Note on BEC in Nonextensive Statistial Mehanis
Kwok Sau Fa 1
and E.K. Lenzi 2
1
Departamentode Fsia, UniversidadeEstadualde Maringa,
Av. Colombo5790,87020-900Maringa-PR,Brazil
2
CentroBrasileirodePesquisas Fsias,R.XavierSigaud150,
22290-180RiodeJaneiro,Brazil
Reeivedon17Otober,2000. Revisedversionreeivedon8January,2001
ThegeneralizedBose-Einsteindistribution,withinthe dilutegasassumption, inthenonextensive
Tsallisstatistisisworkedwithoutapproximationfor theBose-Einsteinondensation(BEC).The
resultsobtained areomparedwiththereentresults presentedinInt. J.Mod. Phys. B14,405
(2000)byL.Salasnih. Furthermore,inorderto promoteaompleteanalysis fortheBEC inthe
nonextensivesenariowealsondexatexpressionwithinthenormalizedonstraintsinaharmoni
trap.
I Introdution
Inthislast deade,wehavewitnessed agrowing
inter-est intheTsallisstatistis[1,2℄. Thestartingpointof
Tsallis statistisis basedon the nonextensiveentropy
S
q =k(1
P
W
i=1 p
q
i
)=(q 1),wherekisapositive
on-stantandq2R .Inthelimitq!1theusual
Boltzman-Gibbsentropyisreovered. Ithasbeenappliedinmany
situationssuhas,Levy-typeanomaloussuperdiusion
[3℄,Eulerturbulene[4℄,self-gravitatingandorrelated
systems [5℄, anomalous relaxation through
eletron-phonon interation [6℄, ferrouid-likesystems [7℄, and
among others[8℄. Inpartiular,theBose-Einsteinand
Fermi-Dira distributions have been extensively
ana-lyzedintheTsallisframework[6,9-16℄.Further,wean
mentionthespeiheatobtainedfor 4
He[17℄whihis
in agreementwiththeexperimentalresultsreportedin
[18℄. Forthemostoftheseasestheanalysesinvolving
thesegeneralizedBose-EisnteinandFermi-Dira
distri-bution have been restrited for free systems, i.e., the
interation between the partiles is absent. However,
important questions about the omplete solution for
the Bose-Einstein distributionandits impliations for
thethermodynamis ofthesephysial systemsarestill
presentin this formalism. In this ontext, it may not
be outof plae to mention herethe generalized
Bose-Einstein distributionin thedilutegasassumption[13℄,
givenby,
hn()i
q =
1
1=(q 1)
; (1)
anditsexpansionatrstorderin (q 1)
hn()i
q =
1
e
( )
1 +
1
2 (q 1)
2
( )
2
e
( )
e
( )
1
2 :
(2)
Theequation (2)hasbeenapplied widely in the
liter-ature[6, 12,13,14,15℄(see[8℄andreferenestherein)
forobtainingthermodynamialquantitiesthat emerge
from it. In this way, a omplete solution
involv-ing Eq.(1) an be useful for quantitative and
qual-itative analysis of systems in the nonextensive
on-text. Furthermore, the development based on Eq.(1)
an also be useful in the desription of the systems
with(multi)fratalstruture(similar situationin
met-als[19℄has been analyzedin [16℄), systemswith
long-timememories,andamong others. Anotherimportant
pointisabouttheonstraintsandtherelationbetween
theLagrangeparameter andtemperatureinthe
for-mulationsofTsallisstatistis[2℄.
Thus,wepropose,inthisworktondtheomplete
solutionforthe BECtransition temperature
onsider-ingD dimension and q >0, within thedilute gas
as-sumption. Then,weompareit withtheapproximate
resultspresentedintheliterature,andinpartiularthe
Salasnih'swork[12℄,byvaryingthevaluesofqandD,
and as a onsequeneto hek the validity of the
ap-proximation(2). Inaddition,weshowthattheresults
obtainedbyusingEq.(2)arenotaurateformost
situ-ationswhenomparedwithourones. Also,topromote
afuture disussioninvolvingtheonstraints,Lagrange
ob-tain the Bose-Einstein distribution by taking into
a-ountthenormalizedonstraints[2℄.
Inordertofailitateouranalysis,letusreallsome
results evaluated in [12℄ with Eq.(2). The quantities
evaluated in [12℄ are: the BEC transition
tempera-ture, theondensedfrationand theenergyper
parti-letothehomogeneousgas,thegasinaharmonitrap
andtherelativistihomogeneousgasin D-dimensional
spae. Fornon-interatingbosons,thetotalnumberof
partileisevaluatedbythefollowingexpression
N= Z
1
0
d()hn()i
q
; (3)
where () is the density of states. In a D-dimensional box of volume V, the densities of states for
homoge-neousgas and gas in harmoni trapare given respetivelyby () =f(V=)[m=(2~ 2
)℄ D=2
g= (D=2) and () =
f[=(~!)℄ D
g=( (D)),where ! isthegeometriaverageofthetrapfrequeniesand (x)isthegammafuntion.
