Entropy and occupancy. We argued in this chapter that the entropy of the cosmic black body radiation has not changed with time because the number of photons in each mode has not changed with time, although the frequency of each mode has decreased as the wavelength has increased with the expansion of the universe. Establish the implied connection between entropy and occupancy of the modes, by showing that for one mode of frequency ω the entropy is a function of the photon occupancy ⟨s⟩ only:
σ=⟨s+1⟩log⟨s+1⟩−⟨s⟩log⟨s⟩.(58)
It is convenient to start from the partition function.