Boltzmann distribution

E46139

concept in statistical mechanics probability distribution

The Boltzmann distribution is a fundamental probability distribution in statistical mechanics that describes how particles or states are populated over different energy levels at thermal equilibrium.

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All labels observed (5)

Label	Occurrences
Boltzmann distribution canonical	10
Boltzmann statistics	4
Boltzmann factor	3
Gibbs distribution	2
Maxwell–Boltzmann factor	1

How this entity was disambiguated

This entity first appeared as the object of triple T364198 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Boltzmann distribution
Context triple: [Boltzmann machines, basedOn, Boltzmann distribution]

A. Maxwell–Boltzmann statistics
Maxwell–Boltzmann statistics is a classical statistical framework in physics that describes the distribution of speeds or energies among distinguishable, non-quantum particles in thermal equilibrium.
B. Bose–Einstein statistics
Bose–Einstein statistics is a quantum statistical framework that describes the distribution and collective behavior of indistinguishable bosons, underpinning phenomena such as Bose–Einstein condensation.
C. Fermi–Dirac statistics
Fermi–Dirac statistics is the quantum statistical framework that describes the distribution and behavior of indistinguishable fermions, such as electrons, which obey the Pauli exclusion principle.
D. Boltzmann constant
The Boltzmann constant is a fundamental physical constant that links temperature to energy at the particle level, playing a central role in statistical mechanics and thermodynamics.
E. Boltzmann–Gibbs entropy in statistical mechanics
Boltzmann–Gibbs entropy in statistical mechanics is the standard measure of disorder or uncertainty in a system, quantifying how many microscopic configurations correspond to a given macroscopic state and forming the basis of classical equilibrium statistical mechanics.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Boltzmann distribution
Target entity description: The Boltzmann distribution is a fundamental probability distribution in statistical mechanics that describes how particles or states are populated over different energy levels at thermal equilibrium.

A. Maxwell–Boltzmann statistics
Maxwell–Boltzmann statistics is a classical statistical framework in physics that describes the distribution of speeds or energies among distinguishable, non-quantum particles in thermal equilibrium.
B. Bose–Einstein statistics
Bose–Einstein statistics is a quantum statistical framework that describes the distribution and collective behavior of indistinguishable bosons, underpinning phenomena such as Bose–Einstein condensation.
C. Fermi–Dirac statistics
Fermi–Dirac statistics is the quantum statistical framework that describes the distribution and behavior of indistinguishable fermions, such as electrons, which obey the Pauli exclusion principle.
D. Boltzmann constant
The Boltzmann constant is a fundamental physical constant that links temperature to energy at the particle level, playing a central role in statistical mechanics and thermodynamics.
E. Boltzmann–Gibbs entropy in statistical mechanics
Boltzmann–Gibbs entropy in statistical mechanics is the standard measure of disorder or uncertainty in a system, quantifying how many microscopic configurations correspond to a given macroscopic state and forming the basis of classical equilibrium statistical mechanics.
F. None of above. chosen

Statements (49)

Predicate	Object
instanceOf	concept in statistical mechanics ⓘ probability distribution ⓘ
appliesTo	systems in thermal equilibrium ⓘ
approximationOf	Bose–Einstein distribution at high temperature and low density ⓘ Fermi–Dirac distribution at high temperature and low density ⓘ
assumes	classical distinguishable particles in many applications ⓘ
contrastsWith	Bose–Einstein distribution ⓘ linked to: Bose–Einstein statistics Fermi–Dirac distribution ⓘ linked to: Fermi–Dirac statistics
dependsOn	Boltzmann constant ⓘ energy of the state ⓘ temperature ⓘ
describes	probability of a system occupying a state with given energy at thermal equilibrium ⓘ
field	physics ⓘ statistical mechanics ⓘ thermodynamics ⓘ
hasComponent	Boltzmann factor exp(-E/(k_B T)) ⓘ
hasMathematicalForm	P(E) ∝ exp(-E/(k_B T)) ⓘ
implies	higher energy states are less populated than lower energy states ⓘ
namedAfter	Ludwig Boltzmann ⓘ
normalizationCondition	sum over all states of P(E_i) = 1 ⓘ
relatedConcept	Boltzmann entropy formula ⓘ linked to: Boltzmann–Gibbs entropy in statistical mechanics grand canonical ensemble ⓘ microcanonical ensemble ⓘ
relatedTo	Boltzmann factor ⓘ linked to: Boltzmann distribution Gibbs distribution ⓘ linked to: Boltzmann distribution Maxwell–Boltzmann distribution ⓘ linked to: Maxwell–Boltzmann statistics canonical ensemble ⓘ partition function ⓘ
temperatureEffect	higher temperature increases population of higher energy states ⓘ
usedFor	calculating average energy ⓘ calculating entropy in canonical ensemble ⓘ calculating heat capacity ⓘ deriving macroscopic thermodynamic quantities ⓘ population of electronic energy levels ⓘ population of energy levels in gases ⓘ population of molecular rotational states ⓘ population of molecular vibrational states ⓘ
usedIn	astrophysics ⓘ chemical equilibrium calculations ⓘ laser physics ⓘ molecular simulations ⓘ plasma physics ⓘ reaction rate theory ⓘ spectroscopy ⓘ
usesSymbol	E for energy ⓘ T for absolute temperature ⓘ k_B for Boltzmann constant ⓘ
validWhen	system is ergodic ⓘ system is in contact with a heat bath at fixed temperature ⓘ

How these facts were elicited

Referenced by (20)

Full triples — surface form annotated when it differs from this entity's canonical label.

Boltzmann machines → basedOn → Boltzmann distribution ⓘ

Maxwell–Boltzmann statistics → relatedTo → Boltzmann distribution ⓘ

Ludwig Boltzmann → notableIdea → Boltzmann distribution ⓘ

Boltzmann–Gibbs entropy → maximizationYields → Boltzmann distribution ⓘ

subject linked to: Boltzmann–Gibbs entropy in statistical mechanics

Boltzmann distribution → relatedTo → Gibbs distribution ⓘ