Consider $N$ rods in a plane with length $2l$ restricted to rotate by an angle $\theta$ leading to excluded volume $\Omega (\theta) = l^2(\theta + \sin\theta)$. Under the assumption that the phase space volume corresponding to the rotational freedom is proportional to $\theta$, maximizing the entropy leads to the following constraint on the density: $$n=\frac{2}{l^2} \frac{1}{\theta(2+\cos\theta)+\sin\theta}$$ such that there is a minimum density at some critical angle $\theta_c$. There are two related claims about this system I am not understanding: (1) that a phase transition occurs at $\theta_c$, and (2) that the local entropy maximum becomes "unstable" at $\theta_c$.

I am wondering how to justify either of these claims, and how the two are related. Wouldn't a singularity in $n$ be a more appropriate way to characterize a phase transition? How does a minimum characterize it instead? What makes an entropy maximum unstable and how does this imply a phase transition?

This is all inspired by problem 5.7 in Kardar's Statistical Physics of Particles.