Thermodynamics and Statistical Mechanics

Thermo & Stat Mech - Spring 2006 Class 18

1

Thermodynamics and Statistical Mechanics

Statistical Distributions


2

Multiple Outcomes

NN

N

N

NNN

Nw

ii

i

!

!

!!!

!

321

Distinguishable particles


3

Degenerate States

NN

N

gN

NNN

gggNw

n

jj

n

j j

Nj

NNN

B

j

1

1321

321

!!

!!!

! 321

Suppose there are gj states that have the same energy.


4

Boltzmann Statistics (Classical)

UN

NN

N

gN

NNN

gggNw

n

jjj

n

jj

n

j j

Nj

NNN

B

j

1

1

1321

321

!!

!!!

! 321


5

Most Probable Distribution

n

jj

n

jjj

n

jjjB

n

jj

n

jjjB

B

n

j j

Nj

BB

NNNgNNw

NgNNw

w

N

gNww

j

111

11

1

lnln!lnln

!ln ln!lnln

0ln :Instead

!! where0


6


n

j j

jjB

n

j j

jjjjjB

n

jjjjB

N

gNw

N

NNNgNw

NgNNw

1

1

1

ln)(ln

)1ln)(ln(ln

)1ln(ln!lnln


7

Constraints (Lagrange Multipliers)

0)()(ln)(

0ln

0ln

111

11

11

n

jjj

n

jj

n

j j

jj

n

jjj

n

jjB

n

jjj

n

jj

B

NNN

gN

NNw

UNNN

w


8


jj

j

jj

j

n

jj

j

jj

n

jjj

n

jj

n

j j

jj

g

N

N

g

N

gN

NNN

gN

ln

0ln

0ln)(

0)()(ln)(

1

111


9

Boltzmann Distribution

stateper particles ofNumber

ln

jj

j

jj

j

feg

N

g

N

j


10

Quantum Statistics

Indistinguishable particles.

1. Bose-Einstein – Any number of particles per state. Particles with integer spin:0,1,2, etc

2. Fermi-Dirac – Only one particle per state: Particles with integer plus ½ spin: 1/2, 3/2, etc


11

Bose-Einstein

At energy i there are Ni particles divided among gi states. How many ways can they be distributed? Consider Ni particles and gi – 1 barriers between states, a total of Ni + gi – 1 objects to be arranged. How many arrangements?


12

Bose-Einstein

n

jj

n

jj

n

jjjBE

n

j jj

jjnBE

jj

jjj

gNgNw

gN

gNNNNw

gN

gNw

111

121

)!1ln(!ln)!1ln(ln

)!1(!

)!1(),,(

)!1(!

)!1(


13

Bose-Einstein

n

jjjj

n

jjjj

n

jjjjjjjBE

n

jj

n

jj

n

jjjBE

ggg

NNN

gNgNgNw

gNgNw

1

1

1

111

)]1()1ln()1[(

]ln[

)]1()1ln()1[(ln

)!1ln(!ln)!1ln(ln


14

Bose-Einstein

n

j j

jjjBE

n

j j

jj

jj

jjjjjBE

jj

n

jjjjjjjBE

N

gNNw

N

NN

gN

gNgNNw

gg

NNgNgNw

1

1

1

lnln

ln)1(

)1()1ln(ln

)]1ln()1(

ln)1ln()1[(ln


15


01ln

0ln

0ln

1

11

jj

j

n

jj

j

jjj

n

jjj

n

jjBE

N

g

N

gNN

NNw


16

Bose-Einstein

jj

j

j

j

j

j

jj

j

feg

N

eN

ge

N

g

N

g

j

jj

1

1

11

01ln


17


j

j

efe

g

N

g

N

jj

j

jj

j

1

ln


18

Fermi-Dirac

At energy i there are Ni particles divided among gi states, but only one per state. gi Ni.

How many ways can the Ni occupied states be selected from the gi states?


19

Fermi-Dirac

n

jjj

n

jj

n

jjFD

n

j jjj

jnFD

jjj

jj

NgNgw

NgN

gNNNw

NgN

gw

111

121

)!ln(!ln!lnln

)!(!