Toalulate theBEC transition temperature T
q
, with = 0,wesubstitute Eq.(2) and () into (3), and we
obtain
kT
q =
2~ 2
m
(N=V) 2=D
=[(D=2)℄ 2=D
h
1+ q 1
2
(D=2+2)(D=2+1)
(D=2)(D=2) i
2=D
; (4)
forthehomogeneousgas,and
kT
q =
~!N 1=D
[(D)℄ 1=D
h
1+ q 1
2
(D+2)(D+1)
(D)(D) i
1=D
; (5)
foragasinaharmonitrap. (x)istheRiemann-funtion. Theenergyisalulatedbythefollowingexpression
E= Z
1
0
d()hn()i
q
: (6)
Theresultsare
E
kT
q =
VD
2
mkT
q
2~ 2
D=2
(D=2+1)
1+ q 1
2
(D=2+3)(D=2+2)
(D=2+1)(D=2+1)
; (7)
forthehomogeneousgas,and
E
kT
q =
kT
q
~!
D
D(D+1)
1+ q 1
2
(D+3)(D+2)
(D+1)(D+1)
; (8)
foragasin aharmonitrap.
Fortherelativistigas,thetotalnumberofpartilesisnotonservedduetotheprodutionofantipartiles,but
thedierenebetweenthenumberNofpartilesandthenumberN ofantipartilesisonserved,i.e.,Q=N N =
R
1
0
d()[hn()i
q
hn()i
q
℄ ; wherehn()i
q
isobtainedfromhn()i
q
byreplaing! . Thedensityofstates
for the relativisti gas is given by () = (V2 D=2
)=(2~ 2
) D
(D=2)) 2
m 2
4
(D 2)=2
. For ultrarelativisti
regionkT >>m 2
,theritialtemperatureatwhih BECours orrespondstojj=m 2
. ExpandingQat rst
orderin yields
kT
q =
(2~) D
(D=2)jQj
4V D=2
(D)(D 1)m
2
1=(D 1)
1+ q 1
2
(D 1)D(D)
(D 1)
1=(D 1)
: (9)
TheEqs.(4),(5),(6),(7)and(9)aretheresultsobtainedin[12℄,byusingEq.(2). Similarresultsforfreepartiles
were also obtainedin [15℄. Now, wealulate again thequantities abovebyusing thedistribution (1), insteadof
(2). Todoso,wenotethat thedistribution(1)anbewrittenasasum,i.e.,
hn()i
q =
1
[1+(q 1)( )℄ 1
q 1
1
= 1
X
[1+(q 1)( )℄
n=(q 1)
Substituting () and Eq.(10) into (3) and taking into aount the ut-o [2℄ for q < 1, we obtain the ritial
temperatureas
mkT
q
2~ 2
= 8
>
>
>
<
>
>
>
:
V
N P
1
n=1 (1 q)
D =2
( n
1 q +1)
( n
1 q +1+
D
2 )
2
D
; 0<q<1
V
N P
1
n=1 (q 1)
D =2
( n
q 1 D
2 )
( n
q 1 )
2
D
; q>1
(11)
forthehomogeneousgas,and
kT
q
=(~!)N 1=D
"
1
X
n=1 D
Y
l=1 1
n+(1 q)l #
1=D
q>0; (12)
for agasin aharmonitrap. Note that Eq. (12)is validfor(q 1)<1=D. Inaddition, theenergyisalsoeasily
obtainedfromEq.(6)
E=V
kT
q =
8
>
<
>
: P
1
n=1
((1 q)mk T
q )
D =2+1
D
( n
1 q +1)
2mk T
q (2~
2
) D =2
( n
1 q +2+
D
2 )
; 0<q<1
P
1
n=1
((q 1)mk Tq) D =2+1
D
( n
q 1 D
2 1)
2mk Tq(2~ 2
) D =2
( n
q 1 )
; q>1
(13)
forthehomogeneousgas,and
E
kT
q =D
kT
q
~!
D 1
X
n=1 D+1
Y
l=1 1
n+(1 q)l
q>0; (14)
d
foragasin aharmonitrap. TheEq. (14)isvalid for
(q 1) < 1=(1+D). We note that our results dier
from those obtainedby using thedistribution(2). To
see that losely, we quote T
q
with dierent values of
D, forq =1:1 (see Table1). Wealso plotthe ritial
temperatureinfuntionofq. Forhomogeneousgasthe
onvergeneoftheseriesofEq. (11)isslowforD=3.
