Tsallis Nonextensive Statistics Research Articles

Non-extensive effects on the QCD equation of state and fluctuations of conserved charges within Polyakov quark meson model

The influence of non-extensive Tsallis statistics on the hadron phase structure has been investigated using the Polyakov-quark-meson (PQM) model. The analysis examines the non-extensive effects on the temperature dependence of PQM order parameters, thermodynamic quantities related to the quantum chromodynamics (QCD) equation of state, and fluctuations of conserved charges at varying chemical potentials. The results show that non-extensive effects have the most significant deviations near the crossover region. The pseudo-critical temperature T χ (μ B ) is not a universal constant and decreases with increasing non-extensive q parameter. The chiral phase diagram of the PQM model indicates a decrease in the behavior of the (T χ − μ B ) plane with increasing non-extensive q parameter. The PQM model exhibits good qualitative agreement with lattice QCD calculations. Moreover, these findings suggest the existence of a Tsallis limit, which serves as an alternative to the Stefan–Boltzmann limit for the massless ideal gas. The critical endpoint exhibits lower temperature but higher chemical potential with increasing non-extensive q parameter. Overall, this study highlights the importance of non-extensive Tsallis statistics in characterizing the quark-hadron phase structure of the PQM model and contributes to a deeper understanding of non-extensive effects in the quark-hadron phase transition.

Role of chemical potential at kinetic freeze-out using Tsallis non-extensive statistics in proton-proton collisions at the Large Hadron Collider

Role of chemical potential at kinetic freeze-out using Tsallis non-extensive statistics in proton-proton collisions at the Large Hadron Collider

Pressure Distribution Inside Nucleons in a Tsallis-MIT Bag Model.

We present a phenomenological framework based on the MIT bag model to estimate the pressure experienced by quarks and gluons inside nucleons. This is accomplished by implementing non-extensive Tsallis statistics for the two-component system. In this model of hadrons, the strong interaction generates correlations effectively described by the q-Tsallis parameter. The resulting hadron pressure exhibits general agreement with recent calculations derived from Lattice QCD. Additionally, we compared this pressure with data extracted from deep virtual Compton scattering experiments and gravitational form factor analyses. The extended bag model provides an alternative interpretation of bag pressure in terms of the q-Tsallis parameter. Consequently, the MIT bag model can be expressed without requiring the inclusion of the bag pressure parameter.

Thermodynamic properties and phase diagram of quark matter within non-extensive Polyakov chiral SU (3) quark mean field model

In the present study, we applied Tsallis non-extensive statistics to investigate the thermodynamic properties and phase diagram of quark matter in the Polyakov chiral SU(3) quark mean field model. Within this model, the properties of the quark matter were modified through the scalar fields , vector fields , ϕ, and Polyakov fields Φ and at finite temperature and chemical potential. Non-extensive effects were introduced through a dimensionless parameter q, and the results were compared to those of the extensive case ( ). In the non-extensive case, the exponential in the Fermi-Dirac (FD) function was modified to a q-exponential form. The influence of the q parameter on the thermodynamic properties, pressure, energy, and entropy density, as well as trace anomaly, was investigated. The speed of sound and specific heat with non-extensive effects were also studied. Furthermore, the effect of non-extensivity on the deconfinement phase transition as well as the chiral phase transition of and s quarks was explored. We found that the critical end point (CEP), which defines the point in the phase diagram where the order of the phase transition changes, shifts to a lower value of temperature, , and a higher value of chemical potential, , as the non-extensivity is increased, that is, 1.

Nonextensive Behavior of Stellar Rotation in the Galactic Disk Components

The solar neighborhood is a unique stellar astrophysical laboratory formed by a variety of stars from different origins. In particular, two of the most notable populations known are the thick and thin disk stars, each characterized by distinct chemical compositions, ages, kinematics, and origins. Based on Tsallis nonextensive statistics, we investigate the observed distribution of the projected rotational velocity of the thin and thick disk component stars. Through Bayesian inference, our results show that the distributions of the Galactic disk populations selected from both kinematic and chemical criteria follow a nonextensive behavior, where non-Gaussian statistics provide a more accurate representation. We also observed an anticorrelation between the entropic index q and the age of disk components confirming the interpretation of initial angular momentum memory loss scaled by the parameter q. In contrast, a subextensivity case with q > 1 was found for the old high-α metal-rich subgroup hαmr, and due to their distinguished rotational behavior and atypical subextensive regime, we infer that thick disk and hαmr stars are, in fact, distinct objects. Our results also suggest that the rotational velocities of stars are defined not only by their evolutionary spin-down processes but also by their birth sites.

