Tsallis Distribution Function Research Articles

An Extensive Analysis of Tsallis Statistics: π±, K± Mesons, and pp¯ Baryon in Inelastic pp Collisions

This study explores the inelastic doubly differential transverse momentum spectra of the primary charged particles, (π++π−), (K++K−) and (pp¯), as a function of observables associated with underlying event (UE) at s=13TeV. The particle production is measured on the basis of different angular regions like toward, transverse and away, elucidated with respect to the direction of leading particle of an event. To study the thermal freeze-out parameters, the non-extensive Tsallis distribution function is used to extract the temperature Teff and chemical potential μ, which provide a basis to explain the QCD matter. The Tsallis distribution function describes transverse momentum spectra in pseudorapidity region of |η|<0.8. It is observed that effective temperature Teff changes from away to towards and forward region.

Analysis of [formula omitted] spectra for [formula omitted] mesons in [formula omitted]–[formula omitted] collisions at 200 GeV, using PYTHIA and Tsallis function

The ϕ(1020)0 mesons produced by asymmetric Cu–Au collisions at sNN=200GeV in the pseudorapidity region of |η|<0.35 is studied in relation to the transverse momentum (pT). The nuclear modification factor (RAB) is also calculated for ϕ(1020)0 mesons. To validate the model’s predictions, the simulations are compared to the experimental data from PHENIX using the PYTHIA8 and tuned to color reconnection mode. A systematically close relationship exists between PYTHIA8 predictions and the PHENIX data. PYTHIA CR1 mode predictions do not agree well with PHENIX experimental data. In addition to the RAB predictions, we investigated the ratio between the production of ϕ(1020)0 mesons to K∗(892)0 mesons and Λ baryons in Cu–Au collisions at sNN=200GeV. A comparison of the ϕ(1020)0 mesons yield produced by symmetric Au–Au and Cu–Cu collisions at sNN=200GeV are conducted using PYTHIA8 for comparison against asymmetric Cu–Au collisions. In this study, we also fit thermodynamically consistent Tsallis distribution functions for ϕ(1020)0 mesons onto experimental RHIC (PHENIX) pT spectra obtained from Cu–Au collisions at sNN=200GeV energy.

Intermittency in the Magnetic Hump in the VLISM

Voyager 1 has been moving through the very local interstellar medium (VLISM) for ∼1 solar cycle, from 122.58 au on 2012/DOY 238 (August 25) to 158.5 au on 2023.0. A magnetic hump was observed, beginning with an abrupt increase (jump) in the magnetic field strength B and proton density N by a factor of 1.35 and 1.36, respectively, in the course of ∼8 days, ending at ∼2020/DOY 147, after which it remained above the pre-jump levels for 2.6 yr, and it is still increasing. Unlike the three previous jumps, which were followed by a slow decrease in B, B in the magnetic hump continued to increase to a maximum value ∼0.56 nT at ∼2021/DOY 146.7. This paper discusses the intermittency of the magnetic field components and strength in the magnetic hump, and compares them with previous values in the VLISM. We consider the intermittency of the increments of B and its components observed on a scale of one day. The distribution function (the Tsallis distribution or q-distribution function) of the increments of hourly averages of the components of B had a Gaussian core associated with randomness in the observations, and it had two symmetric tails associated with the observed intermittency. The parameters q-dBR1, q-dBT1, and q-dBN1 increased slightly to a broad maximum with q = ∼1.4 between 2018 and 2019, and then decreased to q ∼ 1 (corresponding to no intermittency) in 2022. The Z-component of the magnetic field is within 4 ± 4 counts (0.02 nT ± 0.02 nT).

Determination of Tsallis parameters for K∗(892)± mesons in inelastic pp, pPb and PbPb collisions

Doubly differential transverse momentum spectra of strange mesons [Formula: see text] and [Formula: see text] within rapidity interval of [Formula: see text] are studied by using Monte Carlo event generators in pp, pPb and PbPb collisions at different center-of-mass energies. The predictions of simulations are compared to the ALICE data. The fitting of doubly differential transverse momentum spectra of ALICE data is performed by using Tsallis distribution function. To check the quality of the fitting, chemical potential is also set to zero value. The extracted values of the non-extensive parameter ([Formula: see text]), a chemical potential [Formula: see text] and effective Tsallis temperature [Formula: see text] for different particle species have been obtained from the thermodynamically consistent Tsallis distribution function. It is observed that the effective Tsallis temperature increases at different center-of-mass energies. The predictions of the MC event generators, i.e. PYTHIA8 CR and HIJING 2.0 are compared to the ALICE data, deviations are observed which may be due to the reshuffling of parton colors and string fragmentation.

