Tsallis statistics and thermofractals: applications to high energy and hadron physics
Eugenio Megias, Evandro Andrade, Airton Deppman, Arnaldo Gammal, Debora P. Menezes, Tiago Nunes da Silva, Varese S. Timóteo
备注：
16 pages, 8 figures. Talk given by E.Megias at the 10th International Conference on New Frontiers in Physics (ICNFP 2021), 23 Aug - 7 Oct 2021, Kolymbari, Crete, Greece
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 non extensive statistics proposed by C. Tsallis has found wide applicability, being present even in the description of experimental data from high energy collisions. A system with a fractal structure in its energy-momentum space, named thermofractal, was shown to be described thermodynamically by the non extensive statistics. Due to the many common features between thermofractals and Hagedorn's fireballs, this system offers the possibility to investigate the origins of non extensivity in hadronic physics and in QCD. In this regard, the investigation of the scaling properties of thermofract…
A system presenting fractal structure in its thermodynamical functions is introduced, and it is shown that Tsallis statistics is the correct framework for describing the thermodynamical aspects of such fractal. Its Haussdorf dimension and its Lipshitz-Hölder exponent are determined in terms of the entropic index $q$. The connections with the intermittency in experimental data is discussed. The thermodynamical aspects of the thermofractal is related to the microscopic interaction of its components through the S-matrix.
The non extensive aspects of $p_T$ distributions obtained in high energy collisions are discussed in relation to possible fractal structure in hadrons, in the sense of the thermofractal structure recently introduced. The evidences of self-similarity in both theoretical and experimental works in High Energy and in Hadron Physics are discussed, to show that the idea of fractal structure of hadrons and fireballs have being under discussion for decades. The non extensive self-consistent thermodynamics and the thermofractal structure allow one to connect non extensivity to intermittence and possibl…
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…
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…
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 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…
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…
In the present paper, the Tsallis statistics in the grand canonical ensemble was reconsidered in a general form. The thermodynamic properties of the nonrelativistic ideal gas of hadrons in the grand canonical ensemble was studied numerically and analytically in a finite volume and the thermodynamic limit. It was proved that the Tsallis statistics in the grand canonical ensemble satisfies the requirements of the equilibrium thermodynamics in the thermodynamic limit if the thermodynamic potential is a homogeneous function of the first order with respect to the extensive variables of state of the…
We explore a hadron resonance gas model using Tsallis non-extensive distribution to study the energy dependence of the product of the moments, $Sσ$ and $κσ^2$ of net-proton multiplicity distributions of published STAR data in \auau collisions at relativistic heavy-ion collider (RHIC) energies. While excellent agreements are found between model predictions and measurements of $Sσ$ and $κσ^2$ of most peripheral collisions and $Sσ$ of most central collisions, the $κσ^2$ for most central collisions deviates significantly from the predictions particularly at \sqsn = 19.6 GeV and $27$ GeV. This coul…
The origin of non-extensive thermodynamics in physical systems has been under intense debate for the last decades. Recent results indicate a connection between non-extensive statistics and thermofractals. After reviewing this connection, we analyze how scaling properties of Yang-Mills theory allow the appearance of self-similar structures in gauge fields. The presence of such structures, which actually behave as fractals, allows for recurrent non-perturbative calculations of vertices. It is argued that when a statistical approach is used, the non-extensive statistics is obtained, and the Tsall…
