Intuitive understanding of the derivation of the Rayleigh-Jeans law

Question

I know the Rayleigh-Jeans law and how the formula predicts UV catastrophe. Without getting into the exact derivation, I am trying to get some intuitive understanding of it by using some of the broad arguments that they used, based on whatever I found on the net and various books looks like they applied statistical physics method to waves.

They assumed that total energy of radiation is equally distributed among all possible frequencies. I get that. I guess I am struggling with the 2nd argument which was there is no limit on the number of modes of vibrations that can excited? Why would there be no limit? Can someone please explain?

Answer 1

They assumed the hypotheses of kinetic theory and considered waves of light inside a metallic cage. These waves are standing waves and so the frequencies that the light takes are quantized and so they have an integer number $ n $ associated. If you consider $ n_x $, $ n_y $ and $ n_z $ as the integer numbers for every coordinate then they found that the frequency was given by the sum of the squares of these numbers $ n_x^2 + n_y^2 + n_z^2 $ which forms a sphere. The distribution of frequencies then is given by the surface of the sphere of a given radius $ f $ and so the distribution is given by the square of the frequency $ f^2 $. This is why the Rayleigh-Jeans distribution is a parabola.

It's not very intuitive why this is the case but their derivation goes along those lines. The very important fact is that they found that the distribution of frequencies is not homogeneous and depends on the square of the frequencies, so higher frequencies are likelier than lower frequencies. That's how I picture it at least. If higher frequencies contribute more to the distribution then it is easy to see how at higher frequencies the energy that the blackbody realeases goes to infinity.

Answer 2

I guess I am struggling with the 2nd argument which was there is no limit on the number of modes of vibrations that can excited? Why would there be no limit? Can someone please explain?

This follows from properties of the standard Fourier series expansion of function on a real interval; in order for the expansion to be able to reproduce variations of the original function (here, electric field) on a very short length scale, very high frequency waves have to be included in the Fourier sum. If variations on arbitrarily short length scale are to be captured by the Fourier series, then the Fourier sum over frequencies has to go over arbitrarily high frequencies $kf_0$, $k=1,2,...\infty$.

So the series terms corresponding to arbitrarily high frequencies (and thus infinite number of frequencies and terms) follows from the assumption that the physical electric field can have features on arbitrarily small scale. This may not be true physically, and there may be a short length beyond which there is no such features, and then the number of required frequencies would be finite, and the UV catastrophe avoided. However, nobody has found evidence for existence of such length; electric field seems to be able to have any frequency, even arbitrarily high one.

However, there is a good physical reason to expect suppression of the Fourier expansion coefficients after some very high frequency in case of equilibrium radiation hold in a real cavity; such cavity can't hold radiation of very high frequencies inside, because no cavity can be perfectly reflective to radiation of arbitrarily high frequencies. Even polished metal cavity will partially pass through radiation of high enough frequency (X-rays, gamma radiation), so achieving equilibrium at this frequency in the cavity would be very difficult; the cavity transparency will work against that.