The Maxwell (or Maxwell-Boltzmann) distribution gives the distribution of speeds of molecules in thermal equilibrium as given by statistical mechanics. Defining , where is the Boltzmann constant, is the temperature, is the mass of a molecule, and letting denote the speed a molecule, the probability and cumulative distributions over the range are
(1)
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(2)
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(3)
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using the form of Papoulis (1984), where is an incomplete gamma function and is erf. Spiegel (1992) and von Seggern (1993) each use slightly different definitions of the constant .
It is implemented in the Wolfram Language as MaxwellDistribution[sigma].
The th raw moment is
(4)
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giving the first few as
(5)
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(6)
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(7)
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(8)
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(Papoulis 1984, p. 149).
The mean, variance, skewness, and kurtosis excess are therefore given by
(9)
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(10)
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(11)
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(12)
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The characteristic function is
(13)
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where is the erfi function.