PHY 712 Electrodynamics 10-10:50 AM MWF Olin 107 Plan for Lecture 31:
PHY 770 -- Statistical Mechanics 11 AM – 12:15 & 12:30-1:45 P M TR Olin 107
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Transcript of PHY 770 -- Statistical Mechanics 11 AM – 12:15 & 12:30-1:45 P M TR Olin 107
PHY 770 Spring 2014 -- Lecture 10-11 12/18/2014
PHY 770 -- Statistical Mechanics11 AM – 12:15 & 12:30-1:45 PM TR Olin 107
Instructor: Natalie Holzwarth (Olin 300)Course Webpage: http://www.wfu.edu/~natalie/s14phy770
Lecture 10-11 -- Chapter 5Equilibrium Statistical Mechanics
Canonical ensemble
Probability density matrix Canonical ensembles; comparison with
microcanonical ensembles Ideal gas Lattice vibrations
PHY 770 Spring 2014 -- Lecture 10-11 22/18/2014
PHY 770 Spring 2014 -- Lecture 10-11 32/18/2014
PHY 770 Spring 2014 -- Lecture 10-11 42/18/2014
PHY 770 Spring 2014 -- Lecture 10-11 52/18/2014
Microscopic definition of entropy – due to Boltzmann
nEknENS NB ,ln),,( N
Consider a system with N particles having a total energy E and a macroscopic parameter n.
denotes the multiplicity of microscopic states having the same parameters. Each of these states are assumed to equally likely to occur.
nEN ,N
Alternative description of entropy in terms of probability density (attributed to Gibbs)
ˆTr ˆln( )BS k
PHY 770 Spring 2014 -- Lecture 10-11 62/18/2014
Probability density -- continued
Normalization: Tr 1
Average value of : Tr
ˆ
ˆ X X X
2 2
3 3 3 3 3 32 2
Classical treatment (in 3 dimensions): , ,... , , ,... , )
Tr . . .
(
..
ˆ
.N N
N Nd d d d
t
d d
1 1
1 1
r r r p p p
r r r p p p
PHY 770 Spring 2014 -- Lecture 10-11 72/18/2014
23 3
3
23 3
3
3 /23 /2
3 32
Classical microstate distribution:
1( , , ) ! 2
1 ! 2
2
!
1
N N iN
i
N N iN
i
N NN
N N
pE V N d r d p EN h m
pd r d p EN h m
V mEN h
N
Example of classical treatment of microstate analysis of N three dimensional particles in volume V with energy E
3 /2
3 32
21( , , )! 1
NN
N
mEVE V NN h
N
PHY 770 Spring 2014 -- Lecture 10-11 82/18/2014
Example of classical treatment of microstate analysis of N three dimensional particles in volume V with energy E -- continued
Rough statement of equivalence of Boltzmann’s and Gibbs’ entropy analysis for this and similar cases:
Boltzmann: ( , , ) ln , ,
1Gibbs: , ,
1 1 ( , , ) Tr ln, , , ,
=
ˆ
n
l
ˆ ˆl
B
B B
B
i
S E V N k E V N
E V N
S E V N k kE V N E V N
k
N
N
N NN
n , ,E V NN
PHY 770 Spring 2014 -- Lecture 10-11 92/18/2014
Quantum representation of density matrix
ˆ
Probability amplitude: | ( )| ( ) ˆSchroedinger equati
Hamil
on: | ( )
ˆDensity operator: ( ) | ( )
tonian operator
|
(
:
)
i
ii
i i ii
Ht
ti H t
tt t t
,
ˆ in terms of eigenstates | :ˆ ˆ ˆˆ ˆ
Evaluation of ave
Tr
rage value of op
( ) | ( ) |
erator
||i
i j j ii j
O a
O t O a t a a O a
PHY 770 Spring 2014 -- Lecture 10-11 102/18/2014
Microcanonical ensemble:
Consider a closed, isolated system in equilibrium characterized by a time-independent Hamiltonian with energy eigenstates . This implies that the density matrix is constant in time and is diagonal in the energy eigenstates:
H| nE
2 3
Note that, in this case:
ˆ ˆ ˆˆ ˆ ˆ| ( ) | | | | |ˆ ˆ| ln | l
1 1Hint for | 1| 1: ln( ) 1 1 1 ..
