Simple question about the definitions, but I can't find it explicitly stated anywhere.
If you have a two level system described by $n_i = \frac{N}{Z}e^{-\beta E_i}$ and calculate the heat capacity
$$ c_v = \bigg( \frac{\partial U}{\partial T} \bigg)_{N,V} = \frac{\partial}{\partial T} NE₁\frac{e^{-\beta E_1}}{1 + e^{-\beta E_1}} = \frac{1}{Z²} Nk_B(\beta E_1)² e^{-\beta E_1}$$
Where:
$ T $ is temperature
$ \beta = \frac{1}{k_BT}$
$n_i$ is the number of particles in energy level $i$.
$ Z $ is the partition function
You get a peak at low temperature. In fact, I think this is true (I graphed it) for any system with a finite number of states of discrete energy. But I don't understand how a system with finite states can have a temperature. I understand temperature as the average translational kinetic energy per particle. When there are other degrees of freedom available, addition energy added to the system is partitioned among this translational energy and other modes like vibration and rotation. So it takes more total energy added to increase the average translational $E_K$ by the same amount.
If you have translational motion (around the centre of mass frame), then there are an infinite number of energy levels available, unless the velocities are bounded.
So by "two-level system" do we actually mean two states other than the translational energies? Like a cloud of electrons (non-interacting... neutrons?) with spin up and down in a magnetic field.