M-B, B-E, and F-D comparisons statistical physics

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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 mechanicsand 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 distributionor Gibbs ensemble, are used to analyzethe statistical
properties of classical systems.
quantum statisticsis 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
indistinguishabilityof 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
bosonsand 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 anyspin which are seperated in the assembly and can be
distinguishedfrom one another .the moleculesof the gas are particles of this
kinds
2.Identical particles of zeroorintegerspin which can not be distinguishedfrom
one another .thease particles are known as Bosons.they do notobey pauli 's
exclusion principles.photons, alpha particles etc.
3.Identical particles of halfintegerspin which cannot be distinguishedfrom one
another these particles obeyPauli's exclusion principles. ex. Electrons…
Three kinds of identical particles:

7. InquantummechanicsPauliexclusionprinciplestates
thattwoormoreidenticalparticleswithhalf-integer
spins(i.e.fermions)cannotsimultaneouslyoccupythe
samequantumstatewithinaquantumsystem.This
principlewasformulatedbyAustrianphysicistWolfgang
Pauliin1925forelectrons,andlaterextendedtoall
fermionswithhisspin–statisticstheoremof1940.
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. andBoltzmann later, in the 1870.
Maxwell–Boltzmann distribution is a result of thekinetic theory of gaseswhich provides a
simplified explanation of many fundamental gaseous properties,
includingpressureanddiffusion, their particles are Identical with distinguishedfrom one
another,
Maxwell–Boltzmann statistics
Maxwell–Boltzmann statistics is used to derive theMaxwell–Boltzmann distribution of
an ideal gas.
In Maxwell–Boltzmann statistics the configuration of particleAin state 1and particleBin
state 2is different from the case in which particleBis in state 1 and particleAis in state 2.

9. •The motion of molecules is
extremely perplexed
•Any individual molecule is
colliding with others at an
enormousrate
❖Typically at a rate of a billion
timesper 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 particlesin a system at
temperature T, having energy εis
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:
•εiis the energy of thei-thenergy level
•⟨Ni⟩is the average number of particles in
the set of states with energy εi
•giis thedegeneracy of energy leveli, that is,
the number of states with energyεi
•μ is thechemical potential,
•kis theBoltzmann constant,
•Tis absolutetemperature
…(1)

11. •It is applicableto an isolated gasof 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
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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) bySatyendra
Nath Bose who recognized that a collectionof identical
andindistinguishable particlescan be distributed in this
way.
The idea was later extendedbyAlbert Einstein in
collaboration with Bose.
Bose–Einstein statistics apply only to the particles that do
not follow thePauli exclusion principlerestrictions.
Particles that follow Bose-Einstein statistics are
calledbosons, which have zero orinteger values ofspin.
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 ofbosons 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
Theabilityofabodytoradiateiscloselyrelatedtoitsability
toabsorbradiation.Abodyataconstanttemperatureisin
thermalequilibriumwithitssurroundingsandmustabsorb
energyfromthematthesamerateasitemitsenergy.
Aperfectblackbodyisoneinwhichabsorbscompletelyall
theradiation,incidentonit.
Soitisaperfectemitterandaperfectabsorber.
One of the applications of Bose-Einstein spectrum distribution

17. Fermi-Dirac (F-D) statistics
Inquantummechanicsisoneofthetwopossiblewaysin
whichasystemofindistinguishableparticlescanbe
distributedamongasetofenergystates.
eachoftheavailablediscretestatescanbeoccupiedby
onlyoneparticle.SoitisobeyPauliexclusionprinciple.
Itisaccountsfortheelectronstructureofatomsinwhich
electronsremaininseparatestatesratherthancollapsing
intoacommonstate.
Thetheoryofthisstatisticalbehaviorwasdeveloped
(1926–27)bythetwophysicists:EnricoFermiandPaul
AdrienMauriceDirac.
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 andobeying Pauli exclusion principle, applicable to electronsand
elementary particles.
•Fermi–Dirac statistics applies to identical andindistinguishable particles
withhalf-integer spin (1/2, 3/2, etc.), calledfermions.
•Fermi–Dirac statistics is most commonly applied toelectrons, a type of
fermion withspin1/2
Fermi-Dirac (F-D) statistics (2)

19. For a system of identical fermions in thermodynamic equilibrium, the average number of
fermions is given by theFermi–Dirac (F–D) distribution.
Fermi-Dirac (F-D) distribution law
…(3)
Equation (3): represents Fermi-Dirac
distribution law
Where:
kBis theBoltzmann constant
Tis the absolutetemperature
εiis the energy of the single-particle statei, and
μis thetotal chemical potential
g

20. Particlesare 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

21. Verification of exclusion principle

22. Verification of exclusion principle (2)

23. Fermi-DiracBose-EinsteinMaxwell-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