n
..2 3
n n n n n n nn nn
n n nn
x
E t O E E E E O E
x x x x
O
E E
1
ˆ ˆln
ˆ ˆ= lnmax
B
N
B nn nnn
S k
k
Tr
PHY 770 Spring 2014 -- Lecture 10-11 112/18/2014
Microcanonical ensemble -- continued
1
1
1
1
ln
General analysis of probability density matrix elements :
Find which maximizes
ˆ ˆ
ˆ
ˆ ˆ
ˆ ˆ ˆ
ˆ
and satisfies 1
ln 0
ln
max
max
max
max
N
B nn nnn
nn
N
nn nnn
N
B nn nn nnn
N
nn B Bn
S k
S
k
k k
1
0
1exp 1 (constant)=
l
ˆ
ˆ
ˆ ˆn lnmax
nn
nnB max
N
B nn nn B maxn
k N
S k k N
PHY 770 Spring 2014 -- Lecture 10-11 122/18/2014
Summary of results for microcanonical ensemble – Isolated and closed system in equilibrium with fixed energy E:
1
1
Equilibrium implies that ln is maximum
1Analysis
ˆ
finds = where
ln
ˆ
ˆ
max
max
N
B nn nnn
N
nn nnmax
B max
S k
E EN
S k N
Now consider a closed system which can exchange energy with surroundings – canonical ensemble
Two viewpoints• Optimization with additional constraints• Explicit treatment of effects of surroundings
PHY 770 Spring 2014 -- Lecture 10-11 132/18/2014
Canonical ensemble – derivation from optimization
Find 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 r xp ˆˆ eTB
Z Zk
H
PHY 770 Spring 2014 -- Lecture 10-11 142/18/2014
Canonical ensemble from optimization – continued
Tr Tr
ˆexpˆ
ˆˆ ˆ ˆln ln
ln
l
=
n
B
B BB
BB
B
Hk
Z
S k k H Zk
k Zk
k
E
EZ S
Recall that for a closed system (fixed number of particles), the Helmholz f
ln for
ree energy is given by: 1 1T and B
A U TS
E S U TS T E Uk Z
PHY 770 Spring 2014 -- Lecture 10-11 152/18/2014
Canonical ensemble from optimization – summary of results
ˆexp
ˆˆ where T
Tr
r exp
ln Z
ˆ ˆln
ˆˆTr
B
B
B
B
TT
Hk HZ T
Z T k
A k T T
S k
U H
PHY 770 Spring 2014 -- Lecture 10-11 162/18/2014
Canonical ensemble -- explicit consideration of effects of “bath” Eb and “system” Es:
Eb
Es
Analogy (thanks to H. Callen, Thermodyanmics and introduction to thermostatistics)
bath: system:
PHY 770 Spring 2014 -- Lecture 10-11 172/18/2014
Canonical ensemble -- explicit consideration of effects of “bath” Eb and “system” Es; analogy with 3 dice
bath: Eb
system: Es
For every toss of the 3 dice, record all outcomes with a sum of 12Etot as a function of the red dice representing Es.
Es Eb Ps
1 5+6,6+5 2/25
2 4+6,6+4,5+5 3/25
3 3+6,6+3,4+5,5+4 4/25
4 2+6,6+2,3+5,5+3,4+4 5/25
5 1+6,6+1,2+5,5+2,3+4,4+3 6/25
6 1+5,5+1,2+4,4+2,3+3 5/25
PHY 770 Spring 2014 -- Lecture 10-11 182/18/2014
Canonical ensemble -- explicit consideration of effects of “bath” Eb and “system” Es:
Eb
Es
''
Estimation of probabilty function:
b tot ss
b tots
s
E EE E
NNP
PHY 770 Spring 2014 -- Lecture 10-11 192/18/2014
Canonical ensemble (continued)
''
Probability that system is in microstate :
ln ln
ln ln
tot
tot s b s tot
b tot ss
b tot ss
s b tot s
bb tot s
E
E E E E Es
E EE E
C E E
EC E E
E
NP N
P NNN
PHY 770 Spring 2014 -- Lecture 10-11 202/18/2014
Canonical ensemble (continued)
ln ln
ln ln
tot
s b tot s
bb tot s
E
C E E
EC E E
E
P NNN
,
ln ( ) 1Recall that: tot
