Analytic results for the Tsallis thermodynamic variables

Abstract

We analytically investigate the thermodynamic variables of a hot and dense system, in the framework of the Tsallis non-extensive classical statistics. After a brief review, we start by recalling the corresponding massless limits for all the thermodynamic variables. We then present the detail of calculation for the exact massive result regarding the pressure -- valid for all values of the $q$-parameter -- as well as the Tsallis $T$-, $\mu$- and $m$- parameters, the former characterizing the non-extensivity of the system. The results for other thermodynamic variables, in the massive case, readily follow from appropriate differentiations of the pressure, for which we provide the necessary formulas. For the convenience of the reader, we tabulate all of our results. A special emphasis is put on the method used in order to perform these computations, which happens to reduce cumbersome momentum integrals into simpler ones. Numerical consistency between our analytic results and the corresponding usual numerical integrals are found to be perfectly consistent. Finally, it should be noted that our findings substantially simplify calculations within the Tsallis framework. The latter being extensively used in various different fields of science as for example, but not limited to, high-energy nucleus collisions, we hope to enlighten a number of possible applications.Comment: 9 page

Hadron transverse momentum distributions of the Tsallis normalized and unnormalized statistics

“…See Appendix B. Note that there is a limiting value of q at which the zeroth term approximation and the Tsallis statistics on the whole become divergent [25,20].…”

“…Power-law functions are used to describe the experimental data for the transverse momentum distribution of particles produced in proton-proton and heavy-ion collisions at the LHC and RHIC energies [1,2,3,4,5,6,7,8]. Now the phenomenological transverse momentum distribution [9,10,11] inspired by the Tsallis statistics [12] has gained much attention and it is successfully used for the description of the experimental data on high-energy proton-proton reactions [13,14,15,16,17,18,19,20,21,22,23,24,25,26] and relativistic heavy-ion collisions [27,28,29,30,31,32]. However, it has some dificulties in relation to the fundamental principles of thermodynamics and statistical mechanics.…”

Non-extensivity of the QCD pT-spectra

“…Finally, in order to conclude, we recall that we have established a connection between the hadronic distributions and the Tsallis distributions at very high transverse momentum domain where dominance of hard scattering has been assumed. From the dimensional analysis we obtain the upper bound of the Tsallis q parameter to be 5/4 = 1.25 which is below the one q < 4/3 ≈ 1.33 proposed in [21] from the argument of convergence of the Tsallis thermodynamic quantities (like the number density for example). We however notice that while specifically fitting high p T data (up to 20 GeV) from transverse momenta spectra at 7 TeV, such as that given by the ALICE collaboration [24], we consistently obtain the same value for the q parameter independent of the choice for the high p T range lower bound.…”

“…which leads to q = 5/4 = 1.25 (9) This above relation provides a natural upper bound to the Tsallis q parameter, which is below the upper bound q < 4/3 ≈ 1.33 required from thermodynamic consistency as pointed out in [21]. We can also find out the Tsallis tempertaure corresponding to this upper bound with the help of Table 1 of Ref.…”

The Parameters of The Tsallis Distribution at the LHC

“…For larger values of q the integrals become divergent [19]. The above power law has been used to fit the p T spectra of charged particles measured by the ATLAS [20] and CMS [21] collaborations in [14].…”