The Hagedorn spectrum, nuclear level densities and first order phase transitions

An exponential mass spectrum, like the Hagedorn spectrum, with slope 1/TH was interpreted as fixing an upper limiting temperature TH that the system can achieve. However, thermodynamically, such spectrum indicates a 1st order phase transition at a fixed temperature TH. A much lower energy example is the log linear level nuclear density below the neutron binding energy that prevails throughout the nuclear chart. We show that, for non-magic nuclei, such linearity implies a 1st order phase transition from the pairing superfluid to an ideal gas of quasi particles.