Tsallis Statistics Research Articles

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.

Analysis of kinetic freeze-out temperature and transverse flow velocity in nucleus–nucleus and proton–proton collisions at same center-of-mass energy

Transverse momentum spectra of different types of identified charged particles in central Gold–Gold (Au–Au) collisions and inelastic (INEL) or nonsingle diffractive (NSD) proton–proton (pp) collisions at the Relativistic Heavy Ion Collider (RHIC), as well as in central and peripheral Lead–Lead (Pb–Pb) collisions, and INEL or NSD pp collisions at the Large Hadron Collider (LHC) are analyzed by the blast-wave model with Tsallis statistics. The model results are approximately in agreement with the experimental data measured by STAR, PHENIX and ALICE Collaborations in special transverse momentum ranges. Kinetic freeze-out (KFO) temperature and transverse flow velocity are extracted from the transverse momentum spectra of the particles. It is shown that KFO temperature of the emission source depends on mass of the particles, which reveals the mass-dependent KFO scenario in collisions at RHIC and LHC. Furthermore, the KFO temperature and transverse flow velocity in central nucleus–nucleus (AA) collisions are larger than in peripheral collisions, and both of them are slightly larger in peripheral nucleus–nucleus collisions or almost equivalent to that in proton–proton collisions at the same center-of-mass energy which shows their similar thermodynamic nature.

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.

Remarks on the phenomenological Tsallis distributions and their link with the Tsallis statistics

From the Tsallis unnormalized (or Tsallis-2) statistical mechanical formulation, Büyükkiliç et al (1995 Phys. Lett. A 197, 209) derived the expressions for the single-particle distribution functions (known as the phenomenological Tsallis distributions) for particles obeying the Maxwell–Boltzmann, Bose–Einstein and the Fermi–Dirac statistics using the factorization approximation. In spite of the fact that this paper was published long time ago, its results are still extensively used in many fields of physics, and it is considered that it was this paper that established the connection between the phenomenological Tsallis distributions and the Tsallis statistics. Here we show that this result is incorrect: the mistake lies in the fact that the probability distribution function was derived from the maximum entropy principle using the definition of the generalized expectation values (of the Tsallis-2 statistics), but the single-particle distribution functions were calculated on the basis of this probability distribution of the Tsallis-2 statistics using the standard definition of the expectation values of the Tsallis normalized (or Tsallis-1) statistics. Considering the definition of the expectation values for thermodynamic quantities which is consistent with the definition of expectation values of the constraints in the maximum entropy principle of the Tsallis-2 formulation, we have proved that the single-particle (classical and quantum) distribution functions in the factorization approximation differ from the well-known phenomenological Tsallis distributions.

Phenomenological Tsallis distribution from thermal field theory

Classical and quantum Tsallis distributions have been widely used in many branches of natural and social sciences. But, the quantum field theory of the Tsallis distributions is relatively a less explored arena. In this paper, we derive the expression for the thermal two-point functions in the Tsallis statistics with the help of the corresponding statistical mechanical formulations. We show that the quantum Tsallis distributions used in the literature appear in the thermal part of the propagator much in the same way the Boltzmann–Gibbs distributions appear in the conventional thermal field theory. As an application of our findings, we calculate the thermal mass in the [Formula: see text] scalar field theory within the realm of the Tsallis statistics.

Kinetic approach on the DIA wave propagation in a non‐extensive distributed dusty plasma

AbstractThe Landau damping of the dust ion‐acoustic wave (DIAW) in a dusty plasma with non‐extensive distributed components is analysed relying on the kinetic approach. The electron, ion, and dust particles are effectively modelled by the non‐extensive distribution function of the Tsallis statistics. For a collisionless plasma with different values of plasma components indices, the general dispersion relation is achieved, and the non‐extensivity effects on the frequency, as well as the Landau damping of the DIAW, are studied. We show that for , the preliminary results of the Maxwellian plasma are obtained. The decrease of wave damping is achieved by increasing the coefficient q index and the ion‐to‐electron density ratio. The damping rate also increases with an increasing ion‐to‐electron temperature ratio.

