I am currently revising quantum gases, and a small but confusing thought experiment has been bugging me for a while.
I understand the bookwork stuff on photons and how a photon gas in a blackbody cavity follows Bose-Einstein statistics. I also understand how to obtain key quantities like the energy density, peak wavelength etc.
Now, my thought experiment is as such: suppose a confine a gas of atoms that emit photons in a box. Given that the atoms follow a Maxwell-Boltzmann distribution, does the emission follow the Maxwell-Boltzmann distribution as well, and is there an eventual transition to the Bose-Einstein distribution?
Additionally, what distributions to I use to calculate the usual variables of the system? If, for example, I cut a hole in the box and observe photons that escape through the hole, these photons’ frequencies $\omega$ should be Doppler shifted corresponding to the velocities of the atoms $v$. If I wish to calculate the mean frequency for example, should I integrate over $v^2 \exp(-mv^2/2k_BT)$, or $g(v)f(v)$ from the usual density of states calculation with $v$ being obtained from $\omega$ via the Doppler shift formula, which in turn is obtained from the wavevector? I ask in particular for $\omega$ since I understand this is quite an important concept in determining the structure of astrophysical bodies from the spectral lines.
Apologies if this seems like a trivial thing, but it’s been boggling my mind!