thermo & stat mech - spring 2006 class 18 1 thermodynamics and statistical mechanics statistical...
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Thermo & Stat Mech - Spring 2006 Class 18
1
Thermodynamics and Statistical Mechanics
Statistical Distributions
Thermo & Stat Mech - Spring 2006 Class 18
2
Multiple Outcomes
NN
N
N
NNN
Nw
ii
i
!
!
!!!
!
321
Distinguishable particles
Thermo & Stat Mech - Spring 2006 Class 18
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.
Thermo & Stat Mech - Spring 2006 Class 18
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
Thermo & Stat Mech - Spring 2006 Class 18
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
Thermo & Stat Mech - Spring 2006 Class 18
6
Most Probable Distribution
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
Thermo & Stat Mech - Spring 2006 Class 18
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
Thermo & Stat Mech - Spring 2006 Class 18
8
Most Probable Distribution
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
Thermo & Stat Mech - Spring 2006 Class 18
9
Boltzmann Distribution
stateper particles ofNumber
ln
jj
j
jj
j
feg
N
g
N
j
Thermo & Stat Mech - Spring 2006 Class 18
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
Thermo & Stat Mech - Spring 2006 Class 18
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?
Thermo & Stat Mech - Spring 2006 Class 18
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(
Thermo & Stat Mech - Spring 2006 Class 18
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
Thermo & Stat Mech - Spring 2006 Class 18
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
Thermo & Stat Mech - Spring 2006 Class 18
15
Constraints (Lagrange Multipliers)
01ln
0ln
0ln
1
11
jj
j
n
jj
j
jjj
n
jjj
n
jjBE
N
g
N
gNN
NNw
Thermo & Stat Mech - Spring 2006 Class 18
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
Thermo & Stat Mech - Spring 2006 Class 18
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?
Thermo & Stat Mech - Spring 2006 Class 18
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
)!(!
!),,(
)!(!
!
Thermo & Stat Mech - Spring 2006 Class 18
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
Thermo & Stat Mech - Spring 2006 Class 18
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
Thermo & Stat Mech - Spring 2006 Class 18
22
Constraints (Lagrange Multipliers)
01ln
0ln
0ln
1
11
jj
j
n
jj
j
jjj
n
jjj
n
jjFD
N
g
N
NgN
NNw
Thermo & Stat Mech - Spring 2006 Class 18
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
Thermo & Stat Mech - Spring 2006 Class 18
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
Thermo & Stat Mech - Spring 2006 Class 18
25
Boltzmann Distribution
j
j
egeN
feg
N
g
N
jj
jj
j
jj
j
stateper particles ofNumber
ln
Thermo & Stat Mech - Spring 2006 Class 18
26
Boltzmann Distribution
n
jj
jj
n
jj
n
jj
n
jj
jj
j
j
j
j
j
eg
egNN
eg
Ne
egeNN
egeN
1
1
11
Thermo & Stat Mech - Spring 2006 Class 18
27
Partition Function
Z
egNN
Zeg
eg
egNN
j
j
j
j
jj
n
jj
n
jj
jj
FunctionPartition 1
1
Thermo & Stat Mech - Spring 2006 Class 18
28
Boltzmann Distribution
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
Thermo & Stat Mech - Spring 2006 Class 18
29
Ideal Gas
01
2/3
22
2/3
22
)(
2
4)(
1
2
4)(
dgeegZ
dmV
dg
dmV
dg
n
jj
j
Thermo & Stat Mech - Spring 2006 Class 18
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
Thermo & Stat Mech - Spring 2006 Class 18
31
Gamma Function
2
1
23
0
21
21
21
21
23
0
123
0
21
dxex
nnn
dxexndxex
x
xnx
Thermo & Stat Mech - Spring 2006 Class 18
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
Thermo & Stat Mech - Spring 2006 Class 18
33
Boltzmann Distribution
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
Thermo & Stat Mech - Spring 2006 Class 18
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
Thermo & Stat Mech - Spring 2006 Class 18
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.
Thermo & Stat Mech - Spring 2006 Class 18
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
Thermo & Stat Mech - Spring 2006 Class 18
37
Quantum Results
Dirac-Fermi
Einstein-Bose 1
1
kT
jj
j
j
ee
fg
N
Thermo & Stat Mech - Spring 2006 Class 18
38
Chemical Potential
Dirac-Fermi
Einstein-Bose 1
1
1
1
kTkT
jj
j
jj
eee
fg
NkT