Physics Letters A 372 (2008) 1574–1578

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More accurate theory for Bose–Einstein condensation fraction

Shyamal Biswas

Department of Theoretical Physics, Indian Association for the Cultivation of Science Jadavpur, Kolkata-700032, India

Received 4 May 2007; received in revised form 1 August 2007; accepted 4 October 2007

Available online 9 October 2007

Communicated by J. Flouquet

Abstract

Bose–Einstein statistics is derived in the thermodynamic limit when the ratio of system size to thermal de Broglie wavelength goes to infinity.However, according to the experimental setup of Bose–Einstein condensation of harmonically trapped Bose gas of alkali atoms, the ratio near thecondensation temperature (To) is 30–50. And, at ultralow temperatures well below To, this ratio becomes comparable to 1. We argue that finitesize as well as the ultralow temperature induces corrections to Bose–Einstein statistics. From the corrected statistics we plot condensation fractionversus temperature graph. This theoretical plot satisfies well with the experimental plot [A. Griesmaier et al., Phys. Rev. Lett. 94 (2005) 160401].© 2007 Elsevier B.V. All rights reserved.

PACS: 05.30.Jp; 03.75.Hh; 05.40.-a

Bose–Einstein condensation (BEC) is a topic of high ex-perimental [1] and theoretical [2] interest. Within the last tenyears several thousand works were done on this topic. Gener-ally inter particle interaction is responsible for a phase tran-sition. But BEC type of phase transition occurs entirely dueto the Bose–Einstein (BE) statistics. BE statistics is derivedin the thermodynamic limit when the ratio of system size tothermal de Broglie wavelength (λT ) goes to infinity. In the ther-modynamic limit, for 0 < T � To, the condensation fraction ofharmonically trapped ideal Bose gas obey the law [1 − ( T

To)3].

However, according to the experimental setup of Bose–Einsteincondensation of harmonically trapped Bose gas, the ratio nearTo is 30–50. And, at ultralow temperatures (T ), well below To,this ratio becomes comparable to 1. In this situation, our famil-iar Bose–Einstein statistics needs a correction. As a correction,we introduce nonextensivity to the statistics. This nonextensiv-ity along with the finite size correction and two body interac-tion modify the condensation fraction. We shall show that ourmodified fraction would satisfy the experimental plot of con-densation fraction.

In the following sections we have shown the use of thermo-dynamic limit in the derivation of standard BE statistics. Then

E-mail address: tpsb@iacs.res.in.

0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2007.10.009

we semi-qualitatively discuss the BEC of Ideal Bose gas whichsatisfy the thermodynamic limit. This qualitative discussions isextended for trapped ideal Bose gas. Then we show how thecondition of thermodynamic limit is not satisfied in the BECexperiment. After this discussions we introduce Tsallis type ofgeneralized (nonextensive) BE statistics [3]. Then we justifya redefinition of nonextensive q-parameter [4,5]. We redefinethe q-parameter by temperature and system size. We keep thenonextensivity as a correction to the BE statistics. We also dis-cuss the finite size correction. Finally we shall introduce twobody weak interactions. Like Giorgini, Piteavskii and Stringariwe shall perturbatively treat the interaction and take the resultof Hartree–Fock (HF) approximation for this interaction [6].Considering finite size corrections, two body interaction andnonextensive corrections we plot the condensation fraction withtemperature.