B b b
V N bE
k E S EE E T
N
/
1ln ln
' s B
s b tot sB
E k Ts
C E Ek T
C e
P N
P
PHY 770 Spring 2014 -- Lecture 10-11 212/18/2014
'
/
/
/
'
'1
where: "partition function"
s B
s B
s B
E k Ts
E k T
E k T
s
C e
eZ
Z e
PCanonical ensemble (continued):
' '/ 1
' '
Calculations using the partition function:
where s B s
B
E k T Ek T
s s
Z e e β
PHY 770 Spring 2014 -- Lecture 10-11 222/18/2014
Canonical ensemble continued – average energy of system:' '/
' '' '
1 1
1 ln
s B sE k T Es s s
s s
E E e E eZ Z
Z ZZ
Heat capacity for canonical ensemble:
2
22 2
2 2 2 2
222
1
1 ln 1 1 1
1
s sV
B
B B
s sB
E EC
T k T
Z Z Zk T k T Z Z
E Ek T
PHY 770 Spring 2014 -- Lecture 10-11 232/18/2014
First Law of Thermodynamics for canonical ensemble (T fixed)
' ''
' ' ' '' '
'' ' '
' '
(internal energy)
s s ss
s s s s ss s
ss s s
s s
E E U
d E E d dE
dEE d dVdV
P
P P
P P- Pressure associated with state s
' ''
s s ss
dU d E E d P dV P
PHY 770 Spring 2014 -- Lecture 10-11 242/18/2014
' ''
First law of thermodynamics:
s ss
dU TdS PdV TdS E d P
''
' ' '
' ' ' ' '' ' '
1where
1 1note that: ln ln ln
ln ln
sEs
s s s
s s B s s ss s s
eZ
E Z Z
E d k T Z d d
P
P P
P P P P
=0
' ''
s s ss
dU d E E d P dV P
PHY 770 Spring 2014 -- Lecture 10-11 252/18/2014
' '
' ' ' '' '
' ' ' '' '
0 ' ''
/'
ln
ln ln
ln
1 1where s s B
s s B s ss s
B s s B s ss s
B s ss
E E k Ts
TdS E d k T d
dS k d d k
S S k
e eZ Z
P P P
P P P P
P P
P
PHY 770 Spring 2014 -- Lecture 10-11 262/18/2014
Canonical ensemble – summary and further results
' '/
' '
,,
Partition function:
( , ) ( , )
ln lnln ( , )
ln
ln
ln
s B sE k T E
s s
T NV N
s
s s
s
Z e e Z T V Z V
Z Zd Z V d dVV
d Z E d P dV
d Z E d E P dV
d Z E TdS
Note: Using first law: sd E dU TdS PdV
PHY 770 Spring 2014 -- Lecture 10-11 27
( , , ) ln , ,i B iA T V N k T Z T V N
2/18/2014
ln
ln
ln
ln ( , ) Helmholz Free Energy
s
sB
sB
B s
d Z E TdS
Ed k Z d S
T
Ek Z S
Tk T Z E TS U TS A T V
PHY 770 Spring 2014 -- Lecture 10-11 282/18/2014
, , ,
, ,
, , , ,
ln lnln
ln
ln
i i i
i i
j j
BB B
V N V N V N
BT N T N
Bi iT V N T V N
i
k T Z ZA S k Z k TT T T
ZA P k TV V
ZA k TN N
( , , ) ln , ,i B iA T V N k T Z T V N
Summary of relationship between canonical ensemble and its partition function and the Helmholz Free Energy:
PHY 770 Spring 2014 -- Lecture 10-11 292/18/2014
Canonical ensemble in terms of probability density operator
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 10-11 302/18/2014
Example: Canonical distribution for free particles
2
2
0
Classical canonical distribution for free particles of mass moving in dimensions in box of length
1 i
i
i
pmdN dN
dNr L
dN
Nm d L
Z(T,V,N) d r d p e N!h
L
2
/2
2
2
21 !
dN /BdN
dNdN B
mk TN!h
mk TLN h
3
3 /2
2
3 /2
232
For 3,
2( , , )!
Compare with microcanonical ensemble:
2( , , )! 1
NNB
NN
N
d L V
mk TVZ T V N N h
V mEE V NN h
N
PHY 770 Spring 2014 -- Lecture 10-11 312/18/2014
Example: Canonical distribution for free particles -- continued
3 /2
2
3/2
2
3 2
,
/
2
2( , , )!
