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

PHY 770 Spring 2014 -- Lecture 12 12/20/2014

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 12 -- Chapter 5Equilibrium Statistical Mechanics

Canonical ensemble

Magnetic effects Ising model

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


Canonical ensemble – example including magnetic effectsConsider a system of N particles in a box of volume V , treated in the semiclassical limit. Since we are in the semiclassical limit, we can use the relationship

1

11 1

2

1 1( ) 1(mag)

ˆ ˆwhere Tr exp Tr exp

ˆ ˆ ˆ2

here, denotes the magnetic field strength denotes the magnetic moment factor

1=

!

N

N

B

trans

Z Z TTN

HZ T Hk

H sB H HmB

T

p

1 denotes the intrinsic spin2s


Canonical ensemble – example including magnetic effects -- continued

1 1( ) 1(mag) 1( ) 1( )

1( ) 3

/2 /21( )

3

, ,,

2

ˆ ˆTr exp

/ 2

1 2 / 2!

cosh

cosh

ln 3 1 tanh / 222

trans trans mag

transT

B Bmag

N

NN

T

NB

V N BV N

Z T H H Z T Z T

VZ T

Z T e e B

VZ T BN

Z TT N BU Nk

UC

B

T

,

22

, ,

3 sech2 2 2

lnMagnetization: tanh

2 2

B B

B N

T V

B

N

N B B

T Z T B

k Nk

k NMB



22

, ,3 sech2 2 2V N B B BC Nk BN Bk

B=1

B=5 B=10



, ,

ln tanh

2 2B N

T V N

T Zk BNMB

T


Statistical mechanics of the Ising modelSpin ½ system with competing effects of nearest-neighbor interactions and spin alignment energy in a magnetic field

2412

Spin alignment contribution (convenient redefinition of " ")

where 1, spin alignment energy

9.28 10 / (for an electr

ˆ

on)

i ii

B

B

s

s B s B

g J T

H

1

1 2 3

1

1

1 1 1 1

11

Partition function for for non-interactingˆ term al spins:

one

N

ii

N

B s

Ns s s s

NNB

s

B

s

N

Z e

e Z

H


Calculation of Z1

)cosh(21

11

1 BeeeZ BB

s

Bs

Thermodynamic functions:

1 1

1

2 2

ln ln ln 2cosh( )

ln tanh

sech

N

B

A kT Z NkT Z NkT B

ZU N N B B

UC kN B BT


Statistical mechanics of the Ising model -- continuedSpin ½ system with competing effects of nearest-neighbor interactions and spin alignment energy in a magnetic field

, ( )

Nearest-neighbor interaction m:

ˆ

terN

iint ji j nn

ijs sH ò

3 71 2 654

e12 e23 e34 e45 e56 e67

, ( ) 1

ˆ ˆ

Ising mod :

ˆ

el

Ising int B

N N

i ji j nn i

ij iB sH sH H s

ò


, ( )

ˆN

i ji j n

in

nt s sH ò

2

2

4cosh

ln tanh

Z e e e e

ZU

e e e e e

e e

Ising model – effects of interaction term alone for N=2; eij=e

ˆ intH ò ò ò ò


2 22 ( , )

2 cosh 2 2

B BZ B e e e e

e B e

e e e e

e e

e

Ising model -- full model for N=2; eij=e2 2

, ( ) 1Ising

ˆi j

i j ii

nn

H Bs s s

ò

ˆ intH ò ò ò ò


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

1 2 3

1 11 1

1 2 2 3 1 1, ,

Tr exp2

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

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

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

N

N N

N i i i ii i

N N N Ns

B

s

B

s s

Z s s s s

f s s f s s f s s f s s

f ff s s

f f

e e

e

B

e

e 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

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

e e

e e

e e

e e

P

Λ

P



1/22 41

1/22 42

cosh sinh

cosh sinh

e e

e

B

e

B

B B

e e

e e

1

2

1,

2

B

e1



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 e

e

e

e

P

1/22 4

nh

sinh e

B

B e


1/22 4

sinh( , , )

sinh

NM T J H

e

B

B e

B

M/N

e=0

e=1

e2


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

e

e

e

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

e

e

Mean field partition function and Free energy:


Consistency condition:

tanhi i Bs s e

s

tanh i Bs e


Mean field solution:

tanhi is s B e

1/22 4

Exact solution:sinh

sinhi

B

B

MsN e e

=1e=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

e

e

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

e

<s>

tanh2 is e


0 0


Define: tanh2

B

s s e


0

20

2

2cosh( / 2)

for 0

for

ln( ) 12

N

N

N

N

s

s

Z

Z

N

s

sU e

e


0 0


Define: tanh2

B

s s e


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

e

e e e


0 0For: tanh2

Note that there is no solution for 1:2

s s e

e


1:2

e

12

e

Define critical temperature

2 cTke



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 ke

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