂
N = z lnY(z,E) , (42)
δ2hNeeiMMCC = z∂∂∂zzz∂∂z lnY(z(cid:12)(cid:12)(cid:12)(cid:12),zE=1) . (43) z(cid:18)∂∂Nze(cid:19)T =z(cid:18)∂∂Nze(cid:19)E+z(cid:0)∂∂Ezk(cid:1)BTTk2BT∂∂E2T(cid:0)∂z∂NTe(cid:1)z. (48)
(cid:12)z=1
(cid:10) (cid:11) (cid:12) (cid:0) (cid:1)
(cid:12) Now, if we observe that
According to Eqs. (42) and (43), the fluct(cid:12)uations of the
number of noncondensed particles can be expressed as
∂N ∂E
k T2 e = z = δN δE ,
δ2Ne MC =z ∂hN∂eziMC , (44) B (cid:18) ∂T (cid:19)z(cid:12)(cid:12)z=1 (cid:18)∂z(cid:19)T(cid:12)(cid:12)z=1 h e iCN(49)
(cid:18) (cid:19)E(cid:12)z=1 (cid:12) (cid:12)
(cid:10) (cid:11) (cid:12) and (cid:12) (cid:12)
(cid:12)
where derivative of the mean number o(cid:12)f noncondensed
particles should be taken before setting z = 1 in Eq. ∂E
k T2 = δ2E , (50)
t(h42e)flfuocrtuhNateiioMnCs.ofInthaesniummilbarermoafnnnoenrc,oonndeencasendrpeparrteiscelnest B (cid:18)∂T(cid:19)z(cid:12)z=1 CN
(cid:12) (cid:10) (cid:11)
in the canonical ensemble (cid:12)
we arrive at the desired r(cid:12)elation stated by Eq. (10).
∂ N Thus, we have expressed the microcanonical fluctua-
δ2N =z h eiCN . (45)
e CN ∂z tions in terms of the quantities calculated in the canon-
(cid:10) (cid:11) (cid:18) (cid:19)E(cid:12)(cid:12)z=1 ical ensemble. Moreover, the particle-energy correlation
Now we are ready to establish the relatio(cid:12)(cid:12)n between fluc- hδNeδEiCN andfluctuationsofthetotalenergy δ2E CN
tuations in the microcanonical and canonical ensemble. that enter this relation, according to Eqs. (49) and
(cid:10) (cid:11)
In the following we show that identity (10), derived pre- (50) can be determined directly from the mean number
viously for an ideal gas, holds also in the case of an in- of noncondensed particles N and the mean energy E.
e
teractinggas. Webasicallyrepeatthestepsofderivation We calculate the latter quantities in a similar manner to
for the ideal gas. First, we assume that the mean num- the canonicalfluctuations, representingthem interms of
ber of noncondensed particles in both consideredensem- contour integrals containing spectral Zeta function, and
bles coincide. This must be true in the thermodynamic summing the contributions from all the poles of the in-
limit, where we expect that both ensembles are equiva- tegrand. This yields
lentwithrespectto thebasicthermodynamic quantities.
Abecrcoorfdninognctoontdheinssaesdsupmarpttiicolens,wbye dNen,ontoetthdeismtinegaunisnhuimng- hNei=−4π25t23ζH(12,1+η)+ 21ζE(1)−π2√2α t,
e
between microcanonical and canonical ensembles. We (cid:16) (cid:17)(51)
choose energy E and fugacity z as a pair of state vari-
ables and take the total differential of N and
e
dNe = ∂∂Nze dz+ ∂∂NEe dE. (46) hE/∆i=− η2ζH(12,1+η)−2ηζH(−21,1+η)
From the total di(cid:18)fferent(cid:19)iaEl dN (cid:18)it is st(cid:19)razightforward to +(cid:2)ζH(−32,1+η) 8π27t52 + E∆0 − 41π5(2α)32 −t,
e
(52)
derive the following relation (cid:3)
z ∂Ne =z ∂Ne +z ∂Ne ∂E (47) where η = α/2πt, and E0 is the ground-state energy.
