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 +=
Byeong-Joo Leewww.postech.ac.kr/~calphad
"'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−
∑=
Byeong-Joo Leewww.postech.ac.kr/~calphad
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
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
Byeong-Joo Leewww.postech.ac.kr/~calphad
−=
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:
Byeong-Joo Leewww.postech.ac.kr/~calphad
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
Byeong-Joo Leewww.postech.ac.kr/~calphad
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 ∑−=
0lnln max =−= ∑ ii ndnWd
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:
Byeong-Joo Leewww.postech.ac.kr/~calphad
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
,
Byeong-Joo Leewww.postech.ac.kr/~calphad
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
==π
Byeong-Joo Leewww.postech.ac.kr/~calphad
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
Byeong-Joo Leewww.postech.ac.kr/~calphad
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 ∑−=
0lnln max =−= ∑ ii ndnWd
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 ,
,
~ ∑>=<
Byeong-Joo Leewww.postech.ac.kr/~calphad
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 =
Byeong-Joo Leewww.postech.ac.kr/~calphad
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
Byeong-Joo Leewww.postech.ac.kr/~calphad
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)
Byeong-Joo Leewww.postech.ac.kr/~calphad
▷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
−−+
Byeong-Joo Leewww.postech.ac.kr/~calphad
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
Byeong-Joo Leewww.postech.ac.kr/~calphad
!)!(
!
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
=∑
Byeong-Joo Leewww.postech.ac.kr/~calphad
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−
=
−
=
−
=∑∑∑=
Byeong-Joo Leewww.postech.ac.kr/~calphad
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
Byeong-Joo Leewww.postech.ac.kr/~calphad
)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
Byeong-Joo Leewww.postech.ac.kr/~calphad
1
1
1)()(
)()(
)(
+=
+==
−−−
−−−−
µεβµεβ
µεβ
εεee
efn DFDF
1
1
1)(
)()(
)(
−=
−=
−−−
−−−
µεβµεβ
µεβ
εee
ef EB
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