Discrete Cavity Radiation vs Continuous Blackbody radiation, violation of the 2nd law of thermodynamics?

Planck’s law states that emission of light is

$$ I_nu(hbaromega) = g(hbaromega) f(hbaromega, T) $$

Let’s say that $I_nu$ is the energy emitted from the surface of the blackbody per unit area per into per unit solid angle per unit time.

If the blackbody is emitting into free empty space we know that the density of photon states is,

$$ g(hbaromega) = frac{2pi}{c^2h^3} left(hbaromegaright)^2 $$

And because photons are bosons $f(hbaromega, T)$ is the Bose-Einstein distribution.

So two blackbody cavities radiating into free space will eventually come into equilibrium with each other by exchanging blackbody radiation over all wavelengths,

If you now place a filter between the cavities which only passes energy at a single photon frequency.

This does not change anything fundamental because they can still exchange energy and will eventually reach the same temperature.

It’s just like modifying the density of states with a delta function,

$$ I_nu(hbaromega) = delta(hbaromega_f)g(hbaromega) f(hbaromega, T) $$

I think you argument and conclusion are correct. A cavity which radiates only at discrete frequencies won't behave as blackbody radiator and will put out more energy than it receives at those discrete frequencies.

It is the assumption that is wrong - the idea that perfectly reflecting cavity with equilibrium radiation radiates at discrete frequencies. Equilibrium radiation means all frequencies are possible, not just some discrete ones.

The idea there are only waves of discrete frequencies inside the cavity probably comes from the usual derivation on Rayleigh-Jeans or Planck's formula, where the field is expanded into Fourier series.

Fourier series has the property that in expressing the function of position $x$, only sine waves with integer multiples of a fundamental wave number $frac{pi}{L}$ are present, where $L$ is size of the region where we seek to express the function as Fourier series. This region is usually taken to be the whole inside of the cavity, but nothing prevents us from taking a bigger box with side length $2L$.

With twice as big integrating region dimensions, we get twice as dense wave numbers and twice as dense corresponding frequencies $omega_{nlm} = frac{pi csqrt{n^2+l^2+m^2} }{2L}$. Instead of a fundamental frequency (lowest) at $frac{pi csqrt{3}}{L}$ we get fundamental frequency at $frac{pi csqrt{3}}{2L}$, which is lower. Behold, radiation at lower frequency appeared, just due to using a different integrating region!

It is clear that position of singularities and their strengths are an artifact of the particular finite integrating region in the Fourier series method. They are correct for the used region, but there are infinitely many other choices.

If we use Fourier integral expansion instead of the Fourier series expansion, there is no $L$ in the formulae, and no discreteness in Fourier amplitude $tilde{E}_x(k,l,m)$ as function of continuous wavenumbers $k,l,m$. It all becomes unique continuous quantities.

So equilibrium radiation inside a perfectly reflecting cavity is not (except near the cavity walls) physically different from that of a bigger cavity or radiation of a cavity that has walls made of blackbody at the same temperature.