PHY 341/641 Thermodynamics and Statistical Physics Lecture 29users.wfu.edu/natalie/s12phy341/lecturenote/Lecture29.pdf · 2012-04-09 · 4/8/2012 PHY 341/641 Spring 2012 -- Lecture

4/8/2012 PHY 341/641 Spring 2012 -- Lecture 29 1

PHY 341/641 Thermodynamics and Statistical Physics

Lecture 29

Review (Chapters 5-7 in STP)

• Magnetic systems; Ising model • Fermi statistics • Bose statistics • Phase transformations • Chemical equilibria


Second exam: April 9-13 -- student presentations 4/30, 5/2 (need to pick topics)


TJgBsBs

B

iii

/1028.9

energyalignment spin ,1 where:sMicrostate

2421 −×−=≡

≡±=−=

µµ

µµε

( )NN

s

Bs

s

sB

sssN

Ze

eZN

N

ii

11

1111

1

1

1

321

=

=

∑=

∑

∑∑∑∑

±=

±=

±=±=±=

=

βµ

βµ

Review of statistical mechanics of spin ½ systems -- Chapter 5 in STP Fist consider system with independent particles in a magnetic field:


Calculation of Z1

)cosh(21

11

1 BeeeZ BB

s

Bs βµβµβµβµ =+== −

±=∑

Thermodynamic functions:

( ) ( )

( )

( ) ( )BBkNTE

C

BBNZNE

BNkTZNkTZkTF

B

N

βµβµ

βµµβ

βµ

22

1

11

sech

tanhln)cosh(2lnlnln

=

∂

∂=

−=∂

∂−=

−=−=−=


Magnetic field dependence of Z: ( )

∑=

=

=

N

ii

NN

sM

BBNTZ

1

:ionMagnetizat)cosh(2),,(

µ

βµ

BZ

e

ess

s

Bss

Bs

i ∂∂

==∑∑

±=

±= 1

1

11 ln1

1

1

1

1

βµβµ

βµ

( )

( )BNBF

BM

BFBNM

TT

βµβµχ

βµµ

222

2

sech

tanh

=

∂∂

−=

∂∂

≡

∂∂

−==⇒


Magnetization and susceptibility of independent spin ½ particles

βµB

M/M0

χ/χ0


Independent particle system

∑∑==

−≡−=N

ii

N

iis sHBsE

11

:sMicrostate

µ

Interacting particle system – Ising model

( )( )∑

∑∑

++

=

++−=

−−=

iiiiis

N

ii

N

nnjijis

ssHsJsE

sHssJE

121

1

1)(,

:dimension oneFor

:sMicrostate

Effects of interactions between particles:


Partition function for 1-dimensional Ising system of N spins with periodic boundary conditions (sN+1=s1)

( )

( ) ( )

( ) ( ) T≡

≡

−−−

−=

≡

++=

−−

−+

+−

= =++

∑

∑ ∑ ∑

HJJ

JHJ

NNNNssss

s

N

i

N

iiiiiN

eeee

ffff

ssf

ssfssfssfssf

ssHssJZ

N

βββ

βββ

ββ

)1,1()1,1()1,1()1,1(

)',(

:where

),(),(),(),( 2

exp

11322,,

1

1 111

321


( ) ( )

( ) ( )

≡

=

=

−−

−+

+−

+∑

∑

HJJ

JHJ

ssssssss

ssssss

NNNNssss

N

eeee

TTTTT

ssfssfssfssfZ

NN

N

N

βββ

βββ

T

:where

),(),(),(),(

154

321

433221

321

,,

11322,,

1

1-dimensional Ising system of N spins with periodic boundary conditions (sN+1=s1) (continued)

( )NNZ TTr=



)(Tr)(Tr)(Tr 3. 2.

.