Before going into further discussion let us recall the use ofthermodynamic limit in the derivation of our familiar BE sta-tistics. The prescription of the derivation of BE statistics is asfollows [7]. The partition function for this system of N particlesis

(1)Z(N) =∑R

e−(n1ε1+n2ε2+n3ε3+···)/kT ,

S. Biswas / Physics Letters A 372 (2008) 1574–1578 1575

where εi is single particle energy level and ni is the numberof particles in the ith single particle state. For Bose gas ni =0,1,2,3, . . . R represent all possible number distribution ({ni})subject to the constraint

∑∞i=0 ni = N . For this constraint the

evaluation of Z(N) without approximation is impossible. SinceZ(N) is a rapidly increasing function we choose a exponen-tially decreasing function eμN/kT so that Z(N) is maximizedat a suitable negative value of μ. Thus we can approximatelywrite

(2)∑N ′

Z(N ′)eμN ′/kT = Z(N)eμN/kT �N ′,

where �N ′ is the width of the Z(N ′) distribution about themaximum Z(N). μ is called the chemical potential. In the ther-modynamic limit, N → ∞ and length scale L → ∞. Fromthe first condition of the thermodynamic limit we can write�N ′N

→ 0. Using �N ′N

→ 0, we can arrive at the relation

(3)lnZ = −μN/kT −∑

i

ln(1 − e−(εi−μ)/kT

).

The average no. of particles in the single particle state (i) is n̄i =∑R nie

−(n1ε1+n2ε2+n3ε3+···)/kT∑R e−(n1ε1+n2ε2+n3ε3+···)/kT . Using the second condition of ther-

modynamic limit (L → ∞) we can write εi+1−εi

kT→ 0 (since

εi ∼ h̄2i2

mL2 ). From this condition we can write n̄i = −kT ∂ lnZ∂εi

.So, in the thermodynamic limit we have the BE statistics as

(4)n̄i = 1

e(εi−μ)/kT − 1.

Let us now introduce the BEC of ideal Bose gas. We con-sider a noninteracting many particle system of Bose gas con-taining N alkali atoms. The system is in equilibrium with itssurroundings at temperature (T ). The mass of a particle is m.The length and the volume of the system are L and V (= L3)

respectively. Thermal de Broglie wave length of a single parti-

cle is λT =√

2πh̄2

mkT, where k is the Boltzmann constant. Average

separation of particles is l = ( VN

)1/3. In the classical limit wemust have

(5)l

λT

� 1.

From Eq. (5) we can easily write

(6)kT

2πh̄2

mL2

� N2/3.

At sufficiently low temperatures when lλT

∼ 1, the gas be-comes degenerate and quantum correction is necessary. BECoccur at the onset of this degeneracy. So, for T ∼ To wehave kTo

2πh̄2

mL2

∼ N2/3. However the condition of thermodynamic

limit (L/λT → ∞) is also valid at this temperature. For finitesystem the thermodynamic limit is realized as L/λT � 1. How-ever, at sufficiently ultralow temperatures L/λT can be compa-rable to 1. In this situation, the BE statistics needs a correction.

The usual condition of thermodynamic limit is not properlysatisfied in the case of the experimental setup of BEC of 3d

harmonically trapped Bose gas [1]. In the semiclassical approx-imation the number density at the distance (r) from the center

of the trap is [8] n̄(r) ∼ e− mω2r22kT , where 1

2mω2r2 is the trap po-tential and ω is the angular trap frequency. The length scale of

this 3d trapped Bose gas is LT ∼√

2kT

mω2 . Putting this expres-sion of length in Eq. (6), we get the conditions of classical limitas kT

h̄ω� N1/3, and the thermodynamic limit (LT /λT → ∞)

as kTh̄ω

→ ∞. Eventually we get the condensation temperature

for this system as To ∼ h̄ωk

N1/3. More precisely the thermo-dynamic limit for this system is realized as ω → 0, N → ∞and Nω3 = const. Exact calculation with thermodynamic limitshow that [8]

(7)LT =√

h̄

mω

√2ζ(4)kT

ζ(3)h̄ω

and [8,9]

(8)To = h̄ω

k

[N

ζ(3)

]1/3

.