∂z ∂z ∂E ∂z Takingintoaccountthe leadingorderterms inEqs. (51)
(cid:18) (cid:19)T (cid:18) (cid:19)E (cid:18) (cid:19)z(cid:18) (cid:19)T
and (52), after some algebra we obtain the following re-
According to Eq. (45), l.h.s. of Eq. (47) at z = 1 sult for the difference between the canonicaland the mi-
represents the canonical fluctuations of the number of crocanonicalfluctuations
δ2N δ2N = 4π3/2 3ζH −21,1+η −ηζH 12,1+η +√η 2t3/2 . (53)
e CN− e MC − 4η3ζ 3,1+η +12η2ζ 1,1+η 36ηζ 1,1+η +20ζ 3,1+η
H 2 (cid:0) (cid:0) H 2 (cid:1) − (cid:0) H −(cid:1)2 (cid:1) H −2
(cid:10) (cid:11) (cid:10) (cid:11)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
Since t3/2 = (k T/∆)3/2 NT3/2, and η remains con- fluctuations scales normally with the number of parti-
B
∼
stantinthethermodynamiclimit,thedifferencebetween
8
cles. Thus, the microcanonical fluctuations, similarly to
the canonical ones, exhibit anomalous behavior. Fur- 30
thermore, the relative difference between δ2N and 25
e CN
δ2Ne MC tendstozerointhethermodyna(cid:10)miclim(cid:11) it,and 25 20 canonical
fluctuationsinbothensemblesbecomeequalforlargeN.

(cid:10) (cid:11)
Let us investigatethe behavior ofmicrocanonicalfluc- 15
microcanonical
tuations in the two limits: k T gn and k T gn. > 20
To this end, we can expand EBq. (≫53) for largeBand≪small Ne 10 = 10

valuesofη. Intheregimeα/2πt 1,the leadingbehav- < 0 2 4 6 8
≪ 15
ior of fluctuations in the canonical and microcanonical
ensemble is governedby canonical
10
δ2Ne CN ≃12(cid:20)ζE(2)+ √π22α(cid:21)t2+ζ(12)(πt)32, (54) 5 = 2.4 microcanonical
(cid:10) (cid:11) 1 π2
δ2N ζ (2)+ t2 (55) 0 2 4 6 8
e MC ≃2 E √2α k T/
(cid:20) (cid:21) B
(cid:10) (cid:11) ζ(1)2 3ζ(3)
+π3/2 2 − 5 2 t3/2, (56) FIG. 3: Fluctuations of the number of noncondensed atoms
ζ(1) in the canonical and microcanonical ensemble for a three di-
2
mensional box with periodic boundary conditions. Plots are
We notice that in this limit, the difference between en- shown for α = 2.4 and α = 10 (inset), where α = 2Na/πL.
sembles is given in the leading order by the ideal gas Depicted are: the analytical result of Eqs. (36) and (53)
result (cf Eqs. (8) and (14)). In the opposite limit: for the canonical and microcanonical fluctuations, respec-
tively,thecanonical fluctuationscalculated numerically from
α/2πt 1, we obtain
≫ Eq. (31), and the microcanonical fluctuations evaluated nu-
δ2Ne CN ≃21ζE(2)t2−π√2αt+ 14π2(2α)23, (57) mpeerraitcuarlleyiswsitchaletdheinhetlhpeoufnritescuorfr∆enc=e2aπlg2o~r2i/thmmLs2.. The tem-
(cid:10)δ2Ne(cid:11)MC ≃21ζE(2)t2− 1127π√2αt+ 14π2(2α)23. (58)
In(cid:10)this r(cid:11)egime, the microcanonical fluctuations differ
from the canonical ones by the prefactor of the finite
size correction term.
Inordertoverifypredictionsoftheanalyticresults,we result in terms of the temperature, we calculate the mi-
performed numerical calculations in the microcanonical crocanonical temperature TMC from the entropy of the
ensemble. The microcanonical partition functions were system: 1/TMC = ∂S(N,E)/∂E. In our case, the en-
computed with the help of recurrence algorithms [6, 28]. tropy is given by S(N,E) = kBlogY(1,E), where the
From the partition functions, we calculate the statistics r.h.s. depends on the total number of particles only
of elementary excitations, and finally the fluctuations of through the energy spectrum [36].
