### Maxwell-Boltzmann distribution

(distribution of velocities of gas particles)

The Maxwell-Boltzmann distribution is a probability density function of the possible velocities of the particles making up gas at a constant temperature in a container. It is defined by the Maxwell-Boltzmann equation, which relates the fraction of particles expected to be within a range of velocities to the velocity, the temperature of the gas and the mass of each particle. 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 as to how closely they match the ideal, so it can be a good approximation, a coarse one, or a poor one. One form (distribution of velocities):

```f(u,v,w) du dv dw = (m/2πkT)3/2exp(-Eke/kT) du dv dw
```

Where:

• u, v, w - rectangular coordinates of the possible velocities.
• f(u,v,w) - probability density over the velocity space of <u,v,w>.
• m - mass of each particle.
• k - Boltzmann constant.
• T - temperature of the gas.
• Eke - kinetic energy at the velocity <u,v,w>.
```Eke = m(u²+v²+w²)/2
```

(This form, giving the velocity distribution in rectangular coordinates is symmetric around zero, as is the associated momentum distribution; the associated speed and kinetic energy distributions are not symmetric and always non-negative.) 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;σ) = (x/σ²)exp(-x²/2σ²)
```
• x - random variable.
• σ - a scale parameter.
• f(x;σ) - 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 the Maxwell-Boltzmann distribution and the Rayleigh distribution (in the above forms) 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 randomizing 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.

(physics)
https://en.wikipedia.org/wiki/Maxwell-Boltzmann_distribution
https://en.wikipedia.org/wiki/Rayleigh_distribution

Referenced by pages:
atmospheric escape
Boltzmann equation
Boltzmann Transport Equation (BTE)
electron pressure
Gamow peak
Jeans escape
kappa distribution
Landau damping
Lorentzian distribution
solar wind
stellar dynamics
suprathermal
thermal bremsstrahlung
thermal emission
Vlasov-Poisson equation

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