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Repair Quantum Collection B backlink template

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{{Quantum book backlink|Statistical mechanics and kinetic theory}}
'''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>
[[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.]]

== Maxwell–Boltzmann distribution ==

In the classical limit:

<math>f(E) = e^{-\beta(E-\mu)}</math><ref name="MIT562">https://ocw.mit.edu/courses/5-62-physical-chemistry-ii-spring-2008/2351f20e4727ae0a7e03ccaca02452d7_08_562ln08.pdf</ref>

Valid when quantum degeneracy is negligible.<ref name="MIT562"/>

== Bose–Einstein distribution ==

For bosons:

<math>f(E) = \frac{1}{e^{\beta(E-\mu)} - 1}</math><ref name="TongQG"/>

Bosons can accumulate in low-energy states, leading to Bose–Einstein condensation.<ref name="MITBose">https://ocw.mit.edu/courses/8-08-statistical-physics-ii-spring-2005/resources/the_bose_gas/</ref>

== Fermi–Dirac distribution ==

For fermions:

<math>f(E) = \frac{1}{e^{\beta(E-\mu)} + 1}</math><ref name="TongQG"/>

The Pauli exclusion principle limits occupation to one particle per state.<ref name="MITFermi">https://ocw.mit.edu/courses/8-08-statistical-physics-ii-spring-2005/3d0cf2cb43a2b62f92089db14e8e2904_the_fermi_gas.pdf</ref>

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

== Classical limit ==

When <math>e^{\beta(E-\mu)} \gg 1</math>, both quantum distributions reduce to:

<math>f(E) \approx e^{-\beta(E-\mu)}</math><ref name="MIT562"/>

== Chemical potential ==

The chemical potential <math>\mu</math> controls particle number.

* For fermions: <math>\mu \to E_F</math> at low temperature
* For bosons: <math>\mu \leq E_0</math>

These constraints determine quantum gas behavior.<ref name="TongQG"/>

== 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.<ref name="TongQG"/>

== Applications ==

Quantum distribution functions are essential in:

* classical gases and kinetic theory<ref name="MIT562"/>
* electron behavior in solids<ref name="MITFermi"/>
* photons and phonons<ref name="MITBose"/>
* quantum many-body systems<ref name="TongQG"/>

==See also==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

==References==
{{reflist|3}}
{{Author|Harold Foppele}}
[[Category:Quantum mechanics]]
[[Category:Statistical mechanics]]

{{Sourceattribution|Quantum Distribution functions|1}}

Physics:Quantum Distribution functions - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template