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Equation of states for Bose-Einstein condensation
Masatsugu Sei Suzuki
Department of Physics, SUNY at Binghamton
(Date: November 06, 2018)
Here we discuss the equation of states for the Bose-Einstein condensation. We use the following
integrals.
],2
3[PolyLog)(
11
22/3
0
zz
ez
xdx
x
],2
5[PolyLog
2
3)(
2
3
11
2
0
2/5
2/3
zz
ez
xdx
x
],2
3[PolyLog
2)(
21
10
2/3
2/1
zz
ez
xdx
x
],2
3[PolyLog
4
3)(
4
3
11
0
2/3
2/3
zz
ez
xdx
x
with z as the fugacity, and
)()( 2/5 zTkgnP BQ , 3/2( , ) ( )QN T z gVn z .
34149.1)2
5()1(2/5 z , 61238.2)
2
3()1(2/3 z
5/2
3/2
( 1)0.513512447
( 1)
z
z
Definition:
],2
5[PolyLog)(2/5 zz , ],
2
3[PolyLog)(2/3 zz
((Series expansion))
.....5583322
)(5432
2/3 zzzz
zz around 0z
.....525323924
)(5432
2/5 zzzz
zz around 0z
((Derivative))
)(1
)( 2/12/3 zz
zdz
d , )(
1)( 2/32/5 z
zz
dz
d
((Note))
The condition for the occurrence of Bose-Einstein condensation is that
1/3(2.6138) 1.3775th av avd d
This form tells that the thermal de Broglie wavelength th must exceed 1.38 times the average
interatomic separation. In short, the thermal wave packets must overlap substantially. The Bose-
Einstein condensation occurs below
2/322
( )2.612
E
B
nT n
mk
ℏ.
Since
3/2( ) ( 1) 2.61238 ( )Q E Q En gn T z gn T
with g = 1.
(i) The average interatomic separation between atoms, d
3
1n
d or
1/3
1d
n
where n is the number density n:
(ii) The thermal de Broglie wave length th
3/2
3 2
1( )
2
BQ
th
mk Tn T
ℏ
(quantum concentration)
1/2
22th
Bmk T
ℏ (de Broglie wavelength)
1. Pressure and number for the Boson system
The pressure is given by
,T
G
VP
The grand potential G is
0
2
3
)(3
3
)(
)1ln(4)2(
]1ln[)2(
]1ln[
k
k
k
k
k
zedkkgV
Tk
edgV
Tk
eTk
B
B
BG
We use the following dispersion relation
22
2k
m
ℏ
k,
2/1
2
2
ℏ
mk ,
12
2
12/1
2
ℏ
mdk
So we have
0
2/3
22
0
2/3
23
)1ln(2
4
)1ln(2
2
4
)2(
zedmgV
Tk
zedmgV
Tk
PV
B
B
G
ℏ
ℏ
The integration by parts gives
)()(
11
2)(
3
2
11
)()2
(2
)2(
3
2
11
)(23
2
1123
2
2/5
0
2/3
0
2/32/3
22
2/3
0
2/32/5
32
2/3
0
2/3
32
2/3
zTkgVn
dx
ez
xTkgVn
dx
ez
xTk
TmkgV
dx
ez
xTk
gVm
d
ez
gVm
BQ
xBQ
xB
B
xB
G
ℏ
ℏ
ℏ
Then the pressure P is obtained as
)()( 2/5
,
zTkgnV
P BQ
T
G
This is valid in both the gas and the condensed phase, because particles with zero momentum do
not contribute to the pressure. Since z = 1 in the condensed phase, the pressure becomes
independent of number density.
5/2( ) ( 1)Q
B
Pgn T z
k T
The number is given by
)(
11
2
12
),(
2/3
0
0
)(32
2/3
,
zgVn
ez
dxxgVn
e
dgVm
zTN
Q
xQ
VT
G
ℏ
The number density is given by
)(),(
),( 2/3 zgnV
zTNzTn Q
where nQ is the quantum concentration,
2/3
2
2/3
2
2
2
h
TmkTmkn BB
Q
ℏ
We note that
( , 1) 2.61238 ( )Qn T z n T
The total particle number is a sum of 0)1,( nzTn , where 0n is the number density in the
ground state.
)1,(0 zTnnn
The Bose-Einstein condensation temperature (Einstein temperature) is defined as
( ( ), 1) 2.61238 ( ( ))E Q En n T n z n T T n
So ( )ET n depends on the number density as
2 2
2/3 2/32( ) ( )
2.61238E
B
nT n n
mk m
ℏ ℏ
t T TE
n t,z 1
n t,0 z 1n0
n
0.2 0.4 0.6 0.8 1.0 1.2 1.4
0.2
0.4
0.6
0.8
1.0
n
T E n
0.5 1.0 1.5 2.0 2.5 3.0
0.5
1.0
1.5
2.0
Fig. Plot of 3/2
2
(2.61238)( ) ( )
2
BE E
mkT n T n
ɶ
ℏ as a function of the number density n.
