Editing Maxwell–Boltzmann distribution (section)

===Maxwell–Boltzmann statistics===
{{main|Maxwell–Boltzmann statistics#Derivations|Boltzmann distribution}}
The original derivation in 1860 by [[James Clerk Maxwell]] was an argument based on molecular collisions of the [[Kinetic theory of gases]] as well as certain symmetries in the speed distribution function; Maxwell also gave an early argument that these molecular collisions entail a tendency towards equilibrium.<ref name=MaxwellA/><ref name=MaxwellB/><ref>{{Cite journal | last1 = Gyenis | first1 = Balazs | doi = 10.1016/j.shpsb.2017.01.001 | title = Maxwell and the normal distribution: A colored story of probability, independence, and tendency towards equilibrium | journal = Studies in History and Philosophy of Modern Physics | volume = 57 | pages = 53–65 | year = 2017| arxiv = 1702.01411 | bibcode = 2017SHPMP..57...53G | s2cid = 38272381 }}</ref> After Maxwell, [[Ludwig Boltzmann]] in 1872<ref>Boltzmann, L., "Weitere studien über das Wärmegleichgewicht unter Gasmolekülen." ''Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften in Wien, mathematisch-naturwissenschaftliche Classe'', '''66''', 1872, pp. 275–370.</ref> also derived the distribution on mechanical grounds and argued that gases should over time tend toward this distribution, due to collisions (see [[H-theorem]]). He later (1877)<ref>Boltzmann, L., "Über die Beziehung zwischen dem zweiten Hauptsatz der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung respektive den Sätzen über das Wärmegleichgewicht." ''Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften in Wien, Mathematisch-Naturwissenschaftliche Classe''. Abt. II, '''76''', 1877, pp. 373–435. Reprinted in ''Wissenschaftliche Abhandlungen'', Vol. II, pp. 164–223, Leipzig: Barth, 1909. '''Translation available at''': http://crystal.med.upenn.edu/sharp-lab-pdfs/2015SharpMatschinsky_Boltz1877_Entropy17.pdf {{Webarchive|url=https://web.archive.org/web/20210305005604/http://crystal.med.upenn.edu/sharp-lab-pdfs/2015SharpMatschinsky_Boltz1877_Entropy17.pdf |date=2021-03-05 }}</ref> derived the distribution again under the framework of [[statistical thermodynamics]]. The derivations in this section are along the lines of Boltzmann's 1877 derivation, starting with result known as [[Maxwell–Boltzmann statistics]] (from statistical thermodynamics). Maxwell–Boltzmann statistics gives the average number of particles found in a given single-particle [[Microstate (statistical mechanics)|microstate]]. Under certain assumptions, the logarithm of the fraction of particles in a given microstate is linear in the ratio of the energy of that state to the temperature of the system: there are constants <math>k</math> and <math>C</math> such that, for all <math>i</math>,
<math display="block">-\log \left(\frac{N_i}{N}\right) = \frac{1}{k}\cdot\frac{E_i}{T} + C.</math>
The assumptions of this equation are that the particles do not interact, and that they are classical; this means that each particle's state can be considered independently from the other particles' states. Additionally, the particles are assumed to be in thermal equilibrium.<ref name="StatisticalPhysics" /><ref>{{ cite book | last = Parker | first = Sybil P. | title = McGraw-Hill Encyclopedia of Physics | date = 1993 | publisher = McGraw-Hill | isbn = 978-0-07-051400-3 | edition = 2nd}}</ref>

This relation can be written as an equation by introducing a normalizing factor:
{{NumBlk||<math display="block">
\frac{N_i} N =
\frac{
    \exp\left(-\frac{E_i}{k_\text{B}T}\right)
}{
    \displaystyle \sum_j \exp\left(-\tfrac{E_j}{k_\text{B}T}\right)
}</math>|{{EquationRef|1}}}}

where:
* {{mvar|N<sub>i</sub>}} is the expected number of particles in the single-particle microstate {{mvar|i}},
* {{mvar|N}} is the total number of particles in the system,
* {{mvar|E<sub>i</sub>}}  is the energy of microstate {{mvar|i}},
* the sum over index {{mvar|j}} takes into account all microstates, 
* {{mvar|T}} is the equilibrium temperature of the system,
* {{math|''k''<sub>B</sub>}} is the [[Boltzmann constant]].
The denominator in {{EquationNote|1|equation 1}} is a normalizing factor so that the ratios <math>N_i:N</math>  add up to unity — in other words it is a kind of [[partition function (statistical mechanics)|partition function]] (for the single-particle system, not the usual partition function of the entire system).

Because velocity and speed are related to energy, Equation ({{EquationNote|1}}) can be used to derive relationships between temperature and the speeds of gas particles. All that is needed is to discover the density of microstates in energy, which is determined by dividing up momentum space into equal sized regions.