Self-consistency principle in high energy Physics
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The self-consistency principle was established by
Rolf Hagedorn Rolf Hagedorn (20 July 1919 – 9 March 2003) was a German theoretical physicist who worked at CERN. He is known for the idea that hadronic matter has a "melting point". The Hagedorn temperature is named in his honor. Early life Hagedorn's y ...
in 1965 to explain the thermodynamics of fireballs in
high energy physics Particle physics or high energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standa ...
collisions. A thermodynamical approach to the high energy collisions first proposed by E. Fermi.


Partition function

The partition function of the fireballs can be written in two forms, one in terms of its density of states, \sigma(E), and the other in terms of its mass spectrum, \rho(m). The self-consistency principle says that both forms must be asymptotically equivalent for energies or masses sufficiently high (asymptotic limit). Also, the density of states and the mass spectrum must be asymptotically equivalent in the sense of the weak constraint proposed by Hagedorn as : log rho(m) log sigma(E). These two conditions are known as the ''self-consistency principle'' or ''bootstrap-idea''. After a long mathematical analysis Hagedorn was able to prove that there is in fact \textstyle \rho(m) and \textstyle \sigma(E) satisfying the above conditions, resulting in : \rho(m) = a m^ exp(\beta_o m) and : \sigma(E) = b E^ exp(\beta_o E) with \textstyle a and \textstyle \alpha related by :\alpha=\frac. Then the asymptotic partition function is given by : Z_q(V_o,T)=\bigg(\frac\bigg)^-1 where a singularity is clearly observed for \beta\beta_o. This singularity determines the limiting temperature \textstyle T_o=1/\beta _o in Hagedorn's theory, which is also known as
Hagedorn temperature The Hagedorn temperature, ''T''H, is the temperature in theoretical physics where hadronic matter (i.e. ordinary matter) is no longer stable, and must either "evaporate" or convert into quark matter; as such, it can be thought of as the "boiling p ...
. Hagedorn was able not only to give a simple explanation for the thermodynamical aspect of high energy particle production, but also worked out a formula for the hadronic mass spectrum and predicted the limiting temperature for hot hadronic systems. After some time this limiting temperature was shown by N. Cabibbo and G. Parisi to be related to a
phase transition In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states o ...
, which characterizes by the deconfinement of quarks at high energies. The mass spectrum was further analyzed by
Steven Frautschi Steven C. Frautschi (; born December 6, 1933) is an American theoretical physicist, currently professor of physics emeritus at the California Institute of Technology (Caltech). He is known principally for his contributions to the bootstrap theory ...
.


Q-exponential function

The Hagedorn theory was able to describe correctly the experimental data from collision with center-of-mass energies up to approximately 10 GeV, but above this region it failed. In 2000 I. Bediaga, E. M. F. Curado and J. M. de Miranda proposed a phenomenological generalization of Hagedorn's theory by replacing the exponential function that appears in the partition function by the q-exponential function from the Tsallis non-extensive statistics. With this modification the generalized theory was able again to describe the extended experimental data. In 2012 A. Deppman proposed a non-extensive self-consistent thermodynamical theory that includes the self-consistency principle and the non-extensive statistics. This theory gives as result the same formula proposed by Bediaga et al., which describes correctly the high energy data, but also new formulas for the mass spectrum and density of states of fireball. It also predicts a new limiting temperature and a limiting entropic index.


References

{{Reflist Particle physics Nuclear physics Principles