Maxwell Distribution

The Maxwell distribution, also known as the Maxwell-Boltzmann distribution, describes the distribution of speeds of molecules in a gas in thermal equilibrium.

The Maxwell distribution is defined by its Probability Density Function (PDF):

where $x\ge 0$ and $a>0$ is the scale parameter. The scale parameter $a$ is proportional to the square root of the temperature and inversely proportional to the mass of the particles.

The Cumulative Distribution Function (CDF) is:

where $\mathrm{erf}$ is the error function.

• Statistical mechanics uses the distribution to model molecular speeds in ideal gases.

• Plasma physics employs it to describe particle velocities in plasmas.

• Astronomy applications include modeling stellar velocities in globular clusters.

• Materials science uses it for analyzing particle size distributions.

Notable properties include:

• The distribution is always right-skewed.

• The most probable speed (mode) differs from the mean speed.

• The distribution has zero probability density at x = 0.

• If X, Y, and Z are independent normal variables with mean 0 and variance $a^2$, then $\sqrt{X^2+Y^2+Z^2}$ follows a Maxwell distribution with parameter a.

• The Maxwell distribution is related to the chi distribution with 3 degrees of freedom.

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• The square of a Maxwell-distributed variable follows a gamma distribution.

The Maxwell distribution is implemented by the MaxwellDistribution class. It has one constructor that takes the value of the scale parameter. The code below creates a Maxwell distribution with scale parameter equal to 3:

var maxwell = new MaxwellDistribution(3.0);

Dim maxwell = New MaxwellDistribution(3.0)

No code example is currently available or this language may not be supported.

let maxwell = new MaxwellDistribution(3.0)

MaxwellDistribution provides static Sample methods for generating random values. The preferred method uses IRandomSource:

Sample

For compatibility, Random overloads are also available. See Introduction to Random Sources for details on creating random sources. The second parameter is the scale parameter of the distribution that is to be sampled.

var random = new Pcg64();
double sample = MaxwellDistribution.Sample(random, 3.0);

Dim random = New Pcg64()
Dim sample = MaxwellDistribution.Sample(random, 3.0)
'

No code example is currently available or this language may not be supported.

let random = new Pcg64()
let sample = MaxwellDistribution.Sample(random, 3.0)

• Maxwell, J. C. (1860, 1927). Illustrations of the dynamical theory of gases, Scientific Papers, 1, 377-410. Paris: Librairies Scientifiques Hermann.

• Boltzmann, L. (1878). Weitere Bemerkungen iiber einige Probleme der mechanischen Warmetheorie, Wiss. Abh., 2, 250-288.

• "Statistical Mechanics" by R.K. Pathria and Paul D. Beale

• "Introduction to Modern Statistical Mechanics" by David Chandler

Other Resources