the number of noncondensed particles from
Fig. 3 presents fluctuations of the number of noncon-
δ2Ne MC = u2k+vk2 u2q+vq2 densed particles in the canonicaland microcanonicalen-
k,q6=0 semble, calculated for α = 2.4 and α = 10 (inset). The
(cid:10) (cid:11) X (cid:2)(cid:0) (cid:1)(cid:0) (cid:1)
( nˆ nˆ nˆ nˆ )] highest temperature presented in the plot corresponds
k q MC k MC q MC
× h i −h i h i to T/T 0.5 for N = 1000 atoms. The figure shows
+4 u2v2 nˆ nˆ + nˆ + 1 , C ≈
k k h k −kiMC h kiMC 2 the analytic results of Eqs. (36) and (53) for the canon-
kX=6 0 (cid:0) (cid:1) ical and microcanonical fluctuations, respectively. They
(59) are compared with the canonical fluctuations calculated
numerically from Eq. (31) and with the microcanoni-
where nˆk = aˆ†kaˆk. This relation can be derived starting cal fluctuations evaluated from the recurrence relations
fromthedefinitionofY(z,E),andperformingdifferenti- for the microcanonical partition functions. The analyt-
ation according to Eq. (43). The mean number nˆk MC ical predictions for both ensembles agree very well with
h i
of elementary excitations with energy εBk in the micro- the numerical calculations. We note that, the missing
canonical ensemble is given by low temperature part of the numerical curve for the mi-
crocanonical fluctuations is due to large uncertainty in
Φ nˆ δ(E Uˆ−1Hˆ Uˆ) Φ
nˆ = {nk} {nk} k − B {nk} . determination of the microcanonical temperature at low
h kiMC P (cid:10) (cid:12) Y(1,E) (cid:12) (cid:11) energies of the system. On the other hand, we see that
(cid:12) (cid:12) (60) in the system of a moderate size (N 1000) the canon-
∼
Analogous expression holds for the product of the occu- ical and microcanonical fluctuations can be clearly dis-
pations numbers nˆ nˆ In order to express the final tinguished.
k q MC
h i
9
IV. WEAKLY INTERACTING TRAPPED GAS where f = u v , y = r/R, ξ = ~ω/(2µ),
ν± ν ν
±
P(l+1/2,0)(x)denotetheJacobipolynomialandY (θ,φ)
n lm
A. Fluctuations in the canonical ensemble is the spherical harmonic. In the above formula n =
0,1,2,...is the radialquantumnumber,l andm arethe
In this section we calculate the canonical fluctuations quantum numbers of the angular part of the wave func-
ofaweaklyinteractinggasconfinedinasphericallysym- tion, and l = 0 when n = 0. Since we consider only the
6
metric harmonic trap of frequency ω. For a homoge- lowestcollectivemodeswithenergiesEnl µ,infurther
≪
neous system we have demonstrated that the particle- calculations we can neglect the amplitude fν−, and use
number-conserving approach leads to the same value of the following approximation: uν vν fν+/2.
≈ ≈
fluctuationsasthetraditional,nonconservingBogoliubov We calculate the fluctuations substituting the oper-
method. Since the trapped gas is more complicated to ator of the number of noncondensed particles Nˆe =
treatanalytically,inthefollowingweapplythestandard d3rΨˆ†(r)′Ψˆ(r)′intohδ2Nei=hNˆe2i−hNˆei2. Thisyields
Bogoliubovtheorygeneralizedforaninhomogeneoussys-
R
tem,limitingouranalysistotheleadingorderbehaviorin 2
the thermodynamic limit. Similarly to the homogeneous δ2Ne CN ≈8 d3ru∗ν(r)uλ(r) hνhλ (64)
case we restrict our considerations to the temperatures νλ (cid:18)Z (cid:19)
(cid:10) (cid:11) X
much smaller than the critical temperature, neglecting where we retain only the leading order term, and h =
ν
the influence of the thermal depletion on the excitation (eβεν 1)−1. In derivation of Eq. (64) we assume that
−
spectrum. In addition, we assume the dilute-gas regime, elementary excitations are populated according to the
whichallowsto neglectthe effects ofthe quantumdeple- grand-canonical statistics: nˆ nˆ = nˆ nˆ
ν λ CN ν CN λ CN
h i h i h i
tion. Furthermore, we assume that the number of atoms for ν = λ, nˆ2 =2 nˆ 2 + nˆ , with nˆ =ˆb†ˆb .
