Editing Maxwell–Boltzmann distribution (section)

{{distinguish|Maxwell–Boltzmann statistics}}
{{Short description|Specific probability distribution function, important in physics}}
{{about|particle energy levels and velocities|system energy states|Boltzmann distribution}}
{{Probability distribution
| name       = Maxwell–Boltzmann distribution
| 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}{a^3} \, \exp\left(\frac{-x^2}{2a^2} \right)</math> 
(where {{math|exp}} is the [[exponential function]])
| cdf        = <math>\operatorname{erf}\left(\frac{x}{\sqrt{2} a}\right) -\sqrt{\frac{2}{\pi}} \, \frac{x}{a} \, \exp\left(\frac{-x^2}{2a^2} \right)</math>
(where {{math|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}} \approx 0.48569</math>
| kurtosis   = <math>\gamma_2=\frac{4(-96+40\pi-3\pi^2)}{(3 \pi - 8)^2} \approx 0.10816</math>
| entropy    = <math>\ln\left(a\sqrt{2\pi}\right)+\gamma-\frac{1}{2}</math>
| mgf        =
| char       =
}}
In [[physics]] (in particular in [[statistical mechanics]]), the '''Maxwell–Boltzmann distribution''', or '''Maxwell(ian) distribution''', is a particular [[probability distribution]] named after [[James Clerk Maxwell]] and [[Ludwig Boltzmann]].

It was first defined and used for describing particle [[speed]]s in [[ideal gas|idealized gases]], where the particles move freely inside a stationary container without interacting with one another, except for very brief [[collision]]s in which they exchange energy and momentum with each other or with their thermal environment. The term "particle" in this context refers to gaseous particles only ([[atoms]] or [[molecules]]), and the system of particles is assumed to have reached [[thermodynamic equilibrium]].<ref name="StatisticalPhysics">{{ cite book | last = Mandl | first = Franz | title = Statistical Physics | date = 2008 | publisher = John Wiley & Sons | isbn = 978-0471915331 | edition = 2nd |series = Manchester Physics | location = Chichester }}</ref> The energies of such particles follow what is known as [[Maxwell–Boltzmann statistics]], and the statistical distribution of speeds is derived by equating particle energies with [[kinetic energy]].

Mathematically, the Maxwell–Boltzmann distribution is the [[chi distribution]] with three [[degrees of freedom]] (the components of the [[velocity]] vector in [[Euclidean space]]), with a [[scale parameter]] measuring speeds in units proportional to the square root of <math>T/m</math> (the ratio of temperature and particle mass).<ref>{{ cite book | last1 = Young |first1 = Hugh D. | last2 = Friedman | first2 = Roger A. | last3 = Ford | first3 = Albert Lewis | last4 = Sears | first4 = Francis Weston | last5 = Zemansky | first5 = Mark Waldo | title = Sears and Zemansky's University Physics: With Modern Physics |date = 2008 | publisher = Pearson, Addison-Wesley | isbn = 978-0-321-50130-1 | edition = 12th | location = San Francisco }}</ref>

The Maxwell–Boltzmann distribution is a result 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), [[Rita G. Lerner|R.G. Lerner]], G.L. Trigg, VHC publishers, 1991, {{isbn|3-527-26954-1}} (Verlagsgesellschaft), {{isbn|0-89573-752-3}} (VHC Inc.)</ref> The Maxwell–Boltzmann distribution applies fundamentally to particle velocities in three dimensions, but turns out to depend only on the speed (the [[Magnitude (mathematics)|magnitude]] of the velocity) of the particles. A particle speed probability distribution indicates which speeds are more likely: a randomly chosen particle will have a speed selected randomly from the distribution, and is more likely to be within one range of speeds than another. The kinetic theory of gases 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 [[exchange interaction]]s) that can make their speed distribution different from the Maxwell–Boltzmann form. However, [[Rarefaction|rarefied]] gases at ordinary temperatures behave very nearly like an ideal gas and the Maxwell speed distribution is an excellent approximation for such gases. This is also true for ideal [[plasma (physics)|plasmas]], which are ionized gases of sufficiently low density.<ref>N.A. Krall and A.W. Trivelpiece, Principles of Plasma Physics, San Francisco Press, Inc., 1986, among many other texts on basic plasma physics</ref>

The distribution was first derived by Maxwell in 1860 on [[heuristic]] grounds.<ref name="MaxwellA">Maxwell, J.C. (1860 A): ''Illustrations of the dynamical theory of gases. Part I. On the motions and collisions of perfectly elastic spheres.  The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science'', 4th Series, vol.19, pp.19–32. [https://www.biodiversitylibrary.org/item/53795#page/33/mode/1up]</ref><ref name="MaxwellB">Maxwell, J.C. (1860 B): ''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.  The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science'', 4th Ser., vol.20, pp.21–37. [https://www.biodiversitylibrary.org/item/20012#page/37/mode/1up]</ref> Boltzmann later, in the 1870s, carried out significant investigations into the physical origins of this distribution. The distribution can be derived on the ground that it maximizes the entropy of the system. A list of derivations are:

# [[Maximum entropy probability distribution#Distributions with measured constants|Maximum entropy probability distribution]] in the phase space, with the constraint of [[Conservation of energy|conservation of average energy]] <math>\langle H \rangle = E;</math>
# [[Canonical ensemble]].