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
PHY 770 Spring 2014 -- Lecture 12 22/20/2014
PHY 770 Spring 2014 -- Lecture 12 32/20/2014
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
PHY 770 Spring 2014 -- Lecture 12 42/20/2014
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
PHY 770 Spring 2014 -- Lecture 12 52/20/2014
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
PHY 770 Spring 2014 -- Lecture 12 62/20/2014
Canonical ensemble – example including magnetic effects -- continued
22
, ,3 sech2 2 2V N B B BC Nk BN Bk
B=1
B=5 B=10
PHY 770 Spring 2014 -- Lecture 12 72/20/2014
Canonical ensemble – example including magnetic effects -- continued
, ,
ln tanh
2 2B N
T V N
T Zk BNMB
T
PHY 770 Spring 2014 -- Lecture 12 82/20/2014
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
PHY 770 Spring 2014 -- Lecture 12 92/20/2014
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
PHY 770 Spring 2014 -- Lecture 12 102/20/2014
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
PHY 770 Spring 2014 -- Lecture 12 112/20/2014
, ( )
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
PHY 770 Spring 2014 -- Lecture 12 122/20/2014
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
PHY 770 Spring 2014 -- Lecture 12 132/20/2014
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
PHY 770 Spring 2014 -- Lecture 12 142/20/2014
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
PHY 770 Spring 2014 -- Lecture 12 152/20/2014
1-dimensional Ising system of N spins with periodic boundary conditions (sN+1=s1) (continued)
)(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
PHY 770 Spring 2014 -- Lecture 12 162/20/2014
1-dimensional Ising system of N spins with periodic boundary conditions (sN+1=s1) (continued)
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
PHY 770 Spring 2014 -- Lecture 12 172/20/2014
1-dimensional Ising system of N spins with periodic boundary conditions (sN+1=s1) (continued)
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
PHY 770 Spring 2014 -- Lecture 12 182/20/2014
1-dimensional Ising system of N spins with periodic boundary conditions (sN+1=s1) (continued)
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
PHY 770 Spring 2014 -- Lecture 12 192/20/2014
1/22 4
sinh( , , )
sinh
NM T J H
e
B
B e
B
M/N
e=0
e=1
e2
PHY 770 Spring 2014 -- Lecture 12 202/20/2014
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
PHY 770 Spring 2014 -- Lecture 12 212/20/2014
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:
PHY 770 Spring 2014 -- Lecture 12 222/20/2014
Consistency condition:
tanhi i Bs s e
s
tanh i Bs e
PHY 770 Spring 2014 -- Lecture 12 232/20/2014
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:
PHY 770 Spring 2014 -- Lecture 12 242/20/2014
, ( ) 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
PHY 770 Spring 2014 -- Lecture 12 252/20/2014
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:
PHY 770 Spring 2014 -- Lecture 12 262/20/2014
Extension of mean field analysis to more complicated geometries -- continued
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
PHY 770 Spring 2014 -- Lecture 12 272/20/2014
0 0
Consistency condition for =0:
Define: tanh2
B
s s e
Extension of mean field analysis to more complicated geometries -- continued
0
20
2
2cosh( / 2)
for 0
for
ln( ) 12
N
N
N
N
s
s
Z
Z
N
s
sU e
e
PHY 770 Spring 2014 -- Lecture 12 282/20/2014
0 0
Consistency condition for =0:
Define: tanh2
B
s s e
Extension of mean field analysis to more complicated geometries -- continued
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
PHY 770 Spring 2014 -- Lecture 12 292/20/2014
0 0For: tanh2
Note that there is no solution for 1:2
s s e
e
Extension of mean field analysis to more complicated geometries -- continued
1:2
e
12
e
Define critical temperature
2 cTke
PHY 770 Spring 2014 -- Lecture 12 302/20/2014
Extension of mean field analysis to more complicated geometries -- continued
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