000

0000

type theof

ation transformaby eddiagonaliz becan matrix symmetricAny 1.:algebralinear from tricksSome

1

111

2

1

1

ΛΛΛTUTTTUTTTTTUTUTUUTUUTTTT

ΛTUU

T

==

=

≡=

−

−−−

−

nλ

λλ

Nn

NNNN λλλλ 321)(Tr ++=⇒ T



( ) ( )

( ) ( )

( ) ( )[ ] ( ) ( )[ ]

( ) NNNN

JJ

JJ

HJJ

JHJ

ZeHHe

eHHe

eeee

21

2/1422

2/1421

2

1

Tr

sinhcosh

sinhcosh

00

:case In this

λλ

ββλ

ββλ

λλ

ββ

ββ

βββ

βββ

+==

+−=

++=

=

≡

−

−

−−

−+

T

Λ

T



( )

( ) ( )[ ][ ]( )

( )[ ] 2/142

2/142

1

1

21

1

2121

sinhsinh),,(

sinhcoshln

ln

1lnlnln),,(

1Tr

J

J

N

N

NNNNN

N

eHHN

HFHJTM

eHHkTNJ

NkT

kTNkTZkTHJTF

Z

β

β

ββ

ββ

λ

λλλ

λλλλλ

−

−

+=

∂∂

−=

++−−=

−≈

+−−=−=

+=+== T


( )( )[ ] 2/142sinhsinh),,(

JeHHNHJTM

βββ

−+=

βH

Μ/Ν βJ=0

βJ=1

βJ=2


( )

∑

∑

∑∑

∑∑

=

=

==

==+

−≡

+−=

−−=

−−=

N

iieff

N

iii

N

ii

N

iii

MFs

N

ii

N

iiis

sH

sHsJ

sHssJE

sHssJE

1

1

11

111

:energy macrostate fieldMean

:energy macrostateExact Mean field approximation for 1-dimensional Ising model


( ) ( )

( )[ ]HsJesZ

s

HsJH

HNkTZNkTZkTF

is

sHii

ieff

effMFNMFMF

i

ieff +==

+=

−=−=−=

∑ − β

β

β tanh1:conditiony Consistenc

)cosh(2lnlnln

1

11

Mean field partition function and Free energy:


( )[ ]HsJs ii += βtanh

:solution fieldMean ( )

( )[ ] 2/142sinhsinh

:solutionExact

JieH

HNMs

βββ

−+=≡

β=1 J=1

H

is

One dimensional Ising model with periodic boundary conditions:


Summary of qualitative results of Ising model systems : Exact solutions of one-dimensional system show no phase

transitions Exact solution by Onsager for two-dimensions system

shows phase transitions as a function of T in zero magnetic field

Mean field approximation shows phase transitions for all dimensions as a function of T in zero magnetic field

Mean field approximation for fixed T as a function of magnetic field is qualitatively similar to exact solution for one dimension


HJqmHsJqH

HsJH

jeff

q

jjeff

+≡+=

+≡ ∑=1

Self-consistency condition for mean field treatment for general system with q nearest neighbors

( )( )

( )( )HJqmHF

Nm

ZkTNF

HJqmeZs

sHeff

+=∂∂

−=

−=

+== ∑±=

β

ββ

tanh1

ln

cosh2

1

11

1

1


( )( )HJqmm += βtanh

5.0=Jqβ

1=Jqβ2=Jqβ 5.1=Jqβ

1 :0at for solution

trivial-nonfor Condition

≥=

JqHm

β

Mean field self-consistency condition:

Mean field solutions exhibit “critical behavior” (phase transition) at βcJq=1.

H=0


Mean field self-consistency condition for H=0: ( )( )Jqmm βtanh=

m

T/Tc

( )( )

=

==

=

mTTmJqmm

Jq

c

c

c

tanhtanhtanh

1 :Define

βββ

β

>

≤

= for 0

for tanh

c

cc

TT

TTmTT

m


Summary of results for mean field treatment of Ising model

( )( )

( )( )

( )( )

( )( ) JqJqmmqJNkC

JqJqmJqmm

JqmmJqJqmm

mJqmNkTEC

JqmmNJqmE

H

N

NN

NN

βββ

βββ

ββ

ββ

ββ

β

−=

−=

∂∂

∂∂

+=

∂∂

∂∂

=

∂∂

=

=−=

=

2

2222

2

2

2

2

cosh2

cosh

sech

2

:capacityHeat tanh where

:0for energy Internal


Behavior of magnetic susceptibility (in scaled units):

( )( )

( )( )

( )( )