2( , , ) ln ( , , ) 1 ln
25( , , ) ln2
NNB
BB B
B BV
B
N
mk TVZ T V N N h
mk TVA T V N k T Z T V N k TNN h
mk TVS T V N Nk NkN h
AT
PHY 770 Spring 2014 -- Lecture 10-11 322/18/2014
Canonical ensemble of indistinguishable quantum particlesIndistinguishable quantum particles generally must obey specific symmetrization rules under the exchange of two particle labels (see Appendix D of your textbook)
, 1 2 1 2, ..... ... ... , ..... ... ...| |i j i j iN Njk k k k k kk k kP k Bose-Einstein particles Fermi-Dirac particles
1 2
General notation: denotes permutation operator
denotes 0 or 1 for even or odd permutations
, ... denotes sym
metrized (anti s
|
N
P
p
k k k
1 2 1 2
ymmetrized wavefunction)1, ... , ...|
!| p
N NP
k kk k k kN
P
PHY 770 Spring 2014 -- Lecture 10-11 332/18/2014
Canonical ensemble of indistinguishable quantum particles
Example: 3-body ideal gas confined in large volume V with momenta k1, k2,and k3,
3 3 3 1 2 2 3 1
2 1 3 3 2 3 1
1 2 1 2
1 2
1, , , ,3!
| , | , | , | ,
, | , | , , | , ,
k k k k
k k
k k k k k k k k
k k k k k kk
/(2 )31 3
3/
2 3
2
ˆ( ) Tr[exp( / )] e
is the thermal wavel
Partition function for single particle:
2 where " "ength
Bmk TB
TT
B
VT H k T d ph
mk T
Z
VVh
2p
3 1 2 3 1 2 3ˆ( ) Tr[ , , | exp( / ) | , , ]
Partition function for three particles:
BT H k TZ k k k k k k
PHY 770 Spring 2014 -- Lecture 10-11 342/18/2014
Canonical ensemble of indistinguishable quantum particles
Example: 3-body ideal gas confined in large volume V with momenta k1, k2,and k3,
3 3
3
3 1 1 1 1
3 23 3
3 3
semiclassic
/2 3/2
3
l 3a
2
1( ) 3 23! 2 3
1 =3!
Note that |
3 212 3
13!
T T
T
T
T TZ T Z T Z T Z Z
VV V
V
1 2 1 2k k k k
PHY 770 Spring 2014 -- Lecture 10-11 352/18/2014
Reduced single particle density matrix and the Maxwell-Boltzmann distribution
2
1 2 1 2,..
1ˆ ˆ(
Reduced single particle density matrix:
) | | , ... , ...( 1)!
| |N
R N NnN
1 1 R 1
k kkk k k kkk k k
2
ˆexp
ˆ ˆˆˆ where Tr exp , 2
B i
iB
TH
k HZ TZ T k mT
H
p
3
3 2 21
We have previously shown in the semi-classical limit:
( , 1!
( ) ex
)
p2
T
TR
B
NV
N
NV mk
Z T N
nT
1kk
PHY 770 Spring 2014 -- Lecture 10-11 362/18/2014
Reduced single particle density matrix and the Maxwell-Boltzmann distribution -- continued
3 2 2 21
33
3
( ) exp where 2 2
Note that ( ) ( )2
=
Maxwell-Boltzmann velocity distribution:
TR T
B B
R R
hNV mk T mk
V d k N
N d v
T
n n
F
n
1
1
1k
kk
k k
v
3/2 2
ex p 2 2
B B
mk
mFk T T
vv
PHY 770 Spring 2014 -- Lecture 10-11 372/18/2014
Reduced single particle density matrix and the Maxwell-Boltzmann distribution -- continued
3/2 2
Maxwell-Boltzmann velocity distribution:
e 2
2
xpB B
mFk kT
mT
vv
T=100
T=500T=1000
v
24 v F v
PHY 770 Spring 2014 -- Lecture 10-11 382/18/2014
Canonical ensemble example: Einstein solid revisitedConsider N independent harmonic oscillators (in generalN=3 x number of lattice sites) with frequency w
1 2
1
0
1
1 2 1 21
/2
1
0 0
10
1
Hamiltonian:
( , )
1ˆ ˆ2
1ˆ... | ...2
1ˆ = |
exp |
exp |12
( , )
Nn n n
N N
n
N
ii
N
N i Ni
Z T N
H n
n n n n n