Tsallis statistics and generalized uncertainty principle

It has been argued that non-Gaussian statistics provide a natural framework to investigate semiclassical effects in the context of Planck-scale deformations of the Heisenberg uncertainty relation. Here we substantiate this point by considering the Unruh effect as a specific playground. By working in the realm of quantum field theory, we reformulate the derivation of the modified Unruh effect from the generalized uncertainty principle (GUP) in the language of the nonextensive Tsallis thermostatistics. We find a nontrivial monotonic relation between the nonextensivity index q and the GUP deformation parameter beta , which generalizes an earlier result obtained in quantum mechanics. We then extend our analysis to black hole thermodynamics. We preliminarily discuss our outcome in the broader context of an effective description of Planck-scale gravitational physics based on Tsallis theory.

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.

An evidence of triple kinetic freezeout scenario observed in all centrality intervals in Cu–Cu, Au–Au and Pb–Pb collisions at high energies

Transverse momentum spectra of π +, K +, p, , Λ, Ξ or and Ω or or in copper–copper (Cu–Cu), gold–gold (Au–Au) and lead–lead (Pb–Pb) collisions at 200 GeV, 62.4 GeV and 2.76 TeV respectively, are analyzed in different centrality bins by the blast wave model with Tsallis statistics. The model results are approximately in agreement with the experimental data measured by BRAHMS, STAR and ALICE Collaborations in special transverse momentum ranges. Kinetic freeze out temperature, transverse flow velocity and kinetic freezeout volume are extracted from the transverse momentum spectra of the particles. It is observed that and Ω or or have larger kinetic freezeout temperature followed by K +, and Λ than π + and p due to smaller reaction cross-sections of multi-strange and strange particles than non-strange particles. The present work reveals the scenario of triple kinetic freezeout in collisions at BRAHMS, STAR and ALICE Collaborations, however the transverse flow velocity and kinetic freezeout volume are mass dependent and they decrease with the increasing rest mass of the particle. In addition, the kinetic freezeout temperature, transverse flow velocity and kinetic freezeout volume are decreasing from central to peripheral collisions while the parameter q increase from central to peripheral collisions, indicating the approach of quick equilibrium in the central collisions. Besides, the kinetic freezeout temperature and kinetic freezeout volume are observed to be larger in larger collision system which shows its dependence on the size of the interacting system, while transverse flow velocity increase with increasing 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.

Tsallis Log-Scale-Location Models. Moments, Gini Index and Some Stochastic Orders

In this article we give theoretical results for different stochastic orders of a log-scale-location family which uses Tsallis statistics functions. These results describe the inequalities of moments or Gini index according to parameters. We also compute the mean in the case of q-Weibull and q-Gaussian distributions. The paper is aimed at analyzing the order between survival functions, Lorenz curves and (as consequences) the moments together with the Gini index (respectively a generalized Gini index). A real data application is presented in the last section. This application uses only the survival function because the stochastic order implies the order of moments. Given some supplementary conditions, we prove that the stochastic order implies the Lorenz order in the log-scale-location model and this implies the order between Gini coefficients. The application uses the estimated parameters of a Pareto distribution computed from a real data set in a log-scale-location model, by specifying the Kolmogorov–Smirnov p-value. The examples presented in this application highlight the stochastic order between four models in several cases using survival functions. As direct consequences, we highlight the inequalities between the moments and the generalized Gini coefficients by using the stochastic order and the Lorenz order.

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.

Study of Proton, Deuteron, and Triton at 54.4 GeV

Transverse momentum spectra of proton, deuteron, and triton in gold-gold (Au-Au) collisions at 54.4 GeV are analyzed in different centrality bins by the blast wave model with Tsallis statistics. The model results are approximately in agreement with the experimental data measured by STAR Collaboration in special transverse momentum ranges. We extracted the kinetic freeze-out temperature, transverse flow velocity, and freeze-out volume from the transverse momentum spectra of the particles. It is observed that the kinetic freeze-out temperature is increasing from the central to peripheral collisions. However, the transverse flow velocity and freeze-out volume decrease from the central to peripheral collisions. The present work reveals the mass dependent kinetic freeze-out scenario and volume differential freeze-out scenario in collisions at STAR Collaboration. In addition, parameter q characterizes the degree of nonequilibrium of the produced system, and it increases from the central to peripheral collisions and increases with mass .

