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{{Short description|Quantum Collection topic on Quantum Distribution functions}}
{{Quantum book backlink|Statistical mechanics and kinetic theory}}
{{Quantum book backlink|Statistical mechanics and kinetic theory}}
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'''Quantum distribution functions''' describe the average occupation of energy states in a many-particle system at thermal equilibrium. They distinguish classical from quantum statistical behavior.
'''Quantum distribution functions''' describe the average occupation of energy states in a many-particle system at thermal equilibrium. They distinguish classical from quantum statistical behavior.


For a state of energy <math>E</math>, the occupation depends on particle type.<ref name="TongQG">https://www.damtp.cam.ac.uk/user/tong/statphys/statmechhtml/S3.html</ref>
For a state of energy <math>E</math>, the occupation depends on particle type.<ref name="TongQG">https://www.damtp.cam.ac.uk/user/tong/statphys/statmechhtml/S3.html</ref>
[[File:Quantum_distribution_functions.jpg|thumb|400px|Comparison of Maxwell–Boltzmann, Bose–Einstein, and Fermi–Dirac distribution functions showing how equilibrium occupation depends on energy.]]
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[[File:Quantum_distribution_functions.jpg|thumb|280px|Quantum Distribution functions.]]
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== Maxwell–Boltzmann distribution ==
== Maxwell–Boltzmann distribution ==

Revision as of 14:02, 17 May 2026


← Back to Statistical mechanics and kinetic theory Book I

Quantum distribution functions describe the average occupation of energy states in a many-particle system at thermal equilibrium. They distinguish classical from quantum statistical behavior.

For a state of energy E, the occupation depends on particle type.[1]

Quantum Distribution functions.

Maxwell–Boltzmann distribution

In the classical limit:

f(E)=e−β(E−μ)[2]

Valid when quantum degeneracy is negligible.[2]

Bose–Einstein distribution

For bosons:

f(E)=1eβ(E−μ)−1[1]

Bosons can accumulate in low-energy states, leading to Bose–Einstein condensation.[3]

Fermi–Dirac distribution

For fermions:

f(E)=1eβ(E−μ)+1[1]

The Pauli exclusion principle limits occupation to one particle per state.[4]

At low temperature, the distribution approaches a step function at the Fermi energy.

Classical limit

When eβ(E−μ)≫1, both quantum distributions reduce to:

f(E)≈e−β(E−μ)[2]

Chemical potential

The chemical potential μ controls particle number.

  • For fermions: μ→EF at low temperature
  • For bosons: μ≤E0

These constraints determine quantum gas behavior.[1]

Physical interpretation

The three distributions reflect different statistics:

  • Maxwell–Boltzmann → classical limit
  • Bose–Einstein → state clustering
  • Fermi–Dirac → exclusion principle

These differences produce distinct macroscopic phenomena.[1]

Applications

Quantum distribution functions are essential in:

  • classical gases and kinetic theory[2]
  • electron behavior in solids[4]
  • photons and phonons[3]
  • quantum many-body systems[1]

See also

Table of contents (217 articles)

Index

Full contents

References

  1. ↑ 1.0 1.1 1.2 1.3 1.4 1.5 https://www.damtp.cam.ac.uk/user/tong/statphys/statmechhtml/S3.html
  2. ↑ 2.0 2.1 2.2 2.3 https://ocw.mit.edu/courses/5-62-physical-chemistry-ii-spring-2008/2351f20e4727ae0a7e03ccaca02452d7_08_562ln08.pdf
  3. ↑ 3.0 3.1 https://ocw.mit.edu/courses/8-08-statistical-physics-ii-spring-2005/resources/the_bose_gas/
  4. ↑ 4.0 4.1 https://ocw.mit.edu/courses/8-08-statistical-physics-ii-spring-2005/3d0cf2cb43a2b62f92089db14e8e2904_the_fermi_gas.pdf
Author: Harold Foppele

Source attribution: Quantum Distribution functions

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