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{{Short description|Fermi–Dirac statistics in quantum matter}}
{{Short description|Quantum statistics for fermions}}
{{Quantum matter backlink|Atoms}}
{{Quantum matter backlink|Atoms}}
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'''Fermi–Dirac statistics''' describe the occupation of quantum states by fermions. They are central to electrons in atoms, metals, semiconductors, neutron matter, and many-body systems.
[[File:Quantum_Fermi_Dirac_statistics_educational_yellow.png|thumb|right|Fermi-Dirac statistics fill one-particle states subject to the Pauli exclusion principle.]]


== See also ==
'''Fermi–Dirac statistics''' describe the occupation of quantum states by identical [[Physics:Quantum fermion|fermions]]. They apply to particles with half-integer spin, including [[Physics:Quantum electron|electrons]], [[Physics:Quantum neutrino|neutrinos]], [[Physics:Quantum quark|quarks]], and composite fermions such as many atoms and nuclei.
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also/Matter}}
 
== Description ==
Fermions obey the [[Physics:Quantum Pauli exclusion principle|Pauli exclusion principle]]: no two identical fermions may occupy the same quantum state. For a system in thermal equilibrium, the average occupation of a one-particle state with energy <math>E</math> is given by the Fermi-Dirac distribution:
 
<math>f(E)=\frac{1}{\exp\left((E-\mu)/k_{\mathrm B}T\right)+1}</math>
 
where <math>\mu</math> is the chemical potential, <math>k_{\mathrm B}</math> is the Boltzmann constant, and <math>T</math> is temperature.
 
== Physical meaning ==
At absolute zero, states below the Fermi energy are filled and states above it are empty. At nonzero temperature the boundary is smoothed, but the exclusion principle still limits each available state to one fermion of a given quantum state.
 
This statistical rule explains the structure of electron shells in atoms, the behavior of electrons in metals and semiconductors, degeneracy pressure in white dwarfs and neutron stars, and the existence of [[Physics:Quantum Fermi surfaces|Fermi surfaces]] in condensed matter systems.
 
== Historical names ==
* [[Biography:Enrico Fermi|Enrico Fermi]] developed the statistical treatment of particles obeying the exclusion principle.
* [[Biography:Paul Dirac|Paul Dirac]] independently developed the same statistics in quantum theory.
* [[Biography:Wolfgang Pauli|Wolfgang Pauli]] formulated the exclusion principle on which the statistics depends.
 
== Related concepts ==
* [[Physics:Quantum fermion]]
* [[Physics:Quantum Pauli exclusion principle]]
* [[Physics:Quantum electron]]
* [[Physics:Quantum Fermi surfaces]]
* [[Physics:Quantum Bose–Einstein statistics]]


== References ==
== References ==
{{reflist|3}}
{{reflist|3}}
* {{Cite journal |last=Fermi |first=E. |title=Sulla quantizzazione del gas perfetto monoatomico |journal=Rendiconti Lincei |year=1926 |volume=3 |pages=145-149}}
* {{Cite journal |last=Dirac |first=P. A. M. |title=On the Theory of Quantum Mechanics |journal=Proceedings of the Royal Society A |year=1926 |volume=112 |issue=762 |pages=661-677 |doi=10.1098/rspa.1926.0133}}
* {{Cite web |title=Fermi-Dirac statistics |url=https://www.britannica.com/science/Fermi-Dirac-statistics |website=Encyclopaedia Britannica |access-date=2026-05-23}}


{{Author|Harold Foppele}}
{{Author|Harold Foppele}}
{{Sourceattribution|Physics:Quantum Fermi–Dirac statistics|1}}

Revision as of 10:03, 23 May 2026

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Fermi-Dirac statistics fill one-particle states subject to the Pauli exclusion principle.

Fermi–Dirac statistics describe the occupation of quantum states by identical fermions. They apply to particles with half-integer spin, including electrons, neutrinos, quarks, and composite fermions such as many atoms and nuclei.

Description

Fermions obey the Pauli exclusion principle: no two identical fermions may occupy the same quantum state. For a system in thermal equilibrium, the average occupation of a one-particle state with energy E is given by the Fermi-Dirac distribution:

f(E)=1exp⁡((E−μ)/kBT)+1

where μ is the chemical potential, kB is the Boltzmann constant, and T is temperature.

Physical meaning

At absolute zero, states below the Fermi energy are filled and states above it are empty. At nonzero temperature the boundary is smoothed, but the exclusion principle still limits each available state to one fermion of a given quantum state.

This statistical rule explains the structure of electron shells in atoms, the behavior of electrons in metals and semiconductors, degeneracy pressure in white dwarfs and neutron stars, and the existence of Fermi surfaces in condensed matter systems.

Historical names

  • Enrico Fermi developed the statistical treatment of particles obeying the exclusion principle.
  • Paul Dirac independently developed the same statistics in quantum theory.
  • Wolfgang Pauli formulated the exclusion principle on which the statistics depends.

References


Author: Harold Foppele


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