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).