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The **Maxwell-Boltzmann distribution**
is a probability density function of the
possible velocities of particles of gas
at a constant temperature in a container
that yields the fraction of particles expected
to be in a range of velocities.
It is a mathematical function modeling
a "classical" (non-quantum-mechanical) system,
based on idealized characteristics, e.g.,
an ideal gas, with no influence from various physical
factors, e.g., gravity.
Actual physical systems vary in how closely
they match the ideal, so it can be a good
approximation, a coarse one, or a poor one.

One form:

f(u,v,w)dudvdw = (m/2πkT)^{3/2}exp(-E_{ke}/kT)

Where:

- f(u,v,w)dudvdw - probability density over the velocity space of <u,v,w>.
- m - mass of each particle.
- k - Boltzmann constant.
- T - temperature of the gas.
- E
_{ke}- kinetic energy at the velocity; <u,v,w>.

E_{ke}= m(u²+v²+w²)/2

The Maxwell-Boltzmann distribution with its exponential component
and 3/2 exponent seems inevitable since a gas in 3d space would seem
to need a distribution that does not depend upon choice of axis
directions. The **Rayleigh distribution** is a similar distribution
for two dimensions:

f(x;ρ) = (x/ρ²)exp(-x²/2ρ²)

- x - random variable.
- ρ - a scale parameter.
- f(x;ρ) - probability density function.

(The Maxwell-Boltzmann distribution is equivalent, but besides being 3d, is written in terms of the physics it is modeling, i.e., based on particle mass, temperature, and the Boltzmann constant.) A distribution describing just two of the three components of the gas particle movements will be a Rayleigh distribution. The Rayleigh distribution has other uses, and can be used in modeling disks (e.g., circumstellar disks), as well as modeling the overall patterns of orbits of numerous bodies in a plane, such as asteroid belts.

Both are essentially multi-dimensional generalizations of the
**normal distribution** assuming identical variance in all dimensions
and zero centering. A gas (i.e., bunch of moving particles) in a
theoretical 1d space (like beads on a slippery wire) might be modeled
as such a normal distribution, but without the effects of glancing
bounces, would seem to stick to whatever distribution it began with,
i.e., with no tendency to evolve toward a normal distribution.

atmospheric escape

Boltzmann equation

Gamow peak

Jeans escape

stellar dynamics