ByeongByeong--Joo Lee Joo Lee · 2013-02-19 · Rotational motion Byeong-Joo Lee ... =∑ − + +...

Byeong-Joo Leewww.postech.ac.kr/~calphad

ByeongByeong--Joo LeeJoo Lee

POSTECH POSTECH -- MSEMSE

[email protected]@postech.ac.kr

Properties of the Partition FunctionProperties of the Partition Function

"' srrs EEE +=

))(("'"

,

'

,

")'( ∑∑∑∑ −−−−+− ===s

E

r

EE

sr

E

sr

EE srsrsr eeeeeQβββββ

"'QQQ =

For a system consists of two parts A’ and A” which interacts only weakly

with each other.

→ Two different distinguishable groups of particles

→ Two different sets of degree of freedom of the same groups of particles

"ln'lnln QQQ +=


"'QQQ = "ln'lnln QQQ +=

'2)'2ln(ln2

3'2 SsVTkNS o ≠

++=

'2"' SSSS =+=

Gibbs Paradox: Consider combination of two identical part with N’ and V’

++== osVTkNSS 'lnln2

3'"'

!N

QQ→

+++= 1lnln2

3os

N

VTNkS

System Energy vs. Particle EnergySystem Energy vs. Particle Energy

Neglecting inter-particle interaction

),()( TVqqandVaa == εε

L+++= cbaE εεε

,/ kT

j

ajaeq

ε−

∑= L,/ kT

j

bjbeq

ε−

∑=


L++

++=⋅+−

+−+−+−

kT

kTkTkT

ba

ba

bababa

e

eeeqq

/)(

/)(/)(/)(

21

131211

εε

εεεεεε

NqQ =

L⋅⋅⋅== −∑ cba

kTE

i

qqqeQ i /

Independent, identical, distinguishable particle system

Independent, identical, indistinguishable particle system!N

qQ

N

=

Independence of Modes of Energy StorageIndependence of Modes of Energy Storage

L+++=vibrottrans aaaa εεεε

)(8 2

2

2

2

2

22

c

n

b

n

a

n

m

h zyxi ++=ε

)1)((8 22

2

+= jjmr

hi πε

νε hii )1( +=

Translational motion

Motion in a Potential Field

Rotational motion


−=

222

42

8 nh

emz

o

i εε

LL

⋅⋅⋅==+++−∑ vibrottrans

kT

i

a qqqeq vibarotatransa /)( εεε

−=

r

zeU

2

kT

i

trans eq//ε−∑=

νε hii )1( +=Motion in a Potential Field

Motion of an electron in a Coulomb Potential

Ensemble for Isolated System Ensemble for Isolated System –– Microcanonical ensembleMicrocanonical ensemble

ii nnNNW ln~

ln~

ln ∑−=

0lnln max =−= ∑ ii ndnWd

0=∑dn 0=∑ nαδ

!!!

!~

21 k

innn

NnW

L=

i

i

nN ∑=~Number of NVE systems:


0=∑ idn 0=∑ inαδ

0ln =+αinα−= eni

EHatstatesofnumber

Ne

==−

~α

NVE

i

iN

np

Ω==

1~

!!!

!~

21

~

k

iN nnn

Nn

L=Ω

Microcanonical Ensemble Microcanonical Ensemble –– EntropyEntropy

]ln~

ln~

[ ii nnNNk ∑−=

ii

i

ii

i

N ppkN

n

N

nk

N

SS ln~ln~~

~

∑∑ −=

−==

NNkS ~~ lnΩ=

>−≡< ipkS ln


NVE

NVE

kkS Ω=Ω

−= ln1

ln

>−≡< ipkS ln

Computation of ΩNVE and state density g(E) is critical

but generally difficult

System in contact with a heat reservoir System in contact with a heat reservoir –– Canonical ensembleCanonical ensemble

ii nnNNW ln~

ln~

ln ∑−=


0=∑dn 0=∑ nαδ

ii

i

i EpEU ∑>=≡< ii

i

EnUN ∑=~

i

i

nN ∑=~Number of NVT systems with Ei:

Average Energy of a system:


