Derivation of Rayleigh-Jeans law

[dEdt]

All the derivations of the Rayleigh-Jeans 'Law' I've seen assume that the electromagnetic radiation is enclosed in a cube. I'm trying to derive the law using less arbitrary circumstances. That is, by starting with the equation $$U=\int \left[ \frac{\epsilon_0}{2}E^2 + \frac{1}{2\mu_0}B^2 \right]dV,$$
then taking the Fourier transform of the electric and magnetic fields, appling Parseval's theorem, and finally using the equipartition theorem I hope to calculate the spectral energy density. Unfortunately I'm having trouble filling in the details, and would appreciate some help. Thanks.

 

[Jano L.]

Fourier transform is an integral over the whole real axis. Do you really want to consider thermal radiation in the whole space? Usually, cavity is assumed. If cavity is cuboid, one can use standard Fourier series with sines. If cavity has more complicated shape, one can use generalized Fourier series with eigenfunctions of the cavity (these are not sine functions of position, but they are still sine functions of time.)

 

[dEdt]

Do you really want to consider thermal radiation in the whole space?

Yes. Or at least, the whole space outside of the blackbody in question. I know that the cavity approach is standard (and easier), but I'd like to be able to do it this way.

 

[WannabeNewton]

It's impossible to help you unless you post the details of the calculations you have thus far. Incidentally I'm having trouble seeing how you would use Parseval's identity when you don't have countably many normal modes of the electromagnetic field to work with since you aren't imposing periodic box boundary conditions. It may be that there is a generalized form of Parseval's identity; the one I know of is only for the relationship between Fourier transforms and sums.

 

[dauto]

It's impossible to help you unless you post the details of the calculations you have thus far. Incidentally I'm having trouble seeing how you would use Parseval's identity when you don't have countably many normal modes of the electromagnetic field to work with since you aren't imposing periodic box boundary conditions. It may be that there is a generalized form of Parseval's identity; the one I know of is only for the relationship between Fourier transforms and sums.

Yes, it generalizes to $$\int_{-\infty}^\infty | x(t) |^2 \, dt = \int_{-\infty}^\infty | X(f) |^2 \, df, $$ where X(f) is the Fourier transform of x(t).

 

[dEdt]

It's impossible to help you unless you post the details of the calculations you have thus far. Incidentally I'm having trouble seeing how you would use Parseval's identity when you don't have countably many normal modes of the electromagnetic field to work with since you aren't imposing periodic box boundary conditions. It may be that there is a generalized form of Parseval's identity; the one I know of is only for the relationship between Fourier transforms and sums.

My approach is as follows: as Dauto mentioned, $$\int (\phi(\mathbf{x}))^2 dV = \int |\tilde{\phi}(\mathbf{k})|^2 d^3 k,$$ where the tilde denotes the Fourier transform.

Applying this equation to the electromagnetic energy formula, we get that
$$U=\int \left[ \frac{\epsilon_0}{2} |\tilde{\mathbf{E}}(\mathbf{k})|^2 + \frac{1}{2\mu_0} |\tilde{\mathbf{B}}(\mathbf{k})|^2 \right] d^3 k.$$

If I could rearrange this integral into something like $$U=\int \left[ \mbox{something that looks like a harmonic oscillator} \right] d\nu,$$ then I would be a happy man because I could apply the equipartition theorem (or at least some 'continuous extension' of it) to derive the Rayeigh-Jeans law. But I'm having trouble with that last step.

 

[Born2bwire]

I don't have the derivation on hand so I do not know if this is simply repeating the typical textbook derivation, but why not simply assume a cavity of finite size and take its definition via the Fourier series. Then take the volume of the cavity to infinity at a suitable point. You would have to introduce an infinitesimal loss when doing so to remove the incoming wave solution in the cavity mode. In essence, you are simply finding the vacuum modes. This is something that I have done for derivations like the Casimir force.

 

[dauto]

I think you need to write the fields in terms of the potentials A and ø.

 

[Jano L.]

Yes. Or at least, the whole space outside of the blackbody in question. I know that the cavity approach is standard (and easier), but I'd like to be able to do it this way.

Equilibrium radiation spectral formula gives radiation energy per unit frequency interval. One of the assumptions behind its derivation is

(reasonable for low but wrong for high frequencies)
1) that the radiation is enclosed in a perfectly reflecting cavity of finite volume

or

2) unphysical but mathematically similar condition that the field in space repeats the pattern of the field in a finite-sized cuboid

Both cases allow for expansion of the field into Fourier series and lead to finite number of modes (oscillators) per unit frequency interval. Multiplicating by finite average energy of such oscillator, this in turn leads to finite spectral density of Poynting energy density (R-J or Planck function) - just divide the energy of oscillators per unit frequency interval by the volume of the cavity/cuboid.

However, if the region where the field is considered is infinite right from the beginning and no periodic conditions are imposed, the field cannot be expanded into Fourier series. It may be expandable into Fourier integral, but then the number of independent modes per unit frequency interval is infinite.

There is no obvious way to ascribe energy to such continuous mode or interval of modes, except for reverse-engineering the desired spectral function or returning to finite volume. In case of the Rayleigh-Jeans or Planck spectral function, one continuous mode has to be ascribed zero energy. Derivation of the spectral function in line with the Rayleigh-Jeans procedure does not work for infinite volume.

Physically, this is not much of a problem since there is little reason to think radiation in the whole space is equilibrium radiation. And I would like to say that the calculation method of Rayleigh-Jeans is of limited value even for finite cavities, as it does not (and should not) work for high frequencies.