ee
A
n
n
N
n
T
n
n w
w
w
w
w
( , )) 1ln( ln2
12 1
B B
N
k Z T N N k e
eU A TS A T
T T
A NeT
w
w
w
w
w
Consistent with result from microcanonial ensemble
PHY 770 Spring 2014 -- Lecture 10-11 392/18/2014
Lattice vibrations for 3-dimensional latticeExample: diamond lattice
Ref: http://phycomp.technion.ac.il/~nika/diamond_structure.html
More realistic model of lattice vibrations
PHY 770 Spring 2014 -- Lecture 10-11 402/18/2014
0
0
2
0 0 0,
Atoms located at the positions:
Potential energy function near equilibrium:
1 2 a
a a a
a a a a b ba b
a b
R
R R u
R R R R R RR R
0
2
0, , ,
2
0, , , ,
Define:
so that12
1 1Lagrangian: ,2 2
a
abjk a b
j k
a a ab bj jk k
a b j k
a a a a ab bj j a j j jk k
a j a b j k
D
u D u
L u u m u u D u
RR R
R
PHY 770 Spring 2014 -- Lecture 10-11 412/18/2014
0
2
0, , , ,
,
0
1 1,2 2
Equations of motion:
Solution form:1
Details: where denotes
a
a a a a ab bj j a j j jk k
a j a b j k
a ab ba j jk k
b k
i t ia aj j
a
a a a
L u u m u u D u
m u D u
u t A em
w
q R
R τ T τ
unique sites and denotes replicas T
PHY 770 Spring 2014 -- Lecture 10-11 422/18/2014
q
q
q Tq
T
ττq
w
w
curves" dispersion" Find sites. atomic unique
overonly issummation heequation t In this
:equations Eigenvalue
:Define
,
2
kb
bk
abjk
aj
i
ba
iabjkab
jk
AWA
emm
eDW
ba
PHY 770 Spring 2014 -- Lecture 10-11 432/18/2014
B. P. Pandey andB. Dayal, J. Phys. C 6, 2943 (1973)
PHY 770 Spring 2014 -- Lecture 10-11 442/18/2014
B. P. Pandey andB. Dayal, J. Phys. C 6, 2943 (1973)
PHY 770 Spring 2014 -- Lecture 10-11 452/18/2014
Lattice vibrations – continuedClassical analysis determines the normal mode frequencies and their corresponding modes
In general, for a lattice with M atoms in a unit cell, there will be 3M normal modes for each q . While the normal mode analysis for and the normal mode geometries are well approximated by the classical treatment, the quantum effects of the vibrations are important. The corresponding quantum Hamiltonian is given by:
w q
w q
1 denotes number operator2
Eigenvalues of are 0, 1, 2, ..
ˆ ˆ
..
ˆ where
ˆ
H n n
n
w
q q qq
q
PHY 770 Spring 2014 -- Lecture 10-11 462/18/2014
Lattice vibrations continued1 denotes number operator2
Eigenvalues of are 0, 1, 2, ..
ˆ ˆ
..
ˆ where
ˆ
H n n
n
w
q q qq
q
/2
ˆ
0
Partition function:
1( ) Tr exp2
1 =
n
H nZ T e
ee
w
w
w
q
q
q
q qq
q
Average energy associated with lattice vibrations
1 1 11
ln(2
)2
ZE ne w
w w
qq q qq q
PHY 770 Spring 2014 -- Lecture 10-11 472/18/2014
Lattice vibrations continued
3
3
Average energy associated with lattice vibrations1 12 1
1 1 2
(
=2 1
1 1 = 2 1
)
E
V d q
e
e
de
g
w
w
w
w
w
w w w
q
q
q
3
3
Phonon density of states
( =
Note tha
) ( )2
( ) 3t:
V d q
g M
g
d
w w w
w w
q
PHY 770 Spring 2014 -- Lecture 10-11 482/18/2014
Lattice vibrations continued
Debye model for g(w
2 2
2 3 3 3
0
3
30
/ 4
3 20
9)
( )
1 =9M 8
Heat capacity
9 where 1
2 1 for ( = 2
0 otherwise
1 1 2 1
1
D
D
D
DD t D
DD
D
D
T T xB
D xD
l
M
E g
Mk x e
Vg c c
de
de
C dx Te
w
w
w
w
w w w ww w
w w w
w www
w
/D D Bkw