Tsallis statistics and thermofractals: Applications to high energy and hadron physics

We study the applications of nonextensive Tsallis statistics to high energy and hadron physics. These applications include studies of [Formula: see text] 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.

Analysis of properties of the charged-particle jets in pp collisions at 13 TeV using non-extensive Tsallis statistics

Analysis of properties of the charged-particle jets in pp collisions at 13 TeV using non-extensive Tsallis statistics

Study of midrapidity pt distributions of identified charged particles in Xe+Xe collisions at (snn)1/2 = 5.44TeV using non-extensive Tsallis statistics with transverse flow

The midrapidity transverse momentum [Formula: see text] distributions of the charged pions and kaons, protons, and antiprotons, measured by ALICE Collaboration at nine centrality groups of Xe[Formula: see text]Xe collisions at [Formula: see text], have been reproduced quite well using simultaneous (combined) minimum [Formula: see text] fits with the simple (non-consistent) as well as thermodynamically consistent Tsallis function with included transverse flow. The parameters [Formula: see text], [Formula: see text], and [Formula: see text] extracted in this work using both consistent and non-consistent Tsallis function with included transverse flow demonstrate the similar dependencies on collision centrality [Formula: see text]. The obtained non-extensivity parameter [Formula: see text] values decrease systematically for all studied particle species with increasing Xe[Formula: see text]Xe collision centrality, indicating an increase in degree of system thermalization with increasing centrality of heavy-ion collisions. The average transverse flow velocity demonstrates significantly different growth rates in regions [Formula: see text] and [Formula: see text], and parameter [Formula: see text] stays constant within uncertainties in [Formula: see text] range in Xe[Formula: see text]Xe collisions at [Formula: see text]. It is argued that [Formula: see text] could possibly be a threshold border value for a crossover transition from a dense hadronic state to the QGP phase (or mixed phase of QGP and hadrons) in Xe[Formula: see text]Xe collisions at [Formula: see text]. Depletion (enhancement) of [Formula: see text] ratio at low [Formula: see text] (intermediate [Formula: see text]) has been observed in this work in Xe[Formula: see text]Xe collisions at [Formula: see text], which agrees with the similar results of ALICE Collaboration obtained recently in high-energy Xe[Formula: see text]Xe and [Formula: see text] collisions at the LHC. Analyzing and reflecting the extracted [Formula: see text] versus [Formula: see text] and [Formula: see text] versus [Formula: see text] dependencies, we have verified that the depletion (enhancement) of baryon-to-meson ratio at low [Formula: see text] (intermediate [Formula: see text]) values with increasing [Formula: see text] is due to radial flow effects.

Nonextensive Tsallis statistics in Unruh effect for Dirac neutrinos

Flavor mixing of quantum fields was found to be responsible for the breakdown of the thermality of Unruh effect. Recently, this result was revisited in the context of nonextensive Tsallis thermostatistics, showing that the emergent vacuum condensate can still be featured as a thermal-like bath, provided that the underlying statistics is assumed to obey Tsallis prescription. This was analyzed explicitly for bosons. Here we extend this study to Dirac fermions and in particular to neutrinos. Working in the relativistic approximation, we provide an effective description of the modified Unruh spectrum in terms of the q-generalized Tsallis statistics, the q-entropic index being dependent on the mixing parameters sin theta and Delta m. As opposed to bosons, we find q>1, which is indicative of the subadditivity regime of Tsallis entropy. An intuitive understanding of this result is discussed in relation to the nontrivial entangled structure exhibited by the quantum vacuum for mixed fields, combined with the Pauli exclusion principle.

Perspective on Tsallis statistics for nuclear and particle physics

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 nonintuitive manner for them to be so. Nevertheless, the Boltzmann limit of the Tsallis distribution is extremely efficient in representing a wide variety of single-particle distributions in high-energy proton–proton, proton–nucleus and nucleus–nucleus collisions with only three parameters, one of them being the so-called nonextensitivity parameter [Formula: see text]. This distribution interpolates between an exponential at low transverse energy, reflecting thermal equilibrium, to a power law at high transverse energy, reflecting the asymptotic freedom of Quantum Chromodynamics (QCD). It should not be viewed as a fundamental new parameter representing nonextensive behavior in these collisions.

