Nonextensive Statistics Research Articles

Extension of Yager's negation of a probability distribution based on Tsallis entropy

The negation of probability distribution becomes an important topic since some problems are burdensome to deal with directly. Inspired by Yager's negation of probability distribution, an extension model to measure the negation of a probability distribution is proposed using the idea of a nonextensive statistic based on Tsallis entropy. Proofs show that the proposed extension of negation of probability distribution converges to the maximum Tsallis entropy. The proposed model may extend Yager's method to consider the influences of the correlations in a system, which gives the different convergent routes. Some numerical simulation results are used to illustrate the effectiveness of the proposed methodology.

The nonextensive Bose-Einstein condensation and photon gas with parameter transformation

The nonextensive boson system is revisited with the statistical ensemble theory, where the fundamental grand canonical distribution is derived from the equiprobability principle. By recourse to the approach of parameter transformation, one familiar and concise statistical formula for bosons is deduced, in an accurate fashion. Then the Bose-Einstein condensation phenomenon is re-discussed. The results show that the critical temperature is dependent on the nonextensive parameter $ \nu$ , and in the generalized expression of the heat capacity for the condensated phase of boson systems there exists one additional term, obeying the T3 law. By use of the statistical formula, the nonextensive photon gas is also researched. The internal energy and heat capacity for the nonextensive photon gas exhibit a similar dependence on the temperature to the classical photon gas, apart from a coefficient correction dependent on the parameter $ \nu$ . The Gibbs function for the nonextensive photon gas is still zero, showing that the photon field is also at thermal equilibrium, like the situation of classical photon field. The entropy of photon field can be calculated through the integral of the nonextensive quantum statistics formula and can also be derived from the original definition of Tsallis entropy, by recourse to the direct parameter transformation. This seems to indicate the validity of the treatment technique for the nonextensive quantum systems.

Analysis on hadron spectra in heavy-ion collisions with a new non-extensive approach

The transverse momentum spectra of identified charged hadrons stemming from high energy collisions at different beam energies are described by a new non-extensive distribution, the Kaniadakis κ-distribution, with respect to the constraints in non-extensive quantum statistics. All fittings are also compared with the Tsallis distributions as well as the usual Boltzmann–Gibbs one. χ2/ndf is also used to test the fitting goodness of all functions. Our results show that these different non-extensive approaches can be well applied in high energy collisions rather than the classical one. The Kaniadakis statistics are typically better applied into such systems with both positive and negative particles considered. This provides an alternative non-extensive view to study high energy physics. Analysis on the fitting parameters are present as well. The similar relationships of all functions remind us of the further understanding of the non-extensivity.

Violation of Wiedemann-Franz Law for hot hadronic matter created at NICA, FAIR and RHIC energies using non-extensive statistics

We present here the computation of electrical and thermal conductivity by solving the Boltzmann transport equation in relaxation time approximation. We use the $q$-generalized Boltzmann distribution function to incorporate the effects of non-extensivity. The behaviour of these quantities with changing temperature and baryochemical potential has been studied as the system slowly moves towards thermodynamic equilibrium. We have estimated the Lorenz number at NICA, FAIR and the top RHIC energies and studied as a function of temperature, baryochemical potential and the non-extensive parameter, $q$. We have observed that Wiedemann-Franz law is violated for a non-extensive hadronic phase as well as for an equilibrated hadron gas at high temperatures.

Emergence of Tsallis statistics as a consequence of invariance

For non-equilibrium systems in a steady state we present two necessary and sufficient conditions for the emergence of q-canonical ensembles, also known as Tsallis statistics. These conditions are invariance requirements over the definition of subsystem and environment, and over the joint rescaling of temperature and energy. Our approach is complementary to the notions of Tsallis non-extensive statistics and Superstatistics.

