What's Wrong with the Equipartition Theorem

Why should all the modes get the same energy in the first place? It's an important question, and let's start by looking at a case where it works.

In a diatomic gas like hydrogen, we consider the molecules to be tiny rigid dumbbells. Let's assume that they start off zipping every which way, but that none of them are spinning. It's easy to see that this situation cannot last long. Any kind of glancing collision will result in some energy being converted into rotational motion. As an extreme case, we can show a situation where the conversion is 100%. Imagine two molecules speeding towards each other with each of their axes perpendicular to the direction of motion. Two of the atoms collide - not four, in general - and at the moment of collision the other two atoms, at opposite ends of their respective dumbbells, are unaffected. If the two atoms collide exactly head on, they will each instantly reverse direction - because that is what billiard balls do.

The kinetic energy of this system has been converted from 100% translational to 100% rotational. You can see it must be so because at the moment the molecules begin to separate, their component atoms have equal and opposite velocities. Therefore, they are spinning, and their average forward motion is zero. It's not hard to imagine that eventually the motion in a gas is randomly distributed between all its different possible forms.

But what about the vibrations? Shouldn't a good clean hit between stiff molecules set each of them ringing? Indeed, that is what equipartition tells us, and in an ideal world that's just what would happen. But as pointed out in the last essay, the measuremen of specific heats shows that the vibrational modes are not in fact active for common diatomic gasses.

Why not? I think it's pretty obvious that the stumbling block is going to be that "good clean hit". Atoms are not dumbbells. When two atoms collide there is no meeting of hard surfaces. As they approach they influence each other with electric forces. It is a gradual process...very fast on our time scales, but compared to the vibrational frequency of a hydrogen molecule - well, it's easy to imagine that it might be very much slower indeed.

It's not a billiard ball collision and that's why the vibrational mode is not excited. So maybe there is no need to worry about the discrepancy in specific heats. In fact, when the temperatures get high enough, so the velocities are high enough, so the collisions take place on a more compressed time scale...voila, the molecules start to vibrate after all.

What's more, if we can explain away the vibrations, maybe we can explain away the ultraviolet catastrophe. The vibrating hydrogen molecule happens to be electrically neutral, so it doesn't influence the cavity modes one way or the other. But to drive the standing waves inside the metal box, you need some means of converting the thermal energy of the molecules into electromagnetic energy. You do that by getting charges to vibrate. Hydrogen doesn't work so well, but any composite molecule like CN or NO will do. They are polarized to some extent, so when they vibrate they become tiny antennas, pumping electromagnetic energy into the field.

But if they don't vibrate, then there is no high-frequency energy, and there is no ultraviolet catastrophe. The idea that they would have to vibrate was based on a literal reading of the equipartion theorem. We've made the case that there is no mechanism for equipartition when the collisions are "soft", as they must be in the real world. Have we thenb solved the problem that was the ruin of classical physics at the end of the nineteenth century?

That's something we'll take up in our next posting.