The Bose-Einstein and Fermi-Dirac statistics of the identified hadrons were verified on the basis of the transverse momentum distributions of bosons and fermions created in the $pp$ collisions at high energies using the Tsallis-factorized statistics and the two-component distributions of the Boltzmann-Gibbs statistics. The main differences between the parameters of the Tsallis-factorized statistics and the Boltzmann-Gibbs statistics were identified. The results of the Boltzmann-Gibbs statistics are reasonable and suggest that the soft and hard hadrons may be produced from the two different mac…
The transverse momentum ($p_{T}$) distribution of strange hadrons ($K_{S}^{0}$ and $Λ$) and multi-strange hadrons( $Ξ$ and $Ω$) measured in p$-$p, p$-$Pb, and Pb$-$Pb collisions at LHC energies have been studied for different multiplicity classes using Tsallis-Weibull (or q$-$Weibull)formalism. The distribution describes the measured $p_{T}$ spectra for all multiplicity (or centrality)classes. The multiplicity dependence of the extracted parameters are studied for the mentioned collisions systems. The $λ$ parameter was observed to increase systematically with the collision multiplicity and fol…
We present a study of transverse momentum ($p_{T}$) spectra of unidentified charged particles in pp collisions at RHIC and LHC energies from $\sqrt{s}$ = 62.4 GeV to 13 TeV using Tsallis/Hagedorn function. The power law of Tsallis/Hagedorn form gives excellent description of the hadron spectra in $p_{T}$ range from 0.2 to 300 GeV/$c$. The power index $n$ of the $p_T$ distributions is found to follow a function of the type $a+b/\sqrt {s}$ with asymptotic value $a = 5.72$. The parameter $T$ governing the soft bulk contribution to the spectra remains almost same over wide range of collision energ…
A thermal model, based on the Tsallis distribution and blast-wave model, is proposed to compute hadron double-differential spectra $d^{2}N/dp_{T}dy$ in $pp$ (also high energy $p\overline{p}$) collisions. It successfully describes the available experimental data on pion and quarkonia ($ϕ$, $J/ψ$, $ψ(2S)$, $Υ$ family) production at energies from $\sqrt{s} = 5$ GeV to the LHC ones. Simple parametrizations for the $\sqrt{s}$ dependence of the model parameters are provided allowing predictions for the yields of these particles at new collision energies. The model can be used also for the pion Bose-…
The transverse momentum distribution of identified hadrons from recent years are analyzed within the thermodynamically consistent formulation of non-extensive statistics. A wide range of center-of-mass energies and average event multiplicities are studied for various hadron species. We demonstrate that the average event multiplicity is a key variable in the study of high-energy collisions and a smooth transition in the description from small to large systems is possible. For this purpose the non-extensive statistical approach is more than appropriate. The validity of the non-extensive descript…
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…
Moments play a crucial role in investigating the characteristics of charged particle multiplicities in high energy interactions. The success of any model which describes the multiplicity data can be understood well by studying the normalised and factorial moments of that distribution.~The Tsallis model is one of the most successful models which describes the multiplicity spectra, transverse momentum ($p_T$) spectra very precisely in high energy interactions.~In our previous work we have used the Tsallis $q$-statistics to describe the multiplicity distributions in leptonic and hadronic collisio…
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 speed of sound ($c_s$) is studied to understand the hydrodynamical evolution of the matter created in heavy-ion collisions. The quark-gluon plasma (QGP) formed in heavy-ion collisions evolves from an initial QGP to the hadronic phase via a possible mixed phase. Due to the system expansion in a first order phase transition scenario, the speed of sound reduces to zero as the specific heat diverges. We study the speed of sound for systems, which deviate from a thermalized Boltzmann distribution using non-extensive Tsallis statistics. In the present work, we calculate the speed of sound as a f…
In this article, we will present a systematic analysis of transverse momentum spectra of the strange hadron in different multiplicity events produced in pp collision at $\sqrt{s}$ = 7 TeV, pPb collision at $\sqrt{s_{NN}}$ = 5.02 TeV and PbPb collision at $\sqrt{s_{NN}}$ = 2.76 TeV. The differential freeze out scenario of strange hadron $K^{0}_{s}$ assumed while analyzing the data using a Tsallis distribution which is modified to include transverse flow. The $p_{T}$ distributions of strange hadron in different systems are characterized in terms of the parameters namely, Tsallis temperature ($T$…