issufficientlylarge: Na/aho 1,whereaho = ~/(mω) This i6s equhivaνlieCnNt withh uνsiiCnNg thhe MνiDCNE for an iνdeal νgaνs,
≫
istheharmonicoscillatorlength,andthecondensatecan andresultsfromassumptionthatatsufficientlylowtem-
p
be described in the Thomas-Fermi (TF) regime. This peratures the number of elementary excitations can be
is particularly convenient since in this regime the an- considered as unconstrained. Furthermore, in analogy
alytic solutions of the Bogoliubov-de Gennes equations to the homogeneous gas we expect that the main con-
are known [29]. tribution to the fluctuations comes from the low-energy
In the Bogoliubov theory the field operator is decom- phononlike modes, and in the thermodynamic limit we
posedintothecondensatewavefunctionΨ0 andthenon- can approximate hnl by (βεnl)−1. The overlap integral
condensed part Ψˆ′: Ψˆ(r)=Ψ0(r)+Ψˆ(r)′. The approxi- d3ru∗ν(r)uλ(r) is calculated in Appendix D. Substitut-
mateHamiltonianwithneglectedthird-andfourth-order ing the result of integration into Eq. (64), we arrive at
terms in Ψˆ′ is diagonalized by the Bogoliubov transfor- Rthe final result for the canonical fluctuations [10]
mation
µ 2 k T 2
Ψˆ′(r)=Xν (cid:16)uν(r)ˆbν −vν∗(r)ˆb†ν(cid:17), (61) with (cid:10)δ2Ne(cid:11)CN =2A(cid:16)~ω(cid:17) (cid:18) ~Bω (cid:19) , (65)
where ˆb , ˆb† are the creation and annihilation opera-
ν ν 2l+1 2(n+1)2 n+l+ 3 2
tors of the elementary excitations, while the amplitudes = 2
ueqνu(ra)t,iovnνs(.rI)natrheesTolFutrioegnismoef,tthhee cBoongdoelniusabtoev-ddeensGiteynhnaess A Xnl εnl "ε2n+1,l 2n+l+ 25(cid:0)2−1 2n(cid:1)+l+ 52 2
typical ”inverted parabola” shape with the condensate h(cid:0) ε2 +(cid:1)1 2 i(cid:0) (cid:1)
+ nl 2 .
wave function Ψ0(r) = µ(1−r2/R2)/g, which van- ε2 2n+l(cid:0)+ 1 2 2(cid:1)n+l+ 5 2#
ishes outside the condensate radius R = 2µ/(mω2). nl 2 2
p (66)
The chemical potential is determined by the normaliza- (cid:0) (cid:1) (cid:0) (cid:1)
tion of Ψ (r) and in the TF approximationpis given by In the above equation n,l = 0,1,2,... and l = 0 for
0 6
µ=(15Na/a )2/5~ω/2. The low-energycollective exci- n = 0. The numerical calculation of Eq. (66) yields
ho
tationsofthe condensateinthe TFregimehaveenergies A 0.56. We observe that in the thermodynamic limit
[29, 30] (N≃ , ω 0, and Nω3 = const) the fluctuations
→ ∞ →
exhibit anomalous scaling, similarly to the behavior in
Enl =~ωεnl =~ω(2n2+2nl+3n+l)1/2, (62) a homogeneous system. Moreover, the T2 temperature
dependence of fluctuations becomes universal, both for
which is valid for E µ. On the other hand, the
nl
≪ the homogeneous and trapped gases.
amplitudes u and v of those modes are described by
ν ν
[29]
√4n+2l+3 1 y2 ±1/2 B. Fluctuations in the microcanonical ensemble
f = −
nlm± R3/2 ξε
(cid:20) nl (cid:21) Tocalculatethe microcanonicalfluctuations inatrap-
ylP(l+1/2,0)(1 2y2)Y (θ,φ), (63)
× n − lm ped gas we employ the thermodynamic relation (10). In
10
section IIIC we have shown that this identity applies mogeneous and trapped Bose gases. For a homogeneous
also to an interacting gas. The proof is quite general, system we apply the particle-number-conserving formu-
and in the case of a trapped gas the only difference with lation of the Bogoliubov theory, and obtain the analyti-
respect to the homogeneous system is the form of the cal results for the fluctuations in the canonical and mi-
operatorsNˆ andHˆ ,thatenterthedefinitionofX(z,β) crocanonical ensemble, including the corrections due to
e B
and Y(z,E). the finite size of the system. Our derivation is based on
According to Eqs. (11) and (12), the particle-energy contour integral representation in terms of spectral Zeta
correlation and the fluctuations of the system’s energy, function. We determine the poles of Zeta functions re-
canbedirectlycalculatedfromthemeannumberofnon- sultingfromthe Bogoliubovspectrum, anddeveloptheir
condensed particles and the mean energy. The former asymptotic expansions. This allows us to find analytical
quantity can be evaluated as the expectation value of formulasforthemeannumberofnoncondensedparticles,
the operator Nˆ = d3rΨˆ†(r)′Ψˆ(r)′, where in place of its fluctuations and the mean energy, which are valid for
e
Ψˆ(r)′ we substitute decomposition (61). For ξ 1 we arbitrary ratio of the temperature to the chemical po-
use the approximatiRon u v f /2, obtain≪ing the tential. We determine the microcanonical fluctuations
ν ν ν+
following result ≈ ≈ from the thermodynamic identity which relates them to
the quantities calculated in the canonicalensemble. Our
N =2 h d3r u (r)2+N , (67) analysis shows that both the microcanonical and canon-
e ν ν qd
h i | |
ν Z ical fluctuations exhibit anomalous scaling, and in the
X
thermodynamic limit fluctuations in these two ensem-
where N stands for the quantum depletion. Since N
qd qd
bles become equal. A similar behavior is observed for a
does not depend on the temperature, its exact value is
trapped gas. In this case we carry out our calculations
not important for the subsequent derivation. Now, we
withinthestandardBogoliubovtheory,andevaluateonly
utilize the results of Appendix D for the overlapintegral
the leading order behavior in the thermodynamic limit.
