Bose statistics and classical fields

0.24 MB·English

by  Emilia Witkowska

Bose statisti
s and
lassi
al (cid:28)elds
1 1,2 2,3
1 Emilia Witkowska , Mariusz Gajda and Kazim3ie2r/z46Rz¡»ewski
2Institute of Physi
s, Polish A
ademy of S
ien
es, Aleja Lotników , 02-668 Warsaw, Poland
3 Fa
ulty of Mathemati
s and S
ien
es, Cardinal Stefan Wyszy«ski Univ3e2r/s4it6y, Warsaw, Poland
Center for Theoreti
al Physi
s, Polish A
ademy of S
ien
es, Aleja Lotników , 02-668 Warsaw, Poland
(Dated: January 13, 2009)
Classi
al (cid:28)elds
ounterpartof theideal Bosegasstatisti
s ina trapisinvestigatedbyperforming

al
ulations in the
anoni
al ensemble. There exists the optimal
ut-o(cid:27) whi
h allows to mat
h the
9 full probability distribution of the
ondensate population by its
lassi
al
ounterpart. Universal
0 s
aling of that
ut-o(cid:27)with temperature and dimensionality is derived.
0
2 E = n ǫ
n i
sAotftthheeeenled
torfoXmIaXgn
eetnit
u(cid:28)ryeldeq,uthileibtrhiuemorythoefrmaosody
naallmed- wBohseeregsaisngilne-apahrtair
mleoenni
ertgriaespǫojf=en~eωrgjy[8℄. InPorjderjtjo,
a
J bla
k body radiation, su(cid:27)ered for the persistent ultra- obtain the probability distribution we (cid:28)rst
onsider par-
3 violet divergen
e. It was Max Plan
k [1℄ who
ured the tition fun
tions. The
aNnoni
al ensemble partition fun
-
1 problemintrodu
inga
on
eptoflightquantalater
alled tion for a system with ex ex
ited bosons and tempera-
T
photons. With thehelpofphotonsthe energy
arriedby ture of 1D harmoni
os
illator is [10℄:

] high frequen
y degrees of freedom was tamed. ∞ ∞
r
he preMliomrienathryanid2ea0syoefaBrsoslaet[e3r℄Einitnrsotdeuin
e[d2℄s,tasttiimstui
laaltepdrobpy- Zex(Nex,β)=nX1=0nX2=0...e−βEδNex,Nex (1)
t
o erties of the ideal gas of indistinguishable massive par- ∞
n j N = n
t. ti
les that are now
alled bosons. In parti
ular it was where j arepopulations of th state, ex j=1 j is
a shownthat entirelydue to the indistinguishability, when a number of bosons and β = 1/kBT. KroneP
ker delta
m N

ooled, bosons tend to assemble at the ground state of fun
tion in (1) enfor
es the
ondition of ex thermal
d- the binding potential at relatively high temperature. bosons. In termsNof these fun
tions proPba1Dbi(lNity ,dβis)tr=i-
ex ex
n This phenomena
alled Bose-Einstein
ondensation bution of having thermalNbosons is
Z (N ,β)/Z Z = Z (N ,β)
o was(cid:28)nallyrealizedintheseminalexperimentsin1995[4℄. ex ex N with N Nex=0 ex ex . Intro-
[c Of
ourserealisti
theoryhadtodepartfromtheidealgas δdu
in=g a1n i2nπteegixr(aal−rbe)pdrxesentPation of the delta fun
tion
and wasto a

ount forintera
tions. While a well known a,b 2π 0 and
al
ulating the integral we
1 Gross-Pitaevskii equation
ombined with small ex
ita- obtain: R
v tions - the Bogoliubov approximation [5℄ - forms a good ∞ ∞
1 1
69 ebxapsiesriomfeanntaslyresqisuinreeaar mabestohloudtevzaelridoftoermhpiegrhaetrurtee,mmpearnay- P1D(Nex,β)= ZN Xj=1e−βjNexYl6=j 1−e−β(ǫl−ǫj) . (2)
7
tures. Several groups proposed a
lassi
al (cid:28)eld approxi-
1
. mation as an e(cid:27)e
tive approa
h to nonzero temperature Similarprobabilitydistribution
anbe
al
ulatedforthe
1
Bose gas [6℄. This is like traveling the road from Jaynes
lassi
al
ounterpart. In this
ase the
anoni
al ensem-
0
9 to Max Plan
k ba
kward in time -ignoring the granular ble partition fun
tion must takeinto a