0=∑ idn 0=∑ inαδ

0ln =++ ii En βα iE

i eenβα −−=

0=∑ iidnE

i

i

E

i

E

ii

e

e

N

np β

β

−

−

∑== ~

0=∑ iidnEβ

iE

i

e

Ne

βα

−−

∑=

~

i

i

E

i

E

i

i

ie

eE

EU β

β

−

−

∑

∑>==<

Canonical Ensemble Canonical Ensemble –– Free Energy & Thermodynamic RelationsFree Energy & Thermodynamic Relations

dNT

dVT

PdU

TdS

µ−+=

1

kT

1=β

QkUkQEpkpkS ii

i

i ln)ln(ln +=−−−>=−≡< ∑ ββ

βkTU

S

NV

==

∂∂ 1

,


QkTSTUFNVT ln−=−≡

β∂∂

−=Q

Uln

TNTN V

QkT

V

FP

,,

ln

∂∂

=

∂∂

−=

NVNVNV

V

Qk

T

U

T

UC

,

2

22

,,

ln

∂

∂=

∂∂

∂∂

=

∂∂

=β

ββ

β

NVNV T

QkTZk

T

FS

,,

lnln

∂∂

+=

∂∂

−=

Canonical Ensemble Canonical Ensemble –– Ideal GasIdeal Gas

QkTFNVT ln−=

V

NkT

V

FP

TN

=

∂∂

−=,

2/3

2

2

!!

NNN

NVTh

mkT

N

V

N

qQ

==π


NkTQ

U2

3ln=

∂∂

−=β

NkT

UC

NV

V2

3

,

=

∂∂

=

+++=

∂∂

−= o

NV

sTN

VNk

T

FS ln

2

3ln

2

5

,

Canonical Ensemble Canonical Ensemble –– relation with Microcanonical Ensemblerelation with Microcanonical Ensemble

β∂∂

−=><≡Q

EUln

2

222 11

ββ

∂

∂==>< −∑

Q

QeE

QE iE

i

i

βββββ ∂∂

−=

∂∂

∂∂

=

∂∂

−−∂

∂=><−><

UQ

Q

Q

Q

Q

QEE

1112

2

222


βββββ ∂ ∂∂ ∂∂ QQQ

VCkTT

UkT

UEEE 22222)( =

∂∂

=∂∂

−=><−>>=<∆<β

NU

CkT

E

E V 1)(22

∝=><

>∆<

Open System with a heat reservoir Open System with a heat reservoir –– Grand Canonical ensembleGrand Canonical ensemble

ii nnNNW ln~

ln~

ln ∑−=


0=∑dn 0=∑ nαδ

iri

ri

i EpEU ,

,

∑>=≡< iri

ri

EnUN ,

,

~ ∑=ri

ri

nN ,

,

~ ∑=Number of µVT systems with Ei & Nr:

Average Energy of a system:

Average # of ptl. of a system:rri

ri

NpN ,

,

∑>=<rri

ri

NnNN ,

,

~ ∑>=<


0, =∑ ridn 0, =∑ rinαδ

0ln , =−++ riri NEn γβα ri NE

ri eenγβα +−−=,

0, =∑ riidnE

ri

ri

NE

ri

NEri

rie

e

N

np

γβ

γβ

+−

+−

∑==

,

,

, ~

0, =∑ riidnEβ

ri NE

ri

e

Ne

γβα

+−−

∑=

,

~

ri

ri

NE

ri

NE

i

ri

e

eE

U γβ

γβ

+−

+−

∑

∑=

,

,

0, =∑ rirdnN 0, =∑ rirdnNγ

ri

ri

NE

ri

NE

r

ri

e

eN

N γβ

γβ

+−

+−

∑

∑>=<

,

,

Grand Canonical Ensemble Grand Canonical Ensemble –– Thermodynamic RelationsThermodynamic Relations

dNT

dVT

PdU

TdS

µ−+=

1

kT

1=β

GGriri

ri

ri QkNkUkQNEpkpkS ln)ln(ln ,

,

, +><−=−+−−>=−≡< ∑ γβγβ

βkTU

S

NV

==

∂∂ 1

,

γµ

kTN

S−=−=

∂∂

kT

µγ =


N

NVT

N

N

NVT

N

NE

ri

G zQeQeeQ ri ∑∑∑ === −)(

,

βµβµβ

µµµ NdPdVSdTNddNSdTTdSdUd −−−=−−−−=Φ

TN VU ∂ , kT

)(ln PVNTSUQkT VTG −=Φ≡−−=− µµ

kTezFugacity

µ

=

µ,VTS

∂Φ∂

−=µ,TV

P

∂Φ∂

−=VT

N,

∂Φ∂

−=µ

Grand Canonical Ensemble Grand Canonical Ensemble –– Ideal GasIdeal Gas

VTGVT kTzqQkT 1ln −=−=Φ µ

zqN

VT

N

N

NVT

N

GVTezq

NzQQ

⋅

==

=⋅== ∑∑ 1)(!

