# Relativistic maxwell-boltzmann-distribution

## Main Question or Discussion Point

In thermodynamics (ignoring relativistic effects) you can use the maxwell-boltzmann-distribution to find the average speed of the gas particles.
$v^2=\frac{8kT}{\pi m}$

But there are high Temperatures that would have average speeds > c.
Are there distributions that describe gases with an average speed of 0.5 relativistically?

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mfb
Mentor
Well, you have to replace the nonrelativistic formula with the relativistic one.

The average squared velocity will go towards c^2, and all particles are always slower than c for every temperature.

bcrowell
Staff Emeritus
Gold Member
Well, you have to replace the nonrelativistic formula with the relativistic one.
...which to me is not very illuminating, since it's given without any derivation and written in terms of a goofy choice of variable.

Can't you just take the partition function and put in the relativistic expression for the energy? I.e.:

$$e^{-\beta\sqrt{m^2+p^2}}$$

This is in units with c=1, and beta is the inverse temperature.

Bill_K