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Maxwell Distribution

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MaxwellDistribution

The Maxwell (or Maxwell-Boltzmann) distribution gives the distribution of speeds of molecules in thermal equilibrium as given by statistical mechanics. Defining a=sqrt(kT/m), where k is the Boltzmann constant, T is the temperature, m is the mass of a molecule, and letting x denote the speed a molecule, the probability and cumulative distributions over the range x in [0,infty) are

P(x)=sqrt(2/pi)(x^2e^(-x^2/(2a^2)))/(a^3)
(1)
D(x)=(2gamma(3/2,(x^2)/(2a^2)))/(sqrt(pi))
(2)
=erf(x/(sqrt(2)a))-(xe^(-x^2/(2a^2)))/asqrt(2/pi),
(3)

using the form of Papoulis (1984), where gamma(a,x) is an incomplete gamma function and erf(x) is erf. Spiegel (1992) and von Seggern (1993) each use slightly different definitions of the constant a.

It is implemented in the Wolfram Language as MaxwellDistribution[sigma].

The nth raw moment is

mu_n^'=(2^(1+n/2)a^nGamma(1/2(3+n)))/(sqrt(pi)),
(4)

giving the first few as

mu^'=2asqrt(2/pi)
(5)
mu_2^'=3a^2
(6)
mu_3^'=8a^3sqrt(2/pi)
(7)
mu_4^'=15a^4
(8)

(Papoulis 1984, p. 149).

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

mu=2asqrt(2/pi)
(9)
sigma^2=(a^2(3pi-8))/pi
(10)
gamma_1=(2sqrt(2)(5pi-16))/((3pi-8)^(3/2))
(11)
gamma_2=-(4(96-40pi+3pi^2))/((3pi-8)^2).
(12)

The characteristic function is

phi(t)=i{atsqrt(2/pi)-e^(-a^2t^2/2)(a^2t^2-1)×[sgn(t)erfi((a|t|)/(sqrt(2)))-i]},
(13)

where erfi(z) is the erfi function.

See also

Exponential Distribution, Normal Distribution, Rayleigh Distribution

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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.

Referenced on Wolfram|Alpha

Maxwell Distribution

Cite this as:

Weisstein, Eric W. "Maxwell Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MaxwellDistribution.html

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