Impact of nonextensivity on the transport coefficients of a magnetized hot and dense QCD matter

We have studied the impact of the nonextensivity on the transport coefficients related to charge and heat in thermal QCD. For this purpose, the electrical (sigma _textrm{el}), Hall (sigma _textrm{H}), thermal (kappa ) and Hall-type thermal (kappa _textrm{H}) conductivities are determined using the kinetic theory approach in association with the nonextensive Tsallis statistical mechanism. The effect of nonextensivity is encoded in the nonextensive Tsallis distribution function, where the deviation of the parameter q from 1 signifies the degree of nonextensivity in the concerned system. The thermal and electrical conductivities are found to increase with the introduction of nonextensivity, which means that the deviation of the medium from thermal equilibrium enhances both charge and heat transports. With the magnetic field, the deviations of sigma _textrm{el}, sigma _textrm{H}, kappa and kappa _textrm{H} from their respective equilibrated values increase, whereas these deviations decrease with the chemical potential. We have also studied how the extent of the nonextensivity modulates the longevity of magnetic field. Present work is further extended to the study of some observables associated with the aforesaid transport phenomena, such as the Knudsen number and the elliptic flow within the nonextensive Tsallis framework.

Correlations among parameters of the Tsallis distribution and Hagedorn function with embedded transverse flow in proton–proton collisions at (s)1/2 = 7 and 13 TeV

Correlations among parameters of the Tsallis distribution and Hagedorn function with embedded transverse flow in proton–proton collisions at (s)1/2 = 7 and 13 TeV

Hadron gas in the presence of a magnetic field using non-extensive statistics: a transition from diamagnetic to paramagnetic system

Non-central heavy-ion collisions at ultra-relativistic energies are unique in producing magnetic fields of the largest strength in the laboratory. Such fields being produced at the early stages of the collision could affect the properties of Quantum Chromodynamics matter formed in the relativistic heavy-ion collisions. The transient magnetic field leaves its reminiscence, which in principle, can affect the thermodynamic and transport properties of the final state dynamics of the system. In this work, we study the thermodynamic properties of a hadron gas in the presence of an external static magnetic field using a thermodynamically consistent non-extensive Tsallis distribution function. Various thermodynamical observables such as energy density (ϵ), entropy density (s), pressure (P) and speed of sound (c s) are studied. Investigation of magnetization (M) is also performed and this analysis reveals an interplay of diamagnetic and paramagnetic nature of the system in the presence of a magnetic field of varying strength. Further, to understand the system dynamics under equilibrium and non-equilibrium conditions, the effect of the non-extensive parameter (q) on the above observables is also studied.

Pseudorapidity dependence of the bulk properties of hadronic medium in pp collisions at 7 TeV

The measured charged particle p_T spectra in proton-proton collisions obtained by the CMS experiment at CERN is compared with the simulation results of EPOS–LHC and Pythia8.24 models at 7 TeV center-of-mass energy. The Pythia8.24 model describes the experimental data very well, particularly in the high p_T region. The model also predicts the p_T spectra for |eta | < 2.4 at 0 le p_Tle 6 text {GeV/}c. The EPOS–LHC model underpredicts the p_T spectra from 0.1 to 2 text {GeV/}c in all eta bins for about 20% and the p_T spectrum from 0.1 to 4.2 text {GeV/}c for |eta | < 2.4 by about 15% while reasonably predicts well for p_T > 4.2 text {GeV/}c within the experimental errors. Furthermore, to get information about collective properties of the hadronic matter, modified Hagedorn function with embedded transverse flow velocity and thermodynamically consistent Tsallis distribution functions are used to fit the experimental data and simulated results. The values of chi ^2/ndf show that the functions fit the data and simulation results well. The parameter extracted by the functions: beta _T, T_0, and T_{eff} decreases with increasing eta . The decrease in beta _T with increasing eta is due to the large energy deposition in lower rapidity bins producing rapid expansion due to large pressure gradient resulting quick expansion of the fireball. Similarly, large energy transfer in the lower pseudo-rapidity bin results in higher degree of excitation of the system which results larger values of T_0 and T_{eff}. The values of the fit constant N_0 increase with eta where the values of N_0 extracted from Pythia8.24 are closer to the data than the EPOS–LHC model. The Pythia8.24 model has better prediction than the EPOS–LHC model which might be connected to its flow-like features and color re-connections resulting from different Parton interactions in the initial and final state.

