en>Rjwilmsi: Journal cites:, added 2 DOIs, added 3 issue numbers using AWB (9513)

2013-10-09T07:09:56Z

Journal cites:, added 2 DOIs, added 3 issue numbers using AWB (9513)

New page

The '''self-consistency principle''' was established by [[Rolf Hagedorn]] in 1965 to explain the thermodynamics of [[Fireball concept|fireballs]] in [[high energy physics]] collisions. A thermodynamical approach to the high energy collisions first proposed by [[Enrico Fermi|E. Fermi]].<ref>E. Fermi. Prog. Theor. Phys. 5 (1950) 570.</ref>

==Partition function==
The partition function of the [[Fireball concept|fireballs]] can be written in two forms, one in terms of its density of states, <math>\sigma(E)</math>, and the other in terms of its mass spectrum, <math>\rho(m)</math>.

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<ref>R. Hagedorn, Suppl. Al Nuovo Cimento 3 (1965) 147.</ref> as

:<math> log[\rho(m)]= log[\sigma(E)] </math>.

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 <math>\textstyle \rho(m) </math> and <math>\textstyle \sigma(E)</math> satisfying the above conditions, resulting in

:<math> \rho(m) = a m^{-5/2} exp(\beta_o m) </math>

and

:<math> \sigma(E) = b E^{\alpha -1} exp(\beta_o E) </math>

with <math>\textstyle a </math> and <math>\textstyle \alpha </math> related by

:<math>\alpha=\frac{a V}{(2 \pi \beta)^{3/5}}</math>.

Then the asymptotic partition function is given by

:<math> Z_q(V_o,T)=\bigg(\frac{1}{\beta - \beta _o }\bigg)^{\alpha}-1 </math>

where a singularity is clearly observed for <math>\beta</math> →<math>\beta_o</math>. This singularity determines the limiting temperature <math>\textstyle T_o=1/\beta _o</math> in Hagedorn's theory, which is also known as [[Hagedorn temperature]].

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 [[Nicola Cabibbo|N. Cabibbo]] and [[Giorgio Parisi|G. Parisi]] to be related to a [[phase transition]],<ref>N. Cabibbo and G. Parisi, Phys. Lett. 59B (1975) 67.</ref> which characterizes by the deconfinement of [[quark]]s at high energies. The mass spectrum was further analyzed by [[Steven Frautschi]].<ref>S. Frautschi, Phys. Rev. D 11 (1971) 2821.</ref>

==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]]<ref>I. Bediaga, E.M.F. Curado and J.M. de Miranda, Physica A 286 (2000) 156.</ref> proposed a phenomenological generalization of Hagedorn's theory by replacing the exponential function that appears in the partition function by the [[Tsallis_statistics#q-exponential|q-exponential]] function from the [[Constantino Tsallis|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]]<ref>A. Deppman, Physica A 391 (2012) 6380.</ref> 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.
== See also ==
*[[Particle physics]]
== References ==
{{Reflist}}

[[Category:Particle physics]]
[[Category:Nuclear physics]]
[[Category:Principles]]

2-hydroxy-dATP diphosphatase - Revision history

en>Rjwilmsi: Journal cites:, added 2 DOIs, added 3 issue numbers using AWB (9513)