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
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 possibility of formation of Bose-Einstein Condensation (BEC) is studied in $pp$ collisions at $\sqrt s$ = 7 TeV at the Large Hadron Collider. A thermodynamically consistent non-extensive formulation of the identified hadron transverse momentum distributions is used to estimate the critical temperature required to form BEC of charged pions, which are the most abundant species in a multi-particle production process in hadronic and nuclear collisions. The obtained results have been contrasted with the systems produced in Pb-Pb collisions to have a better understanding. We observe an explicit …
Blast-wave-Tsallis-power model for $p_{\rm T}$-spectra and elliptic flow $v_2$ of hadrons in collisions of identical nuclei at energies available at the Large Hadron Collider
Joint Institute for Nuclear Research（俄罗斯联合核子研究所）；A.I.Alikhanyan National Science Laboratory (YerPhI) Foundation（阿·伊·阿里哈尼扬国家科学实验室（叶尔皮）基金会）
We investigate the identified hadrons transverse momentum ($p_{\rm T}$) spectra in Pb-Pb (Pb-Pb, Xe-Xe, p-Pb) collisions at $\sqrt{s_{\rm NN}}=$ 2.76 (5.02, 5.44, 5.02) TeV in the framework of Tsallis-blast wave (TBW) model with a linear transverse velocity profile and with a constant velocity profile. In this model, the Tsallis temperature ($T$), the average radial flow velocity ($\langle β\rangle$) and the degree of non-equilibrium ($q$) of the system are common for all hadrons when a combined fit is performed to the $p_{\rm T}$ spectra of different particles at a given centrality. It is fou…
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.
In the framework of a multi-source thermal model at the partonic-level, we have analyzed transverse momentum spectra of hadrons measured by the ALICE Collaboration in proton-proton ($pp$ or $p$-$p$) collisions at the center-of-mass energy of $\sqrt{s}=7$ and 13 TeV, proton-lead ($p$-Pb) collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV, and lead-lead (Pb-Pb) collisions at $\sqrt{s_{\rm NN}}=2.76$ TeV. For meson(baryon), the contributions of two(three) constituent quarks are considered, in which each quark contributes to hadron transverse momentum to obey the revised phenomenological Tsallis transvers…
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…
In the present work, we study the recent collision energy and multiplicity dependence of the charged particle transverse momentum spectra as measured by the ALICE collaboration in $pp$ collisions at $\sqrt{s}$ = 5.02 and 13 TeV using the non-extensive Tsallis distribution and the Boltzmann-Gibbs Blast Wave (BGBW) model. A thermodynamically consistent form of the Tsallis distribution is used to extract the kinetic freeze-out parameters from the transverse momentum spectra of charged particles at mid-rapidity. In addition, a comprehensive study of fitting range dependence of transverse momentum …
The thermodynamic properties of matter created in high-energy heavy-ion collisions have been studied in the framework of the non-extensive Tsallis statistics. The transverse momentum ($p_{\rm T}$)~spectra of identified charged particles (pions, kaons, protons) and all charged particles from the available experimental data of Au-Au collisions at the Relativistic Heavy Ion Collider (RHIC) energies and Pb-Pb collisions at the Large Hadron Collider (LHC) energies are fitted by the Tsallis distribution. The fit parameters, $q$ and $T$ measure the degree of deviation from an equilibrium state and th…
A systematic analysis of transverse momentum and rapidity distributions measured in high-energy proton - proton (pp) collisions for energies ranging from 53 GeV to 7 TeV using Tsallis thermodynamics is presented. The excellent description of all transverse momentum spectra obtained in earlier analyses is confirmed and extended. All energies can be described by a single Tsallis temperature of 68 +/- 5 MeV at all beam energies and particle types investigated (43 in total). The value of the entropic index, q, shows a wider spread but is always close to q approx 1.146. These values are then used t…
Final-state effects on the transverse-momentum spectra of charged hadrons in $p+Pb$ and $Pb+Pb$ collisions at $\sqrt {s_{\rm{NN}}}~=~5.02$~TeV using the modified Tsallis distribution
