Equipartition of energy in the modes of a resonant cavity

[Guilherme Franco]

TL;DR

Why was energy expected to be equally distributed between the modes of a resonant cavity? Why couldn't they be independent like in a string?

I think the answer for this may be straightforward, but I don't see anywhere that explains this from the scratch:

A large resonant cavity with a small hole is used to approximate an ideal black body.

I understand the conditions for the modes inside the cavity. But there are two points that aren't clear to me:

1. I don't understand why it was considered that the energy should be equipartitioned between those modes. Because I don't see a reason why they couldn't be independent. At least not if it was ideally reflecting body inside. In that case, just like in ideal vibrating stings, there could be no exchange of energy between the modes and the spectrum of the light inside it would be just like the spectrum of the light entering it. I think the story has to do with the body not being perfectly reflecting and being in equilibrium with the modes inside the cavity. But even then: Why couldn't it just stay in equilibrium with the modes that has already being formed by the light that entered the cavity? Is the equipartition being mediated by the material portion of the cavity?
2. Why exactly does particular solution serves as a model for entirely solid radiators? Does this EM field modes exist inside opaque materials?

 

[sophiecentaur]

Summary: Why was energy expected to be equally distributed between the modes of a resonant cavity? Why couldn't they be independent like in a string?

I don't understand why it was considered that the energy should be equipartitioned between those modes. Because I don't see a reason why they couldn't be independent.

I imagine that it's assumed that there is a lot of coupling between modes, for practical reasons due to interaction with the surface. At lower frequencies (microwave cavities, for instance) I think the energy in the mode that was excited in an unloaded (ideal highly conductive) cavity would not couple to other modes very quickly.

 

[nasu]

The equipartition of energy is seen in a system at thermal equilibrium. This is the necessary condition and is assumed for a black body cavity.

 

[Henryk]

Summary: Why was energy expected to be equally distributed between the modes of a resonant cavity? Why couldn't they be independent like in a string?

Because I don't see a reason why they couldn't be independent

The reason is that no modes are perfectly isolated. There is always some coupling between the nodes. It is also true for a string. A violin always sounds like a violin because the ration of vibrations between different modes is at equilibrium.

 

[sophiecentaur]

The reason is that no modes are perfectly isolated. There is always some coupling between the nodes. It is also true for a string. A violin always sounds like a violin because the ration of vibrations between different modes is at equilibrium.

The apparent conflict between the dominance of the fundamental in strings and things and the black body continuous spectrum is a practical one.
A string or an air / microwave cavity will not sustain the energy from an impulse for many cycles and the modes which are examined are pretty low order. That also implies that, for a continuous exciting waveform, the observed vibrations will be dominated by selected frequencies of the input waveform (any others will be at a much lower level).
In the case of a practical black body cavity, it has been excited by a continuous noise like source - such as white light or a hot wall. That will excite a continuum of modes (very high order, when we're talking about IR spectrum of the fields inside).