ount
ontinuous
0
hara
terofatomsandstipulatingthatamountofatoms values of populations while energy levels are quantized
: may be
hanging in a
ontinuous way. Understandably aspreviously. Forthe
lassi
al
ounterpartssumsarere-
v
equipartition of the energy between the modes leads to pla
ed by integral and the
anoni
al ensemble partition
i
X theultravioletdivergen
ejustasitwasfor
lassi
alradi- fun
tion is:
ar a
utito-no.(cid:27)Htoen

uer,etthheeapurtohbolresmin.tDroid(cid:27)uer
eenat ashuothrtorwsaavnedlesnogmthe Zex(Nex,β)= d2α d2α...e−βEδ(Nex−Nex),
Z π Z π
ofthem di(cid:27)erentlyindi(cid:27)erentpapers
hoosethevalueof
the
ut-o(cid:27) [6, 7℄. |α |2 ∈ (0,∞) E (3=)
j
where amplitudes , the energy
ItisthepurposeofthisRapidCommuni
ationtoshow, Kmax|α |2ǫ N = Kmax|α |2
based on exa
tly soluble models of the ideal gas, that Pj=1 j j and ex Pj=1 j . NKotmi
aex, that
there exists the optimal
ut-o(cid:27) that allows to mat
h the we in
lude a (cid:28)nite numbers of states up to only.
full probability distribution of the number of
ondensed Populations of the higher energy states are set to zero
atomsbyits
lassi
al(cid:28)elds
ounterpart. Wegiveas
aling and not taken into a

ount within the
lassi
al approx-
ofthis
ut-o(cid:27)withtemperatureanddimensionalityofthe imation. δT(ahe−ibn)te=gra1l re∞predsexnetixa(tai−onb)of the Dira
delta
bindingpotential. Ofparti
ularinterestisthepopulation fun
tion 2π R−∞ simpli(cid:28)es
al
ula-
of the last state retained in the optimal
lassi
al (cid:28)elds tions whi
h give:
approximation. It depends on dimensionality but not on 1 Kmax Kmax 1
temperature. P1D(Nex,β)= e−βjNex .
ZN Xj=1 Yl6=j β(ǫl−ǫj) (4)
We start our analysis with the one-dimensional (1D)
2
2.5
1DHLΒPN,ex 0000....000000110505ææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææ ÑΩΒKmax 112...050 èèèèáèèèáèèèèáèèèáôèèèáèèèáèèèáôèèèáèèèèáèèôèáèèèáèèèáèèôèáèèèáèèèáèèôèáèèèáèèèèáôèèèáèèèáèèèáôèèèáèèèáèèôèáèèèáèèèáèèôèèáèèèáèèèáôèèèáèèèáèèèôáèèèáèèèáèèôèáèèèèáèèèáèôèèáèèèáèèèáôèèèáèèèáèèèôáèèèáèèèáèèôèáèèèáèèèáèôèèáèèèáèèèôáèèèáèèèáèèôèáèèèáèèèáèôè
50 100 150 200
0.0 0.2 0.4 0.6 0.8 1.0
N
ex
T(cid:144)T
c
Nex
FIG.1: (Coloronline)Probabilitydistributionofhaving
bosons outside the ground state of a 1D trap. Points repre- FIG. 2: (CTol/oTrconline) Solution of the equation (10) as a
sent the exa
t quantum~dωisβtKribmuatxion=w0h.9ile lines are its
las- fun
tionNo=f 103 HforN3D=h2ar·m10o4ni
(cid:3)traNp=an6d·t1o0t5al•number of
~siω
aβlK
mouaxnt=erp1arts with ~ωβKmax =(cid:21)1d.1ot-dashed line, bosons ( ), ~ωKm(axβ), ( ). Dashed
(cid:21) solid lineaNnd=1000 ~ωβ(cid:21) d=as0h.0e4dline. line represents a value of given by the expression
Total numberof bosons is and at . (11).
3.0
Note, that this formula leads to the equipartition of en-
hnjiǫj = kBT 2.5
ergy in every single-parti
le mode, i.e. ,
hnji Β
wǫjh.erTehis feaistuthreeimse
ahnaroa