With the consideration of thermodynamic limit, total number ofexcited particles for T � To are Ne = [ kT

h̄ω]3ζ(3) and the num-

ber of condensed particles are No = N [1− ( TTo

)3]. In the exper-imental setup the typical value of [10] N , ω and m are of theorder of 50 000, 2645 s−1 and 52 amu. With these experimen-tal parameters, at To the length scale as expressed in Eq. (7) isLT ∼ 56×10−4 mm and λTo = 2.88×10−4 mm and their ratiois LT /λT = 32.2. However, for T ∼ h̄ω

k, a large fraction of par-

ticle come down to the ground state and the length scale of the

system becomes ∼√

h̄mω

∼ 6.76 × 10−4 mm. At these ultralowlow temperatures, the thermal de Broglie wavelength becomescomparable to the system size. At these ultralow temperaturesthe usual theory of statistical mechanics of finite system is notproperly valid. So, for h̄ω

k� T � To, we seek a ultralow tem-

perature as well as finite size correction to BE statistics.To quantify the correction arising from ultralow tempera-

tures, let us start from Tsallis type of generalized Bose–Einsteinstatistics which is expressed as

(9)n̄i = 1

[1 + (q − 1)(εi−μ)

kT] 1

(q−1) − 1

where q is a hidden variable [3,11]. The relative probabilitythat E be the total energy of a system is given by Boltzmannfactor e−E/kT . In Tsallis statistics this factor is replaced by

1(1+(q−1)E/kT )1/(q−1) . As q → 1, we get back Boltzmann sta-tistics. However, Tsallis statistics [12] is applied to equilib-rium [11,13–15] as well as to nonequilibrium [5,16,17] sys-tems. Tsallis statistics is rederived as one of the superstatis-tics [4] of a nonequilibrium system. In this theory the hiddenvariable (q) of Tsallis statistics is equated with system para-meter and q is no longer a variable. In the theory of super-statistics [4] 2/(q − 1) is redefined as effective no. of degreesof freedom. In the theory of dynamical foundation of nonex-tensive statistical mechanics, this effective no. of degrees of

1576 S. Biswas / Physics Letters A 372 (2008) 1574–1578

freedom is equated as [5] (3 − q)/(q − 1). However, for fi-nite equilibrium system we equate q with a system parameter.We will equate q − 1 with λT /LT so that in the thermody-namic limit (LT /λT � 1) of finite system we can go to theusual Bose–Einstein statistics. Fig. 1 shows experimental andtheoretical plot of condensation fraction with temperature. Inthis figure we see that a negative shift of condensation fractionis necessary to satisfy the theoretical plot with the experimen-tal data. In the theoretical plot there is finite size correction andthe correction due to two body interactions. We shall see thatthe generalized BE statistics with (q − 1) ∝ h̄ω

kTwill give rise to

a significant negative shift of condensation fraction with tem-perature. This significant shift along with finite size correctionand the correction due to the two body interaction might satisfythe experimental results. For the trapped Bose gas we shall startfrom Eq. (9) and put q − 1 = h̄ω

αkTto get

(10)n̄i = 1

[1 + h̄ωαkT

(εi−μ)kT

] αkTh̄ω − 1

,

where α is an arbitrary constant. We shall determine this α fromthe experimental result.

We are considering a finite system. For this finite system thecondensation temperature would not be To. For this system thecondensation temperature is Tc which must be close to To. Fora system of Bose gas in isotropic harmonic trap, the single par-ticle energy levels are εj = ( 3

2 + j)h̄ω (j = 0,1,2,3, . . .). ForT � Tc, we have μ = 3

2 h̄ω. So for T � Tc, from Eq. (10) weget the average number of particles in j th state as

(11)n̄j = 1

[1 + j

αt2 ]αt − 1

where t = kTh̄ω

. Obviously, in the thermodynamic limit of thisfinite system, we have t � 1. Since for t ∼ 1, the thermal deBroglie become comparable to the system size, this expressionof n̄j is not valid for t � 1. For a 3 − d isotropic harmonic

oscillator the density of states is (j2+3j+2)2 . So, from Eq. (11)

