We investigate the transverse momentum ($p_{\rm T}$) spectra of identified hadrons in minimum-bias proton-proton (pp) collisions at a centre-of-mass energy ($\sqrt{s}$) of 0.9, 2.76, 5.02, 7 and 13 TeV in the framework of Tsallis-blast wave (TBW) model. It is found that the model describes well the particle spectra up to 10 GeV/c. The radial flow ($\langle β\rangle$) increases with the collision energy. The degrees of non-equilibrium ($q$) and the Tsallis temperature parameter ($T$) show a similar behaviour, but with a much weaker trend. With this dependence of the freeze-out parameters on the collision energy, we evaluate $\langle β\rangle$, $T$ and $q$ in pp collisions at $\sqrt{s}=$ 8 and 14 TeV and predict the particle spectra at these two energies. Moreover, in order to investigate the multiplicity dependence of the freeze-out parameters, the TBW model is extended to the spectra at different charged-particle multiplicity classes in pp collisions at $\sqrt{s}=$ 7 and 13 TeV. It is observed that at both energies the radial flow increases with the multiplicity while the degree of non-equilibrium shows an opposite behaviour, which is similar to that observed in proton-nucleus (pA) and nucleus-nucleus (AA) collisions at the Large Hadron Collider (LHC) energies. However, the Tsallis temperature parameter increases with the multiplicity, which is opposite to the trend in pA and AA collisions. At similar multiplicities, the radial flow in pp collisions is stronger than those in pA and AA collisions, indicating that the size of the colliding system has significant effects on the final state particle dynamics. Finally, we apply an additional flow correction to the Tsallis temperature parameter and find that the doppler-corrected temperature parameter almost scales with the multiplicity in a uniform way, despite the difference in the colliding system and collision energy.
The Tsallis statistics was applied to describe the experimental data on the transverse momentum distributions of hadrons. We considered the energy dependence of the parameters of the Tsallis-factorized statistics, which is now widely used for the description of the experimental transverse momentum distributions of hadrons, and the Tsallis statistics for the charged pions produced in $pp$ collisions at high energies. We found that the results of the Tsallis-factorized statistics deviate from the results of the Tsallis statistics only at low NA61/SHINE energies when the value of the entropic par…
The analytical expressions for the ultrarelativistic transverse momentum distributions of the Tsallis and the Tsallis-$2$ statistics were obtained. We found that the transverse momentum distribution of the Tsallis-factorized statistics, which is now largely used to describe the experimental transverse momentum spectra of hadrons measured in $pp$ collisions at LHC and RHIC energies, in the ultrarelativistic case is not equivalent to the transverse momentum distributions of the Tsallis and the Tsallis-$2$ statistics. However, we revealed that this distribution exactly coincides with the transver…
The exact analytical formulas for the transverse momentum distributions of the Bose-Einstein, Fermi-Dirac and Maxwell-Boltzmann statistics of particles with nonzero mass in the framework of the Tsallis normalized and Tsallis unnormalized (also known as Tsallis-1 and Tsallis-2) statistics have been consistently derived. The final exact results were expressed in terms of the series expansions in the integral representation. The zeroth term approximation to both quantum and classical statistics of particles has been introduced. We have revealed that the phenomenological classical Tsallis distribu…
In the present work, we have found that the phenomenological Tsallis distribution (which nowadays is largely used to describe the transverse momentum distributions of hadrons measured in $pp$ collisions at high energies) is consistent with the basis of the statistical mechanics if it belongs to the $q$-dual nonextensive statistics instead of the Tsallis one. We have defined the $q$-dual statistics based on the $q$-dual entropy which was obtained from the Tsallis entropy under the multiplicative transformation of the entropic parameter $q\to 1/q$. We have found that the phenomenological Tsallis…
The energy dependence of Tsallis statistics parameters is presented for charged pions produced at beam energies ranging between 6.3 GeV and 7 TeV. It is found that deviations from Boltzmann statistics are monotonically growing with beam energy. This may be attributed to the dominance of low $x$ at higher energies leading to the power-like NLO QCD spectra for which Tsallis statistics may serve as effective theory. At the same time, for larger $x$ at lower beam energies the Gaussian falloff of transverse-momentum-dependent distributions is crucial and correspondent effective description is provi…
The study of higher-order moments of a distribution and its cumulants constitute a sensitive tool to investigate the correlations between the particle produced in high energy interactions. In our previous work we have used the Tsallis $q$ statistics, NBD, Gamma and shifted Gamma distributions to describe the multiplicity distributions in $π^-$ -nucleus and $p$ -nucleus fixed target interactions at various energies ranging from P$_{Lab}$ = 27 GeV to 800 GeV. In the present study we have extended our analysis by calculating the moments using the Tsallis model at these fixed target experiment dat…
The distribution characteristic of final-state particles is one of significant parts in high energy nuclear collisions. The transverse momentum distribution of charged particles carries essential evolution information about the collision system. Tsallis statistics is used to investigate the transverse momentum distribution of charged particles produced in Xe-Xe collisions at 5.44 TeV. On the basis, we reproduce the nuclear modification factor of the charged particles. The calculated results agree approximately with the experimental data measured by the ALICE Collaboration.
