# Negative temperature as limiting case for two-level system

For N particles in a two-energy-level system, with energy levels being 0 and $\Delta$, the internal energy tends to $N \Delta /2$ as temperature tends to infinity. At this point, the occupancy of each energy level is equal, and any further increase in internal energy would lead to a decrease in the entropy of the system.

Temperature is defined as $T = (\frac{\partial U}{\partial S})_V$. Using this definition, for a state with equal occupancies described above, the temperature should be negative. Where is the contradiction?

• With equal occupancies, the temperature is infinite. It is the population inversion (higher occupancy for a higher-energy state) that makes temperature negative. There is no contradiction. – Ivan Neretin Apr 8 '17 at 7:35
• and inversion is very common, for example in NMR experiments and in lasers – porphyrin Apr 8 '17 at 8:00
• – Mithoron Apr 8 '17 at 17:43

I think you would need to describe the derivative $\frac{\partial U}{\partial S}$ at that point, which should be infinity by your logic since the entropy is at a maximum with respect to internal energy ($\frac{\partial S}{\partial U}=0$). This is obtained using $$\frac{\partial U}{\partial S}=\frac{1}{\frac{\partial S}{\partial U}}$$