Can Maxwell's law of distribution of velocities be used to determine a value for absolute heat?

Question

I'm currently reading about Maxwell's law of distribution of velocities, and the thought occurred to me that I could use this to calculate the maximum temperature that an atom could reach.

My theory is that there is a temperature at which atoms would exhibit movement equal to $c$, and hence could not move any faster.

Using the average velocity formula to calculate $T$ where $\bar{u} = c$ for a hydrogen atom (Wolfram Alpha):

$\bar{u} = \sqrt{8RT\over\pi M} \\\bar{u} = c \\ M = 2 \\ c = \sqrt{8RT\over2\pi} \\ T = 8.494352186809165 \times 10^{12} K$

Given that I haven't found any evidence of this theory online, I assume that I've made a mistake.

Is this correct or not, and why?

An interesting side note to think about is that under this theory, an atom heated to this temperature would have infinite mass.

There is no maximum temperature (see also this search of this site). The Maxwell-Boltzmann distribution is a non-relativistic approximation and will need to be modified for relativistic energies. — John Rennie, Sep 25 '15 at 16:56
@JohnRennie, so are you saying that $M$ is going to increase at such high energies, thus balancing out the increase of $\bar{u}$ such that $\bar{u} = c$ only when $M = \infty$? That's clever. So... is this a dup, should I delete, or what? — Bardi Harborow, Sep 25 '15 at 17:14
You might want to take a look at the Maxwell-Jüttner distribution — Kyle Kanos, Sep 25 '15 at 17:17
Note also that atoms cannot have velocity of $c$, they can only asymptotically approach it. — Kyle Kanos, Sep 25 '15 at 17:18
@KyleKanos, sorry, I didn't express it as an upper bound properly. I know you can't actually get to $c$. — Bardi Harborow, Sep 25 '15 at 17:24
@Bardi: I suspected that was the case, but I wanted to ensure you clear on that point. — Kyle Kanos, Sep 25 '15 at 17:25
The key difference is that while velocity is limited in relativity, energy is not. So as you heat the gas up to relativistic levels, atoms can have more and more energy without going much faster. — zeldredge, Sep 25 '15 at 17:36

Floris · Accepted Answer · 2015-09-25T18:26:59.470

Comments already hinted at this, but the simple answer is that when you wrote the average velocity as a function of temperature, you were using a non-relativistic approximation (which is valid for most "everyday" situations, but not at extreme temperatures). Furthermore, since the particles have a mean velocity $\bar u$, you can't simply set that equal to $c$ as many particles would have then to go faster than the mean... whichever way you look at it, your analysis doesn't work.

Instead, you can use the more general equipartition theorem that states that every degree of freedom of a particle in a gas has an energy of $\frac12 kT$

You could write an expression for the kinetic energy in one dimension:

$$E = \frac{p^2}{2m} = \frac12 kT$$

Note that I got rid of "velocity" and instead expressed the result in terms of momentum. As you know, momentum will continue to increase as a particle gets closer to the speed of light, because it will become more massive.

How best to calculate the "mean" velocity becomes tricky - the velocity distribution is no longer following the classical distribution. The actual velocity distribution at relativistic velocities is the Maxwell-Jüttner distribution which converges to the Maxwell-Boltzmann distribution at non-relativistic velocities (tip of the hat to Kyle Kanos for pointing this out).

All of which is a long winded way to say "There is no upper limit on temperature".

although the momentum distribution is... huh? The momentum follows the Maxwell-Juttner distribution which isn't classical. — Kyle Kanos, Sep 25 '15 at 18:15

score 1 · Answer 2 · answered Sep 25 '15 at 17:52

Maxwell's distribution of velocities has been derived through classical arguments. It does not take into account relativistic effects. You don't need to evaluate anything to see that; just observe that it accepts arguments (velocities components) over the whole real line and does not have a cut-off at the speed of light. So it gives a non-zero probability even for superluminal velocities

Can Maxwell's law of distribution of velocities be used to determine a value for absolute heat?

2 Answers2