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

Put $x=\beta ϵ$. The one-particle canonical partition function isand independence gives $Z=z^N$. Therefore the independent symmetric three-level system hasand, using part (c),As $T\to \infty$, $x\to 0$, all three levels become equally likely andAs $T\to 0^{+}$, every particle occupies the nondegenerate ground level $-ϵ$, and