Local thermal stability against energy exchange with a fixed-temperature reservoir requires $C_J>0$. Since $s\geq0$, this means
$$
2-2s-s^2>0
\quad\Longleftrightarrow\quad
s<\sqrt3-1.
$$
Equivalently,
$$
\boxed{\sqrt{2\sqrt3-3}<\frac{|a|}{M}<1}.
$$
The lower endpoint is the divergent-heat-capacity transition. The extremal endpoint $|a|=M$ has $T_H=0$ and $C_J=0$ and is obtained only as a limiting case.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 311 / 4 / c / iv / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-311/4/c/iv/solution