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References: Add persondata short description using AWB

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== Non-extensive self-consistent thermodynamics ==

The non-extensive self-consistent thermodynamical theory keeps the concept of [[Fireball concept|fireball]] for high-energy particle collisions at the same time that uses the [[Constantino Tsallis|Tsallis]] [[Tsallis statistics|non-extensive thermodynamics]].<ref>A. Deppman, Physica A 391 (2012) 6380.</ref>
Fireballs lead naturally to the bootstrap-idea, or self-consistency principle, just as it happens in the case of the Boltzmann statistics used by [[Rolf Hagedorn|Hagedorn]].<ref>R. Hagedorn, Suppl. Al Nuovo Cimento 3 (1965) 147.</ref> Assuming that the distribution function gets modifications due to possible symmetrical change, [[Abdel Nasser Tawfik]] applied the non-extensive concepts on high-energy particle production.<ref>A. Tawfik, Int. J. Theor. Phys, 51, 1396-1407 (2012); arXiv:1011.6622 [hep-ph].</ref><ref>A. Tawfik, Prog. Theor. Phys. 126, 279-292 (2011)</ref>

The motivation to use the non-extensive statistics from Tsallis<ref>C. Tsallis, J Stat Phys 52, 479-487, 1988.</ref> comes from the results obtained by Bediaga et al.,<ref>I. Bediaga, E.M.F. Curado and J.M. de Miranda, Physica A 286 (2000) 156.</ref> who showed that with the substitution of the [[Boltzmann]] [[Boltzmann factor|factor]] which appears in the Hagedorn's theory by the [[Tsallis_statistics#q-exponential|q-exponential function]], it was possible to recover the good agreement between calculation and experiment, even at energies as high as those achieved at [[LHC]], with <math>q>1</math>.

==Non-extensive entropy for ideal quantum gas==
The starting point for the theory is entropy for a non-extensive quantum gas of bosons and fermions, as proposed by Conroy, Miller and Plastino,<ref>J.M. Conroy, H.G. Miller, A.R. Plastino, Phys. Lett. A 374 (2010) 4581–4584.</ref> which is given by <math>S_q=S_q^{FD}+S_q^{BE}</math> where <math>S_q^{FD}</math> is the non-extensive version of the [[Fermi-Dirac statistics|Fermi-Dirac]] entropy and <math>S_q^{BE}</math> is the non-extensive version of the [[Bose–Einstein statistics|Bose-Einstein]] entropy.

As shown by Conroy, Miller and Plastino<ref>J.M. Conroy, H.G. Miller and A.R. Plastino, Physics Letters A 374 (2010) 4581–4584.</ref> and also by Cleymans and Worku,<ref>J. Cleymans and D. Worku, J. Phys. G 39 (2012) 025006.</ref> the entropy just defined leads to occupation numbers formulaes
that reduces to the one used by Bediaga et al. and by C. Beck,<ref>C. Beck, Physica A 286 (2000) 164–180.</ref> and shows the power-like tails present in the distributions found in High Energy Physics experiments.

== Non-extensive partition function for ideal quantum gas ==
Using the entropy defined above, the [[Partition function (statistical mechanics)|partition function]] results to be
:<math> \ln[1+Z_q(V_o,T)]=\frac{V_o}{2\pi^2}\sum_{n=1}^{\infty}\frac{1}{n}\int_0^{\infty}dm \int_0^{\infty}dp \, p^2 \rho(n;m)[1+(q-1)\beta \sqrt{p^2+m^2}]^{-\frac{nq}{(q-1)}} \,.</math>
Since experiments have shown that <math>q>1</math>, this restriction is adopted.

Another way to write the non-extensive partition function for a fireball is
:<math> Z_q(V_o,T)=\int_0^{\infty}\sigma(E)[1+(q-1)\beta E]^{-\frac{q}{(q-1)}} dE\,,</math>
where <math>\sigma(E)</math> is the density of states of the fireballs.

==Self-consistency principle==
The self-consistency 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
:<math> log[\rho(m)]= log[\sigma(E)] </math>,
in the limit of <math>m,E</math> sufficiently large.

The self-consistency can be asymptotically achieved by choosing<ref>A. Deppman, Physica A 391 (2012) 6380.</ref>
:<math> m^{3/2} \rho(m)=\frac{\gamma}{m}\big[1+(q_o-1) \beta _o m\big]^{\frac{1}{q_o -1}}=\frac{\gamma}{m}[1+(q'_o-1) m]^{\frac{\beta _o}{q'_o -1}}</math>
and
:<math>\sigma(E)=bE^a\big[1+(q'_o-1)E\big]^{\frac{\beta _o}{q'_o -1}}\,,</math>
where <math>\gamma</math> is a constant and <math>q'_o-1=\beta _o (q_o-1)</math>. Here, <math>a,b,\gamma</math> are arbitrary constants. For <math>q' \rightarrow 1</math> the two expressions above approach the corresponding expressions in the Hagedorn's theory.

==Main results==
With the mass spectrum and density of states given above, the asymptotic form of the partition function results to be
:<math> Z_q(V_o,T) \rightarrow \bigg(\frac{1}{\beta - \beta _o }\bigg)^{\alpha}</math>
where
:<math>\alpha=\frac{\gamma V_o}{2\pi^2 \beta^{3/2}}\,,</math>
with
:<math> a+1=\alpha=\frac{\gamma V_o}{2\pi^2 \beta^{3/2}} \,.</math>

One immediate consequence of the expression for the partition function is the existence of a limiting temperature <math>T_o=1/\beta _o</math>. This result is equivalent to the important result obtained by Hagedorn.<ref>R. Hagedorn, Suppl. Al Nuovo Cimento 3 (1965) 147.</ref> With these results it is expected that at sufficiently high energy collisions the fireball will present not only a constant temperature, but also constant entropic factor.

==Experimental evidences==

Experimental evidences of the existence of a limiting temperature and of a limiting entropic index can be found in the papers by J. Cleymans and collaborators,<ref>J. Cleymans and D. Worku, J. Phys. G: Nucl. Part. Phys. 39 (2012)
025006.</ref><ref>J. Cleymans, G.I. Lykasov, A.S. Parvan, A.S. Sorin, O.V. Teryaev and
D. Worku, arXiv:1302.1970 (2013).</ref> by Isaac Sena and A. Deppman.<ref>I. Sena and A. Deppman, Eur. Phys. J. A 49 (2013) 17.</ref><ref>I. Sena and A. Deppman, AIP Conf. Proc. 1520, 172 (2013) -
arXiv:1208.2952v1.</ref>

==References==
{{Reflist}}

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

Octave Boudouard - Revision history

en>Waacstats: /* References */Add persondata short description using AWB