2. Outline of Statistical Mechanics
2
Objective: To treat an ensemble of particles or system in a statistical or probabilistic fashion, in
which we are concerned with the most probable values of the properties of the ensemble without
investigating in detail what the value of that properties may be for a particular particle at any time.
We limit our discussion to an assembly of identical particles that are independent of each other and
can only interact via instantaneous processes that conserve energy and momentum.
Phase space: A six dimensional phase space is defined by the coordinates (x,y,z,kx,ky,kz) where
x,y,z refer to real space and kx,ky,kz refer to momentum space.
Basic Postulate :The apriori probability for a system to be in any quantum state is the same for all
quantum states of the system. This is true only when there are no dynamical restrictions. The apriori
probabilities will be modified by external constraints, such as the number of particles is a constant and
the total energy is constant.
Different types of systems considered:
Distinguishable particles (Fermions when spin is not considered)
Indistinguishable particles that obey Pauli exclusion principle (Fermions)
The role of indistinguishability of elementary particles (Bosons)
3.
dE
E
g
E
f
E
N
dE
E
N
dE
E
N
E
)
(
)
(
)
(
1
)
(
)
(
)
(
)]
(
2
[
)]
(
2
[
2
1
2
2
i
i
i
i
i
i q
V
m
p
q
mV
p
m
H
3
Thus the type of distribution function depends upon
whether the number of particles is constant or not
whether particles are indistinguishable
whether particles are quantum like or not
The Distribution Function and DOS
Definitions:
f(E) is the probability that a state at energy E is occupied by a particle
g(E)dE is the number of available states in the energy range E and E+dE
Number of particles between E and E+dE is given by N(E)dE=f(E)g(E)dE
The average value of any quantity α is given by
In general α can be a function of the system coordinates qi (x,y,z for i=1,2,3) and pi (px,py,pz).
For example the Hamiltonian of a system is in general,
4. 4
Calculation of the 3D Density of states function
We solve the 3D SWE for free particles confined in a box with dimensions x0,y0,z0.The TISE is then of the
form
)
.
.
.
(
.
2
2
2
2
2
2
)
,
,
(
0
)
0
(
0
2
2
z
x
x k
z
k
y
k
x
i
r
ik
Ae
Ae
z
y
x
k
V
m
k
m
The Boundary conditions imposed cause the values of kx, ky and kz to be quantized as there is
confinement.
Thus this results in quantization of energy: Only discrete values of energy are allowed. The
allowed values are :
0
2 ,
,
,
,
x
n
k
z
y
x
z
y
x
)
2
0
2
2
0
2
2
0
2
(
2
2
2
)
,
,
(
z
z
n
y
y
n
x
x
n
m
z
n
y
n
x
n
E
5. In momentum space, this Energy can be represented by a sphere, whose radius is given by
2
2
mE
k
dk
k
dV 2
4
5
If we draw another sphere E+dE then the volume of the shell between these two spheres will be
The number of quantum states found in this volume of momentum space is
dk
k
V
V
dk
k
dE
E
g 2
2
3
2
2
)
2
(
4
)
(
For free particle, dk
m
k
dE
m
k
E
2
2
2
;
2
E
m
V
dE
mE
m
V
dE
E
g
2
3
2
2
2
2
2
2
2
2
2
)
(
E
m
V
E
g
2
3
2
2
2
2
)
(
Including corrections for spin, we see that since only the volume appears, the same DOS would be obtained
for a 3D system of any shape.
Therefore,
6. Maxwell Boltzmann
Particles are identifiable
and distinguishable
The number of particles is
constant
The total Energy is constant
Spin is ignored
Fermi Dirac
Particles are indistinguishable
Particles obey Pauli principle
Each state can have only one
particle.
Each particle has one half spin
Bose Einstein
Particles are indistinguishable
Particles do not obey Pauli
principle
Each state can have more than
one particle, like phonons and
photons
Particles have integral spin
!
2
!
1
)!
2
1
(
)
2
1
(
)
2
,
1
( 1
N
N
N
N
N
N
N
N
Q N
C
6
E1 E2 E3 ..………….. EN Energy levels
N1 N2 N3 ………………… NN # of particles
N1E1 N2E2 N3E3 ……………. NNEN Energy in each level
U
E
N
N
N
i
i
i
i
i
The distribution which has the maximum probability of occurrence is one which can be realized
in a maximum number of statistically independent ways. This is analogous to putting
numbered objects into a set of numbered containers.