Temperature fluctuations and Tsallis statistics in relativistic heavy ion collisions

In this paper, we study temperature fluctuations in the initial stages of the relativistic heavy ion collision using a multiphase transport model. We consider the plasma in the initial stages after collision before it has a chance to equilibrate. We have considered [Formula: see text] collision with a center-of-mass energy of 200 GeV. We use the nonextensive Tsallis statistics to find the entropic index in the partonic stages of the relativistic heavy ion collisions. We find that the temperature and the entropic index have a linear relationship during the partonic stages of the heavy ion collision. This has already been observed in the hadronic phase. A detailed analysis of the dependence of the entropic index on the system shows that for increasing spacetime rapidity, the entropic index of the partonic system increases. The entropic index also depends on the beam collision energy. The calculation of the entropic index from the experimental data fitting of the transverse momenta deals with the hadronic phase. However, our study shows that the behavior of the entropic index in the initial nonequilibrium stage of the collision is very similar to the behavior of the entropic index in the hadronic stage.

On the Gouy–Chapman–Stern model of the electrical double-layer structure with a generalized Boltzmann factor

The classical treatment of the electrical double-layer (EDL) structure at a planar metal/electrolyte junction via the Gouy–Chapman–Stern (GCS) approach is based on the Poisson equation relating the electrostatic potential to the net mean charge density. The ions concentration in the diffuse layer are assumed to follow the Boltzmann distribution law, i.e. ∝exp(−ψ̃) where ψ̃ is the dimensionless electrostatic potential. However, even in stationary equilibrium in which variables are averaged over a large number of elementary stochastic events, deviations from the mean-value are expected. In this study we evaluate the behavior of the EDL by assuming some small perturbations superposed on top of its Boltzmann distribution of ion concentrations using the Tsallis nonextensive statistics framework. With this we assume the ion concentrations to be proportional to [1−(1−q)ψ̃]1/(1−q)=expq(−ψ̃) with q being a real parameter that characterizes the system’s statistics. We derive analytical expression and provide computational results for the overall differential capacitance of the EDL structure, which, depending on the values of the parameter q can show both the traditional inverse bell-shaped curves for aqueous solutions and bell curves observed with ionic liquids.

Tsallis statistics, fractals and QCD

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.

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.

Asymmetric Tsallis distributions for modeling financial market dynamics

Financial markets are highly non-linear and non-equilibrium systems. Earlier works have suggested that the behavior of market returns can be well described within the framework of non-extensive Tsallis statistics or superstatistics. For small time scales (delays), a good fit to the distributions of stock returns is obtained with q-Gaussian distributions, which can be derived either from Tsallis statistics or superstatistics. These distributions are symmetric. However, as the time lag increases, the distributions become increasingly non-symmetric. In this work, we address this problem by considering the data distribution as a linear combination of two independent normalized distributions — one for negative returns and one for positive returns. Each of these two independent distributions are half q-Gaussians with different non-extensivity parameter q and temperature parameter beta. Using this model, we investigate the behavior of stock market returns over time scales from 1 to 80 days. The data covers both the .com bubble and the 2008 crash periods. These investigations show that for all the time lags, the fits to the data distributions are better using asymmetric distributions than symmetric q-Gaussian distributions. The behaviors of the q parameter are quite different for positive and negative returns. For positive returns, q approaches a constant value of 1 after a certain lag, indicating the distributions have reached equilibrium. On the other hand, for negative returns, the q values do not reach a stationary value over the time scales studied. In the present model, the markets show a transition from normal to superdiffusive behavior (a possible phase transition) during the 2008 crash period. Such behavior is not observed with a symmetric q-Gaussian distribution model with q independent of time lag.

Dynamical characteristics of global stock markets based on time dependent Tsallis non-extensive statistics and generalized Hurst exponents