!),,(

)!(!

!


20

Fermi-Dirac

n

jjjjjjjjjFD

jjjjjj

n

jjjjjjjFD

n

jjj

n

jj

n

jjFD

NgNgNNggw

NgNgNg

NNNgggw

NgNgw

1

1

111

)]ln()(lnln[ln

)]()ln()(

lnln[ln

)!ln(!ln!lnln


21

Fermi-Dirac

n

j j

jjjFD

n

j jj

jjjj

j

jjjFD

n

jjjjjjjjjFD

N

NgNw

Ng

NgNg

N

NNNw

NgNgNNggw

1

1

1

lnln

)(

)()ln(lnln

)]ln()(lnln[ln


22


01ln

0ln

0ln

1

11

jj

j

n

jj

j

jjj

n

jjj

n

jjFD

N

g

N

NgN

NNw


23

Fermi-Dirac

jj

j

j

j

j

j

jj

j

feg

N

eN

ge

N

g

N

g

j

jj

1

1

11

01ln


24

Distributions

Dirac-Fermi 1

1

Einstein-Bose 1

1

Boltzmann 1

jj

j

jj

j

jj

j

feg

N

feg

N

feg

N

j

j

j


25


j

j

egeN

feg

N

g

N

jj

jj

j

jj

j

stateper particles ofNumber

ln


26


n

jj

jj

n

jj

n

jj

n

jj

jj

j

j

j

j

j

eg

egNN

eg

Ne

egeNN

egeN

1

1

11


27

Partition Function

Z

egNN

Zeg

eg

egNN

j

j

j

j

jj

n

jj

n

jj

jj

FunctionPartition 1

1


28


Z

Z

Z

N

U

eg

eg

NUN

eg

egNN

n

jj

n

jjjn

jjj

n

jj

jj

j

j

j

j

ln

1

1

1

1


29

Ideal Gas

01

2/3

22

2/3

22

)(

2

4)(

1

2

4)(

dgeegZ

dmV

dg

dmV

dg

n

jj

j


30

Ideal Gas

0

212/3

2/3

22

0

212/3

2/3

22

0

21

2/3

220

2/3

22

12

4

)()(12

4

2

4)(

2

4)(

dxexmV

Z

demV

Z

demV

dgeZ

dmV

dg

x


31

Gamma Function

2

1

23

0

21

21

21

21

23

0

123

0

21

dxex

nnn

dxexndxex

x

xnx


32

Partition Function for Ideal Gas

N

UZ

CZ

CmVZ

dxexmV

Z x

2

3ln

ln2

3lnln

1

2

2

4

12

4

2/32/3

2/3

22

0

212/3

2/3

22


33


Number Occupation

1

2

3

2

3

Z

Ne

g

Nf

Z

egN

Z

egNN

kT

kTN

U

kT

j

jj

kTjj

j

j

j

j


34

Ideal Gas

2/3

2

2/3

223

32/3

2/3

22

2/3

2/3

22

2

2

8

)2()(

2

2

4

1

2

2

4

h

mkTVZ

h

mkTVkT

mVZ

mVZ


35

Quantum Statistics

When taken to classical limit quantum results must agree with classical. B-E and F-D must approach Boltzmann in classical limit. What is that limit?

Low particle density! Then distinguishability is not a factor.


36

Classical limit

kTβ

ef

ef

ef

g

N

j

j

j

j

j

jj

j

1

Boltzmann as Same 1

1,1For

1

1


37

Quantum Results

Dirac-Fermi

Einstein-Bose 1

1

kT

jj

j

j

ee

fg

N


38

Chemical Potential

Dirac-Fermi

Einstein-Bose 1

1

1

1

kTkT

jj

j

jj

eee

fg

NkT


39

Three Distributions

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.05 0.1 0.15 0.2 0.25

E(eV)

M(T1)

BE(T1)

FD(T1)

Thermodynamics and Statistical Mechanics

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