The number density 0n in the BE condensed phase is expressed by
0
3/2
( , ) ( , 1)
( , 1)[1 ]
( ( ), 1)
( )[1 ]
( ( ))
[1 ]( )
E
Q
Q E
E
n n T n n T z
n T zn
n T n z
n Tn
n T n
Tn
T n
In the vicinity of ( )ET n , 0 ( , )n n T can be approximated by
0
3 3( , ) [ ( )] [ ( ) ]
2 ( ) 2 ( )E E
E E
n nn n T T T n T n T
T n T n (critical behavior)
((Note)) Taylor expansion
0 ( , ( )) 0En n T T n , , 0
3( , ( ))
2 ( )E
E
d nn n T T n
dT T n
3. Calculation of the internal energy U
2
( )3
0
3/2 3/2
2 2 ( )
0
3/2 3/2
( )2 3
0
4(2 ) 1
2
4 1
12
U n
gVk dk
e
gV m d
e
gVm d
e
k
k k
k
k
ℏ
ℏ
or
5/2
3( ) ( )
2Q B
Ugn k T z
V
5/2
3/2
3/2
5/2
3/2
( )3
2 ( )
( 1)3
2 ( 1)
B
B
c
zk T
zUu
V z Tk T
z T
( )
( )
E
E
T T
T T
So U is related to the grand potential as
5/2
3 3( ) ( )
2 2G Q BU gVn k T z .
We get the relation
2
3PV U .
______________________________________________________________________________
4. Equation of state for BE condensation (I)
We consider the variation of the pressure P of the Bose gas with its volume V, keeping the
temperature T fixed. The pressure P is given by
)(),( 2/5
,
zTngkV
zTP QB
T
G
where 1g . and
)(),(
),( 2/3 zgnV
zTNzTn Q .
For 1z ( 0 ), the pressure P is given by
5( , 1) ( ) 1.34149 g
2B Q B QP T z gk Tn k T n
vanishes as 2/5T and is independent of density. This is because only the excited fraction
)1,( zTN has finite momentum and contributes to the pressure. Alternatively, Bose condensation
can be achieved at a fixed temperature by increasing density (reducing volume). The transition
occurs at a specific volume.
The transition temperature ( )ET n depends on the number density n as
3/2
21 1 2( ) 0.38279 0.38279
[ ( )] ( )Q E B E
v nn n T n mk T n
ℏ
or
2 2
2/3 2/3 2/32 2( ) [2.61238 ( )] (2.61238) [ ( )]E
B B
T n v n v nmk mk
ℏ ℏ
Fig. 3/2
2( ) ( )(2.61238)
2
BE E
mkT n T n
ɶ
ℏ vs 1/ .v n
(i) The normal phase ( 0z ) [ ( )]ET T n
The pressure of the gas changes as
5/2( , ) ( )B QP T z gk T n z .
(ii) The BE phase ( )1z [ ( )]ET T n
v 1 n
TE n
0.5 1.0 1.5 2.0 2.5 3.0
1.0
1.5
2.0
2.5
3.0
An appreciable number of particles falls down to the lowest level, this may be called
condensation in momentum space, or Bose condensation.
5/2
5 ( ) ( )
( , , 1) 2 [ ]5( ( ), , 1) ( )
( ) [ ( )] ( )2
B Q
E EB E Q E
gk T n TP T n z T
P T n n z T ngk T n n T n
)( 2/5T
which is independent of the volume 1/v n of the system. Since
2
2/32 ( )( ) [ ]
0.38279E
B
v nT n
mk
ℏ
Using this relation, we have
5/2
3/2 5/2
2
( , ( ), 1) ( ) [ ( ) ] ( 1)
1.34149 ( ) [ ( )] 2
E B E Q E
BB E
P n T n z gk T n n T n z
mkgk T n
ℏ
or
5/2
3/2 5/2
2
( , ( ), 1) ( ) [ ( ) ] ( 1)
1.34149 ( ) [ ( )] 2
E B E Q E
BB E
P n T n z gk T n n T n z
mkgk T n
ℏ
or
1 5/2
2( , ( ), 1) 0.270721 ( ) ( )
2
BE B
mkP n T n z gk v n
ℏ
Fig. Isotherms of the ideal Bose gas. The BE condensation shows up as a first-order phase
transition. T1>T2. For cvv , Pv const.
For ( )v v n , the pressure–volume isotherm is flat, since 0
v
P. The flat portion of isotherms
is reminiscent of coexisting liquid and gas phases. We can similarly regard Bose condensation as
the coexistence of a “normal gas” of specific volume cv , and a “liquid” of volume 0. The vanishing
of the “liquid” volume is an unrealistic feature due to the absence of any interaction potential
between the particles.
