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 has one static (Shared in Visual Basic) method, Sample, which generates a random sample using a user-supplied uniform random number generator. The second parameter is the scale parameter of the distribution that is to be sampled.

var random = new Pcg32();
double sample = MaxwellDistribution.Sample(random, 3.0);

Dim random = new Pcg32()
Dim sample = MaxwellDistribution.Sample(random, 3.0)
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No code example is currently available or this language may not be supported.

let random = new Pcg32()
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