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)
			(2)
			(3)

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)

giving the first few as

			(5)
			(6)
			(7)
			(8)

(Papoulis 1984, p. 149).

The mean, variance, skewness, and kurtosis excess are therefore given by

			(9)
			(10)
			(11)
			(12)

The characteristic function is

{atsqrt(2/pi)-e^(-a^2t^2/2)(a^2t^2-1)×[sgn(t)erfi((a|t|)/(sqrt(2)))-i]}, " src="https://mathworld.wolfram.com/images/equations/MaxwellDistribution/NumberedEquation2.gif" style="height:48px; width:384px" />

(13)

where  is the erfi function.

REFERENCES:

Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 104 and 148-149, 1984.

Spiegel, M. R. Schaum's Outline of 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.