Start with two boxes #1, #2.Let us denote Q(N1,N2) as the number of statistically independent
ways of putting N1+N2 objects in two boxes such that one of them contains N1 and the other
N2 objects. N1 N2
The Maxwell Boltzmann Distribution
7.
n
i
i
N
N
Nn
N
N
N
Q
1
!
)
..
,.........
3
,
2
,
1
(
!
2
!
1
)!
2
1
(
)
2
1
(
)
2
,
1
( 1
V
V
V
V
V
V
V
V
Q V
C
!
2
!
1
!
1
!
)
2
,
1
(
*
)
2
,
1
(
)
2
,
1
,
1
(
V
V
N
N
V
V
Q
N
N
Q
V
V
N
Q
i
N
n
i
i
n
i
i
g
N
N
Nn
N
N
N
Q
1
1
!
)
..
,.........
3
,
2
,
1
(
7
Suppose now that the second container is divided into
2 compartments containing v1 and v2 objects. Thus
N2=v1+v2.
We may think of this as having 3 distinct distributions of N1,V1 and V2 objects. In that case,
Generalizing this expression, we get,
Modeling Degeneracy Now if each of the containers were actually a group of containers, say gi,
then there will be an additional gi
Ni
ways of distributing these Ni particles among the gi containers. Thus the
total number of ways will be modified according as,
const
U
E
N
const
N
N
i
i
i
i
i
We now apply, the other two assumptions, namely
conservation of particles and energy.
Method of Lagrangean multipliers: If one wants to find the
maximum of f under the restriction that some other functions which
remain constant, independent of the choice of
0
f
.
,
0
)
...(
..........
2
)
2
2
2
(
1
)
1
1
1
(
dXn
Xn
Xn
Xn
f
dX
X
X
X
f
dX
X
X
X
f
0
)
(
],
,
1
[
dXi
Xi
Xi
Xi
f
n
i
n equations
8. 8
const
Xn
X
X
const
Xn
X
X
0
0
)
,.....
2
,
1
(
)
,.....
2
,
1
( Total of (n+2)
equations.
n
i
i
n
i
i
i N
g
N
N
Q
1
1
!
ln
ln
!
ln
ln
Now applying the method of Lagrangean
multipliers to lnQ we have,
We have from the definition of Q
Using Stirling’s approximation, we have:
Then,
X
X
X
X
ln
!
ln
n
i
i
i
i
n
i
i
i N
N
N
g
N
N
Q
1
1
)
ln
(
ln
!
ln
ln
n
i
i
i
i
j
n
i
i
i
j
j
N
N
N
N
g
N
N
Q
N 1
1
)
ln
(
ln
0
ln
)
ln(
ln
ln
]
1
1
[ln
ln
j
j
j
j
j
j
N
g
N
g
N
g
0
ln
1
1
n
i
i
i
j
n
i
i
j
j
N
E
N
N
N
Q
N
T
k
E
E
j
j
j
E
j
j
j
j
j
j
j
j
B
j
j
j
e
e
e
e
g
N
E
f
e
g
N
E
g
N
E
N
g
or
)
(
)
ln(
0
)
ln(
,
This particular energy distribution obtained under the
classical assumption of identifiable particles without
considering Pauli principle, is called Maxwell Boltzmann
Distribution.α, can be expressed in terms of total number
of particles.
T
K
E
n
j
j
T
K
E
n
j
j
n
j
j
B
j
B
j
e
g
N
e
e
g
e
N
N
1
1
1
If the energy levels are packed very closely, we can
replace the summation by an integral and so,
dE
e
E
g
N
e
dE
e
E
g
e
dE
E
N
N
T
K
E
T
K
E
B
B
)
(
)
(
)
(
9. 9
Fermi Dirac Distribution: In this case since the particles are indistinguishable, there is only one way of
distributing them among the two boxes. Therefore, Q(N1,N2)=1. Now along with this there are other
constraints like the conservation of particles and Total energy.
N1 N2
const
U
E
N
const
N
N
i
i
i
i
i
Consider now the i th
energy level with degeneracy gi. For this level, the total
#of ways of arranging the particles is:
)!
(
!
)
1
)......(
2
)(
1
(
i
i
i
i
i
i
i
i
N
g
g
N
g
g
g
g
Now the Ni particles can have Ni! Permutations and yet not give rise
to any new arrangement as they are indistinguishable. Therefore we
have to divide the number of possible ways of distributing the
particles by this amount.
n
i i
i
i
i
n
N
g
N
g
N
N
N
N
Q
So
1
3
2
1
)!
