Editing Boltzmann distribution (section)

{{Short description|Probability distribution of energy states of a system}}
{{about|system energy states|particle energy levels and velocities|Maxwell–Boltzmann distribution}}
{{Use American English|date = March 2019}}

[[File:Exponential probability density.svg|upright=1.75|right|thumb|Boltzmann's distribution is an exponential distribution.]]
[[File:Boltzmann distribution graph.svg|upright=1.75|right|thumb|Boltzmann factor {{tmath|\tfrac{p_i}{p_j} }} (vertical axis) as a function of temperature {{mvar|T}} for several energy differences {{math|''ε<sub>i</sub>'' − ''ε<sub>j</sub>''}}.]]

In [[statistical mechanics]] and [[mathematics]], a '''Boltzmann distribution''' (also called '''Gibbs distribution'''<ref name ="landau">{{cite book | author=Landau, Lev Davidovich |author2=Lifshitz, Evgeny Mikhailovich |name-list-style=amp | title=Statistical Physics |volume=5 |series=Course of Theoretical Physics |edition=3 |orig-year=1976 |year=1980 |place=Oxford |publisher=Pergamon Press|isbn=0-7506-3372-7|author-link=Lev Landau |author2-link=Evgeny Lifshitz }} Translated by J.B. Sykes and M.J. Kearsley. See section 28</ref>) is a [[probability distribution]] or [[probability measure]] that gives the probability that a system will be in a certain [[microstate (statistical mechanics)|state]] as a function of that state's energy and the temperature of the system. The distribution is expressed in the form:

:<math>p_i \propto \exp\left(- \frac{\varepsilon_i}{kT} \right)</math>

where {{mvar|p<sub>i</sub>}} is the probability of the system being in state {{mvar|i}}, {{math|exp}} is the [[exponential function]], {{mvar|ε<sub>i</sub>}} is the energy of that state, and a constant {{mvar|kT}} of the distribution is the product of the [[Boltzmann constant]] {{mvar|k}} and [[thermodynamic temperature]] {{mvar|T}}. The symbol <math display="inline">\propto</math> denotes [[proportionality (mathematics)|proportionality]] (see {{section link||The distribution}} for the proportionality constant).

The term ''system'' here has a wide meaning; it can range from a collection of 'sufficient number' of atoms or a single atom{{r|landau}} to a macroscopic system such as a [[Natural gas storage|natural gas storage tank]]. Therefore, the Boltzmann distribution can be used to solve a wide variety of problems. The distribution shows that states with lower energy will always have a higher probability of being occupied.

The ''ratio'' of probabilities of two states is known as the '''Boltzmann factor''' and characteristically only depends on the states' energy difference:

:<math>\frac{p_i}{p_j} = \exp\left( \frac{\varepsilon_j - \varepsilon_i}{kT} \right)</math>

The Boltzmann distribution is named after [[Ludwig Boltzmann]] who first formulated it in 1868 during his studies of the [[statistical mechanics]] of gases in [[thermal equilibrium]].<ref>{{cite journal |last=Boltzmann |first=Ludwig |author-link=Ludwig Boltzmann
|year=1868
|title=Studien über das Gleichgewicht der lebendigen Kraft zwischen bewegten materiellen Punkten
|trans-title=Studies on the balance of living force between moving material points
|journal=Wiener Berichte |volume=58 |pages=517–560
}}</ref> Boltzmann's statistical work is borne out in his paper “On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium"<ref>{{Cite web |url=http://crystal.med.upenn.edu/sharp-lab-pdfs/2015SharpMatschinsky_Boltz1877_Entropy17.pdf |title=Translation of Ludwig Boltzmann's Paper "On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium" |access-date=2017-05-11 |archive-date=2020-10-21 |archive-url=https://web.archive.org/web/20201021205227/http://crystal.med.upenn.edu/sharp-lab-pdfs/2015SharpMatschinsky_Boltz1877_Entropy17.pdf |url-status=dead }}</ref>
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  It would be nice to have a citation here! The origin of the Boltzmann factor isn't entirely clear. According to some authors, Boltzmann's 1968 paper “On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium”
  "Studien über das Gleichgewicht der lebendigen Kraft zwischen bewegten materiellen Punkten" is the origin but I can't find this article at the moment,
  so I cannot confirm.
  For example, this book says so, but uses suspiciously modern terminology
    https://books.google.com/books?id=u13KiGlz2zcC&pg=PA93
  On the other hand, Uffink's "Compendium of the foundations of classical statistical physics" does not seem to indicate quite this equation but rather that Boltzmann's 1968 distribution was the simple Maxwell–Boltzmann distribution (for a classical nonrelativistic gas), modified for particles in a potential.
--> 
The distribution was later investigated extensively, in its modern generic form, by [[Josiah Willard Gibbs]] in 1902.<ref name="gibbs">{{cite book |last=Gibbs |first=Josiah Willard |author-link=Josiah Willard Gibbs |title=Elementary Principles in Statistical Mechanics |year=1902 |publisher=[[Charles Scribner's Sons]] 
|location=New York|title-link=Elementary Principles in Statistical Mechanics }}</ref>

The Boltzmann distribution should not be confused with the [[Maxwell–Boltzmann distribution]] or [[Maxwell–Boltzmann statistics|Maxwell-Boltzmann statistics]]. The Boltzmann distribution gives the probability that a system will be in a certain ''state'' as a function of that state's energy,<ref name="Atkins, P. W. 2010">Atkins, P. W. (2010) Quanta, W. H. Freeman and Company, New York</ref> while the Maxwell-Boltzmann distributions give the probabilities of particle ''speeds'' or ''energies'' in ideal gases. The distribution of energies in a <em>one-dimensional</em> gas however, does follow the Boltzmann distribution.