We perform non-linear analysis on stock market indices using time-dependent extended Tsallis statistics. Specifically, we evaluate the q-triplet for particular time periods with the purpose of demonstrating the temporal dependence of the extended characteristics of the underlying market dynamics. We apply the analysis on daily close price timeseries of four major global markets (S&P 500, Tokyo-NIKKEI, Frankfurt-DAX, London-LSE). For comparison, we also compute time-dependentGeneralized Hurst Exponents (GHE) Hq using the GHE method, thus estimating the temporal evolution of the multiscaling characteristics of the index dynamics. We focus on periods before and after critical market events such as stock market bubbles (2000 dot.com bubble, Japanese 1990 bubble, 2008 US real estate crisis) and find that the temporal trends of q-triplet values significantly differ among these periods indicating that in the rising period before a bubble break, the underlying extended statistics of the market dynamics strongly deviates from purely stochastic behavior, whereas, after the breakdown, it gradually converges to the Gaussian-like behavior which is a characteristic of an efficient market. We also conclude that relative temporal variation patterns of the Tsallis q-triplet can be connected to different aspects of market dynamics and reveals useful information about market conditions especially those underlying the development of a stock market bubble. We found specific temporal patterns and trends in the relative variation of the indices in the q-triplet that distinguish periods just before and just after a stock-market bubble break. Differences between endogenous and exogenous stock market crises are also captured by the temporal changes in the Tsallis q-triplet. Finally, we introduce two new time-dependent empirical metrics (Q-metrics) that are functions of the Tsallis q-triplet. We apply them to the above stock market index price timeseries and discuss the significance of their temporal dependence on market dynamics and the possibility of using them, together with the relative temporal changes of the q-triplet, as signaling tools for future market events such as the development of a market bubble.

Bose-Einstein condensation of pions in proton–proton collisions at the Large Hadron Collider using non-extensive Tsallis statistics

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 dependency of BEC critical temperature and number of particles in the pion condensates on the non-extensive parameter q, which is a measure of degree of non-equilibrium – as q decreases, the critical temperature increases and approaches to the critical temperature obtained from Bose-Einstein statistics without non-extensivity. Studies are performed on the final state multiplicity dependence of number of particles in the pion condensates in a wide range of multiplicity covering hadronic and heavy-ion collisions, using the inputs from experimental transverse momentum spectra.

Event Shape and Multiplicity Dependence of Freeze-Out Scenario and System Thermodynamics in Proton+Proton Collisions at s = 13 TeV Using PYTHIA8

Recent observations of QGP-like conditions in high-multiplicity pp collisions from ALICE experiment at the LHC warrant an introspection whether to use pp collisions as a baseline measurement to characterize heavy-ion collisions for the possible formation of a Quark-Gluon Plasma. A double differential study of the particle spectra and thermodynamics of the produced system as a function of charged-particle multiplicity and transverse spherocity in pp collisions would shed light on the underlying event dynamics. Transverse spherocity, one of the event shape observables, allows to separate the events in terms of jetty and isotropic events. We analyse the identified particle transverse momentum ( p T ) spectra as a function of charged-particle multiplicity and transverse spherocity using Tsallis nonextensive statistics and Boltzmann-Gibbs Blast-Wave (BGBW) model in pp collisions at s = 13 TeV using PYTHIA8 event generator. The extracted parameters such as temperature ( T ), radial flow ( β ), and nonextensive parameter ( q ) are shown as a function of charged-particle multiplicity for different spherocity classes. We observe that the isotropic events approach thermal equilibrium while the jetty ones remain far from equilibrium. We argue that, while studying the QGP-like conditions in small systems, one should separate the isotropic events from the spherocity-integrated events, as the production dynamics are different.

Джинсовская неустойчивость протопланетного околозвездного диска с учетом магнитного поля и излучения в неэкстенсивной кинетике Тсаллиса

Within the framework of the Tsallis non-extensive statistics, a derivation of the Jeans gravitational instability criteria is given for a self-gravitating protoplanetary disk, the substance of which consists of a mixture of a conducting ideal q-gas and modified radiation of a photon gas. The instability criteria are derived from the corresponding dispersion relations written for both neutral disk matter and magnetized plasma with modified blackbody radiation. The thermodynamics of a photon gas is constructed based on the nonextensive quantum entropy Tsallis, which depends on the deformation parameter. It is shown that blackbody radiation can stabilize the state of a nonextensive medium for a purely gaseous disk, and for an electrically conducting disk, the Jeans instability criterion is modified by a magnetic field and radiation pressure only in the transverse regime of propagation of the disturbance wave.

Fractal Structures of Yang–Mills Fields and Non-Extensive Statistics: Applications to High Energy Physics

In this work, we provide an overview of the recent investigations on the non-extensive Tsallis statistics and its applications to high energy physics and astrophysics, including physics at the Large Hadron Collider (LHC), hadron physics, and neutron stars. We review some recent investigations on the power-law distributions arising in high energy physics experiments focusing on a thermodynamic description of the system formed, which could explain the power-law behavior. The possible connections with a fractal structure of hadrons is also discussed. The main objective of the present work is to delineate the state-of-the-art of those studies and show some open issues that deserve more careful investigation. We propose several possibilities to test the theory through analyses of experimental data.