Fig. 1. Condensation fraction ( NoN

) versus temperature ( tto

) plot. The thick linefollows from Eq. (17). The dashed line corresponds to the thermodynamic limitand excludes all the correction terms in Eq. (17). The dotted line correspondsto the finite size correction and the correction due to interaction excluding theultralow temperature correction term in Eq. (17). All the theoretical curves aredrawn according to the following experimental parameters. The dotted pointsare experimental points of Bose–Einstein condensation of 52Cr where [10]N = 50 000, ω = 2645 s−1, m = 52 amu, a = 105aB and To ∼ 700 nK.

we get the total no. of particles in the excited states for t < tc =kTc

h̄ωas

(12)Ne =∞∑

j=1

(j2 + 3j + 2)

2

1

[1 + j

αt2 ]αt − 1.

In this summation the volume term of the density of states (i.e.the term j2/2) dominates over the surface term (i.e. the term3j/2). This surface term gives finite size correction. The thirdterm contribute insignificantly to the calculation of number ofexcited particles. Converting the summation into integration wehave

(13)Ne =∞∫

0

j2 + 3j

2× 1

[1 + j

αt2 ]αt − 1dj.

For t � 1, the denominator of the second factor of Eq. (13)

can be approximated as [1 + j

αt2 ]αt ≈ ej/t − j2

2αt3 ej/t + O( 1t5 ).

With this approximation, Eq. (13) gives the following equation.

Ne =∞∫

0

j2

2

1

ej/t − 1dj +

∞∫0

3j

2

1

ej/t − 1dj

+ 1

2αt3

∞∫0

j2

2

ej/t

(ej/t − 1)2dj

(14)= t3ζ(3) + 3t2

2ζ(2) + 6t2

αζ(4).

Till now we did not consider the inter-particle interaction.To achieve BEC it is necessary to take a very dilute gas. Thegas being very dilute there should be a correction term in theexpression of Ne due to two body scattering. This correctionwithin Hartree–Fock (HF) approximation has been discussedin [6,8]. According to HF approximation, the correction termto the above Ne is 3 × 1.326 a√

h̄/mωN

7/6e = 4.932 a√

h̄/mωt7/2,

where a is the s-wave scattering length. Introducing this termin Eq. (14) we get the more corrected expression of number ofexcited particle as

(15)Ne = t3ζ(3) + 3t2

2ζ(2) + 4.932

a√h̄

mω

t7/2 + 6t2

αζ(4).

At T = Tc, all the particles will be in the excited states [8,9,18].So at T = Tc or at tc = kTc

h̄ω, the number of excited particles is

equal to the total number of particles. So, we must have

(16)N = t3c ζ(3) + 3t2

c

2ζ(2) + 4.932

a√h̄

mω

t7/2c + 6t2

c

αζ(4).

In the thermodynamic limit of the ideal trapped Bose gas[8,9] Ne = t3ζ(3) and its condensation temperature To is suchthat [8,9] to = [ N

ζ(3)]1/3, where to = kTo

h̄ω. Comparing this ex-

pression of to and tc of Eq. (16) we see that tc < to and there isa shift δtc = tc − to of condensation temperature due to the in-clusion of finite size correction term, correction term due to twobody scattering and due to the ultralow temperature correction

S. Biswas / Physics Letters A 372 (2008) 1574–1578 1577

of BE statistics. For t � tc, from Eq. (15) we get the fraction ofnumber of particles in the ground state as

No

N= N − Ne

N

= 1 −[

t

to

]3

(17)

−[

3t2

2ζ(2) + 4.932

a√h̄

mω

t7/2 + 6t2

αζ(4)

]/[t3o ζ(3)

].