(
!
!
)
,.......
,
,
(
,
Now we proceed in the standard fashion, by applying
Stirling’s approximation to lnQ, and then using the
method of Lagrange multipliers to maximize Q.
T
K
E
E
j
j
j
E
j
j
B
E
j
j
B
f
j
j
j
e
g
N
E
f
e
g
N
T
K
e
N
g
Solving
/
)
(
)
(
)
(
1
1
)
(
1
1
1
;
1
)
(
,
Now we impose the other restrictions like
conservation of particles and total energy of the
system and obtain the other two functions to apply
the Lagrange method.
The Fermi Dirac Distribution
Now if the energy levels crowd in a continuum, then
T
K
E
E B
f
e
/
)
(
j
j
1
g(E)dE
g(E)f(E)dE
N(E)dE
g(E)dE
g
and
N(E)dE,
N
The Fermi energy is a function of
temperature and also depends upon the
DOS of the system
10. 10
Bose Einstein Distribution: Consider an array of Ni particles and (gi-1) partitions needed to divide
them into gi groups. The number of ways of permuting Ni particles among gi levels equals the number of
ways of permuting objects and partitions. i.e. (Ni+gi-1)!. Now the particles and the partitions are
indistinguishable, the number of ways of permuting them is
n
i i
i
i
i
BE
g
N
g
N
Nn
N
N
Q
1
1
1
)!
1
(
!
)!
1
(
)
.........
,
(
Now we impose the other restrictions like
conservation of particles and total energy of the
system and obtain the other two functions to apply
the Lagrange method.
Now we proceed in the standard fashion, by applying
Stirling’s approximation to lnQ, and then using the
method of Lagrange multipliers to maximize Q.
1
1
)
(
1
1
1
;
1
)
(
,
/
)
(
)
(
e
e
g
N
E
f
e
g
N
T
K
e
N
g
Solving
T
K
E
j
j
j
E
j
j
B
E
j
j
B
j
j
j
The solution for the case where the total number of
particles is not conserved, but only the energy is
conserved can easily be obtained by setting α=0
The Bose Einstein Distribution
11. 11
Applications: Maxwell Boltzmann statistics of an ideal gas
We now discuss the properties of an ideal gas of free particles, for which
E
m
V
E
g
2
3
2
2
2
2
)
(
For this particular case,
I
N
dE
e
E
g
N
e
T
K
E
B
0
)
(
and the integral is evaluated as a gamma function as,
2
3
2
2
3
2
2
0
2
3
2
2
0
2
3
2
2
0
)
2
(
2
2
2
2
2
2
2
2
)
(
T
mK
V
N
e
T
mK
V
du
e
u
T
mK
V
I
dE
e
E
m
V
dE
e
E
g
I
B
B
u
B
T
K
E
T
K
E
B
B
To summarize, the MB distribution function for an ideal
gas is
T
K
E
B
B
e
T
mK
V
N
E
f
2
3
2
)
2
(
2
)
(
The distribution of particle density with energy is given by
dE
e
E
T
K
N
dE
e
E
T
K
N
dE
E
N
dE
e
T
mK
V
N
E
m
V
dE
E
f
E
g
dE
E
N
T
K
E
B
T
K
E
B
T
K
E
B
B
B
B
2
3
)
(
2
)
4
(
4
)
(
)
2
(
2
2
2
)
(
)
(
)
(
2
3
2
2
3
2
2
3
2
2
Now the total internal energy of the
system can be got as follows
T
NK
U
du
e
u
du
e
u
du
e
u
T
K
T
K
N
U
dE
e
E
E
T
K
N
dE
E
EN
U
B
u
u
u
B
B
T
K
E
B
B
2
3
;
4
3
)
2
3
(
2
3
2
3
)
2
5
(
)
(
)
(
2
)
(
2
)
(
0
0
2
3
0
2
3
0 0
2
3
2
3
2
3
T
K
U B
2
3
Thus the average energy of the
particles of the system is given
by,
12. 12
Applications, contd : Our aim is to derive the equations of state for an ideal Boltzmann gas from the
dynamical properties and from the distribution. For this purpose, we need to convert the energy distribution
function into a velocity distribution, using the relation, ,for free particles and parabolic
dispersion. Now we have,
dv
e
v
T
K
m
N
mvdv
ve
m
T
K
N
dv
v
N
mvdv
dE
T
K
mv
B
T
K
mv
B
B
B 2
2
2
2
2
3
2
2
3
)
2
(
4
2
)
(
2
)
(
,
m
k
mv
E
2
2
1 2
2
2
This distribution expresses the # of
particles whose velocities lie in the
range v and v+dv.