Two approaches that prove divergence free nature of non-local gravity

This paper is an attempt to study the thermodynamics of the structure formation in the large scale universe in the non local gravity using Boltzmann statistics and the Tsallis statistics. The partition function is obtained in both the approaches and the corresponding thermodynamics properties are evaluated. The important thing about the paper is that we surprisingly get the divergence free integrals and thus stress upon the fact that the nonlocal gravity is the singularity free model of gravity.

Tsallis Statistics in High Energy Physics: Chemical and Thermal Freeze-Outs

We present an overview of a proposal in relativistic proton-proton (pp) collisions emphasizing the thermal or kinetic freeze-out stage in the framework of the Tsallis distribution. In this paper we take into account the chemical potential present in the Tsallis distribution by following a two step procedure. In the first step we used the redudancy present in the variables such as the system temperature, T, volume, V, Tsallis exponent, q, chemical potential, μ, and performed all fits by effectively setting to zero the chemical potential. In the second step the value q is kept fixed at the value determined in the first step. This way the complete set of variables T,q,V and μ can be determined. The final results show a weak energy dependence in pp collisions at the centre-of-mass energy s=20 TeV to 13 TeV. The chemical potential μ at kinetic freeze-out shows an increase with beam energy. This simplifies the description of the thermal freeze-out stage in pp collisions as the values of T and of the freeze-out radius R remain constant to a good approximation over a wide range of beam energies.

Nuclear modification factor in Pb+Pb collisions at using Boltzmann’s Transport Equation with Tsallis statistics

In this work the transverse momentum spectra and nuclear modification factor for heavy ion collisions are derived using Tsalllis non-extensive statistics in relaxation time approximation. The Boltzmann-Gibbs Blast Wave (BGBW) function is used as the equilibrium function and Tsallis function is used as the initial distribution function while solving the Boltzmann transport equation in the relaxation time approximation. The framework is used to analyse the experimental data for particles like pions, kaons, protons, D 0 meson and J/Ψ produced in Pb+Pb collisions at at the LHC, CERN. We find that our proposed equation of state describes the experimental data successfully.

Extensive and nonextensive statistics in seismic inversion

Seismic inversion is a central procedure for estimating subsurface physical parameters from observed data. In geophysical applications, the seismic inversion is usually formulated as an optimisation problem that aims to minimise the difference between modelled and observed data through the Gauss’ error law, which is linked to the extensive Boltzmann–Gibbs (BG) statistics. However, this approach is known to be sensitive to non-Gaussian errors, especially to outliers in the data set. Therefore, error laws determined by non-Gaussian statistics are essential for robust seismic inversion. In this way, we present a comparative study of seismic inversions using the extensive statistics of Rényi and also on the nonextensive statistics of Tsallis and Kaniadakis. In particular, we consider a classical seismic inversion problem so-called Post-Stack Inversion (PSI), which analyses the interaction between seismic waves and subsurface reflectivity to infer geological structures. Considering a realistic subsurface reflectivity model taking into account spike-noisy data, the numerical results show that the PSI based on generalised statistics outperforms the PSI based on standard BG statistics. We note that, the best results are for the Tsallis case in which the Lévy–Gnedenko central-limit theorem is valid (q>53).

Fluctuation theorems in [formula omitted]-canonical ensembles

Generalizations of statistical mechanics such as Tsallis statistics and superstatistics, among others, describe the occurrence of power-law distributions in long-range interacting systems and complex systems in diverse areas, particularly the so-called q-exponential family of distributions. In this work we present the use of fluctuation theorems for q-canonical ensembles as a powerful tool to readily obtain statistical properties. In particular, we have obtained strong conditions for the possible values of q depending on the density of states of the system.