View source for Maxwell–Boltzmann distribution

{{Hide in print|1={{Probability distribution
| name       = Maxwell–Boltzmann
| type       = density
| pdf_image  = [[File:Maxwell-Boltzmann distribution pdf.svg|325px]]
| cdf_image  = [[File:Maxwell-Boltzmann distribution cdf.svg|325px]]
| parameters = <math>a>0</math>
| support    = <math>x\in (0;\infty)</math>
| pdf        = <math>\sqrt{\frac{2}{\pi}} \frac{x^2 e^{-x^2/\left(2a^2\right)}}{a^3}</math>
| cdf        = <math>\operatorname{erf}\left(\frac{x}{\sqrt{2} a}\right) -\sqrt{\frac{2}{\pi}} \frac{x e^{-x^2/\left(2a^2\right)}}{a} </math> where erf is the [[error function]]
| mean       = <math>\mu=2a \sqrt{\frac{2}{\pi}}</math>
| median     =
| mode       = <math>\sqrt{2} a</math>
| variance   = <math>\sigma^2=\frac{a^2(3 \pi - 8)}{\pi}</math>
| skewness   = <math>\gamma_1=\frac{2 \sqrt{2} (16 -5 \pi)}{(3 \pi - 8)^{3/2}}</math>
| kurtosis   = <math>\gamma_2=4\frac{\left(-96+40\pi-3\pi^2\right)}{(3 \pi - 8)^2}</math>
| entropy    = <math>\ln\left(a\sqrt{2\pi}\right)+\gamma-\frac{1}{2}</math>
| mgf=
| char=
}}}}
In [[statistics]] the '''Maxwell–Boltzmann distribution''' is a particular [[probability distribution]] named after [[James Clerk Maxwell]] and [[Ludwig Boltzmann]]. 
It was first defined and used in [[physics]] (in particular in [[statistical mechanics]]) for describing particle speeds in idealized [[gas]]es. In an idealized gas the particles move freely inside a stationary container without interacting with one another, most of the time. Sometimes they [[collision|collide]] and exchange energy and momentum with each other or with their thermal environment. Particle in this context refers to gaseous particles ([[atoms]] or [[molecules]]), and the system of particles is assumed to have reached [[thermodynamic equilibrium]].<ref name="StatisticalPhysics">''Statistical Physics'' (2nd Edition), F. Mandl, Manchester Physics, John Wiley & Sons, 2008, {{ISBN|9780471915331}}</ref>
While the distribution was first derived by Maxwell in 1860 on heuristic grounds,<ref name=Maxwell>See:
*  Maxwell, J.C. (1860) [https://books.google.com/books?id=-YU7AQAAMAAJ&pg=PA19#v=onepage&q&f=false "Illustrations of the dynamical theory of gases. Part I.  On the motions and collisions of perfectly elastic spheres,"] ''Philosophical Magazine'', 4th series, '''19''' :  19-32. 
*  Maxwell, J.C. (1860) [https://books.google.com/books?id=DIc7AQAAMAAJ&pg=PA21#v=onepage&q&f=false "Illustrations of the dynamical theory of gases. Part II.  On the process of diffusion of two or more kinds of moving particles among one another,"] ''Philosophical Magazine'', 4th series, '''20''' :  21-37.</ref> Boltzmann later carried out significant investigations into the physical origins of this distribution.

A particle speed probability distribution indicates which speeds are more likely: a particle will have a speed selected randomly from the distribution, and is more likely to be within one range of speeds than another. The distribution depends on the [[temperature]] of the system and the mass of the particle.<ref>University Physics – With Modern Physics (12th Edition), H.D. Young, R.A. Freedman (Original edition), Addison-Wesley (Pearson International), 1st Edition: 1949, 12th Edition: 2008, ISBN (10-) 0-321-50130-6, ISBN (13-) 978-0-321-50130-1</ref>
The Maxwell–Boltzmann distribution applies to the classical [[ideal gas]], which is an idealization of real gases. In real gases, there are various effects (e.g., [[van der Waals interaction]]s, [[vortex|vortical flow]], [[special relativity|relativistic]] speed limits, and [[quantum mechanics|quantum]] [[exchange interaction]]s) that can make their speed distribution different from the Maxwell–Boltzmann form. However, [[rarefied]] gases at ordinary temperatures behave very nearly like an ideal gas and the Maxwell speed distribution is an excellent approximation for such gases. For this reason, it forms the basis of the [[Kinetic theory of gases]], which provides a simplified explanation of many fundamental gaseous properties, including [[pressure]] and [[diffusion]].<ref>Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3</ref>

==References==
{{reflist}}

[[Category:Probability distributions]]