Noida International University（诺伊达国际大学）；Graphic Era Hill University（格拉菲克时代山岳大学）；Academia Sinica（中央研究院）；Ewing Christain College（尤因基督学院）；Gujarat Arts and Science College（古吉拉特艺术与理科学院）；Central University of South Bihar（比哈尔南部中央大学）
In high-energy collisions the number of the created particles is far less than the thermodynamic limit, especially in small colliding systems (e.g. proton-proton). Therefore final-state effects and fluctuations in the one-particle energy distribution are appreciable. As a consequence the characterization of identified hadron spectra with the Boltzmann\,--\,Gibbs thermodynamical approach is insufficient. Instead particle spectra measured in high-energy collisions can be described very well with Tsallis\,--\,Pareto distributions, derived from non-extensive thermodynamics. Using the Tsallis q-ent…
We evaluate the transport properties such as shear viscosity ($η$), bulk viscosity ($ζ$) and their ratios over entropy density ($s$) for hadronic matter using relativistic non-extensive Boltzmann transport equation (NBTE) in relaxation time approximation (RTA). In NBTE, we argue that the system far from equilibrium may not reach to an equilibrium described by extensive (Boltzmann-Gibbs (BG)) statistics but to a $q$-equilibrium defined by Tsallis non-extensive statistics after subsequent evolution, where $q$ denotes the degree of non-extensivity. We observe that $η/s$ and $ζ/s$ decrease rapidly…
We studied the effects of the Tsallis distribution on the transverse momentum fluctuation in high energy collisions. The parton-hadron duality and the Bose-Einstein type correlation between partons were assumed. The fluctuation was calculated in the boost-invariant picture for the expectation value used in the Boltzmann-Gibbs statistics and for the expectation value used in the Tsallis nonextensive statistics. It was shown that the fluctuation is a function of $η$ which is the ratio of the inverse temperature to the correlation length. We found the following points: (1) the fluctuation depends…
We analyzed the transverse momentum {\ppt} spectra of different particle species containing strange quark {\ks}, {\lam (\alam)} and {\xim ({\axi})} in different centrality intervals at mid-rapidity ($y$) from {\auau} collisions at {\sqrtsNN}= 54.4 GeV. The {\ppt} spectra of these strange hadrons are investigated by Tsallis-like distribution. The Tsallis-like distribution satisfactorily fits the experimental data. The effective temperature, entropy based parameter $q$ and mean {\ppt} extracted from the Tsallis-like distribution function while the kinetic freeze-out temperature ($T_0$) and trans…
In this paper, the experimental (simulated) transverse momentum spectra of negatively charged pions produced at mid-rapidity in central nucleus-nucleus collisions at the Heavy Ion Synchrotron (SIS), Relativistic Heavy Ion Collider (RHIC), and Large Hadron Collider (LHC) energies obtained by different collaborations are selected by us to investigate, where a few simulated data are taken from the results of FOPI Collaboration who uses the IQMD transport code based on Quantum Molecular Dynamics. A two-component standard distribution and the Tsallis form of standard distribution are used to fit th…
We review transverse momentum distributions of various identified charged particles stemming from high energy collisions fitted by various non-extensive distributions as well as by the usual Boltzmann-Gibbs statistics. We investigate the best-fit formula with the obtained $χ^2/ndf$ values. We find that the physical mass and $\sqrt{s}$ scaling become more explicit with heavier produced hadrons in both proton-proton and heavy-ion collisions. The spectral shape parameters, in particular the temperature $T$ and the non-extensive Tsallis parameter $q$, do exhibit an almost linear dependence with th…
We have implemented the Tsallis q-statistics in the Weibull model of particle production known as the q-Weibull distribution to describe the transverse-momentum (pT ) distribution of the charged hadrons at mid-rapidity measured at RHIC and LHC energies. The model describes the data remarkably well for the entire pT range measured in nucleus-nucleus and nucleon-nucleon collisions. The proposed distribution is based on the non-extensive Tsallis q-statistics which replaces the usual thermal equilibrium assumption of hydrodynamical models. The parameters of the distribution can be related to the v…
Nuclear Modification Factor in Pb-Pb and p-Pb collisions at $\sqrt{s_{NN}}$=5.02 TeV at LHC energies using Boltzmann Transport Equation with Tsallis Blast Wave Description
