1. Comparison of Maxwell-Boltzmann, Bose-Einstein,
and Fermi-Dirac statistics
Prepared by:
Mohammad Idres Omer
Supervised by:
Professor: Dr. Tariq A. Abbas
(2023-2024)
Statistical Mechanics
2. Outlines
Introduction to classical and quantum statistics
Three kinds of identical particles
Pauli exclusion principle
Maxwell-Boltzmann(M-B) statistics
Bose-Einstein(B-E) statistics
Fermi-Dirac(F-D) statistics
Verification of exclusion principle
References
3. Introduction to classical and quantum statistics
Classical statistics deals with systems that follow classical mechanics and obey the laws
of classical physics. In classical statistics, systems are described by macroscopic
variables, such as position, velocity, or energy. Classical statistical methods, such as the
Maxwell-Boltzmann distribution or Gibbs ensemble, are used to analyze the statistical
properties of classical systems.
quantum statistics is concerned with systems that follow quantum mechanics, which is
the theory that describes the behavior of microscopic particles, atoms, and molecules.
Quantum statistics take into account the wave-particle duality of particles and the
indistinguishability of identical particles.
4. Quantum systems are described by wave functions that evolve according to the
Schrödinger equation, and the properties of the system are represented by operators and
observables. Quantum statistical methods, such as the Bose-Einstein distribution for
bosons and the Fermi-Dirac distribution for fermions, are used to analyze the statistical
behavior of quantum systems.
Introduction to classical and quantum statistics
5. Quantum statistics:
Developed by Bose, Einstein, Fermi and Dirac, Including two categories:
• Bose-Einstein (B-E) statistics, and
• Fermi-Dirac (F-D) statistics.
Classical statistics
• This branch is based on the classical results of Maxwell-Boltzmann
(M-B) statistics.
6. 1. Identical particles of any spin which are seperated in the assembly and can be
distinguished from one another .the molecules of the gas are particles of this
kinds
2. Identical particles of zero or integer spin which can not be distinguished from
one another .thease particles are known as Bosons .they do not obey pauli 's
exclusion principles .photons, alpha particles etc.
3. Identical particles of half integer spin which cannot be distinguished from one
another these particles obey Pauli's exclusion principles. ex. Electrons…
Three kinds of identical particles:
7. In quantum mechanics Pauli exclusion principle states
that two or more identical particles with half-integer
spins (i.e. fermions) cannot simultaneously occupy the
same quantum state within a quantum system. This
principle was formulated by Austrian physicist Wolfgang
Pauli in 1925 for electrons, and later extended to all
fermions with his spin–statistics theorem of 1940.
Pauli exclusion principle
Fig 1: Wolfgang Pauli during a
lecture in Copenhagen (1929).
Wolfgang Pauli formulated the
Pauli exclusion principle
8. This one was first derived by Maxwell in 1860. and Boltzmann later, in the 1870.
Maxwell–Boltzmann distribution is a result of the kinetic theory of gases which provides a
simplified explanation of many fundamental gaseous properties,
including pressure and diffusion, their particles are Identical with distinguished from one
another,
Maxwell–Boltzmann statistics
Maxwell–Boltzmann statistics is used to derive the Maxwell–Boltzmann distribution of
an ideal gas.
In Maxwell–Boltzmann statistics the configuration of particle A in state 1 and particle B in
state 2 is different from the case in which particle B is in state 1 and particle A is in state 2.
9. • The motion of molecules is
extremely perplexed
• Any individual molecule is
colliding with others at an
enormous rate
❖ Typically at a rate of a billion
times per second
Particle behavior in Maxwell-Boltzmann Distribution
• 1844-1906
• Austrian physicist
• Contributed to:
1. Kinetic Theory of
Gases
2. Electromagnetism
3. Thermodynamics
• Pioneer in statistical
mechanics
Fig 2:
Ludwig Boltzmann
10. Maxwell-Boltzmann Law of energy
According to this law, number of identical and distinguishable particles in a system at
temperature T, having energy ε is
Ni(ε) = (No. of states of energy ε).(average no. of particles in a state of energy ε)
Equation (1): represents the
Maxwell-Boltzmann Law of
energy
where:
• εi is the energy of the i-th energy level
• ⟨Ni⟩ is the average number of particles in
the set of states with energy εi
• gi is the degeneracy of energy level i, that is,
the number of states with energy εi
• μ is the chemical potential,
• k is the Boltzmann constant,
• T is absolute temperature
…(1)
11. • It is applicable to an isolated gas of identical
molecules in equilibrium which satisfied the
conditions, the gas is said to be ideal.
• Maxwell-Boltzmann distribution of the speeds
of an ideal gas particles can be derived from
the Maxwell-Boltzmann statistics and used to
derive relationships between pressure, volume
and temperature.
