Maxwell–Boltzmann
Probability density function

Cumulative distribution function

Parameters

a>0\,

Support

x\in (0;\infty)

pdf

\sqrt{\frac{2}{\pi}} \frac{x^2 e^{x^2/(2a^2)}}{a^3}

CDF

\textrm{erf}\left(\frac{x}{\sqrt{2} a}\right) \sqrt{\frac{2}{\pi}} \frac{x e^{x^2/(2a^2)}}{a} where erf is the Error function

Mean

\mu=2a \sqrt{\frac{2}{\pi}}

Mode

\sqrt{2} a

Variance

\sigma^2=\frac{a^2(3 \pi  8)}{\pi}

Skewness

\gamma_1=\frac{2 \sqrt{2} (16 5 \pi)}{(3 \pi  8)^{3/2}}

Ex. kurtosis

\gamma_2=4\frac{(96+40\pi3\pi^2)}{(3 \pi  8)^2}

Entropy

\ln(a\sqrt{2\pi})+\gamma\frac{1}{2}

In physics, particularly statistical mechanics, the Maxwell–Boltzmann distribution or Maxwell speed distribution describes particle speeds in idealized gases where the particles move freely inside a stationary container without interacting with one another, except for very brief collisions in which they exchange energy and momentum with each other or with their thermal environment. Particle in this context refers to gaseous atoms or molecules, and the system of particles is assumed to have reached thermodynamic equilibrium.^{[1]}
The distribution is a probability distribution for the speed of a particle within the gas  the magnitude of its velocity. This 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.^{[2]}
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 interactions, vortical flow, relativistic speed limits, and quantum exchange interactions) that make their speed distribution sometimes very different from the Maxwell–Boltzmann form. That said, rarefied gases at ordinary temperatures behave very nearly like an ideal gas and the Maxwell speed distribution is an excellent approximation for such gases. Thus, it forms the basis of the kinetic theory of gases, which prov==Taxonavigation==
Selected references
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