tuepriasttiio
nfoorf tthheersmtaaPtle1sDot(afNteeneo,rfgβya) Kmax 2.0
ex

lassi
al system. ProbabilityPdi1sDtr(iNbut,ioβn)s Ω 1.5
ex Ñ
and its
lassi
al
ounterpart are shown in
Fig.1. One
an see that there exists an oKptima=l v1a/lu~eωβof 1.0
max
the
ut-o(cid:27) whi
h in this
ase is equalto .
Forthisoptimalvaluequantumdistributionandits
las- 0.5
si
al
ounterpart are almost indistinguishable. Noti
e, 1 2 3 4
K
max
that other values of lead to a shift of the entire
D
distributionwhileitsshapedoesnot
hangesigni(cid:28)
antly.
Itisinterestingtoanalyzethesystemin2and3spatial
dimensions. Now, the problem is more
omplex sin
e we
have to deal with a degenera
y. There exists ex
ellent F~ωIGK.m3a:xβ(wCitohlodrimonenlisnieo)naSli
tayliDn~ωg. KPoomfinatxthsβea=r
eu0gt.i4-vo7e(cid:27)7n+bpya0r.f5oa6rmm2eDutelar
approximate formulas in the literature and we (cid:28)nd the (11), line is the best linear (cid:28)t: .
result of [9℄ the most useful for our purposes:
1 (N +H/η−1)! η Nex η
P(N ,β)= ex , and are di(cid:27)erent:
ex
ZN (H/η−1)!(Nex)! (cid:18)η+1(cid:19) (5) Kmax 1
H (β,K ) = ,
cl max
Xj6=0 βǫj (8)
with parameters:
∞ Kmax 1
1 η H = .
H(β) = eβǫj −1, (6) cl cl Xj6=0 (βǫj)2 (9)
Xj6=0
∞
1
This
lassi
al version of Eq.(5) is not obvious. We
ηH = .
(eβǫj −1)2 (7)
he
ked the
lassi
al
ounterpart of (5) by
omparing
Xj6=0
with(4)in1D,andwiththesaddlepointsmethod[11℄re-
sultsin3D.Forthesystemswithsmallnumberofbosons,
Please note again that ea
h term in the sum (6) is a as
onsidered numeri
ally here, the maxima of distribu-
j
mean o

upation of the single-parti
le level whi
h in tions
oin
idebut therearesomedeviationsin its width.
hn iǫ =
j j
the
lassi
al version (8) implied equipartition The agreement improves for larger systems.
k T
B
. The simple expression(5) isvalid foranytrapping In order to
ompare the quantum and
lassi
al
oun-
H η
potential whi
h
ontrols fun
tions and through the terpart of the probability distribution (5) we
hoose the
ǫ K
j max
single parti
le energy spe
trum of a trap. Within the
ut-o(cid:27) in su
hawaythatthemaximaofboth dis-
H

