Equations · equation

Boltzmann Distribution

Statistical Mechanics · 1868

Equation: Pᵢ = e^(−Eᵢ/kT) / Z
Field: Statistical Mechanics
Named After: Ludwig Boltzmann
Year: 1868

Symbols

Pᵢ probability of finding the system in state i · Eᵢ that state's energy · k Boltzmann constant · T absolute temperature · Z the partition function, normalizing the sum to 1

Significance

At temperature T, higher-energy states are exponentially rarer — the law behind the thinning atmosphere, the rates of chemical reactions, and why glowing objects shift color with temperature. Arguably the most-used formula in all of physical science.

Timeline

Neutrino Decoupling Big Bang Nucleosynthesis Recombination

Theories

Big Bang Nucleosynthesis Statistical Mechanics Kinetic Theory of Gases

Related

Arrhenius Equation Zeroth Law of Thermodynamics Debye Length Einstein A and B Coefficients Fermi–Dirac Distribution Planck's Law Normal Distribution Saha Ionization Equation

Constants

Boltzmann constant

Spectral Types

T-type Stars (T Dwarfs)

Required By

Saha Ionization Equation Arrhenius Equation

Referenced by

constant

Boltzmann constant

equation

Arrhenius Equation Debye Length Einstein A and B Coefficients Fermi–Dirac Distribution Normal Distribution Planck's Law Saha Ionization Equation Zeroth Law of Thermodynamics

event

Big Bang Nucleosynthesis Neutrino Decoupling Recombination

theory

Big Bang Nucleosynthesis Kinetic Theory of Gases Statistical Mechanics

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From the Equations database in Notion, last edited 2026-08-15. This page is the row, rendered by the build; nothing on it is written by hand.