Foronveniene,weplotT
q
infuntionofq,forD=4
(Fig. 1). For agas in aharmonitrap, we plotT
q in
funtionofq,forD=3(Fig. 2). Asweanseethe
val-uesquoted in Table1,andthetwogures(Figs.1and
2)showlearlythattheritialtemperaturesobtained
byusing(2)arenotauratewhenomparedwithour
results. Evenfor agas in aharmonitrap withsmall
deviation in the nonextensive parameter q = 1:1 and
D = 3, the divergene between our result and
Salas-nih's result [12℄ is remarkable. This is not surprise
beause the expansion made in (2), in the parameter
(q 1); hasinludedthe fator. Thislast fator is
notneessarilysmallso theexpansionofthe
exponen-tialfuntionat rstorder of(q 1) annotbegoodin
general,espeially whenj1 qj andD are inreased.
WeshouldalsostressthatthehangesintheBEC
tran-sitiontemperature(forq>1),fromtheextensiveBose
statistis,aregreaterthanthoseobtainedbyusingthe
distribution (2). For therelativisti gas, in the
ultra-relativistiregionkT m 2
, theritial temperature
byexpandingQatrstorder in isgivenby
kT
q = ~!N
1=D
D=3 D=4 D=5 D=6 D=7 D=8 D=9
Salasnih 0.8144 0.8288 0.8284 0.8266 0.8258 0.8261 0.8273
ourresults 0.7635 0.7342 0.6837 0.6262 0.5639 0.4954 0.4147
Table1. Resultsforritialtemperatureofagasin aharmonitrapsystemwithq=1:1. Theseondrowshows
1
kT
q =
(
4V D=2
(D)m 2
jQj(2~) D
(D=2) 1
X
n=1 D 1
Y
l=1 1
n+(1 q)l )
1
D 1
q>0: (15)
Eq.(15)alsodiersfromthatobtainedbyusingthedistribution(2). Forexample: forq=1:1andD=4weobtain
[(4V 2
m 2
)=(j Qj(h) 4
)℄ 1=3
kT
q
=0:420181from Eq. (15)and [(4V 2
m 2
)=(j Qj ( h) 4
)℄ 1=3
kT
q
=0:448173from
Eq. (9).
Forompleteness,weshowbelowq-meanvalueof partilenumberand internal energyin thenormalized
on-straints[2℄ bytakinginto aount=0
h ^
Ni
q
h ^
Ni
1 =
1+(1 q) ~
U
q
(1 q) ~
= (1)
!
D 8
>
>
<
>
>
:
D; 1
1 q +D;h
^
Ni
1
1+(1 q) ~
Uq
(1 q) ~
= (1)
D
(D+1)
(D)
D; 1
1 q ;h
^
Ni
1
1+(1 q) ~
Uq
(1 q) ~
= (1)
D
(D+1)
(D)
9
>
>
=
>
>
;
; (16)
U
q
(1)
Dh ^
Ni
1 =
1+(1 q) ~
U
q
(1 q) ~
= (1)
!
D+1
(D+1)
(D)
D; 2 q
1 q +D;h
^
Ni
1
1+(1 q) ~
Uq
(1 q) ~
= (1)
D
(D+1)
(D)
D; 1
1 q ;h
^
Ni
1
1+(1 q) ~
Uq
(1 q) ~
= (1)
D
(D+1)
(D)
; (17)
d
where(;;z)= P
1
k =0 z
k
=(k! (k+))with;>
0 ~
==Tr q
,h ^
Ni
1 /(1=
(1)
)
D
and (1)
=1=T
1 where
T
1
is theritialtemperature. Moredetails aboutthe
alulationabove anbe found in [20℄, where the free
partileasehasbeenanalyzed.
Figure.1Weshowtheritialtemperatureversusthe
nonex-tensive parameter q for the homogeneous gas, onsidering
D=4. Itshouldbenotedthatwealsoextendthe
approx-Figure. 2Plotsofritialtemperatureversusthe
nonexten-sive parameter q for a gas ina harmoni trap. Here, the
threedimension(D=3)ase isonsidered.
Insummary, we haveobtained the exatstandard
BEC formula by using the generalized Bose-Einstein
distribution,within thedilutegasassumption,in
Tsal-lis statistis. The omparisonbetweenourresultsand
those obtained in [12℄ shows that the approximation
ondensation[12,14,15℄andamongothers,inthelight
ofourresults,shouldbereanalysed. Theroleofthe
in-terations in theBEC [21,22℄ in Tsallisstatistiswill
be addressed in an other opportunity. Furthermore,
wehavealsofoundtheexpressionsfortheBECwithin
thenormalizedonstraints,in orderto promotea
pos-sible analysis involvingLagrange parameter,
tempera-ture, onstraints and experimental data. Finally, we
believethat this work an beusefulin theanalysesof
future appliations whih involveTsallisstatistisand
BEC,within thedilute gasassumption.
EKLthanksCNPqand PRONEX(Brazilian
agen-ies) fornanialsupport.
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