Boltzmann factor and ratio of number of states When using Boltzmann factor for two states, we see that $$N_1/N_2 = \exp\left(\frac{h\omega}{k_BT}\right)\;.$$ Where $N_1$ is the number of atoms in state with lower energy $E_1$ than atoms in state $N_2$ with energy $E_2$. 
The Boltzmann factor shows that there will always be more atoms in state $N_1$ and $N_2$, no matter how much we turn up the temperature. But if we turn up the temperature more and more, wouldn't we expect at some point that all atoms would want to be in the state with highest energy $E_2$? Since otherwise, where would that energy go from turning up the temperature?
Thank you.
 A: Your intuition is correct: even though $N_2$ can never exceed $N_1$ as $T \to \infty$, something has to happen if we keep putting in more energy. What happens is that the temperature "overflows" and goes to $-\infty$. It then increases as we put more energy in, finally reaching $-0$ when $N_1 = 0$.
The reason this looks unnatural is because temperature $T$ is not the right variable; the inverse temperature $\beta \propto 1/T$ is more fundamental. In this case, the inverse temperature always decreases when we put more energy in; it changes continuously from $+\infty$ to $-\infty$. 
A: The reason why $N_2$ can never exceed $N_1$ in your expression is that a state in which $N_2>N_1$ is a non-equilibrium state, and your expression is not going to allow that case because it is derived from the canonical probability density, which is only valid at equilibrium.
A state in which $N_1>N_2$ is a non-equilibrium state because a system cannot normally remain in an excited state for an infinite time: after a certain finite amount of time, a fluctuation in the electromagnetic field will cause spontaneous emission and the excited state will decay to the lower energy level (until eventually the ground state is reached).
$N_1=N_2$ means that the number of transition from level 1 to level 2 matches the number of transitions from level 2 to level 1, so it is a dynamical kind of equilibrium.
Non-equilibrium situations in which $N_2>N_1$ can be achieved in many ways, a process known as population inversion.