Initial- and Final-State Temperatures of Emission Source from Differential Cross-Section in Squared Momentum Transfer in High-Energy Collisions

The differential cross-section in squared momentum transfer of ρ , ρ 0 , ω , ϕ , f 0 980 , f 1 1285 , f 0 1370 , f 1 1420 , f 0 1500 , and J / ψ produced in high-energy virtual photon-proton ( γ ∗ p ), photon-proton ( γ p ), and proton-proton ( p p ) collisions measured by the H1, ZEUS, and WA102 Collaborations is analyzed by the Monte Carlo calculations. In the calculations, the Erlang distribution, Tsallis distribution, and Hagedorn function are separately used to describe the transverse momentum spectra of the emitted particles. Our results show that the initial- and final-state temperatures increase from lower squared photon virtuality to a higher one and decrease with the increase of center-of-mass energy.

Centrality, transverse momentum and collision energy dependence of the Tsallis parameters in relativistic heavy-ion collisions

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_mathrm{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 the effective temperature of the thermalized system, respectively. The p_mathrm{T} spectra are well described by the Tsallis distribution function from peripheral to central collisions for the wide range of collision energies, from sqrt{s_mathrm{NN}} = 7.7 GeV to 5.02 TeV. The extracted Tsallis parameters are found to be dependent on the particle species, collision energy, centrality, and fitting ranges in p_mathrm{T}. For central collisions, both q and T depend strongly on the fit ranges in p_mathrm{T}. For most of the collision energies, q remains almost constant as a function of centrality, whereas T increases from peripheral to central collisions. For a given centrality, q systematically increases as a function of collision energy, whereas T has a decreasing trend. A profile plot of q and T with respect to collision energy and centrality shows an anti-correlation between the two parameters.

Characterizing Proton-Proton Collisions at the Large Hadron Collider with Thermal Properties

High-multiplicity proton-proton (pp) collisions at the Large Hadron Collider (LHC) energies have created a new domain of research to look for a possible formation of quark–gluon plasma in these events. In this paper, we estimate various thermal properties of the matter formed in pp collisions at the LHC energies, such as mean free path, isobaric expansivity, thermal pressure, and heat capacity using a thermodynamically consistent Tsallis distribution function. Particle species-dependent mean free path and isobaric expansivity are studied as functions of final state charged particle multiplicity for pp collisions at the center-of-mass energy s = 7 TeV. The effects of degree of non-extensivity, baryochemical potential, and temperature on these thermal properties are studied. The findings are compared with the theoretical expectations.

Collective Dynamics Inside Paul Trap With Stray Electric Field Using Nonextensive Distribution Function

The collective dynamics of a collision-less nonneutral plasma is analyzed in the presence of stray electric field inside a Paul trap. Experiments in Paul traps have indicated the presence of power tail distribution functions. Therefore, a Tsallis distribution function represents the initial distribution of charged particles inside the trap. The stroboscopic map in phase space reveals a closed curve, indicating that a carefully chosen initial distribution function will yield a time-periodic distribution function. The time-averaged distribution function is doubled humped indicating instabilities. The position of the double hump and the asymmetry in the spatial variation of temperature inside the trap depends on the magnitude and the orientation of the stray electric field inside the trap.

Systematic analysis of midrapidity transverse momentum spectra of identified charged particles in p + p collisions at (snn)1/2 = 2.76, 5.02, and 7 TeV at the LHC

The experimental transverse momentum spectra of the charged pions and kaons, protons and antiprotons, produced at midrapidity in inelastic [Formula: see text] collisions at [Formula: see text], 5.02, and 7 TeV, measured by ALICE collaboration, are analyzed systematically using the thermodynamically consistent Tsallis distribution function as well as Hagedorn function with the embedded transverse flow. To compare directly the results obtained from combined (simultaneous) fits with the above theoretical model functions of midrapidity spectra of the charged pions and kaons, protons and antiprotons in [Formula: see text] collisions at [Formula: see text], 5.02, and 7 TeV, we use the optimal identical fitting [Formula: see text] ranges [Formula: see text] in [Formula: see text] collisions at [Formula: see text], 5.02, and 7 TeV. The parameter [Formula: see text] increases consistently for all the particle species with increasing [Formula: see text] of [Formula: see text] collisions from 2.76 to 5.02 TeV, remaining practically constant within fit errors in the collision energy range [Formula: see text] TeV, suggesting probably a saturation and limiting behavior of the [Formula: see text] value in [Formula: see text] collisions at [Formula: see text] TeV. Approximate equality of the extracted nonextensivity parameter [Formula: see text] for the pions and kaons ([Formula: see text]) and relation [Formula: see text] are obtained in [Formula: see text] collisions at [Formula: see text], 5.02, and 7 TeV. Very negligible transverse (radial) flow velocity (consistent with zero value within the fit errors) is obtained in [Formula: see text] collisions at [Formula: see text], 5.02, and 7 TeV from fitting the [Formula: see text] distributions of the charged pions and kaons, protons and antiprotons with the Hagedorn formula with the embedded transverse flow in the selected optimal identical [Formula: see text] ranges as well as in the full measured [Formula: see text] ranges.