5. Equation of state for BE condensation (II)
For ( ) 1t n
5/2( , , ) ( ) ( )
B QP n T z gk T n T z
with
3/2 3/2
3/2
( )( ) 1
( 1)
zt n
z
where
vc2 vc1
T2
T1
v
P
P2
P1
Pv5 3
const
Pv const
Pv const
( )( )E
Tt n
T n
The pressure can be rewritten as
5/2
3/2
( )( , , )
( )B
zP P n T z nk T
z
or
5/2
3/2
( )( )
( ) ( )B E
zPVt n
Nk T n z
where N
nV
. We note that
5/2
3/2
( )2
3 ( )B
zPV U Nk T
z
5/2 5/2
3/2
5/2
2
( ) 3 ( )
( 1)( )
( 1)
0.513512 ( )
B E B E
PV U
Nk T n Nk T n
zt n
z
t n
for ( ) 1t n
As shown the figure below, the actual / [ ( )] 2 / [3 ( )]B E B EPV Nk T n U Nk T n vs ( )t n curve follows
the red line from ( ) 0t n up to ( ) 1t n and thereafter departs, tending asymptotically to the
classical limit.
Fig. Plot of / [ ( )] 2 / [3 ( )]B E B EPV Nk T n U Nk T n as a function of ( ) / ( )Et n T T n .
(i) Boson (red: boson line): 5/2/ [ ( )] 0.513512 [ ( )]B EPV Nk T n t n both for ( ) 1t n and
( ) 1t n .
(ii) Extension of the boson line valid for ( ) 1t n to that for 1t (purple line).
(iii) Classic (blue line): / [ ( )] ( )B EPV Nk T n t n .
Fig. P-T diagram of the ideal Bose gas. Note that the space above the transition curve does not
correspond to anything. The condensed phase lies on the transition line itself.
0.5 1.0 1.5 2.0 2.5 3.0t
0.5
1.0
1.5
2.0
2.5
PV NkBTE n
Fig. P-T curve as the number density n is changed as parameter. (M. Kardar, Statistical Physics
of Particles, Cambridge, 2008).
_______________________________________________________________________
APPENDIX-I Gamma and zeta functions
)1()1(
)1(
)1()1(
11
1
)1(
0
)1(
0 0
)1(
00
yy
ky
ky
dxex
e
dxex
e
dxx
k
y
k
y
k
xky
x
xy
x
y
with
7833.1
34149.14
3
)2
5()
2
5(
10
2/3
xe
dxx
31516.2
61238.22
)2
3()
2
3(
10
2/1
xe
dxx
((Note))
0
2/5
2/3
)(2
3
11
2z
ez
xdx
x
)(
11
22/3
0
z
ez
xdx
x
34149.1)2
5()1(2/5 z , 61238.2)
2
3()1(2/3 z
(i) Zeta function
)(x is the zeta function defined by
1
)(k
xkx
with
61238.2)2
3( , 34149.1)
2
5( , 12673.1)
2
7(
(ii) Gamma function
)(x is the gamma function
)()1( xxx , )()1( xxx
)2
1( ,
2
1)
2
3( ,
4
3)
2
3(
2
3)
2
5(
8
15)
2
5(
2
5)
2
7(
APPENDIX II PolyLog (mathematica)
],2
3[PolyLog)(
11
22/3
0
zz
ez
xdx
x
],2
5[PolyLog
2
3)(
2
3
11
2
0
2/5
2/3
zz
ez
xdx
x
],2
3[PolyLog)(2/3 zz
],2
3[PolyLog)(2/3 zz
Clear "Global` " ; f1x
Exp x
z1; f2
x3 2
Exp x
z1;
g1 Integrate f1, x, 0,
1
2PolyLog
3
2, z
g2 Integrate f2, x, 0,
3
4PolyLog
5
2, z
Series PolyLog3
2, z , z, 0, 5
zz2
2 2
z3
3 3
z4
8
z5
5 5O z
6
Series PolyLog5
2, z , z, 0, 5
zz2
4 2
z3
9 3
z4
32
z5
25 5O z
6
______________________________________________________________________
APPENDIX
I taught Phys.411 (511) in 2016 by using the textbook of K. Huang; Introduction to
Statistical Mechanics. I found the following nice figures.
Fig. Isotherms of the Bose gas at two temperatures (M. Kardar, Statistical Physics of Particles,
Cambridge, 2007).
D PolyLog3
2, z , z
PolyLog1
2, z
z
D PolyLog5
2, z , z
PolyLog3
2, z
z
Fig. Phase diagram of Bose-Einstein condensation in the density temperature plane.; 3/2n T
(K. Huang, Introduction to Statistical Physics).
Fig. Qualitative isotherms of the ideal Bose gas. The Bose-Einstein condensation shows up as
a first-order phase transition. (K. Huang, Introduction to Statistical Physics).
Fig. Isotherms of the ideal Bose gas
REFERENCES
M.Kardar, Statistical Physics of Particles (Cambridge, 2007).
K. Huang, Statistical Mechanics, second edition (John Wiley & Sons, 1987).
K. Huang, Introduction to Statistical Mechanics, second edition (Taylor and Francis, 2010).
R.K. Pathria and P.D. Beale, Statistical Mechanics, 3rd edition (Elsevier, 2011).
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