Hagedorn Temperature

The Hagedorn temperature in theoretical physics is the temperature above which the partition sum diverges in a system with exponential growth in the density of states. It is named after German physicist Rolf Hagedorn. Phase transitions (e.g. from a solid state of matter to that of a liquid one) are in general possible only when the system has a higher number of particles than is thought possible. Such behavior surprised many in the world of elementary particle systems. However, as signaled by abundant particle production present in strong interactions, the quark structure of strongly interacting particles allows an infinite number of "degrees of freedom" to be present in finite volume. In other words, a highly relativistic system can produce pairs and thus effectively be of infinite size.

Because of the divergence, it seemed at first impossible to have temperatures above the Hagedorn temperature, which would make it the absolute hot temperature, because it would require an infinite amount of energy. In equations:

This line of reasoning has been improved in work of Hagedorn and Johann Rafelski, where it was shown that instead a phase transition to quark matter occurs.