phy 770 -- statistical mechanics 12:00 * -1:45 p m tr olin 107

PHY 770 Spring 2014 -- Lecture 14 13/18/2014

PHY 770 -- Statistical Mechanics12:00* -1:45 PM TR Olin 107

Instructor: Natalie Holzwarth (Olin 300)Course Webpage: http://www.wfu.edu/~natalie/s14phy770

Lecture 14

Chap. 6 – Grand canonical ensemble Fermi-Dirac distribution function Bose-Einstein distribution

*Partial make-up lecture -- early start time

http://www.wfu.edu/~natalie/s14phy770


Reminder:

Please think about the subject of your computational project – due next week.

Suggestions available upon request.


Examples of grand canonical ensembles – ideal (non-interacting) quantum particles in a cube of length L with periodic boundary conditions -- Fermi-Dirac case

In the absence of a magnetic field, the particle spin does not effect the energy spectrum, and only effects the enumeration of possible states spin (g)

1ˆ ˆ ( )( )

0

( )

, , Tr

1

i i

i i

i

i

g

nH N

g

F

n

D T V e e

e

Z

p p

p

p

p

p

( )

( )

, , ln 1

= ln 1

i

i

i

i

g

BFD

B

T V k T e

k Tg e

p

p

p

p


Fermi-Dirac case -- continued

( )

( ),

, , ln 1

Self-consistent determination of :

1

i

i

ii

FD

FD

B

T V

T V k Tg e

gNe

p

p

p

p

33 2

30

Recall:

is isotropic in4 since 2 2

L Vd p p dp

pp

p


2

( ),

23 /2

0

22

3/23

1

Let : 4 2

Let ( )2

2 2

iiT V

p m

T

FD

TB

gNe

Vg zz e N p dpe z

p Vgx N f zm mk T

pp

2

2

125/2 5/20

0

12 5/23/2 3/20

0

4Here: ( )

( )4 ( )

ln 1 1

1

x

x

zdx x ze

dz zdx x zdze

f z

f zf zz



22

30

3/22 2

Low temperature behavior: for 0 z for 0

4 2

6( 0) 2

m

F

T

VgN p dp

NT

m g V



2

2

2

2

2

2

Keeping more terms in low temperature expansion:

( )

5

1 ...12

3 1 ..ˆ .5 12

2

BF

F

BF

F

BV

F

k

kU N

N

T

TH

kT

T

C



( )

Behavior of occupancy parameter:

1g gzn

e ze

p pp



Examples of grand canonical ensembles – ideal (non-interacting) quantum particles in a cube of length L with periodic boundary conditions -- Bose-Einstein case (assumed to have spin 0)

ˆ ˆ ( )( )

0

( )

, , Tr

1 1

i i

i i

ii

nH N

n

BE T V e eZ

e

p p

p

p

p

p

( )

( )

1, , ln 1

= ln 1

ii

i

i

BE B

B

T V k Te

k T e

p

p

p

p


Bose-Einstein case

( )

0

( )

, ,

1 = 1

i i

i i

ii

n

n

BE T V e

e

Z

p p

p

p

p

p

( ), , = ln 1 i

i

E BB T V k T e p

p

Fermi-Dirac case

1( )

0

( )

, ,

= 1

i i

i i

i

i

g

n

n

g

FD T V e

e

Z

p p

p

p

p

p

( ), , = ln 1

i

i

D BF T V k Tg e p

p

2

5/23

25/2

0

2

, , = ln 1

4where ln(1 )

2

BB

T

T

B

x

B

Ek TVT V k T z g z

g z x dx ze

mk T

2

5/23

25/2

0

2

, , =

4where ln(1 )

2

B

T

x

FD

TB

gk TVT V f z

f z x dx ze

mk T


Bose-Einstein case Fermi-Dirac case22

TBmk T

2

2

25/2

0

23/2

0

4 ln(1 )

4

x

x

g z x dx ze

zg z x dxe z

2

2

25/2

0

23/2

0

4 ln(1 )

4

x

x

f z x dx ze

zf z x dxe z

5/23

3/23

FD

T

T

gkTP f zV

N gkT f zV

5/23

3/23

ln 1

11

BE

T

T

kT kTP z g zV V

N z kT g zV V z


Case of Bose particles Non-interacting spin 0 particles of mass m at low T moving in 3-dimensions in large box of volume V=L3: Assume that each state ek is singly occupied.

2/3

22

33

22

k

2

2222

,,

24

)(

)( 2

2 ,limit In the

3,2,1,, 8

11

mVg

gdkdLmkL

nnnmL

nnnhe

nN

B

Bk

zyxzyx

nnn

kkk

zyx

k


Case of Bose particles at low T

11 )(

11

00

00

e

gdnN

ennN

B

kkk k

NkT

en

nNT

case, In this

small assuming 111

11

1

such that exists solutions consistent a lowfor that Note

0

0


Critical temperature for Bose condensation:

11 )(

00

e

gdnN B

condensate “normal” state

0

2 2

1If ( ) , there is no "condensate"1

The temperature at which the above equality is satisfied iscalled the Einstein condensation temperature . Approximate value of :

24

B

E

E

N d ge

TT

V mN

3/23/2

2 20 0

3/2 2/3 2

2 2

21 11 4 1

2 / 22.612 4 2 2.612

E

Ex

EE

mkTVd dx xe e

mkTV N VN kTm


Case of Bose particles at low T

11 )(

11

00

00

e

gdnN

ennN

B

kkk k

NkT

en

nNT

case, In this

small assuming 111

11

1

such that exists solutions consistent a lowfor that Note

0

0


Critical temperature for Bose condensation:

