I ended my last posting by speculating that the only way to fix things was to find some way of recalculating the entropy. Because no matter how many practical reasons I could find as to why the high-frequency vibrational modes should not be activated in "real" matter, the fundamental problem remains: we can still imagine an ideal gas made of perfect billiard balls joined by flexible rods, and for this theoretical gas, the entropy calculation agrees with the practical outcome: all the modes, vibrational included, must be excited to their full complement of energy. And any "real" gas must have the same equilibrium as this "ideal" gas.

Isn't this something like what Planck did? By quantising the energy modes of the electromagnetic resonances, he fixed things so the high frequency modes went away. Because when you calculate the entropy of these quantised modes, it gives the desired outcome. But there's something very wrong with this approach. If Planck allows the gas itself to behave as an ideal collection of springs and ball bearings, then both thermodynamics and practical reasoning tell us that the high-frequency vibrations must exist. And it doesn't help to recalculate the entropy of the radiation field, because if those molecules are vibrating, then they will (and must) radiate power into those forbidden frequencies.

One hundred years after the fact, it's hard to know exactly what people were thinking back then. Planck had conjured up this miraculous formula that worked perfectly, but nobody (Planck included) knew quite what to make of it. It seems like people were focused on the thermodynamics of the electromagnetic field, so people figured out that by quantising the modes of vibration, and allowing only certain values of the energy at any given frequency, they could make the entropy come out "right" - in other words, consistent with the experimental facts. But what about the entropy of the mechanical oscillators? Wasn't that still a problem?

It would be a number of years before Einstein would propose that the mechanical oscillations should follow similar rules of quantisation. In the meantime, one has to assume that people were content to allow the mechanical vibrations to have their place in the theory, but to somehow insist, against all logic, that they would not contribute to the intensity of the radiation field. To make this work, they had to suspend the laws of electromagnetism! That is the origin of the idea that energy could only be emitted or absorbed in discrete lumps.

There is a much better way out of this dilemma, and that is to attack the problem at its source: the mechanical vibrations. Because if the unwanted mechanical vibrations are supressed, then the electromagnetic field follows suit and we avert the ultraviolet catastrophe. I have already shown how there are many practical reasons to suppose why those vibrations shouldn't happen in "real" atoms. The problem is with the entropy: the entropy calculation shows that those vibrations must assume their alloted share of the energy; and the entropy calculation is backed up by the theoretical example of the imaginary ideal gas made of springs and billiard balls, where you can verify by mechanical reasoning that it works as claimed - that equipartition must prevail.

What I am saying is that if we solve the problem at the mechanical level, we avoid the problem with the electromagnetic field. You can keep the laws of Maxwell and there is no need for quantized lumps of energy. In my next post, I'm going to delve more deeply into just what is required to solve the problem at the mechanical level.

## Sunday, February 20, 2011

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