If I understand correctly, when we looked at Einstein’s model of solids, we considered just one harmonic oscillator (one atom) with some amount of average excitations n_B = \frac{1}{e^{\beta \varepsilon}-1} (do we consider these to also be phonons?).
Why don’t we sum over all available atoms, like we do in Debye’s theory? Is that simply uninteresting, because we do not have some special dispersion relation that would manifest in an expression more complex than E_{tot}=N\varepsilon (where \varepsilon the energy of a single oscillator)?
In the simplest case, we consider all atoms identical, and so we multiply the single atom result with their number. In the lithium exercise of the Einstein model, we considered different atom types.