11 )(

00

e

gdnN B

condensate “normal” state

mVNkTmkTVN

exdxmkTV

edmVN

T

egdN

EE

xE

E

B

E

23/22/3

22

0

2/3

220

2/3

22

0

2612.2/

2612.22

4

112

4112

4

: valueeApproximat . tureon temperacondensatiEinstein thecalled

is satisfied isequality above heat which t re temperatuThe

"condensate" no is there,1

1 )( If


Summary:

2

2

1/22

23

/

23

/

2Define 1

The Landau potential for the Bose system can be written:

4( , , ) ln(1 ) ln 1

4 1

T

T

T

xBE

T L

Bx

T L

z emkT

kT VT V kT z dx x zeλπ

z V zN dx xz λπ e z

0n


Some convenient integrals

12/3

02/5

22/3

12/5

0

22/5

)( 4)(

1ln 4)(

2

2

n

n

x

n

nx

nzzg

dzdz

zezxdxzg

nzzexdxzg

z

g3/2(z)

g5/2(z)


612.2)( that note )(

4)( 4

2 1 where 1

2/3lim1z2/330

/

0

22/33

/

230

2/12

0

22

zgzgλVnN

zezxdx

πzg

λV

zezxdx

λV

πnN

mkTez

zzn

T

L

xTL

xT

T

T

T

g3/2(z)

z

0n


2/3

3

30

2/330

2/330

32/33

2/330

11 For

)1( and 1 ,For

1for solution a has )( and ,For

.6122 )1(

emperatureEinstein t Define

)(

:for Equation

ET

TE

TE

TE

TT

T

TT

λλ

Nn

TT

gλVnNzTT

zzgλVNNnTT

λVg

λVN

zgλVnN

z

E

EE


Nn0

T/TE

3/230

31 1ET

T E

λn TN λ T


http://www.colorado.edu/physics/2000/bec/three_peaks.html

87Rb atoms (~2000 atoms in condensate)

http://www.colorado.edu/physics/2000/bec/three_peaks.html


Other systems with Bose statistics Thermal distribution of photons -- blackbody radiation:

In this case, the number of particles (photons) is not conserved so that =0.

3

45

3

3

3

332

2332

33

158

18

1

:energy radiated ofon Distributi

)(2

11

hckTVE

ed

chV

ed

cVnE

dc

VckkddL

hcke

n

hkk

k

k

k

k k


Blackbody radiation distribution:

T1

T2>T1

T3>T2


Other systems with Bose statistics Thermal distribution of vibrations -- phonons:

In this case, the number of particles (phonons) is not conserved so that =0.

22

13

21

113

particles. allfor directions 3in vibrates frequency lfundamenta thesolid,Einstein For

1

1

ee

kTNk

TE

C

eNE

N

en

k

k k


Other systems with Bose statistics -- continued Thermal distribution of vibrations -- phonons:

TT

xD

k

k

DD

k

edxx

TTNkT

ed

cVE

ckc

en

/

0

33

0

3

32 19

123

).directions 3in same thebe tohere (assumed sound of speed thedenotes where,frequency lfundamenta thesolid, DebyeFor

1

1


Effects of interactions between particles – classical case

1

1 ,

ˆ3

1 2ˆ

ˆˆ ˆ ˆˆ2

1 1Canonical partition function : ( , )

( , ) ...

Tr!

where .N

N NNNi

iji i j

HN NN

VN

N

T

N

T V

T V d

H V T V

d d e

m

Z T e QN

Q

r r

p

r

r

31 1

Grand canonical partition func

( , )

tion : 1 1

!N

N NNN

N

TN

Z T Z T T Ve Q eN


Effects of interactions between particles – classical case -- continued

1

,

1 2

1 2

1

1

,

ˆ

ˆ

2 21

( , ) ....

= ....

....

ˆ ˆ

, ...

N N

iji j

N

N

N

N

N

N N

iji j

VN

V

N N

V

T V d d d e

d d d e

d d d

V

Q

W

r

r r r

r r r

r r r r

r

r r

( 1)

21,

...

ex

, ( 1)

p( ˆ ) 1

N N

N iN

ij

ji j

ij

W f

f V

r r r

r



0

12 6

ijij ij

Vr r

V

exp( )ˆ 1i ij jf V r



1 2 10

23

1 1( , ) = ... , ....!

NN NN

NN

T

Z T V e d d d WN

r r r r r r

( 1)

21,

...

ex

, ( 1)

p( ˆ ) 1

N N

N iN

ij

ji j

ij

W f

f V

r r r

r

2

2

1 2 1 12

3 1 233 12 13

1 =1 Note that:

,

, ,

= 1 1 1

W W

W

f

f f f

r r

r r r



1 2 130

1 2 1

2

230

1 1( , ) = ....!

In terms of cumulant expansion:

1 1( , ) =exp

, ...

, ......!

.

NN NN

N T

T

NZ T V e d d d WN

Z T V e d d d U

r r r r r r

r r r r r r

1 1 1 1

2 1 2 1 1 12 2 21

Typical cluster functions:

, ,

U W

U W W W

r r

r r r r r r

phy 770 -- statistical mechanics 12:00 * -1:45 p m tr olin 107

Documents

partial makeup lecture

bose condensation

fermidirac casein

particles of mass

number of particles

number of particles

condensatenormal state

particle spin