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Maxwell distribution

From phenomenon to model

James Clerk Maxwell obtained the law in 1860 while studying molecular speeds in an ideal gas. It helped connect statistical mechanics and probability.

What to look at

In Phitter the support begins at loc and continues toward positive infinity. alpha controls component scale; the density has an interior peak and a rapidly decaying tail.

Useful relationships

It is a Chi and Generalized Gamma special case. Rayleigh is the two-dimensional analogue; Maxwell–Boltzmann speed distribution is the usual physics name.

Names you may also encounter

It is also called the Maxwell–Boltzmann speed distribution or Maxwell distribution.

When it makes sense

In physics, equilibrium gas-speed histograms are compared with Maxwell–Boltzmann. In engineering, it can describe the norm of three independent Gaussian errors when isotropy is plausible.

What not to assume

Physical interpretation requires isotropy and equilibrium; fitting Maxwell to grouped speeds does not prove those conditions.

Speed appears after forgetting three directions

Maxwell began with three independent Normal velocity components sharing a scale. Speed is the vector length and retains none of the directional signs. This creates the Maxwell law, also called Maxwell Boltzmann for molecular speed.

It is the three-dimensional case of the Chi distribution. One dimension gives Half Normal and two give Rayleigh. Squared speed, after scale adjustment, is Chi Square with three degrees of freedom and connects with kinetic energy. For real gases, thermal equilibrium and isotropy are physical assumptions. For generic magnitudes, fitting the curve does not prove that three equivalent Gaussian components exist.

Decision guide

A good candidate when: after subtracting loc, the variable is the magnitude of three independent, centred Gaussian components with equal variance.

Compare it with: Rayleigh for two components and Chi for another dimension. If drift or unequal variances are present, the Maxwell construction is no longer appropriate.

References

Maxwell Distribution: equations and calculator

Distribution defintion

X∼Maxwell(α,Loc)X\sim\mathrm{Maxwell}\left(\alpha,\text{Loc}\right)

Distribution domain

x∈(Loc,∞)x\in\left(\text{Loc},\infty\right)

Parameters domain and parameters constraints

α∈R+,Loc∈R\alpha\in\mathbb{R}^{+},\text{Loc}\in\mathbb{R}

Cumulative distribution function

FX(x)=erf⁡(x−Loc2α)−2π(x−Loc)e−(x−Loc)2/(2α2)αF_{X}\left(x\right)=\operatorname{erf}\left(\frac{x-\text{Loc}}{\sqrt{2} \alpha}\right) -\sqrt{\frac{2}{\pi}}\frac{(x-\text{Loc}) e^{-(x-\text{Loc})^2/\left(2\alpha^2\right)}}{\alpha}

Probability density function

fX(x)=2π(x−Loc)2e−(x−Loc)2/(2α2)α3f_{X}\left(x\right)=\sqrt{\frac{2}{\pi}}\frac{(x-\text{Loc})^2 e^{-(x-\text{Loc})^2/\left(2\alpha^2\right)}}{\alpha^3}

Percent point function/Sample

FX−1(u)=Loc+α2P−1(1.5,u)F^{-1}_{X}\left(u\right)=\text{Loc}+\alpha\sqrt{2\text{P}^{-1}\left(1.5,u\right)}

Non-central parametric moments

μk′=E[Xk]=∫−∞∞xkfX(x)dx\mu'_{k}=E[X^k]=\int_{-\infty}^{\infty}x^{k}f_{X}\left(x\right)dx

Parametric mean

Mean(X)=μ1′=Loc+2α2π\mathrm{Mean}(X)=\mu'_{1}=\text{Loc}+2\alpha \sqrt{\frac{2}{\pi}}

Parametric variance

Variance(X)=μ2′−μ1′2=α2(3π−8)π\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}=\frac{\alpha^2(3 \pi-8)}{\pi}

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=22(16−5π)(3π−8)3/2\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=\frac{2 \sqrt{2} (16 -5 \pi)}{(3 \pi-8)^{3/2}}

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=4(−96+40π−3π2)(3π−8)2+3\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}=4\frac{\left(-96+40\pi-3\pi^2\right)}{(3 \pi-8)^2}+3

Parametric median

Median(X)=Loc+α2P−1(1.5,12)\mathrm{Median}(X)=\text{Loc}+\alpha\sqrt{2\text{P}^{-1}\left(1.5,\frac{1}{2}\right)}

Parametric mode

Mode(X)=Loc+α2\mathrm{Mode}(X)=\text{Loc}+\alpha\sqrt{2}

Additional information and definitions

Loc:Location parameter\text{Loc}:\text{Location parameter}
α:Scale parameter\alpha:\text{Scale parameter}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
P−1(a,u):Inverse of regularized lower incomplete gamma function\text{P}^{-1}\left(a,u\right):\text{Inverse of regularized lower incomplete gamma function}