ⓘ Nonextensive selfconsistent thermodynamical theory
In experimental physics, researchers have proposed nonextensive selfconsistent thermodynamic theory to describe phenomena observed in the Large Hadron Collider. This theory investigates a fireball for highenergy particle collisions, while using Tsallis nonextensive thermodynamics. Fireballs lead to the bootstrap idea, or selfconsistency principle, just as in the Boltzmann statistics used by Rolf Hagedorn. Assuming the distribution function gets variations, due to possible symmetrical change, Abdel Nasser Tawfik applied the nonextensive concepts of highenergy particle production.
The motivation to use the nonextensive statistics from Tsallis comes from the results obtained by Bediaga et al. They showed that with the substitution of the Boltzmann factor in Hagedorns theory by the qexponential function, it was possible to recover good agreement between calculation and experiment, even at energies as high as those achieved at the LHC, with q> 1.
1. Nonextensive entropy for ideal quantum gas
The starting point of the theory is entropy for a nonextensive quantum gas of bosons and fermions, as proposed by Conroy, Miller and Plastino, which is given by S q = S q F D + S q B E {\displaystyle S_{q}=S_{q}^{FD}+S_{q}^{BE}} where S q F D {\displaystyle S_{q}^{FD}} is the nonextended version of the Fermi–Dirac entropy and S q B E {\displaystyle S_{q}^{BE}} is the nonextended version of the Bose–Einstein entropy.
That group and also Clemens and Worku, the entropy just defined leads to occupation number formulas that reduce to Bediagas. C. Beck, shows the powerlike tails present in the distributions found in high energy physics experiments.
2. Nonextensive partition function for ideal quantum gas
Using the entropy defined above, the partition function results are
ln ^{{\frac {q}{q1}}}dE\,}where σ E {\displaystyle \sigma E} is the density of states of the fireballs.
3. Selfconsistency principle
Selfconsistency implies that both forms of partition functions must be asymptotically equivalent and that the mass spectrum and the density of states must be related to each other by
l o g }^{\frac {\beta _{o}}{q_{o}1}}\,}where γ {\displaystyle \gamma } is a constant and q o ′ − 1 = β o q o − 1 {\displaystyle q_{o}1=\beta _{o}q_{o}1}. Here, a, b, γ {\displaystyle a,b,\gamma } are arbitrary constants. For q ′ → 1 {\displaystyle q\rightarrow 1} the two expressions above approach the corresponding expressions in Hagedorns theory.
4. Main results
With the mass spectrum and density of states given above, the asymptotic form of the partition function is
Z q V o, T → 1 β − β o α {\displaystyle Z_{q}V_{o},T\rightarrow {\bigg }{\frac {1}{\beta \beta _{o}}}{\bigg}^{\alpha }}where
α = γ V o 2 π 2 β 3 / 2, {\displaystyle \alpha ={\frac {\gamma V_{o}}{2\pi ^{2}\beta ^{3/2}}}\,}with
a + 1 = α = γ V o 2 π 2 β 3 / 2. {\displaystyle a+1=\alpha ={\frac {\gamma V_{o}}{2\pi ^{2}\beta ^{3/2}}}\.}One immediate consequence of the expression for the partition function is the existence of a limiting temperature T o = 1 / β o {\displaystyle T_{o}=1/\beta _{o}}. This result is equivalent to Hagedorns result. With these results, it is expected that at sufficiently high energy, the fireball presents a constant temperature and constant entropic factor.
The connection between Hagedorns theory and Tsallis statistics has been established through the concept of thermofractals, where it isshown that non extensivity can emerge from a fractal structure. This result is interesting because Hagedorns definition of fireball characterizes it as a fractal.
5. Experimental evidence
Experimental evidence of the existence of a limiting temperature and of a limiting entropic index can be found in J. Cleymans and collaborators, and by I. Sena and A. Deppman.
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