Why standing waves in Rayleigh-Jeans derivation?

Question

A central part of the Rayleigh-Jeans law is that the number of allowed states follows $dN \sim \nu ^{2} d\nu$ because the allowed frequencies have to be standing waves. Then by equipartition we get that the energy density in a given frequency band increases without bound for increasing $\nu$, which is obviously unphysical.

Regarding a non-cavity solid object emitting thermally, by contrast, I'm picturing a lattice of atoms and electrons oscillating and/or colliding in such a complicated way as to be random. Perhaps there are conduction electrons, perhaps not. Now, I don't believe the phonon motion contributes much to EM radiation (?); instead, I think, electron oscillations (classically; in QM this would be transitions) dominate. Such electron motion would not be characterized by standing waves or frequencies which depending on the dimensions of the solid, surely?

It seems reasonable that at a given temperature, random "dipole" (or whatever) oscillations in a solid object would classically follow something like a Boltzmann distribution with a maximum at a finite frequency. Intuitively, this could lead to an emission with the qualitative shape of the experimental blackbody curve. Is there a classical theory of random thermal oscillations in a lattice which is relevant here?

Answer 1

Standing waves are not a necessary assumption in the Rayleigh-Jeans derivation. When the derivation works with radiation inside a cavity with walls impermeable to radiation, so the waves can't go through the walls, the radiation inside can be expressed as superposition of standing waves.

But the same kind of derivation can be made for imaginary region in space with imagined walls that do not influence the EM field in any way. The field can be expressed as Fourier sum in finite cuboid region of space, over travelling waves - this is sometimes called "periodic boundary conditions" (because Fourier sum produces function that repeats outside the main region). This is just mathematics, it does not impose any condition on the field inside. The rest of the derivation is otherwise the same, and leads to the same result.

Thus in the Rayleigh-Jeans derivation, standing waves is an unimportant detail; the important assumptions are 1) energy is given by the Poynting formulae; 2) EM field is in thermodynamic equilibrium; 3) equipartition is valid for EM field in thermodynamic equilibrium: every quadratic Fourier term in expression of EM energy contributes with average energy $kT/2$.