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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
4/8/2012 PHY 341/641 Spring 2012 -- Lecture 29 2
Second exam: April 9-13 -- student presentations 4/30, 5/2 (need to pick topics)
4/8/2012 PHY 341/641 Spring 2012 -- Lecture 17 3
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:
4/8/2012 PHY 341/641 Spring 2012 -- Lecture 17 4
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
=
∂
∂=
−=∂
∂−=
−=−=−=
4/8/2012 PHY 341/641 Spring 2012 -- Lecture 17 5
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
=
∂∂
−=
∂∂
≡
∂∂
−==⇒
4/8/2012 PHY 341/641 Spring 2012 -- Lecture 29 6
Magnetization and susceptibility of independent spin ½ particles
βµB
M/M0
χ/χ0
4/8/2012 PHY 341/641 Spring 2012 -- Lecture 18 7
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:
4/8/2012 PHY 341/641 Spring 2012 -- Lecture 18 8
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
4/8/2012 PHY 341/641 Spring 2012 -- Lecture 18 9
( ) ( )
( ) ( )
≡
=
=
−−
−+
+−
+∑
∑
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=
4/8/2012 PHY 341/641 Spring 2012 -- Lecture 18 10
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
==
=
≡=
−
−−−
−
nλ
λλ
Nn
NNNN λλλλ 321)(Tr ++=⇒ T
4/8/2012 PHY 341/641 Spring 2012 -- Lecture 18 11
1-dimensional Ising system of N spins with periodic boundary conditions (sN+1=s1) (continued)
( ) ( )
( ) ( )
( ) ( )[ ] ( ) ( )[ ]
( ) 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
4/8/2012 PHY 341/641 Spring 2012 -- Lecture 18 12
1-dimensional Ising system of N spins with periodic boundary conditions (sN+1=s1) (continued)
( )
( ) ( )[ ][ ]( )
( )[ ] 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
4/8/2012 PHY 341/641 Spring 2012 -- Lecture 18 13
( )( )[ ] 2/142sinhsinh),,(
JeHHNHJTM
βββ
−+=
βH
Μ/Ν βJ=0
βJ=1
βJ=2
4/8/2012 PHY 341/641 Spring 2012 -- Lecture 19 14
( )
∑
∑
∑∑
∑∑
=
=
==
==+
−≡
+−=
−−=
−−=
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
4/8/2012 PHY 341/641 Spring 2012 -- Lecture 19 15
( ) ( )
( )[ ]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:
4/8/2012 PHY 341/641 Spring 2012 -- Lecture 19 16
( )[ ]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:
4/8/2012 PHY 341/641 Spring 2012 -- Lecture 19 17
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
4/8/2012 PHY 341/641 Spring 2012 -- Lecture 19 18
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
4/8/2012 PHY 341/641 Spring 2012 -- Lecture 19 19
( )( )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
4/8/2012 PHY 341/641 Spring 2012 -- Lecture 19 20
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
4/8/2012 PHY 341/641 Spring 2012 -- Lecture 20 21
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
4/8/2012 PHY 341/641 Spring 2012 -- Lecture 20 22
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
4/8/2012 PHY 341/641 Spring 2012 -- Lecture 20 23
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
4/8/2012 PHY 341/641 Spring 2012 -- Lecture 22 24
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
4/8/2012 PHY 341/641 Spring 2012 -- Lecture 22 25
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.
4/8/2012 PHY 341/641 Spring 2012 -- Lecture 22 26
Statistics of non-interacting quantum particles
k
k
n :numbers occupation particle Single :states particle Single ε
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
4/8/2012 PHY 341/641 Spring 2012 -- Lecture 22 33
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
4/8/2012 PHY 341/641 Spring 2012 -- Lecture 24 34
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.