A comment on fluctuations and stability limits with application to superheated black holes

We point out that, contrary to signs of heat capacities, thermodynamic fluctuations are simply and unequivocally related to onset of instabilities that show up near critical points. Fluctuation theory is then applied to \Schw
\bh s surrounded by radiation. This shows that slowly evolving \bh s along quasi-equilibrium states in cavities greater than ${10}^6$ Planck length will not evaporate below the critical Hawking limit temperature despite the fact that pure radiation has a much higher entropy. (CQG in press, figure request to okamoto@gprx.miz.nao.ac.jp)