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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)

and the Mean, Variance, Skewness, and Kurtosis are
$\displaystyle \mu$ $\textstyle =$ $\displaystyle 2\sqrt{2\over\pi a}$ (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)

See also Exponential Distribution, Gaussian Distribution, Rayleigh Distribution


Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, p. 119, 1992.

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

© 1996-9 Eric W. Weisstein