Maxwell Distribution

$\begin{figure}\begin{center}\BoxedEPSF{MaxwellDistribution.epsf scaled 1200}\end{center}\end{figure}$

The distribution of speeds of molecules in thermal equilibrium as given by statistical mechanics. The probability and cumulative distributions are

$\displaystyle P(x)$	$\textstyle =$	$\displaystyle \sqrt{2\over\pi} \,a^{3/2} x^2 e^{-ax^2/2}$	(1)
$\displaystyle D(x)$	$\textstyle =$	$\displaystyle {2\gamma({\textstyle{3\over 2}},{\textstyle{1\over 2}}ax^2)\over\sqrt{\pi}},$	(2)

where $\gamma(a,x)$ is an incomplete Gamma Function and $x\in [0,\infty)$ . The moments are

$\displaystyle \mu$	$\textstyle =$	$\displaystyle 2\sqrt{2\over\pi a}$	(3)
$\displaystyle \mu_2$	$\textstyle =$	$\displaystyle {3\over a}$	(4)
$\displaystyle \mu_3$	$\textstyle =$	$\displaystyle 8\sqrt{2\over a^3\pi}$	(5)
$\displaystyle \mu_4$	$\textstyle =$	$\displaystyle {\textstyle{15\over 2}},$	(6)

$\displaystyle \mu$	$\textstyle =$	$\displaystyle 2\sqrt{2\over\pi a}$	(7)
$\displaystyle \sigma^2$	$\textstyle =$	$\displaystyle {3\pi-8\over\pi a}$	(8)
$\displaystyle \gamma_1$	$\textstyle =$	$\displaystyle {8\over 3}\sqrt{2\over 3\pi}$	(9)
$\displaystyle \gamma_2$	$\textstyle =$	$\displaystyle -{\textstyle{4\over 3}}.$	(10)

References

von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 252, 1993.