Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 37A / e

Because the one-particle spectrum is bounded above, a negative temperature is possible. A population-inverted configuration with more particles in the $+ϵ$ level than in the $-ϵ$ level haswhich requires $\beta <0$. Equivalently, above the maximum-entropy energy $E=0$, adding energy reduces the number of compatible microstates, so $\partial S/\partial E=1/T<0$. The limiting configuration with every particle at $+ϵ$ has $E=Nϵ$, zero entropy, and $T\to 0^{-}$.