between the amplitudes u , and replace the summation
ν
byintegration,whichisvalidinthethermodynamiclimit.
This yields [31]
Ourresultshavebeenderivedintheregimeoftemper-
µ k T 2 atures much lower than the critical temperature, where
N =√2ζ(2) B +N . (68)
h ei ~ω ~ω qd the effects of the quantum and thermal depletion on the
(cid:18) (cid:19) populationofthecondensatecanbeneglected. Westress,
A similar method cannot be applied, however, to cal-
however, that generalization of the obtained results for
culate the mean energy of a weakly interacting trapped
the case ofhighertemperatures,canbe done in straight-
gas. In this case main contribution comes from the
forward way in the spirit of Popov theory [26]. To this
boundary of the condensate, where the excitations have
end, for a homogeneous system parameter α should be
single-particle character. Derivation based on the semi-
replaced by α = gn /∆, with the density of condensed
0 0
classicalformulationoftheHartree-Fock-Bogoliubovthe-
atoms n determined in a self-consistent way from the
0
ory,which includes the effects of collective modes inside,
equation for the mean number of noncondensed parti-
and the single-particle excitations outside the conden-
cles. In a similar way one can generalize the results for
sate, yields the following law [31]
the trapped gas. In this case the total number of par-
5 20ζ 7 µ 1/2 k T 7/2 ticles N in equation for µ should be replaced by the
E = µN + 2 ~ω B , (69) number of condensed atoms N , with N calculated in
h i 7 √π ~ω ~ω 0 0
(cid:0) (cid:1) (cid:16) (cid:17) (cid:18) (cid:19) a self-consistent a manner.
Now, with the help of results (68) and (69) we calculate
the difference between the fluctuations in the canonical
and the microcanonical ensemble: Finallywewouldliketocommentontheapplicationof
δ2N δ2N = π9/2 µ 23 kBT 32 tflhuectpuaarttiiocnles-.nFuomrbaerh-ocmonosgeernveinogusmseytshteomd twoechaalvcuelvaetreifithede
e CN− e MC 315ζ 7 ~ω ~ω that both conserving and nonconserving theories lead to
(cid:10) (cid:11) (cid:10) (cid:11) 2 (cid:16) (cid:17) (cid:18) (cid:19)(70) theidenticalresults. Suchinsensibilityofthecondensate
(cid:0) (cid:1)
It is easy to observe that the difference is proportional statistics, can be attributed to the fact, that below the
to N in the thermodynamic limit. Therefore the micro- critical temperature, the condensate acts as a reservoir
canonical fluctuations, similarly to the canonical ones, ofparticlesforthenoncondensedstates. Thus,aslongas
are anomalous, and in the thermodynamic limit predic- the temperature is not too close to the critical tempera-
tions of both ensembles become equivalent. ture, the probability distribution of the number of non-
condensed atoms remains insensitive to the actual num-
ber ofatoms. The totalnumber of atomsinfluences only
V. CONCLUSIONS the energy spectrum in the interacting gas. Therefore,
predictions of conservingandnonconservingtheories are
In this paper we have analyzed the fluctuations of the the same with respect to the statistics of noncondensed
numberofcondensedparticlesinaweaklyinteractingho- part of the system.