Classical & Quantum Statistics

Different physical situation encountered in nature are described by three types of statistics-Maxwell-Boltzmann Statistics, Bose-Einstein Statistics and Fermi-Dirac Statistics

1. Dr.Anjali Devi JS
Contract Faculty, Mahatma Gandhi University, Kottayam, Kerala
Classical & Quantum Statistics

2. Types of Statistics
Different physical situation encountered in nature are
described by three types of statistics:
(1) Maxwell-Boltzmann (or M-B) statistics
(2) Bose-Einstein (or B-E) statistics
(3) Fermi Dirac Statistics
(Classical
Statistics)
(Quantum
Statistics)

3. (1) Maxwell-Boltzmann (or M-B) statistics
Particles are assumed to be distinguishable.
Any number of particles may occupy the same energy level.
Particles obeying M-B statistics are called boltzonsor maxwellons.
Example: Molecules of a gas.

4. (2) Bose-Einstein (or B-E) statistics
Particles are indistinguishable.
Any number of particles may occupy the same energy level.
Particles obeying B-E statistics are called bosons.
These particles have integral spin.
Example: hydrogen (H
2), deuterium (D
2), nitrogen (N
2), helium-4 (He
4
),
photons.

5. (3) Fermi-Dirac (or F-D) statistics
Particles are indistinguishable.
But only one particle may occupy a given energy level.
These particles have half-integral spin.
Particles obeying F-D statistics are called fermions.
Example: protons, electrons, helium-3, and nitric oxide (NO).

6. Fermions are those species whose wave functions are anti-
symmetricwith respect to the exchange of particles .
Bosonsare those species whose wave functions are symmetric with
respect to the exchange of particles .

7. Maxwell Boltzmann Statistics (classical law)
This law states that, the total fixed amount of energy is distributed
among various members of an assembly of identical particles.
??????
0
??????
1
??????
2 Instantaneous configuration
of the system [5,0,0]
A general configuration [N
0,N
1,N
2..] can be achieved in W
different ways, where Wis called weight of the configuration.
W=
??????!
??????0!??????1!??????2!….

8. Question
Calculate the weight of the configuration in which 20 objects are
distributed in the arrangement 0, 1,5, 0, 8,0,3,2,0,1
Configuration =[0,1,5,0,8,0,3,2,0,1]
N =0+1+5+0+8+0+3+2+0+1= 20
Answer: 4.19×10
10
W=
??????!
??????0!??????1!??????2!….

9. Boltzmann Distribution Law
The Maxwell Boltzmann law
The number of particles in the configuration of greatest weight (i.e.,
most probable distribution for a microstate) depends on energy of
the state by ??????
�=
�
??????
�
(�+�??????
??????
)
For the search of maximum value of W (i.e., configuration in the
greatest weight), we must also ensure that configuration also
satisfies the condition:
Constant total energy:
�??????
�??????
�=??????
Constant total number of molecules:
�??????
�=??????
�
�-degeneracy,
�,�-Undetermined multipliers

10. �=
−??????
??????
�??????
�=
1
�??????
�-Boltzmann constant,
T-Absolute temperature
E
F
-
Fermi Energy
�,�-Undetermined multipliers

11. Concepts
Fermi Level: The term used to describe the top of the collection of
electron energy levels at absolute zero temperature.
Electrons are Fermions and by Pauli exclusion principle cannot exist
in identical energy states.
Fermi Energy: This is the maximum energy that an electron can
have in a conductor at 0K. It is given by,
??????
??????=
1
2
��
??????
2
Where �
??????is the velocity of electron at Fermi level.

12. Fermi Dirac Distribution-Derivation
W=
�
??????!
�
??????−�
??????�
??????!�
??????�
??????!
�
�-Degeneracy, �
�-probability of occupying a state at energy ??????
�
The number of possible ways-called configurations-to fit �
��
�electrons in
�
�states, given the restriction that only one electron in �
�states, given the
restriction that only one electron can occupy each state, equals:

13. Thisequationisobtainedbynumberingtheindividualstatesand
exchangingthestatesratherthantheelectrons.Thisyieldsatotal
numberof�
�possibleconfigurations.Howevertheemptystatesareall
identical,weneedtodividebythenumberofpermutationsbetween
theemptystates,asallpermutationscannotbedistinguishedandcan
thereforeonlybecountedonce.Inaddition,allthefilledstatesare
indistinguishablefromeachother,soweneedtodividealsoby
permutationsbetweenthefilledstatesnamely�
��
�!
Fermi Dirac Distribution-Derivation

14. Multiplicity function
The number of possible ways to fit the electrons in
the number of available states is called the
multiplicity function.
The multiplicity function for the whole system is the
product of the multiplicity functions for each energy??????
�
Fermi Dirac Distribution-Derivation

15. Fermi Dirac Distribution-Derivation
Using Stirling’s approximation, one can eliminate the
factorial signs, yielding:
ln??????=
�ln??????
�=
�[�
�ln�
�−�
�1−�
���(�
�−�
��
�)−�
��
����
��
�]
The total number of particles =N
The total energy of these N electrons = E
These system parameters are related to the number of states at
each energy, �
�and the probability of occupancy of each state, �
�,
by:
N=
��
��
�E=
�??????
��
��
�

16. According to basic assumption of statistical thermodynamics, all
possible configurations are equally probable. The multiplicity function
provides the number of configurations for a specific set of occupancy
probabilities, �
�. The multiplicity function sharply peaks the thermal
equilibrium distribution. The occupancy probability in thermal
equilibrium is therefore obtained by finding the maximum of the
multiplicity function, W, while keeping the total energy and the
number of electrons constant.
Maximise logarithm of multiplicity function using Lagrange method:
ln W-�
��
��
�−�
�??????
��
�
Fermi Dirac Distribution-Derivation

17. The number of particles in the configuration of greatest weight (i.e.,
most probable distribution for a microstate) depends on energy of
the state by ??????
�=
�
??????
�
(�+�??????
??????
)
+1
�
�-degeneracy,
�,�-Undetermined multipliers
Fermi Dirac Distribution
Law

18. Example
Let us consider two particles a and A. Let if, these two particles
occupy the three energy levels (1,2,3). The number of ways of
arranging the particles 3
1
=3 (not more than one particle can be in
any one state)
Energy Level Distribution
1 a A -
2 a - A
3 - A a

19. Bose Einstein Distribution Law
Applies to a weakly interacting gas of indistinguishable
bosons with:
Each group ??????has�
�������,�
�-1 possible subgroups,??????
�to
be shared between them.
Constant total energy:
�??????
�??????
�=??????
Constant total number of molecules:
�??????
�=??????
No Pauli Exclusion Principle:??????
�≥0,���??????�??????���

20. Each group ??????has�
�������,�
�-1 possible subgroups,??????
�to
be shared between them.
Number of combination to do this is:
??????
�+�
�−1!
??????
�!�
�−1!
The number of microstates in distribution N
istates
??????=
�
??????
�+�
�−1!
??????
�!�
�−1!
Bose Einstein Distribution Law

21. The number of particles in the configuration of greatest weight (i.e.,
most probable distribution for a microstate) depends on energy of
the state by ??????
�=
�
??????
�
(�+�??????
??????
)
−1
�
�-degeneracy,
�,�-Undetermined multipliers
Bose Einstein Distribution Law