TTm

TTTT

JqT

HJqHJqmH

mHT

HJqmHmJq

Hm

HmHT

HJqmm

cc

c

T

TT

T

−

=

=−+

=

∂∂

=

+

∂∂

+=

∂∂

∂∂

=

+=

2

2

2

cosh

1)0,(

:0For cosh

),(

sech

),(

tanh

χ

βββχ

βββ

χ

β


Behavior of magnetic susceptibility (in scaled units) -- continued:

12

31

1)0,(

131cosh

13 ; and For

relation Weiss-Curie 1

1

1)0,(

0 ;For

cosh

1)0,(

2

2/1

2

TTJqT

TT

TT

TT

JqT

TTm

TT

TT

TTmTTTT

TTJqT

TT

TT

JqT

mTTTTm

TTTT

JqT

c

c

c

c

c

c

c

cccc

c

c

c

c

c

cc

c

−≈

−

−

≈

−+≈

−

≈≈<

−=

−=

=>

−

=

χ

χ

χ


Interacting systems – N identical particles of mass m: Classical treatment

( )

kTmv

NmvmN

ii

mNVNN

VmNN

N

evkT

mv

dvvv

dvevmepd

dvvmpdm

epderdN!h

Z(T,V,N)

epdrdN!h

Z(T,V,N)

ii

ii

i

ii

i

2/22/3

2/2323

233

23)(33

)(233

3

2

22

2

2

24)(

: and between velocity of particle finding ofy Probabilit

4

4

1

1

−

−

−

−

−

−

−

=

+

=∑

==

∑=

∑=

∫∫

∫∫

∫∫

ππ

π

π

β

β

β

β

ββ

P

p

pr

rp

vp

Many particle systems with translational degrees of freedom – Chapter 6 of STP


vkTmudueuduudvv

evkT

mv

u

kTmv

2 where4)()(

24)(

2

2

2

2/22/3

≡==

=

−

−

π

ππ

uPP

P

Maxwell velocity distribution

)(uuP

For classical particles the Maxwell velocity distribution is the same for all particle interaction potentials.


Statistics of non-interacting quantum particles

k

k

n :numbers occupation particle Single :states particle Single ε

0,1 :spin)integer ( particles Fermi 0,1,2,3, :spin)(integer particles Bose

21 =

=

k

k

nn

( ) seTZs

NEG

ss smicrostate allover summing ),(

:systems for thesefunction partition Grand

∑ −−= µβµ


( ) seTZs

NEG

ss smicrostate allover summing ),(

:systems for thesefunction partition Grand

∑ −−= µβµ

( )

( )∑

∏

∏ ∑

∑ ∑

−−

−−

≡

≡

=

==

s

nnkG

kkG

k s

nnG

k k

sksk

sks

skk

sk

skk

sk

eTZ

TZ

eTZ

nNnE

µεβ

µεβ

µ

µ

µ

ε

),( where

),(

),(

,

,


Fermi particle case : ( )

( )µεβ

µεβµ

−−

−−

+=

≡ ∑k

skk

sk

e

eTZs

nnkG

1

),(,

1,0=skn

( )( )

( ) 11

:numbersoccupancy Mean 1lnln

:case for this potentialLandau

k

,k

+=

∂Ω∂

−=

+−=−=Ω

−

−−

µεβ

µεβ

µ k

k

en

ekTZkT

k

kG


Bose particle case :

( )

( )( )µεβ

µεβ

µεβµ

−−

∞

=

−−

−−

−==

≡

∑

∑

ksk

skk

skk

sk

ee

eTZ

n

n

s

nnkG

11

),(

0

,

,4,3,2,1,0=skn

( )( )

( ) 11

:numbersoccupancy Mean 1lnln

:case for this potentialLandau

k

,k

−=

∂Ω∂

−=

−=−=Ω

−

−−

µεβ

µεβ

µ k

k

en

ekTZkT

k

kG


Bose particle case :

( )( )

( )

0or 11 that implies

series geometric theofsummation that theNote

11),(

0,

<<⇒

<

−==

−−

−−

∞

=

−−∑

µ

µ

βµ

µεβ

µεβµεβ

ee

eeTZ

k

ksk

skk

n

nkG

,4,3,2,1,0=skn

Note a detail:


Case of Fermi particles

εk

nk

µ = 10

T = 0

T >> 0


Case of Fermi particles Non-interacting spin ½ particles of mass m at T=0 moving in 3-dimensions in large box of volume V=L3: Assume that each state ek is doubly occupied (due to spin)

( )

( )

∫∑

∑∑

→

→∞→

=++

=

+== −

kdLmkL

nnnmL

nnnhL

enN

k

zyxzyx

nnn

kkk

zyx

k

33

22

k

2

2222

,,

22

2 ,limit In the

3,2,1,, 8

length ofbox ldimensiona-3in particles :Example1

1

π

ε

ε

µεβ

spin degeneracy


Case of Fermi spin ½ particles for T 0 in 3-dimensional box.

( )

3/22

2

2/3

22

22

33

33

32

23

2

34

22

22

for 0for 1

11

=⇒

=

≡=

=

→=

><

≈+

=

∫∑<

−

VN

m

mVN

mk

kLkdLnN

en

F

F

FF

Fk

k

k

kk

k

k

πε

επ

εµ

πππ

µεµε

µε

µεβ


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.

( )

( )

επ

ε

εεπ

ε

ε

µεβ

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


Critical temperature for Bose condensation:

( ) 11 )(

00 −

+= −

∞

∫ µεβεεe

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

πππ

πεε

π

εε

εβ

µεβ

=⇒

≈

−

=

−

≈

−=

∫∫

∫

∞∞

−

∞


Nn0

T/TE

2/30 1 For

−=≤

EE T

TNn

TT


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

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

π

ννπεεπ

ε

εεπ

εδεπ

νωε

νββε

βε

=

−=

−==

=−

→

===−

=

∫∫∑

∫ ∫∫∑


Blackbody radiation distribution:

ν

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 µ=0.

( )2

2

13

21

113

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

1

1

−

=

∂

∂=

+

−=

=−

=

ωβ

ωβ

ωβ

βε

ω

ω

ωωε

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

ω

ωβ

βε

ωωπ

ωωε


Thermodynamic description of the equilibrium between two forms “phases” of a material under conditions of constant T and P -- Chapter 7 in STP

),(),(

),,(:energy Free sGibb' of Review

NV

Pg

NS

Tg

PTgNGPT

dNVdPSdTdGPVFPVTSENPTGG

TP

=

∂∂

−=

∂∂

≡==

++−=+=+−≡=

µµ

µ


Example of phase diagram :


P

T

gA > gB gA < gB

( ) ( )PTgPTg BA ,, =


( ) ( )( ) ( )

( )( )NV

NSdTdP

dPNV

NVdT

NS

NS

dPPg

PgdT

Tg

Tg

PTdgPTdgPTgPTg

B

B

A

A

B

B

A

A

T

B

T

A

P

B

P

A

BA

BA

//

0

0

,,,,

EquationClapeyron -Clausius

∆∆

=⇒

=

−+

−−

=

∂∂

−

∂∂

+

∂∂

−

∂∂

==


( )( )

( )BA

AB

AB

VVTL

dTdP

TLΔS

NVNS

dTdP

−=

=

∆∆

=

:Heat"Latent " theinvolving change phase aFor //

EquationClapeyron -Clausius


( )

TRL

M

AB

M

AB

M

AB

B

AvoMM

A

AB

BA

AB

MABePTPTR

LP

TdT

RL

PdP

TP

RL

dTdPV

kNRP

TRV

KTJ/mole.kgJL

VVTL

dTdP

P(T)

/0

22

36

)( constant )ln(

0

mole)(per

15.373 10740/10257.2waterBvapor A :Example

ecoexistenc gas-liquidfor eapproximat --Equation Clapeyron -Clausius

−=⇒−=

=⇒=

≈

=≈

≥×=×=

≡⇒≡−

=


Other topics • Van der Waals equation of state • Chemical equilibria

Documents

PHY 341/641 Thermodynamics and Statistical Physics Lecture 29users.wfu.edu/natalie/s12phy341/lecturenote/Lecture29.pdf · 2012-04-09 · 4/8/2012 PHY 341/641 Spring 2012 -- Lecture