lassi
al(cid:28)elds formula(5)isstill validbut parameters tributions
oin
ide. The maximum of probability distri-
3
bution (5) is equal to H and equation for optimal Kmax 0.06 HaL
0.020
leads: 0.05 0.015
H(β)=Hcl(β,Kmax). (10) LΒ 0.04 00..000150
In Fig.2 the solution of the above equation is plotted HPN,ex 0.03 0.000100 150 200 250
for 3D geometry and di(cid:27)erent~ωtoKtal nβumber of bosons. 0.02
max
Forlargesystemsthe value of pra
ti
allydoes
T
0.01
not depend on . Some deviations from this
onstant
vTa/lTue
an beTobserved forvery lowrelativetemperatures 0.00
c c
only ( is a
riti
aNl temperature) a~nωdKthis rβegion 0 200 400 600 800 1000
max
de
reaseswithin
reasing . Thevalueof sat-
N
urates at: ex
(~ωKmaxβ)D−1 =(cid:26)1ζ(,D)(D−1)(D−1)!, for DD =≥21, 0.06 HbL 0.015
for , 0.05 0.01
ζ(D) (11) LΒ 0.04 0.005
dwihtieorne
an biestuhseedRaiesmaan
rnitZereitoanfufonr
ttihoeno[~1pω2tK℄i.mmTaahlxi
sh
ooin
e- HPN,ex 0.03 01.00 200 300
of the maximal energy of
lassi
al states . The
0.02

ondition (11) de(cid:28)nes the universal value of the
ut-o(cid:27)
K
maxwhi
hforlargesystems,as
onsideredinthe
lassi- 0.01

al(cid:28)eldsmethod,dependsonlyontemperatureandspa
e
0.00
dimensionality. InFig.4we
ompare3Dquantumproba- 0 200 400 600 800 1000
bility distribution given by (5) with its
lassi
al
ounter-
part where the Kmax was
hosen a

ording to (10). We Nex
seethatnotonlythemaxima
oin
idebutalsotheentire
Nex
distributions are very similar. Therefore, by the appro- FIG.4: (Coloronline)Probabilitydistributionofhaving
(a)
priate
hoi
e of on~eωKpamraaxmeter only, the
ut-o(cid:27) energy b(bo)sons outside thegroundstateofa 3Dtrap and3Dbox
of
lassi
almodes , we
ana

uratelymimi
the potentials. Points are exa
t distribution while solid lines
true quantum statisti
s by the statisti
s of the
lassi
al representNits=op1t0im00al
lassi
al
ounterparts. Total numberof
(cid:28)eldssatisfyingtheequipartitionofenergy. Thisoptimal bosons is . Noti
e, for3Dboxpotentialprobability
D
distributions are wider as
omparedto 3D harmoni
trap.

ut-o(cid:27) s
ales nearly linearly with dimensionality , see
Fig.3.
The same analysis
an be applied for the uniform N
ex
system with peLriodi
boundary
onditions, i.e.
ubi
under the
ondition of l thermal bosoqns.pT(nheq p=rol)ba=-
ibasnoxdthoenfnileǫ=nkgt=h0,±~.21k,2±T/h22,em..s,.i.nwghleeI-rnpeartkthii
s=le
ea2nsπee(r,ngxyt,hnsepy,e
n
utztr)-u/omL(cid:27) ohbpnf(ilniqtqiys≥toaflth)e=a,−vwipni(tg∞nheqtxhl≥ape
(tln
l+yon1=db)i.tolisTo)onhnesonfinNthmeexoadtveheerramigsealpboopsuolnaNtsi,oins
pKamraamxe=te2rπ(isnmdaexte,rnmmianxe,dnmbayx)t/hLeamndaβxt~ihm2eKa
l2onwda/ivt2eimo-vne(
1t0o)r wiqthNwexeightPpN(l=N1ex|N)q= Zex.(NSeuxm,βm)a/ZtiNonoofvhearvainllg Neexx,
de(cid:28)nes its value. For large systems max very atoms withinq , givesthe formulaforthe averagepopu-
weakly depends on temperature similarlβy~a2sKfo2r th/e2mha=r- lation of the mode:
max
mπ(oζn(3i
/2p)o/t4e)n2tia≃l.1I.t3s4v,a0l.u6e8,s0a.t2u9rates at 1 N Nex
in 3D, 2D, 1D respe
- hnqi= e−βǫqlZex(Nex−l).
tively what was found numeri
ally. In Fig. 4 the quan- ZN NXex=0Xl=1 (13)
tumprobabilitydistributionandits
lassi
al
ounterpart
are shown for the optimal
ut-o(cid:27) given by Eq.(10). One
For the
lassi
al distribution, where populations
hange

