ByeongByeong--Joo Lee Joo Lee - POSTECH

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

ByeongByeong--Joo LeeJoo Lee

POSTECH POSTECH -- MSEMSE

[email protected]@postech.ac.kr

Warming UpWarming Up –– Mathematical SkillsMathematical Skills

xxxxxxdxxxx

−≈+−=≈ ∫ ln1lnln)!(ln1

1. Stirling’s approximation

2. Evaluation of the Integral ∫∞

∞−

− dxe x2


π=∫∞

∞−

− dxe x2

2. Evaluation of the Integral ∫ ∞−

π2

1

0

2

=∫∞ − dxe x

adxe ax π

2

1

0

2

=∫∞ −

3. Lagrangian Undetermined Multiplier Method

Basic Concept of Statistical Mechanics Basic Concept of Statistical Mechanics –– Macro vs. MicroMacro vs. Micro

View Point

Macroscopic vs. Microscopic

State


State

Macrostate vs. Microstate

Particle in a Box Particle in a Box –– Microstates of a ParticleMicrostates of a Particle

ψψψ EVm

=+∇− 22

2

h

022

2 =+∇ ψψ Em

h

πnLmE

=2

2

h


2

222

2mL

nEn

hπ=

)(8

),,( 222

2

2

zyxzyx nnnmL

hnnnE ++=

n = 1, 2, 3, …

for 66 : 8,1,1 7,4,1, 5,5,4

System with particles System with particles –– Microstates of a SystemMicrostates of a System


Macrostate / Energy Levels / Microstates Macrostate / Energy Levels / Microstates ––


Scope and Fundamental Assumptions of Statistical Mechanics Scope and Fundamental Assumptions of Statistical Mechanics

▷each probable case defined by (n1, n2, …, nk) to make a macrostate

→ microstate

▷ mental collection of macrostates, quantum mechanically accessible

to a system → ensemble


all microstates in the same energy level have a same probability

Ensemble average = time average

Number of ways of distribution : in Number of ways of distribution : in kk cells with cells with ggii and and EEii

▷ 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

−=∏

Evaluation of the Most Probable Macrostate Evaluation of the Most Probable Macrostate –– BoltzmanBoltzman

)lnln(lnln iiii nngnNNW −+= ∑

0lnln max =

=∑

i

ii

n

gnW δδ

0=∑ nδ 0=∑ nαδ

kn

k

nn

k

gggnnn

NW )()()(

!!!

!21

21

21

LL

=


0=∑ inδ 0=∑ inαδ

0=∑ ii nδε 0=∑ ii nδβε

0ln =++

i

i

i

g

nβεα ieegn ii

βεα −−=

Z

N

eg

Ne

i

i

==∑ −

−βε

αieg

Z

Nn ii

βε−=

Evaluation of the Most Probable Macrostate Evaluation of the Most Probable Macrostate –– BB--E & FE & F--D D

!)!1(

)!1(

ii

ii

i ng

ngW

−−+

=∏!)!1(

)!(

ii

ii

i ng

ngW

−+

=∏→

1−=

iee

gn i

i βεα

Bose-Einstein Distribution


!)!(

!

iii

i

i nng

gW

−=∏

Fermi-Dirac Distribution

1+=

iee

gn i

i βεα

Definition of Entropy and Significance of Definition of Entropy and Significance of ββ

)(Ω= fS )()()( BABABA fffS Ω+Ω=Ω⋅Ω=−

▷ Consider an Isolated System composed of two part in thermal contact.

Equilibrium condition ?

Classical Thermodynamics → maximum entropy (S)

Statistical mechanics → maximum probability (Ω)

▷ S and Ω have a monotonic relation:


Ω= ln'kS

dUkdnP

Nkdnk

P

NdnkndnkdS iiiiiii βεββε 'ln'')ln('ln' =−=−=−= ∑∑∑ ∑

dNT

dVT

PdU

TdS

µ−+=

1

Tk

1' =β →

Tk '

1=β

Calculation of Macroscopic Properties from the Partition FunctionCalculation of Macroscopic Properties from the Partition Function

( )∑∑∑ +=

−=−=Ω= −ZkTnke

Znk

N

nnkkS ii

kT

ii

ii ln/

1ln)ln(ln

/ εε

ZNkT

US ln+=

ZNkTSTUF ln−=−≡

ZNkTZNk

FS

∂+=

∂−=

lnln


VV T

ZNkTZNk

T

FS

∂

∂+=

∂∂

−=ln

ln

VT

ZNkTU

∂

∂=

ln2

TT V

ZNkT

V

FP

∂

∂=

∂∂

−=ln

Ideal MonoIdeal Mono--Atomic GasAtomic Gas

kT

state

eZ//ε−∑=

)(8 2

2

2

2

2

22

c

n

b

n

a

n

m

h zyxi ++=ε

)/()8/()/()8/()/()8/( 222222222 cnmkThbnmkThanmkTh zyx

eeeZ−−−

∑∑∑=

kT

i

ilevel

energy

egZ//ε−∑=


eeeZ ∑∑∑=

z

cnmkTh

y

bnmkTh

x

anmkThdnednedneZ zyx )/()8/(

0

)/()8/(

0

)/()8/(

0

222222222 −∞−∞−∞

∫∫∫=

adxe ax π

2

1

0

2

=∫∞ −

2/3

2222

28

2

8

2

8

2

=

=

h

mkTV

h

mkTc

h

mkTb

h

mkTaZ

ππππ

Ideal MonoIdeal Mono--Atomic Gas Atomic Gas –– Evaluation of Evaluation of kk

++=2

2ln2

3ln2

3lnln

h

mkTVZ

π

V

nRT

VTNk

V

ZTNkP

T

≡⋅=

∂

∂=

1'

ln'

Z 33ln 22 ==

∂=

2/3

2

2

=h

mkTVZ

π


NkTT

NkTT

ZNkTU

V 2

3

2

3ln 22 ==

∂

∂=

Nkh

mkNkVNkTNk

T

PNkTPNkS

V 2

32ln

2

3lnln

2

3lnln

2+

++=

∂

∂+=

π

ov sVRTcs ++= lnln

for 1 mol of gas

Entropy Entropy –– S = S = kk ln Wln W


ByeongByeong--Joo Lee Joo Lee - POSTECH

Documents

Transcript of ByeongByeong--Joo Lee Joo Lee - POSTECH