11

00

2/3

21

2

=h

mkTVq VT

π


µ,VTS

∂Φ∂

−=

µ,TVP

∂Φ∂

−=

VT

N,

∂Φ∂

−=µ

h

ExamplesExamples

)cosh(2 BeeZ BB βµβµβµ =+= −+

)tanh()()( BsPsz

s

z βµµµµ ==∑

)tanh(1

)()( BBZ

ZsPsEE

s

βµµβ

−=∂∂

−==∑

▷ Paramagnetism: ideal two state (up & down)


▷헤모글로빈분자위에서의산소흡착

ri NE

ri

G eeZβµβ−∑=

,

kTN

N

G eeZ /)()( 1 µεµεβ −−−− +==∑

Quantum Statistics Quantum Statistics –– Number of ways of distributionNumber of ways of distribution

▷ Distinguishable without Pauli exclusion principle

N

kgggW )( 21 +++=∑ L

∏==

!!)()()(

!!!

!21

21

21 j

n

jn

k

nn

k n

gNggg

nnn

NW

j

kLL

▷ Indistinguishable without Pauli exclusion principle

for gi with ni)!1( ii ng

−−+


for gi with ni!)!1( ii ng −

!)!1(

)!1(

ii

ii

i ng

ngW

−−+

=∏

▷ Indistinguishable with Pauli exclusion principle

for gi with ni!)!(

!

iii

i

nng

g

−

!)!(

!

iii

i

i nng

gW

−=∏

Quantum Statistics Quantum Statistics –– BB--E & FE & F--D D

!)!1(

)!1(

ii

ii

i ng

ngW

−−+

=∏!)!1(

)!(

ii

ii

i ng

ngW

−+

=∏→

1−=

iee

gn ii βεα

Bose-Einstein Distribution


!)!(

!

iii

i

i nng

gW

−=∏

Fermi-Dirac Distribution

1+=

iee

gn ii βεα

※ Photon Statistics

Quantum Statistics Quantum Statistics –– mean # of particles in a singlemean # of particles in a single--particle state sparticle state s

rr

r

R nnnnE εεεε ∑=+++= L332211

)( 332211 L+++−− ∑∑ == εεεββ nnn

R

E

R

eeQ R

Nnr

r

=∑


RR

s

nnn

sRnnn

R

nnn

s

Rs

Q

Qe

Qe

en

nεβεβ

εεεβεεεβ

εεεβ

∂∂

−=

∂∂

−== +++−+++−

+++−

∑∑

∑111 )(

)(

)(

332211

332211

332211

L

L

L

s

s

Qn

εβ ∂∂

−=ln1

Quantum Statistics Quantum Statistics –– Photon StatisticsPhoton Statistics

)( 332211 L+++−∑= εεεβ nnn

R

eQ

L

=

111Q

L

L

332211

21 ,,

εβεβεβ nnn

nn

eeeQ−−−∑=

L))()(( 33

3

22

2

11

1 000

εβεβεβ n

n

n

n

n

n

eeeQ−

=

−

=

−

=∑∑∑=


1

1

1

ln1

−=

−=

∂∂

−=−

−

ss

s

ee

eQn

s

s βεβε

βε

εβ

L

−

−

−

=−−− 321 1

1

1

1

1

1βεβεβε

eeeQ

)1(lnln reQr

βε−−−= ∑

Quantum Statistics Quantum Statistics –– BoseBose--Einstein/FermiEinstein/Fermi--Dirac StatisticsDirac Statistics

)(

)()(

,

212211),,(

µεβ

βµεεββµβµ

−−

++++−−

∑=

==

∑

∑∑kk

k

i

i

i

n

n

nnnn

n

NE

Ni

G

e

eeeeTVQLL

nk = 0, 1 Fermion

nk = 0, 1, 2, 3, …, ∞ Boson


)1()(

1

)(1

01

µεβµεβ −−

=

−−

==

− +== ∏∑∏ kkk

k

eeQk

n

nk

DF

G

nk = 0, 1, 2, 3, …, ∞ Boson

)(1

)(

01 1

1µεβ

µεβ−−

=

−−∞

==

−

−== ∏∑∏

k

kk

ke

eQk

n

nk

EB

G

Quantum Statistics Quantum Statistics –– BoseBose--Einstein/FermiEinstein/Fermi--Dirac StatisticsDirac Statistics

)1ln(ln)( µεβ −−− +=∑ keQ

k

DF

G

s

s

Qn

εβ ∂∂

−=ln1

)(1

1lnln

µεβ −−−

−=∑

keQ

k

EB

G

1)( −− µεβe


1

1

1)()(

)()(

)(

+=

+==

−−−

−−−−

µεβµεβ

µεβ

εεee

efn DFDF

1

1

1)(

)()(

)(

−=

−=

−−−

−−−

µεβµεβ

µεβ

εee

ef EB

ByeongByeong--Joo Lee Joo Lee · 2013-02-19 · Rotational motion Byeong-Joo Lee ... =∑ − + +...

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