Hagedorn temperature

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In theoretical physics, the Hagedorn temperature is the maximal allowed temperature of certain systems above which the partition sum diverges.

$\lim_{T\rightarrow T_H^-} Tr[e^{-\beta H}]=\infty$

Because of this, it's not possible to have temperatures about the Hagedorn temperature as it will take an infinite amount of energy to do so

$\lim_{T\rightarrow T_H^-}E=\lim_{T\rightarrow T_H^-}\frac{Tr[H e^{-\beta H}]}{Tr[e^{-\beta H}]}=\infty$

This concept was originally proposed in the context of condensed matter physics. The Hagedorn temperature is the temperature at which the Maxwell-Boltzmann factor approximately cancelled a similar exponential growth of the number of states. However, it also has applications in string theory. Depending on the precise type of strings, or particles, and the theory, the Hagedorn temperature may be a limiting temperature or a phase transition may occur at this point.