Tsallis non-extensive statistics and multifractal analysis of the dynamics of a fully-depleted MOSFET nano-device

In this study Tsallis non-extensive statistics and multifractal analysis for the physical description and understanding of the Random Telegraph Noise (RTN), observed in UTBB FD-SOI MOSFET current timeseries is presented. Specifically, we estimate the Tsallis q-triplet and q-entropy production, the Hurst exponent, the multifractal singular spectrum and other topological and dynamical characteristics of this MOSFET-current dynamics as a function of the drain voltage. The analysis shows the existence of microscopic intermittent turbulence and anomalous diffusion processes, underlying the noisy timeseries. The relevant results indicate the existence of a self-organized critical behavior manifested by a percolation mechanism and a fractional transport process. The multifractal character of the dynamics was demonstrated and explained by the maximization of the Tsallis q-entropy function. Final, our analysis revealed the existence of a non-equilibrium topological transition process, taking place when the drain voltage increases and satisfies the basic principles of Tsallis non-extensive statistical theory.

Big Bang Nucleosynthesis and the Lithium Problem

I briefly describe the cosmological lithium problems followed by a summary of our recent theoretical work on the magnitude of the effects of electron screening, the possible existence of dark matter parallel universes and the use of non-extensive (Tsallis) statistics during big bang nucleosynthesis. Solutions within nuclear physics are also discussed and recent measurements of cross-sections based on indirect experimental techniques are summarized.

The nonextensive statistical ensembles with dual thermodynamic interpretations

The nonextensive statistical ensembles are revisited for the complex systems with long-range interactions and long-range correlations. An approximation, the value of nonextensive parameter (1-q) is assumed to be very tiny, is adopted for the limit of large particle number for most normal systems. In this case, Tsallis entropy can be expanded as a function of energy and particle number fluctuation, and thus the power-law forms of the generalized Gibbs distribution and grand canonical distribution can be derived. These new distribution functions can be applied to derive the free energy and grand thermodynamic potential in nonextensive thermodynamics. In order to establish appropriate nonextensive thermodynamic formalism, the dual thermodynamic interpretations are necessary for thermodynamic relations and thermodynamic quantities. By using a new technique of parameter transformation, the single-particle distribution can be deduced from the power-law Gibbs distribution. This technique produces a link between the statistical ensemble and the quasi-independent system with two kinds of nonextensive parameter having quite different physical explanations. Furthermore, the technique is used to construct nonextensive quantum statistics and effectively to avoid the factorization difficulty in the power-law grand canonical distribution.

Parallel Maxwellian relations and their correlations in nonextensive thermodynamics

We presented a prescript in previous works to solve the issue of the definition of temperature in thermodynamics on the basis of the dual interpretation of temperature concept. A nonextensive thermodynamic formalism consisting of two sets of parallel Legendre transformation structures, namely, physical and Lagrange sets, were proposed by providing further dual interpretations of thermodynamic quantities. In this study, we reconstruct Maxwellian and other thermodynamic relations on the basis of these two parallel formalisms. The correlations between the thermodynamic relations are determined, where an equivalent rule is suggested, by recoursing to the link of the Tsallis factor. This rule shows that the Tsallis factor is invariable with a constant volume or entropy in partial derivative calculations. These correlations could be easily tested in nonextensive statistics, whose applications into adiabatic expansions of nonextensive gas produce interesting results. Lagrange internal energy and generalized heat capacity are negative in some small nonextensive systems due to the enhanced nonextensive effect. The negative definite of generalized heat capacity would not lead to instability, as physical heat capacity is the quantity used to govern the thermodynamic process of nonextensive systems.

A Description of Transverse Momentum Distributions in p+p Collisions at RHIC and LHC Energies

It has long been debated whether the hydrodynamics is suitable for the smaller colliding systems such as p+p collisions. In this paper, by assuming the existence of longitudinal collective motion and long-range interactions in the hot and dense matter created in p+p collisions, the relativistic hydrodynamics incorporating with the nonextensive statistics is used to analyze the transverse momentum distributions of the particles. The investigations of the present paper show that the hybrid model can give a good description of the currently available experimental data obtained in p+p collisions at RHIC and LHC energies, except for p and p¯ produced in the range of pT>3.0 GeV/c at s=200 GeV.

The effect of magnetic field on the viscosity in the weakly ionized and magnetized plasma with power-law q-distributions in nonextensive statistics

We study the effect of magnetic field on the viscosity of charged particles in the weakly ionized and magnetized plasma with the power-law q-distributions by using the generalized Boltzmann equation of transport in nonextensive statistics. We derive six viscosity coefficients of electrons and ions in the complex plasma, including the magnetic field introduced four viscosity coefficients, where the q-parameter and magnetic field both play significant roles. By numerical analyses, we show that these viscosity coefficients depend strongly on the magnetic field and q-parameters of the weakly ionized and magnetized plasma, and so have potential applications in low temperature complex plasmas.