an see that not only the maximum but also a shape of
in the
ontinuousway, it is
onvenientto use aprobabil-
both distributions are very
lose.
intHeraevsitngtom
aatl

hueladtethtehepraovbearabgileitypodpiustlaritbiountiohnnsKmitaxisi ooff liint)y/gZdaeetxn(lsNeiateysxt)ilnsbtoesaodn.sIinntmhiosd
eaqse,isthp˜le=preo−bβalbǫqilZiteyx(oNf ehxav−-
while the probability density of having ex-
tohbeta(cid:28)inneadlbmyo
doensoidfetrhineg
plarsosbi
aablilidtiisetsrpib(untqio≥n.l)Iotfa
amnobdee a
tly l bosons in mode q is p˜(nq = l)= −dp˜l/dl.qThen,
q l the(cid:28)nalexpressionfortheaveragepopulationof mode
being o

upied by at least bosons [13℄:
is:
∞ ∞
1
p(n ≥l)= ... ...e−βEδ , 1 N Nex
q Zex(Nex)nX1=0 nXq≥l Nex,Nex hnqicl = ZN Z0 dNexZ0 dle−βǫqlZex(Nex−l).
(12) (14)
4
0.6
HaL Eq.(14) is a
ontinuous version of Eq.(13) and based on
0.5
Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ theEuler-Ma
laurinsummationformulathhneleiad−inhgnteirm≃
q cl q
0.4 0o.f5itsdi(cid:27)eren
eisapproximatelyequalto K
> . Therefore,theaverageo

upationofthe maxmode
<nkmax 0.3 twuimthionnteh.eI
nlaFssigi
.5alt(cid:28)heeldasvmereatgheopdoipsulalartgieorntshhannKtmhaexqiuaannd-
0.2 hn i
è è è è è è è è è è è è è è è Kmax cl for 3D harmoni
trap (a) and 3D box pTo/teTn-
0.1 tial (b) areshown as fun
tions of the temperature c.
Summation of populations over all ex
ited modes must
0.0 N
ex
0.0 0.2 0.4 0.6 0.8 1.0 give a number of ex
ited atoms in both
ases that
T(cid:144)T is why in the
lassi
al
ase the populations have to be
c
1.0 larger. The values of the average o

upations are very
HbL
lose to expe
ted ones in the thermodynami
al limit.
0.8
ŸŸ Ÿ Ÿ Ÿ Ÿ In summary, we
onsidered a
lassi
al
ounterpart of
Ÿ
Ÿ the probability distribution of the ideal Bose gas within
> 0.6
max the
anoni
al ensemble. We found that there exists the
<nk 0.4èè è è è è è è otrpitbiumtaioln
uotf-ot(cid:27)hewh
io
nhdaelnloswatsetopompautl
ahtitohnebpyrobitasbi
lliatyssdi
ias-l
0.2 (cid:28)elds
ounterparts. We found the s
aling of the
ut-o(cid:27)
energies with temperature and dimensionality. This be-
0.0 ing a very simple model it nevertheless sheds some light
0.0 0.2 0.4 0.6 0.8 1.0
ontheoptimal
hoi
eofthe
ut-o(cid:27)forweaklyintera
ting
T(cid:144)Tc bosonsand moresigni(cid:28)
antly on the veryfoundations of
the method itself. We understood now mu
h better that
FIG. 5: (Color online) Average populations hnKmaxi of the (cid:28)nite numberofthe
lassi
al(cid:28)elds
anadequatelyrepre-
Kmax • sent the statisti
al properties of the full quantum many
(cid:28)nal mode (cid:4) Eq.(13) ( ), and for the
lassi
al distribu-
body system of bosons.
tion Eq.(14) ( ), for 3D harmoni
trap (a) and 3D box (b)
potentials. Linesrepresentexpe
tedresultsinthethermody- A
knowledgment The authors are grateful for use-
N =1000
nami
al limit. Numberof atoms . ful dis
ussion with M. Brew
zyk. The authors a
knowl-
edge support by the Polish Government resear
h funds
for 2006-2009.
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