However, in the thermodynamic limit with no inter parti-cle interaction, the expression of this fraction would be [8,9][No

N]T −L = 1 −[ t

to]3. Now, from Eq. (17), due to the correction

terms we get the fractional change in condensation temperatureas

δTc

To

= δtc

to

=[[[[

N −[

3t2c

2ζ(2) + 4.932

a√h̄

mω

t7/2c

+ 6t2c

αζ(4)

]]/ζ(3)

]1/3

−[

N

ζ(3)

]1/3]]/[N

ζ(3)

]1/3

(18)

≈ −1

3

[3t2

c

2ζ(2) + 4.932

a√h̄

mω

t7/2c + 6t2

c

αζ(4)ζ(3)

]/N.

Putting tc ≈ [ Nζ(3)

]1/3 in the above equation we get

δtc

to= − ζ(2)

2[ζ(3)]2/3N−1/3 − 1.326

a√h̄/mω

N1/6

(19)− 2ζ(4)

α[ζ(3)]2/3N−1/3.

From the first term of Eq. (19), we get the Tc shift due to

the finite size correction as [19] δtf −sc

to= − ζ(2)

2[ζ(3)]2/3 N−1/3 =−0.728N−1/3 and for 50 000 particles we get δt

f −sc

to= −1.97%.

From the second term of Eq. (19), we get the Tc shift due

to the correction of two body interaction as [6,8] δt intc

to=

−1.326 a√h̄/mω

N1/6 = −6.61% for ω = 2645 s−1, a = 105aB

and for N = 50 000 [10]. For this setup to = [ Nζ(3)

]1/3 = 34.65.From the third term of Eq. (19) we get the Tc shift due

to ultralow temperature correction of BE statistics as δtultc

to=

− 2ζ(4)

α[ζ(3)]2/3 N−1/3 = − 1α

5.19% for 50 000 particles. Accord-

ing to the experiment δtcto

should be 10% and since δtcto

=δt

f −sc

to+ δt int

c

to+ δtult

c

towe get α = 3.6549. However, since these

shifts are not negligible, in Eq. (8) we have to take higher or-der terms into account. As the corrections are not negligible weshould not put tc = to in Eq. (19). Since tc < to, the percent-age t

f −sc and t int

c shift we calculated will be lowered. UsingEq. (17), from the numerical plot of No with temperature, we

N

get δtf −sc

tc= −1.93%, δt int

c

tc= −5.75%. From Eq. (17), in the nu-

merical plot 10% tc shift will be achieved if we put α = 1.48.Well below Tc, as t approaches to 1, the BE statistics needs

more corrections. Below Tc , as a comparative study of the twostatistics, let us calculate the ratio of specific heat from the twostatistics. As a comparative study we can take ideal gas and wedisregard the surface term in Eq. (13) to get

(20)Ne(t) =∞∑i=1

(αt2)3 �(αit − 3)

�(αit).

As t � 1 [8,9], Ne(t) → t3ζ(3) as expected from BE statistics.The total energy of the system would be

E(t) ∼ (h̄ω)

∞∫0

j3

2

1

[1 + j

αt2 ]αt − 1dj

(21)= 3(h̄ω)

∞∑i=1

(αt2)4 �(αit − 4)

�(αit).

Now the specific heat would be

Cv(t) = k

h̄ω

d

dtE(t)

(22)

=∞∫

0

k

j3(1 + j

αt2 )αt[ 2j

t2(1+ j

αt2)− α log[1 + j

αt2 ]]2[(1 + j

αt2 )αt − 1]2dj.

As t � 1, E(t) → 3h̄ωζ(4)t4 as expected from the BE statis-tics [8,9]. Let us denote the ratio of specific heat calculated fromthe corrected BE statistics and from the BE statistics as

(23)r ′(t) = Cv

k12ζ(4)t3.

Obviously as t → ∞, r ′ → 1. With α = 1.48 numerical plot ofr ′(t) at very low temperatures (1 � t � tc) is shown in Fig. 2.