Nonperturbative thermodynamic geometry of nonextensive statistics

Following our earlier work on the perturbative thermodynamic geometry of nonextensive quantum and classical gases [H. Mohammadzadeh, F. Adli and S. Nouri, Phys. Rev. E 94 (2016) 062118], we study [Formula: see text]-generalized Bose–Einstein, Fermi–Dirac and classical statistics nonperturbatively. We define [Formula: see text]-generalized polylogarithm functions and evaluate thermodynamics quantities such as internal energy and particle number. We construct the thermodynamic geometry of nonextensive Bose (Fermi) ideal gas and show that the thermodynamic curvature is positive(negative) in full physical range as the same as ordinary statistics. Also, we show that the thermodynamic geometry of nonextensive ideal classical gas is flat, similar to the ordinary one. Therefore, the nonextensive parameter does not change the nature of intrinsic statistical interactions. We argue that the nonextensive boson gas might be more stable than the boson gas due to conjectural interpretation of thermodynamic curvature. In the following, we extract the singular points of thermodynamic curvature of nonextensive Bose gas and relate it to the condensation. We evaluate some thermodynamic quantities such as heat capacities, compressibility and [Formula: see text]-dependent phase transition temperature. We show that the heat capacity is not differentiable at critical temperature, [Formula: see text] which is reduced by increasing nonextensive parameter [Formula: see text]. Moreover, the critical temperature and possibility of condensation is investigated for different values of nonextensive parameter in various dimensions.

A q-Extension of Sigmoid Functions and the Application for Enhancement of Ultrasound Images.

This paper proposes the q-sigmoid functions, which are variations of the sigmoid expressions and an analysis of their application to the process of enhancing regions of interest in digital images. These new functions are based on the non-extensive Tsallis statistics, arising in the field of statistical mechanics through the use of q-exponential functions. The potential of q-sigmoids for image processing is demonstrated in tasks of region enhancement in ultrasound images which are highly affected by speckle noise. Before demonstrating the results in real images, we study the asymptotic behavior of these functions and the effect of the obtained expressions when processing synthetic images. In both experiments, the q-sigmoids overcame the original sigmoid functions, as well as two other well-known methods for the enhancement of regions of interest: slicing and histogram equalization. These results show that q-sigmoids can be used as a preprocessing step in pipelines including segmentation as demonstrated for the Otsu algorithm and deep learning approaches for further feature extractions and analyses.

Chiral phase transition within the linear sigma model in the Tsallis nonextensive statistics based on density operator

We studied the chiral phase transition for small [Formula: see text] within the Tsallis nonextensive statistics of the entropic parameter [Formula: see text], where the quantity [Formula: see text] is the measure of the deviation from the Boltzmann–Gibbs statistics. We adopted the normalized [Formula: see text]-expectation value in this study. We applied the free particle approximation and the massless approximation in the calculations of the expectation values. We estimated the critical physical temperature, and obtained the chiral condensate, the sigma mass, and the pion mass, as functions of the physical temperature [Formula: see text] for various [Formula: see text]. We found the following facts. The [Formula: see text]-dependence of the critical physical temperature is [Formula: see text]. The chiral condensate at [Formula: see text] is smaller than that at [Formula: see text] for [Formula: see text]. The [Formula: see text]-dependence of the pion mass and that of the sigma mass reflect the [Formula: see text]-dependence of the condensate. The pion mass at [Formula: see text] is heavier than that at [Formula: see text] for [Formula: see text]. The sigma mass at [Formula: see text] is heavier than that at [Formula: see text] for [Formula: see text] at high physical temperature, while the sigma mass at [Formula: see text] is lighter than that at [Formula: see text] for [Formula: see text] at low physical temperature. The quantities which are functions of the physical temperature [Formula: see text] and the entropic parameter [Formula: see text] are described by only the effective physical temperature defined as [Formula: see text] under the approximations.

