Can a Hagedorn system have a temperature other than $T_C$ or can a thermostat have a temperature other than its own?

TL;DR

This paper discusses the limitations of Hagedorn systems and thermostats, showing that with a Hagedorn spectrum, the partition function cannot be computed and the canonical ensemble is only valid at a fixed temperature.

Contribution

It clarifies the conditions under which a Hagedorn system can be described thermodynamically, emphasizing the fixed temperature constraint in canonical ensembles.

Findings

Partition function cannot be calculated for Hagedorn spectra

Canonical description exists only at a single fixed temperature

Hagedorn systems cannot have a temperature different from the critical temperature

Abstract

This is a note intended to complement our paper (nucl-th/0504010) and addressed to the attention of QGP workers interested in bag models, Hagedorn spectra, and the like. It tries to show that with a Hagedorn-like experimental spectrum the partition function can not be calculated and that a canonical description derived for the microcanonical ensemble exists only for a single, fixed temperature.