PHY 770 -- Statistical Mechanics 12:00 * -1:45 P M TR Olin 107

PHY 770 Spring 2014 -- Lecture 13 12/26/2014

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 13Chap. 5 – Canonical esemble

Partition function Ising model in 1d and mean field approximation

Chap. 6 – Grand canonical ensemble Grand partition function Classical and quantum ideal gases

*Partial make-up lecture -- early start time

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



Summary of results for the canonical ensemble

ˆexp

ˆˆ where Tr exp

ˆ ˆˆln Z exp exp ( )

T ˆ ˆln

ˆˆTr

r

B

B

BB

B

Hk HZ T

Z T k

H AA k T T H Ak

S

U

TT

T

k

H


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

1 2 3

Ising 1 11 1

1 11 1

1 2 2 3 1 1, ,

2

Tr exp2

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

where:(1,1) (1, 1)

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

N

N N

i i i ii i

N N

N i i i ii i

N N N Ns s s s

s s s s

Z s s s s

f s s f s s f s s f

B

B

s s

f ff s s

f f

H

B

B

e e

e e

P


1 2 3

1 2 2 3 3 4 4 5 1

1 2 3

1 2 2 3 1 1, ,

, ,

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

where:

N

N N

N

N N N N Ns s s s

s s s s s s s s s ss

B

B

s s s

Z f s s f s s f s s f s s

P P P P P

e e

e e

P

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

Tr NNZ P



)(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

n

1 2Tr N N N P



1

2

1/22 41

1/22 42

1 2

In this case:

00

cosh sinh

cosh sinh

Tr N N NN

B

B

e e

e e

B B

B B

e e

e e

Z

P

Λ

P



1/22 41

1/22 42

cosh sinh

cosh sinh

e e

e

B

e

B

B B

1

2

1,2

B

1



21 2 1

1

21

1

1

1/22 4

Tr 1

( , , ) ln ln ln 1

ln

ln cosh sinh

si( , , )

NN N N N

N

N

N

Z

A T B kT Z NkT kT

NkT

N kT e

NAM T BB

B B

P

1/22 4

nh

sinh e

B

B


1/22 4

sinh( , , )

sinh

NM T J H

e

B

B

B

M/N

=0

=1

2


I

11 1

1 1

1

s

1

ing

Exact Hamiltonian:

Mean field approximation:

ˆ

ˆ

Ising

N N

i i ii i

N N

i i ii i

N

i ii

N

eff ii

MF

s s s

s s s

H B

H B

Bs s

H s

Mean field approximation for 1-dimensional Ising model


1 1

1

ln ln ln 2cosh( )

Consistency condition:1 tanheff i

i

Neff

eff i

H si i i

s

A kT Z

B

NkT Z NkT H

H s

s s e sZ

B

Mean field partition function and Free energy:


Consistency condition:

tanhi i Bs s

s

tanh i Bs


Mean field solution:

tanhi is s B

1/22 4

Exact solution:sinh

sinhi

B

B

MsN e

=1=1

B

is

One dimensional Ising model with periodic boundary conditions:


, ( ) 1 , ( ) 1

Ising m del:

ˆ

oN N N N

i j i ji j n

Ising ij i in i i j nn i

s s s s sB B sH

ò ò

Extension of mean field analysis to more complicated geometries

1

1 1

Ising model in mean field approximation:

ˆ2 number of nearest ne ighbors

ˆ2

MFIsing i

MFIs

N N

i ji i

N N

jing i eff ii i

s s s

s

B

B s

H

H s H

ò

ò


Extension of mean field analysis to more complicated geometries -- continued

1 1

1

ln ln ln 2cosh( )

2Consistency condition:

1 tanh2

eff i

i

Neff

eff i

H si i i

s

A kT Z NkT Z NkT H

H s

s s e s

B

BZ

Mean field partition function and Free energy:



1

Consistency condition for =0:

1 tanh2

eff i

i

H si i i

s

B

s s e sZ

<s>

tanh2 is


0 0


Define: tanh2

B

s s


0

20

2

2cosh( / 2)

for 0

for

ln( ) 12

N

N

N

N

s

s

Z

Z

N

s

sU


0 0


Define: tanh2

B

s s


2 00

2 2 2 20

20

Heat capacity:

2co

ln( )

( / 2)sh

NN

N N

N

UC N

N

Z ssT

k ss

C


0 0For: tanh2

Note that there is no solution for 1:2

s s


1:2

12

Define critical temperature

2 cTk



220

20

Heat capacity in terms of critical temperature:

2for

cosh

0 f

/

r

/

o

/

cc

N c c

c

Nks TT T

C TT

s T TT

T T

2cT k

CN

TTc


Comment on mean field heat capacity

220

20

Heat capacity in terms of critical temperature:

2for

cosh

0 f

/

r

/

o

/

cc

N c c

c

Nks TT T

C TT

s T TT

T T

0 0 0

0

Note that: tanh tanh2

Your text claims that: ( ) 3

Hint: ( ) 3

c

N c B

cc

c c

Ts s s

T

C T T Nk

T TTs T TT T


Comment about 1-dimensional case

One can show (rigorously) that for one dimensional systems, there can be no phase transitions! (Mean field results are qualitative correct for 2 and 3 dimensions.)


Canonical ensemble – derivation from optimizationFind form of probability density which optimizes S with constraints

Maximize: Tr ln

Constrain: Tr 1 and

ˆ ˆ

ˆˆ ˆ

ˆˆ

Tr

Tr ln ˆ ˆ ˆ

ˆˆ

0

ˆ

1ˆ ˆˆ exp exp

ln 0

exp 1

B

B

B B

B B B

S k

E

k

k k

k k

H

H

H

H Hk Z

Tr 1 Tr ˆˆ exp

ˆˆln Z exp

B

BB

Z ZH

H AA k T Tk

k

T


Generalization: Grand canonical ensemble – derivation from optimization

Find form of probability density which optimizes S with constraints

Maximize: Tr ln

Constrain: Tr 1 and Tr and Tr

T

ˆ ˆ

ˆ ˆˆ ˆ ˆ

ˆ ˆˆ ˆ ˆ ˆ ˆ

ˆ ˆˆ ˆ

1ˆ ˆ ˆ ˆ

r ln 0

ln 0

exˆ expp 1 exp

B

B

B B

B B B B B

S k

E N

k

k k

k

H N

H N

H N

H Nk

HZk k

Nk

ˆ ˆˆ ˆTr lnB B Bk kH N E N Sk

PHY 770 Spring 2014 -- Lecture 13 26

ˆ ˆ( )

ˆ ˆ( )ˆ ˆ( )

ˆ ˆˆ ˆTr

Recall the grand potential:, ,

1

ln

ln

wh

, , , ,

, ,e Tr

ˆT

re

r

B B B

B

H N

H NH N

k k k

T Z

H N E N S

T V U TS N

T TT V k

Z

T V

T V e

ee

ˆ ˆ( )H Ne

2/26/2014

Grand partition function -- continued


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

1

33 3

3

2ˆ | | where

2 2

Single particle Hamiltonian:

; 2 x zy iH l l l l

LL Vd p d k

m

2

p p

p

p p x y z

Extension to multiparticle case:For particles which obey Fermi-Dirac statistics: 1. Each single pa | has an occupancy

rticle momentum eigenstate

mome0 or 1

2. E ntuach n

p

| has a spin degeneracy 2 1,

1

m eig

where (integer

enstate

)+2

g s

s

p


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

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


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

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PHY 770 -- Statistical Mechanics 12:00 * -1:45 P M TR Olin 107