Hadron transverse momentum distributions of the Tsallis normalized and unnormalized statistics

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 distribution (widely used in high energy physics) is equal to the distribution of the Tsallis unnormalized statistics in the zeroth term approximation, but the phenomenological quantum Tsallis distributions (introduced by definition on the basis of the generalized entropy of the ideal gas) do not correspond to the distributions of the Tsallis statistics. We have found that in the ranges of the entropic parameter relevant to the processes of high-energy physics ($q<1$ for Tsallis-1 and $q>1$ for Tsallis-2) the Tsallis statistics is divergent. Therefore, to obtain physical results, we have regularized the Tsallis statistics by introducing an upper cut-off in the series expansion. The exact numerical results for the Bose-Einstein, Fermi-Dirac and Maxwell-Boltzmann statistics of particles in the Tsallis normalized and unnormalized statistics have been obtained. We observed that the exact results of the Tsallis statistics strongly enhanced the production of high-$p_{T}$ hadrons in comparison with the usual phenomenological Tsallis distribution function at the same values of $q$. The $q$-duality of the Tsallis normalized and unnormalized statistics for the massive particles was studied.

Relativistic effects on dust grain charging in non-equilibrium dusty plasma with relativistic Tsallis distribution

The relativistic effects on the dust grain charging are investigated by calculating the relativistic cross section by the orbit-limited motion (OLM) theory and relativistic Tsallis distribution function for currents carried by ions and electrons in nonequilibrium plasma. The dust grain electrical potential equation is derived in the dust grain charging process, and this nonlinear equation is solved by the numerical method. We indicate that the behavior of nonextensivity is in contrast to ions and electrons, and it is found that the density of species in plasma has the most influence on the degree of nonextensivity. We show that the degree of nonextensivity of the electron and the ion has significant effects on the relationship between dust grain electrical potential and dust density. As another result, it is indicated that by increasing the dust density, the dust particles will be seen as a component from ordinary multi-ionic plasma that shows collective behavior. By comparing the results in relativistic and nonrelativistic regimes, it is indicated that in the relativistic regime, the density of dust is limited, and the electrical potential of dust grains also varies, whereas in the nonrelativistic regime, the dust density is not limited, and the dust grain electrical potential tends to be constant in small amounts of dust density.

Radial flow and differential freeze-out in proton-proton collisions at $\sqrt{s} = 7$s=7 TeV at the LHC

We analyse the transverse momentum ($p_{\rm T}$)-spectra as a function of charged-particle multiplicity at midrapidity ($|y| < 0.5$) for various identified particles such as $\pi^{\pm}$, $K^{\pm}$, $K_S^0$, $p+\overline{p}$, $\phi$, $K^{*0} + \overline {K^{*0}}$, and $\Lambda$ + $\bar{\Lambda}$ in proton-proton collisions at $\sqrt{s}$ = 7 TeV using Boltzmann-Gibbs Blast Wave (BGBW) model and thermodynamically consistent Tsallis distribution function. We obtain the multiplicity dependent kinetic freeze-out temperature ($T_{\rm kin}$) and radial flow ($\beta$) of various particles after fitting the $p_{\rm T}$-distribution with BGBW model. Here, $T_{\rm kin}$ exhibits mild dependence on multiplicity class while $\beta$ shows almost independent behaviour. The information regarding Tsallis temperature and the non-extensivity parameter ($q$) are drawn by fitting the $p_{\rm T}$-spectra with Tsallis distribution function. The extracted parameters of these particles are studied as a function of charged particle multiplicity density ($dN_{ch}/d\eta$). In addition to this, we also study these parameters as a function of particle mass to observe any possible mass ordering. All the identified hadrons show a mass ordering in temperature, non-extensive parameter and also a strong dependence on multiplicity classes, except the lighter particles. It is observed that as the particle multiplicity increases, the $q$-parameter approaches to Boltzmann-Gibbs value, hence a conclusion can be drawn that system tends to thermal equilibrium. The observations are consistent with a differential freeze-out scenario of the produced particles.