Since,
We have,
x
T
K
v
m
B
x
x
z
y
x
T
K
v
v
v
m
B
z
y
x
z
y
x
dv
e
T
K
m
N
dv
v
N
dv
dv
dv
e
T
K
m
N
dv
dv
dv
v
v
v
N
B
x
B
z
y
x
2
)
(
2
)
(
2
2
1
2
2
2
2
3
)
2
(
)
(
)
2
(
)
,
,
(
z
y
x dv
dv
dv
dv
v
2
4
The above derived distributions are equilibrium
distributions : If we apply an external field, they
will change shape.
Applications Fermi Dirac Distribution 3D SYSTEM
The DOS is given by E
m
V
E
g
2
3
2
2
2
2
)
(
The Electron density is then
)
(
0
2
1
2
1
2
3
2
2
/
)
(
0
0
2
3
2
2
1
)
(
);
(
2
2
1
1
2
2
1
)
(
)
(
'
f
B
f
u
f
f
B
T
K
E
E
e
du
u
F
F
T
mK
n
e
dE
E
m
dE
E
f
E
g
V
N
n
The total internal energy of the system is given by,
)
(
2
2
1
)
(
2
2
1
2
2
)
(
2
3
2
3
2
2
)
(
0
2
3
2
3
2
2
0 0
/
)
(
2
3
2
2
2
5
f
B
B
u
B
T
K
E
E
F
T
mK
V
T
K
U
e
du
u
T
K
m
V
U
e
dE
E
E
m
V
dE
E
EN
U
f
B
f
13. 13
The average internal energy per particle is then,
)
(
)
(
2
1
2
3
f
f
F
F
KBT
nV
U
N
U
u
At low temperatures, the Fermi Dirac Distribution
may be represented as a sphere in Momentum
space in which all or most of the states of
energy less than Ef are filled , while those
greater than Ef are empty. The equation of the
sphere is
f
z
y
x E
m
p
p
p *
2
2
2
2
For energies much greater than
the Ef, the FD Distribution can be
written as T
K
E
E B
f
e
E
f
/
)
(
)
(
If all the energies of the system are such
that E-Ef>>KBT, the system is called Non
degenerate and the FD system reduces to
a MB system
Applications Fermi Dirac Distribution 2D SYSTEM
The DOS is given by , A being
the area of the 2D container. The total electron number
is given by,
2
)
(
m
A
E
g
)
1
(
ln
)
1
(
ln
)
1
(
ln
1
1
1
)
(
)
(
/
2
2
/
2
/
2
/
2
2
0
/
)
(
2
0
T
K
E
B
D
T
K
E
B
T
K
E
u
B
T
K
E
u
u
B
E
u
B
T
K
E
E
B
f
B
f
B
f
B
f
f
B
f
e
T
mK
n
e
T
mK
A
e
T
mK
A
N
e
du
e
T
mK
A
e
du
T
mK
A
N
e
dE
m
A
dE
E
f
E
g
N
Sheet electron density
Note on Fermi Energy:
As Temperature increases, . This
means that under high temperature limit, the
Fermi Dirac statistics reduces to Maxwell
Boltzmann.
At lower temperatures, the above will occur in
gases where the masses are large
For dense gas of very light particles (free
electrons in a metal), the Fermi Energy is very
large and the condition, E-Ef>>KBT is practically
never satisfied.
In semiconductors, due to peculiar form of the
DOS function, the MB distribution is virtually
always a good approximation to the FD
distribution
)
(T
Ef
![
dE
E
g
E
f
E
N
dE
E
N
dE
E
N
E
)
(
)
(
)
(
1
)
(
)
(
)
(
)]
(
2
[
)]
(
2
[
2
1
2
2
i
i
i
i
i
i q
V
m
p
q
mV
p
m
H
3
Thus the type of distribution function depends upon
whether the number of particles is constant or not
whether particles are indistinguishable
whether particles are quantum like or not
The Distribution Function and DOS
Definitions:
f(E) is the probability that a state at energy E is occupied by a particle
g(E)dE is the number of available states in the energy range E and E+dE
Number of particles between E and E+dE is given by N(E)dE=f(E)g(E)dE
The average value of any quantity α is given by
In general α can be a function of the system coordinates qi (x,y,z for i=1,2,3) and pi (px,py,pz).
