Fermion Boson Reaction: Ratio of Fermions to Bosons at T=0

[LizardCobra]

Homework Statement

Fermions and bosons combine through the reaction

F + F + ΔE = B
(so the creation of a single boson requires 2 fermions and some positive energy).

What is the ratio of fermions to bosons at T = 0?

Homework Equations

2[nF]/[nB] = K(T), where [nF] is the concentration of fermions.

The Attempt at a Solution

I'm not sure how to set this up. We can calculate the number of [fermions or bosons] per volume using the density of states, but I don't know where to go from there.

The analogue for non-quantum particles is to set the chemical potentials equal

2μF + E = μB. After taking the log of the partition functions we can derive the law of mass action, where the reaction constant is the ratio of the single particle partition functions divided by the volume.
I don't know how I would do this for quantum particles now.

 

[mfb]

Free particles in some volume? Okay.

For fermion states < ΔE/2, which occupancy do you expect?
For fermion states > ΔE/2, which occupancy do you expect?

I am surprised that this should be sufficient to find a ratio (numeric value?), but at least it is possible to get some equations about those densities.