Editing Maxwell–Boltzmann distribution (section)

== Distribution function ==
For a system containing a large number of identical non-interacting, non-relativistic classical particles in thermodynamic equilibrium, the fraction of the particles within an infinitesimal element of the three-dimensional velocity space {{math|''d''{{sup|&thinsp;3}}'''v'''}}, centered on a velocity vector <math> \mathbf{v} </math> of magnitude <math>v</math>, is given by
<math display="block"> f(\mathbf{v}) ~ d^3\mathbf{v} = \biggl[\frac{m}{2 \pi k_\text{B}T}\biggr]^{{3}/{2}} \, \exp\left(-\frac{mv^2}{2k_\text{B}T}\right) ~ d^3\mathbf{v}, </math>
where:
*{{mvar|m}} is the particle mass;
*{{math|''k''<sub>B</sub>}} is the [[Boltzmann constant]];
*{{mvar|T}} is [[thermodynamic temperature]];
*<math> f(\mathbf{v}) </math> is a probability distribution function, properly normalized so that <math display="inline">\int f(\mathbf{v}) \, d^3\mathbf{v}</math> over all velocities is unity.

[[Image:MaxwellBoltzmann-en.svg|right|thumb|340px|The speed probability density functions of the speeds of a few [[noble gas]]es at a temperature of 298.15&nbsp;K (25&nbsp;°C). The ''y''-axis is in s/m so that the area under any section of the curve (which represents the probability of the speed being in that range) is dimensionless.]]
One can write the element of velocity space as <math>d^3\mathbf{v} = dv_x \, dv_y \, dv_z</math>, for velocities in a standard Cartesian coordinate system, or as <math>d^3\mathbf{v} = v^2 \, dv \, d\Omega</math> in a standard spherical coordinate system, where <math>d\Omega = \sin{v_\theta} ~ dv_\phi ~ dv_\theta</math> is an element of solid angle and <math display="inline">v^2 = |\mathbf{v}|^2 = v_x^2 + v_y^2 + v_z^2</math>. 

The Maxwellian distribution function for particles moving in only one direction, if this direction is {{mvar|x}}, is
<math display="block"> f(v_x) ~dv_x = \sqrt{\frac{m}{2 \pi k_\text{B}T}} \, \exp\left(-\frac{mv_x^2}{2k_\text{B}T}\right) ~ dv_x, </math>
which can be obtained by integrating the three-dimensional form given above over {{mvar|v{{sub|y}}}} and {{mvar|v{{sub|z}}}}.

Recognizing the symmetry of <math>f(v)</math>, one can integrate over solid angle and write a probability distribution of speeds as the function<ref>{{ cite book | last = Müller-Kirsten | first = H. J. W. | author-link = Harald J. W. Mueller-Kirsten | url = https://www.worldcat.org/title/822895930 | title = Basics of Statistical Physics | date = 2013 | publisher = [[World Scientific]] | isbn = 978-981-4449-53-3 | edition = 2nd | oclc = 822895930 | chapter = 2 }}</ref>

<math display="block"> f(v) = \biggl[\frac{m}{2 \pi k_\text{B}T}\biggr]^{{3}/{2}} \, 4\pi v^2 \exp\left(-\frac{mv^2}{2k_\text{B}T}\right). </math>

This [[probability density function]] gives the probability, per unit speed, of finding the particle with a speed near {{mvar|v}}. This equation is simply the Maxwell–Boltzmann distribution (given in the infobox) with distribution parameter <math display="inline">a = \sqrt{k_\text{B}T/m}\,.</math> 
The Maxwell–Boltzmann distribution is equivalent to the [[chi distribution]] with three degrees of freedom and [[scale parameter]] <math display="inline">a = \sqrt{k_\text{B}T/m}\,.</math>

The simplest [[ordinary differential equation]] satisfied by the distribution is:
<math display="block">\begin{align}
0 &= k_\text{B}T v f'(v) + f(v) \left(mv^2 - 2k_\text{B}T\right), \\[4pt]
f(1) &= \sqrt{\frac{2}{\pi}} \, \biggl[\frac{m}{k_\text{B} T}\biggr]^{3/2} \exp\left(-\frac{m}{2k_\text{B}T}\right);
\end{align}</math>

or in [[unitless]] presentation:
<math display="block">\begin{align}
0 &= a^2 x f'(x) + \left(x^2-2 a^2\right) f(x), \\[4pt]
f(1) &= \frac{1}{a^3} \sqrt{\frac{2}{\pi }} \exp\left(-\frac{1}{2 a^2} \right).
\end{align}</math>
<!--Note that a distribution (function) is not the same as the probability. The distribution (function) stands for an average number, as in all three kinds of statistics (Maxwell–Boltzmann, [[Bose–Einstein statistics|Bose–Einstein]], [[Fermi–Dirac statistics|Fermi–Dirac]]).--> 
With the [[Darwin–Fowler method]] of mean values, the Maxwell–Boltzmann distribution is obtained as an exact result.

[[File:Simulation of gas for relaxation demonstration.gif|thumb|471x471px|Simulation of a 2D gas relaxing towards a Maxwell–Boltzmann speed distribution]]