For example the Hamiltonian of a system is in general,](https://image.slidesharecdn.com/carrierstatisticsmechanicalpriciple-250811050728-575165dc/85/Carrier-Statistics-mechanical-priciple-ppt-3-320.jpg)
![
n
i
i
N
N
Nn
N
N
N
Q
1
!
)
..
,.........
3
,
2
,
1
(
!
2
!
1
)!
2
1
(
)
2
1
(
)
2
,
1
( 1
V
V
V
V
V
V
V
V
Q V
C
!
2
!
1
!
1
!
)
2
,
1
(
*
)
2
,
1
(
)
2
,
1
,
1
(
V
V
N
N
V
V
Q
N
N
Q
V
V
N
Q
i
N
n
i
i
n
i
i
g
N
N
Nn
N
N
N
Q
1
1
!
)
..
,.........
3
,
2
,
1
(
7
Suppose now that the second container is divided into
2 compartments containing v1 and v2 objects. Thus
N2=v1+v2.
We may think of this as having 3 distinct distributions of N1,V1 and V2 objects. In that case,
Generalizing this expression, we get,
Modeling Degeneracy Now if each of the containers were actually a group of containers, say gi,
then there will be an additional gi
Ni
ways of distributing these Ni particles among the gi containers. Thus the
total number of ways will be modified according as,
const
U
E
N
const
N
N
i
i
i
i
i
We now apply, the other two assumptions, namely
conservation of particles and energy.
Method of Lagrangean multipliers: If one wants to find the
maximum of f under the restriction that some other functions which
remain constant, independent of the choice of
0
f
.
,
0
)
...(
..........
2
)
2
2
2
(
1
)
1
1
1
(
dXn
Xn
Xn
Xn
f
dX
X
X
X
f
dX
X
X
X
f
0
)
(
],
,
1
[
dXi
Xi
Xi
Xi
f
n
i
n equations](https://image.slidesharecdn.com/carrierstatisticsmechanicalpriciple-250811050728-575165dc/85/Carrier-Statistics-mechanical-priciple-ppt-7-320.jpg)
![8
const
Xn
X
X
const
Xn
X
X
0
0
)
,.....
2
,
1
(
)
,.....
2
,
1
( Total of (n+2)
equations.
n
i
i
n
i
i
i N
g
N
N
Q
1
1
!
ln
ln
!
ln
ln
Now applying the method of Lagrangean
multipliers to lnQ we have,
We have from the definition of Q
Using Stirling’s approximation, we have:
Then,
X
X
X
X
ln
!
ln
n
i
i
i
i
n
i
i
i N
N
N
g
N
N
Q
1
1
)
ln
(
ln
!
ln
ln
n
i
i
i
i
j
n
i
i
i
j
j
N
N
N
N
g
N
N
Q
N 1
1
)
ln
(
ln
0
ln
)
ln(
ln
ln
]
1
1
[ln
ln
j
j
j
j
j
j
N
g
N
g
N
g
0
ln
1
1
n
i
i
i
j
n
i
i
j
j
N
E
N
N
N
Q
N
T
k
E
E
j
j
j
E
j
j
j
j
j
j
j
j
B
j
j
j
e
e
e
e
g
N
E
f
e
g
N
E
g
N
E
N
g
or
)
(
)
ln(
0
)
ln(
,
This particular energy distribution obtained under the
classical assumption of identifiable particles without
considering Pauli principle, is called Maxwell Boltzmann
Distribution.α, can be expressed in terms of total number
of particles.
T
K
E
n
j
j
T
K
E
n
j
j
n
j
j
B
j
B
j
e
g
N
e
e
g
e
N
N
1
1
1
If the energy levels are packed very closely, we can
replace the summation by an integral and so,
dE
e
E
g
N
e
dE
e
E
g
e
dE
E
N
N
T
K
E
T
K
E
B
B
)
(
)
(
)
(
](https://image.slidesharecdn.com/carrierstatisticsmechanicalpriciple-250811050728-575165dc/85/Carrier-Statistics-mechanical-priciple-ppt-8-320.jpg)
