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Maxwell-Boltzmann distribution

(distribution of velocities of gas particles)

The Maxwell-Boltzmann distribution is a probability density function of the possible velocities of particles of gas at a constant temperature in a container. It is defined by the Maxwell-Boltzmann equation, defining a function yielding the fraction of particles expected to be within 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)du dv dw =  (m/2πkT)3/2exp(-Eke/kT)


Eke = 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)

(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 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.

Further reading:

Referenced by pages:
atmospheric escape
Boltzmann equation
Boltzmann Transport Equation (BTE)
electron pressure
Gamow peak
Jeans escape
solar wind
stellar dynamics
Thermal Bremsstrahlung
thermal emission
Vlasov-Poisson equation