Fig(3): distribution of two particles in 3
cells
Particle distribution in Maxwell-Boltzmann distribution
𝑔𝑛
P =
12. • The number of molecules in the gas is large, and the average separation between the
molecules is large compared with their dimensions
• The molecules move randomly
• Any molecule can move in any direction
• The molecules interact only by short-range forces during elastic collisions,
so the molecules make elastic collisions with the walls of container
• There is no exist external forces, No forces between particles except when they collide.
• All molecules are identical
The kinetic theory of gases
Maxwell–Boltzmann Distribution is a result of the kinetic theory of gases
13. Bose-Einstein (B-E) statistics
This theory was developed (1924–1925) by Satyendra
Nath Bose who recognized that a collection of identical
and indistinguishable particles can be distributed in this
way.
The idea was later extended by Albert Einstein in
collaboration with Bose.
Bose–Einstein statistics apply only to the particles that do
not follow the Pauli exclusion principle restrictions.
Particles that follow Bose-Einstein statistics are
called bosons, which have zero or integer values of spin.
Fig 4
14. Bose-Einstein distribution tell us how many particles have a certain energy. The
formula is:
Bose-Einstein distribution law
Equation (2): represents Bose-Einstein
distribution law
…(2)
g
15. In such a system, there is not a difference between any of these particles, and the particles
are bosons. Bosons are fundamental particles like the photon.
Any number of bosons can exist in the same quantum state of the system
Or Any number of particles can occupy a single cell in the phase space
Particle distribution in Bose-Einstein distribution
Fig(5): distribution of
two particles in 3 cells
16. Fig 6: The cavity walls
are constantly emitting
and absorbing radiation,
and this radiation is
known as black body
radiation
Black Body Radiation
The ability of a body to radiate is closely related to its ability
to absorb radiation. A body at a constant temperature is in
thermal equilibrium with its surroundings and must absorb
energy from them at the same rate as it emits energy.
A perfect black body is one in which absorbs completely all
the radiation, incident on it.
So it is a perfect emitter and a perfect absorber.
One of the applications of Bose-Einstein spectrum distribution
17. Fermi-Dirac (F-D) statistics
In quantum mechanics is one of the two possible ways in
which a system of indistinguishable particles can be
distributed among a set of energy states.
each of the available discrete states can be occupied by
only one particle. So it is obey Pauli exclusion principle.
It is accounts for the electron structure of atoms in which
electrons remain in separate states rather than collapsing
into a common state.
The theory of this statistical behavior was developed
(1926–27) by the two physicists: Enrico Fermi and Paul
Adrien Maurice Dirac.
Fig 7
18. In M-B, or B-E statistics there is no restrictions on the particles to present in any
energy state. but in the case of fermi- Dirac statistics, applicable to particles like
electron and obeying Pauli exclusion principle, applicable to electrons and
elementary particles.
• Fermi–Dirac statistics applies to identical and indistinguishable particles
with half-integer spin (1/2, 3/2, etc.), called fermions.
• Fermi–Dirac statistics is most commonly applied to electrons, a type of
fermion with spin 1/2
Fermi-Dirac (F-D) statistics (2)
19. For a system of identical fermions in thermodynamic equilibrium, the average number of
fermions is given by the Fermi–Dirac (F–D) distribution.
Fermi-Dirac (F-D) distribution law
…(3)
Equation (3): represents Fermi-Dirac
distribution law
Where:
kB is the Boltzmann constant
T is the absolute temperature
εi is the energy of the single-particle state i, and
μ is the total chemical potential
g
20. Particles are indistinguishable, only one particle may be in a given quantum
state
Particle distribution in Fermi-Dirac distribution
Fig(8): distribution of two particles in 3 cells
23. Fermi-Dirac
Bose-Einstein
Maxwell-Boltzmann
*Their particles are
*Their Particles are
*Particles are identical
indistinguishable
indistinguishable
and distinguishable
*Particles obey Pauli
*Particles do not obey
*The number of particles
exclusion principle
Pauli exclusion principle
is constant
*Each state can have
*Each state can have
*The total Energy is
only one particle
more than one particle
constant
*Each particle has one
like phonons and photons
*Spin is ignored
half spin
*Particles have integer spin
Summary and Comparison between M-B ,B-E, and F-D
24. References
• Books
1. Statistical Mechanics: Rigorous Results. David Ruelle. World
Scientific, (1999).
2. Statistical Mechanics of Disorder Systems - A Mathematical
Perspective. Anton Bovier. Cambridge Series in Statistical and
Probabilistic Mathematics, (2006).
3. Statistical thermal physics, Reif. 6th edition (December 30,
2009).
• Online Guides
1. https://web.stanford.edu/~peastman/statmech/statisticaldescr
iption.html#the-maxwell-boltzmann-distribution
2. https://web.stanford.edu/~peastman/statmech/statisticaldescr
iption.html#quantum-statistical-mechanics
