# Distribution of photons emitted by atoms

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!