= Hagedorn temperature
{c}
{title2=$T_H=\beta_H^{-1}$}
{wiki}
If a density of states has high-energy growth proportional to a power of energy times $e^{\beta_HE}$, its canonical Boltzmann sum fails for $\beta<\beta_H$. The Hagedorn temperature is the corresponding threshold. The <open bosonic string mass spectrum> and <large-level degeneracy of an open bosonic string> give $\beta_H=4\pi\sqrt{\alpha'}$ in units with Boltzmann's constant, the speed of light and Planck's constant set to one. Polynomial prefactors affect behavior at the threshold itself; the growth alone does not distinguish a limiting temperature from a phase transition.
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