2/18/2014PHY 770 Spring 2014 -- Lecture 10-111 PHY 770 -- Statistical Mechanics 11 AM – 12:15 & 12:30-1:45 PM TR Olin 107 Instructor: Natalie Holzwarth

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

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





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


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


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


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


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


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


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


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


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


Canonical ensemble from optimization – continued

Tr Tr

êxpˆ

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


Canonical ensemble from optimization – summary of results

êxp

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


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:


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


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


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


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


'

/

/

/

'

'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


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


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


' ''

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


' '

' ' ' '' '

' ' ' '' '

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


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


, , ,

, ,

, , , ,

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:


Canonical ensemble in terms of probability density operator

êxp

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


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


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


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


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


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


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

êxp

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


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


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


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


Lattice vibrations for 3-dimensional latticeExample: diamond lattice

Ref: http://phycomp.technion.ac.il/~nika/diamond_structure.html

More realistic model of lattice vibrations


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


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


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


B. P. Pandey andB. Dayal, J. Phys. C 6, 2943 (1973)


B. P. Pandey andB. Dayal, J. Phys. C 6, 2943 (1973)


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


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


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

qq

q

3

3

Phonon density of states

( =

Note tha

) ( )2

( ) 3t:

V d q

g M

g

d

w w w

w w

q


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

Documents

2/18/2014PHY 770 Spring 2014 -- Lecture 10-111 PHY 770 -- Statistical Mechanics 11 AM – 12:15 & 12:30-1:45 PM TR Olin 107 Instructor: Natalie Holzwarth