Multiplicity dependence of non-extensive parameters for strange and multi-strange particles in proton-proton collisions at $\sqrt{s} = 7$ s = 7 TeV at the LHC

The transverse momentum ($p_{\rm T}$) spectra in proton-proton collisions at $\sqrt{s}$ = 7 TeV, measured by the ALICE experiment at the LHC are analyzed with a thermodynamically consistent Tsallis distribution. The information about the freeze-out surface in terms of freeze-out volume, temperature and the non-extenisivity parameter, $q$, for $K^{0}_{S}$, $\Lambda+\bar{\Lambda}$, $\Xi^{-}+\bar{\Xi}^{+}$ and $\Omega^{-}+\bar{\Omega}^{+}$ are extracted by fitting the $p_{\rm T}$ spectra with Tsallis distribution function. The freeze-out parameters of these particles are studied as a function of charged particle multiplicity density ($dN_{ch}/d\eta$). In addition, we also study these parameters as a function of particle mass to see any possible mass ordering. The strange and multi-strange particles show mass ordering in volume, temperature, non-extensive parameter and also a strong dependence on multiplicity classes. It is observed that with increase in particle multiplicity, the non-extensivity parameter, $q$ decreases, which indicates the tendency of the produced system towards thermodynamic equilibration. The increase in strange particle multiplicity is observed to be due to the increase of temperature and not to the size of the freeze-out volume.

Indication of a Differential Freeze-Out in Proton-Proton and Heavy-Ion Collisions at RHIC and LHC Energies

The experimental data from the RHIC and LHC experiments of invariant pT spectra for most peripheral A+A and p+p collisions are analyzed with Tsallis distributions in different approaches. The information about the freeze-out surface in terms of freeze-out volume, temperature, chemical potential, and radial flow velocity for π+, K+, and p and their antiparticles is obtained. Furthermore, these parameters are studied as a function of the mass of the particles. A mass dependent differential freeze-out is observed which does not seem to distinguish between particles and their antiparticles. Furthermore, a mass-hierarchy in the radial flow is observed, meaning heavier particles suffer lower radial flow. Tsallis distribution function at finite chemical potential is used to study the mass dependence of chemical potential. The peripheral heavy-ion and proton-proton collisions at the same energies seem to be equivalent in terms of the extracted thermodynamic parameters.

Tsallis Distribution-Based Fractional Derivative Method for Lamb Wave Signal Recovery

A new fractional derivative method based on the Tsallis distribution function is developed to recover the Lamb wave signals from the noisy Lamb wave signals. After the fractional derivative of the amplitude spectrum of the noisy Lamb signal at different derivative orders, the quartic polynomial function between the peak amplitude and the derivative order as well as that between the peak frequency and the derivative order is proposed based on the Tsallis distribution function. Then, the characteristic parameters of the amplitude spectrum are deduced by using the proposed polynomial relationship. Finally, the noise-free Lamb wave signal is recovered based on the characteristic parameters. Simulated results indicate that the Lamb wave signals can be recovered in the case of the white noise, the transient noise and the sine wave signal. Experimental results confirm the validity of the method. Consequently, the developed method can recover the Lamb wave signals effectively.

Tsallis q - triplet and Stock Market Indices : The cases of S &amp; P 500 and TVIX

In this study we present results of the evaluation of q− triplet of Tsallis non-extensive statistics concerning two time stock market time series, Standart & Poor’s 500 (S & P) 500 and TVIX. The analysis of the results, support the hypothesis in which economic dynamics from physical point of view correspond to far from equilibrium spatial distributed non-linear dynamics. In particular, the analysis of the stock market time series revealed underlying complex dynamics, indicating clearly that the statistics of the dynamics in the multifractal phase space can be described by Tsallis distribution functions of power law and heavy tails forms. The non-extensive character of the underlying dynamics is related to the existence of long range interactions in space and time, as well as the interaction in many scales. In addition, the detailed analysis of S & P index unraveled the existence of non-equilibrium phase transitions depicted clearly in the variations of Tsallis q-triplet values. These phase transitions are connected with non-equilibrium stationary states of economical dynamics derived from processes of strong self organization which correspond to local maxima of Tsallis entropy, while the changes in the control parameters can induce new phase transitions and shifts to new metaequilibrium steady states of maximizing Tsallis entropy. These phase transitions lead to changes in Tsallis qtriplet values which correspond to multi-fractal changes in the formation of the phase state and an alteration in the phenomenology of the economical system. Finally, these characteristics also indicate the existence of fractional dynamics in the phase space which can be described through fractional Fokker-Planck equations and anomalous diffusion equations. The solutions of these equations are fractional spatiotemporal functions and non-Gaussian distributions functions which fall into the category of Levy distributions and Tsallis distributions.