- Hagedorn temperature
In
theoretical physics , the Hagedorn temperature is the temperature above which thepartition sum diverges in a system with exponential growth in the density of states.:
Because of the divergence, many people come to the incorrect conclusion that it is impossible to have temperatures above the Hagedorn temperature because it would require an infinite amount of
energy . In equations::
This line of reasoning was well known to be false even to Hagedorn. The partition function for creation of Hydrogen-antiHydrogen pairs diverges even more badly, because it gets a finite contribution from energy levels which accumulate at the ionization energy. The states which cause the divergence are spatially big--- the electrons are very far from the protons. The divergence indicates that at a low temperature Hydrogen-antiHydrogen will not be produced, rather proton/antiproton and electron/antielectron. The Hagedorn temperature is only a maximum temperature in the physically unrealistic case of exponentially many species with energy E and finite size.
The concept of exponential growth in the number of states was originally proposed in the context of
condensed matter physics . It was incorporated into high energy physics in the early 1970s bySteven Frautschi and Hagedorn. In hadronic physics, the Hagedorn temperature is the deconfinement temperature, while in string theory, it indicates a phase transition--- the transition at which very long strings are copiously produced. It is controlled by the size of the string tension which is smaller than the Planck scale by the some power of the coupling constant. By adjusting the tension to be small compared to the Planck scale, the Hagedorn transition can be much less than thePlanck temperature . [ [http://www.pbs.org/wgbh/nova/zero/hot.html NOVA | Absolute Zero | Absolute Hot | PBS ] ] Traditional grand-unified string models place this in the magnitude of 1030K, two orders of magnitude smaller than the Planck temperature.References
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