In Fig. 2 we see that at high temperature t � 1, the spe-cific heat behaves well according to our familiar T 3 law. Atto = 34.65, the numerical value of r ′ is 1.15 and at t = 15, thenumerical value of r ′ is 1.39. So at t = 15, there is 39% error in

Fig. 2. Plot of r ′(t) with temperature (t ) in units of kTh̄ω

. This plot is followed

from Eq. (23). The doted line shows the theoretical value of the ratio r ′(t) inthermodynamic limit when ω → 0.

1578 S. Biswas / Physics Letters A 372 (2008) 1574–1578

the specific heat. In this figure we see that this specific heat lawdose not holds fairly well for t � 6. From this figure we also seethat, to achieve the thermodynamical limit, ω → 0 is no longera necessary criterion. If ω ∼ kT

15h̄, the system (3d isotropic har-

monically trapped Bose gas) will correspond to thermodynamiclimit with ∼39% error in the specific heat. Similarly we can saythat to achieve the thermodynamic limit of a system of particlesin a box, the volume need not necessarily tend to infinity. In thisway we can also estimate the minimum volume of a thermody-namical system. In Fig. 2 we see that specific heat for this finitesystem becomes negative at t � 5. In this range of temperaturesLT � λT and the theory of statistical mechanics is not valid. Inspite of that, due to nonextensivity the appearance of negativespecific at these temperatures is not surprising theoretically [20]and experimentally [21].

For smaller ω the system size (LT ) is larger and λT

Lis

smaller. In this case the ultralow temperature correction wouldbe smaller. In the experiment [22] ω = 1140 s−1 is smaller thanthat of the previous case. For this experimental setup, the nu-

merical value of δtf −sc

to= −2.1% and δt int

c

to= −5.5%. From the

experimental data as shown in Fig. 3, we see that δtcto

∼ 8%. So,δtult

c

to= −0.4% and it corresponds to α = 5.5. Due to the larger

system size the ultralow temperature correction is smaller as wesee in Fig. 3.

That BE statistics needs a correction for finite system at ul-tralow temperatures is justified theoretically. But we did nottheoretically justify why Tsallis type of generalized BE is nec-essary for finite system at ultralow temperatures. However,Tsallis type of generalized BE statistics with our redefinedq(= h̄ω

αkT+ 1) parameter satisfy experimental result. While the

generalized BE statistics is theoretically compelling the experi-mental evidence for its presence is somewhat uncertain at thepresent level of accuracy. The errors are such that the ultralow temperature corrections can be absorbed within the exper-imental error. Hence to explore the ultralow temperature cor-

Fig. 3. Condensation fraction ( NoN

) versus temperature ( tto

) plot at a lower fre-quency that in Fig. 1. The thick line follows from Eq. (17). The dashed linecorresponds to the thermodynamic limit and excludes all the correction termsin Eq. (17). The dotted line corresponds to the finite size correction and the cor-rection due to interaction excluding the ultralow temperature correction term inEq. (17). All the theoretical curves are drawn according to the following experi-mental parameters. The dotted points are experimental points of Bose–Einsteincondensation of 87Rb where [22] N = 40 000, ω = 1140 s−1, m = 87 amu,a = 90aB and To = 280 nK.

rections we have to sufficiently reduce the experimental error.Like Figs. 1 and 3, a negative shift of condensation fraction isalso evident in [23,24]. This negative shift can also be explainedwith our corrected BE statistics. We argued that the correctionover the BE statistics is O(λT /LT ). Smaller the ratio smaller isthe correction and it is evident in Figs. 1 and 3.

Here we present an exact calculation of the condensationfraction for Canonical system. However, due to the finiteness,the calculation might differ from that of a microcanonical sys-tem [25]. Although the experimental uncertainties are largerwith respect to these theoretical corrections yet the ultralowtemperature correction might be of interest. However, this ul-tralow temperature correction is valid only for a finite system.How α is to be determined theoretically remains an open ques-tion.

Acknowledgements

Several useful discussions with J.K. Bhattacharjee andKoushik Ray of I.A.C.S. are gratefully acknowledged.

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