The general formalisms of the $q$-dual statistics, the Boltzmann-Gibbs statistics, and three versions of the Tsallis statistics known as Tsallis-1, Tsallis-2, and Tsallis-3 statistics have been considered in the canonical ensemble. We have rigorously proved that the probability distribution of the Tsallis-1 statistics is invariant under the uniform energy spectrum translation at a fixed temperature. This invariance demonstrates that the formalism of the Tsallis-1 statistics is consistent with the fundamentals of the equilibrium statistical mechanics. The same results we have obtained for the p…
This is a concise introduction to the topic of nonextensive Tsallis statistics meant especially for those interested in its relation to high energy proton-proton, proton-nucleus, and nucleus-nucleus collisions. The three types of Tsallis statistics are reviewed. Only one of them is consistent with the fundamental hypothesis of equilibrium statistical mechanics. The single particle distributions associated with it, namely Boltzmann, Fermi-Dirac, and Bose-Einstein, are derived. These are not equilibrium solutions to the conventional Boltzmann transport equation which must be modified in a rather…
We study the non-extensive Tsallis statistics and its applications to QCD and high energy physics, and analyze the possible connections of this statistics with a fractal structure of hadrons. Then, we describe how scaling properties of Yang-Mills theories allow the appearance of self-similar structures in gauge fields, which actually behave as fractals. The Tsallis entropic index, $q$, is deduced in terms of the field theory parameters, resulting in a good agreement with the value obtained experimentally.
The nuclear modification factor is derived using Tsallis non-extensive statistics in relaxation time approximation. The variation of nuclear modification factor with transverse momentum for different values of non-extensive parameter, $q$, is also observed. The experimental data from RHIC and LHC are analysed in the framework of Tsallis non-extensive statistics in a relaxation time approximation. It is shown that the proposed approach explains the $R_{AA}$ of all particles over a wide range of transverse momenta but doesn't seem to describe the rise in $R_{AA}$ at very high transverse momenta.
Nonextensive statistics has attracted attention as a description of particle spectra in nuclear collisions at QCD energies. First, we construct the equation of state by incorporating Tsallis statistics based on the hadron resonance gas and parton gas models. Thermodynamic conditions are found to impose constraints on the $q$-parameter of Tsallis distribution. Next, we apply the equation of state to the relativistic hydrodynamic modeling of nuclear collisions. The Cooper-Frye prescription is consistently modified. Numerical demonstrations indicate that the model may describe charged particle sp…
Experimental results of the transverse momentum distributions of phi mesons and $Ω$ hyperons produced in gold-gold (Au-Au) collisions with different centrality intervals, measured by the STAR Collaboration at different energies (7.7, 11.5, 19.6, 27, and 39 GeV) in the beam energy scan (BES) program at the relativistic heavy ion collider (RHIC), are approximately described by the single Erlang distribution and the two-component Schwinger mechanism. Meanwhile, the STAR experimental transverse momentum distributions of negatively charged particles, produced in Au-Au collisions at RHIC BES energie…
We study the applications of non-extensive Tsallis statistics to high energy and hadron physics. These applications include studies of $pp$ collisions, equation of state of QCD, as well as Bose-Einstein condensation. We also analyze the connections of Tsallis statistics with thermofractals, and address some of the conceptual aspects of the fractal approach, which are expressed in terms of the renormalization group equation and the self-energy corrections to the parton mass. We associate these well-known concepts with the origins of the fractal structure in the quantum field theory.
The latest, high-accuracy identified hadron spectra measurements in high-energy nuclear collisions led us to the investigation of the strongly interacting particles and collective effects in small systems. Since microscopical processes result in a statistical Tsallis-Pareto distribution, the fit parameters $q$ and $T$ are well suited for identifying system size scalings and initial conditions. Moreover, parameter values provide information on the deviation from the extensive, Boltzmann-Gibbs statistics in finite-volumes. We apply here the fit procedure developed in our earlier study for proton…
Particle production in relativistic $pp$($\overline{p}$) and $AA$ collisions at RHIC and LHC energies with Tsallis statistics using the two-cylindrical multisource thermal model
In the framework of a multisource thermal model, we describe experimental results of the transverse momentum spectra of final-state light flavour particles produced in gold-gold (Au-Au), copper-copper (Cu-Cu), lead-lead (Pb-Pb), proton-lead ($p$-Pb), and proton-proton ($p$-$p$) collisions at various energies, measured by the PHENIX, STAR, ALICE, and CMS Collaborations, by using the Tsallis-standard (Tsallis form of Fermi-Dirac or Bose-Einstein), Tsallis, and two- or three-component standard distributions which can be in fact regarded as different types of "thermometers" or "thermometric scales…
We derive the analytical expressions for the first and second order terms in the hadronic transverse momentum spectra obtained from the Tsallis normalized (Tsallis-1) statistics. We revisit the zeroth order quantum Tsallis distributions and obtain the corresponding analytical closed form expressions. It is observed that unlike the classical case, the analytical closed forms of the zeroth order quantum spectra do not resemble the phenomenological distributions used in the literature after $q \to q^{-1}$ substitution, where $q$ is the Tsallis entropic parameter. However, the factorization approx…
Comparing effective temperatures in standard, Tsallis, and q-dual statistics from transverse momentum spectra of identified light charged hadrons produced in gold--gold collisions at RHIC energies