Nonextensive triplets in stock market indices

Stock market indices are one of the most investigated complex systems in econophysics. Here we extend the existing literature on stock markets in connection with nonextensive statistical mechanics. We explore the nonextensivity of price volatilities for 34 major stock market indices between 2010 and 2019. We discover that stock markets follow nonextensive statistics regarding equilibrium, relaxation and sensitivity. We find nonextensive behavior in stock markets for developed countries, but not for developing countries. Distances between nonextensive triplets suggest that some stock markets might share similar nonextensive dynamics, while others are widely different. The current findings strongly indicate that the stock market represents a system whose physics is properly described by nonextensive statistical mechanics. Our results shed light on the complex nature of stock market indices, and establish another formal link with the nonextensive theory.

A strategy based on non-extensive statistics to improve frame-matching algorithms under large viewpoint changes

In recent decades, methods to find invariant points in digital images, called fiducial points, have gained great attention, mainly due to the demands of several applications in computer vision and image processing, such as the geometric matching of global structures, objects or specific regions. Among the most well known approaches are algorithms like SIFT, HOG, SURF, and their variations as A-SIFT, PCA-SIFT, surrounded by many others. Though the number of researches demonstrating the efficiency of such methods is undoubtedly enormous, the vast majority compares their performances only on pairs of images with little changes in view perspectives, objects or specific regions of the scenes. Thereby, the study of this type of technique under large viewpoint changes, called here LVC, has received little attention from the researchers. On the other hand, with the aim of filtering points of interest, most techniques have used the traditional extensive statistics. However, methods for image processing based on a new type of statistics, called non-extensive statistics, have shown to be efficient in several applications. In this paper, we present a new method, called q-SIFT, based on the non-extensive Tsallis statistics, to find fiducial points in a sequence of frames of videos under large viewpoints changes. We experimentally show the efficiency of the proposed method in video databases and propose new measurement metrics for this type of algorithm.

The current density and transport coefficients in the fully ionized plasma with q-distributions in nonextensive statistics

We study the current density and transport coefficients in the fully ionized plasma with the q-distributions in nonextensive statistics and in strong magnetic field. By using the generalized Boltzmann transport equation in nonextensive statistics, we derive the current density and the expressions of the transport coefficients, including the conductivity, the thermoelectric coefficient, the Hall coefficient, and the Nernst coefficient. It is shown that these new transport coefficients have been generalized to the nonequilibrium complex plasmas with q-distributions in nonextensive statistics, which depend strongly on the q-parameters and when we take the limit q→1, they perfectly return to those for the plasma based on a Maxwellian distribution.

A Generalized Vitali Set from Nonextensive Statistics

We address a generalization of the Vitali set through a deformed translational property that stems from a generalized algebra derived from the nonextensive statistics. The generalization is based on the so-called $q$-addition $x\oplus_{q} y=x+y+(1-q)xy$ for rational values of $q$, where the ordinary formalism is recovered when the control parameter $q \to 1$. The generalized Vitali set is non-measurable for all rational parameter $\frac{1}{2}<q\leq1$, but in the limit $q\rightarrow\frac{1}{2}$ the non-measurability cannot be guaranteed. Furthermore, assuming measurability when $q\rightarrow\frac{1}{2}$, then this must be positive.

Non-extensive statistics in free-electron metals and thermal effective mass

We have applied the non-extensive statistical mechanics to free electrons in several metals to calculate the electronic specific heat at low temperature. In this case, the Fermi–Dirac (FD) function is modified from its Boltzmann–Gibbs (BG) form, with the exponential part going to a q-exponential, in its non-extensive form. In most cases, the non-extensive parameter, q, is found to be greater than unity to produce the correct thermal effective mass, m∗, of electrons. The ratio m∗∕m is found to show a nice systematic dependence on q. Results indicate, electrons in metals, in the presence of long range correlations are reasonably well described by Tsallis statistics.

Condensation of nonextensive ideal Bose gas and critical exponents

We investigate the condensation of a three dimensional nonextensive ideal Bose gas. We use the distribution function of nonextensive Bose statistics and define the q-generalized polylogarithm functions and obtain the internal energy, the particle number and some useful thermodynamics response functions for q≥1. We show that the discontinuity of the slop of heat capacity with respect to the temperature is reduced by increasing the nonextensive parameter. Therefore, approximately for q>1.4 values of nonextensive parameter, the condensation does not occur. Also, we calculated the specific heat at constant pressure, the isothermal compressibility and the thermodynamic curvature, that behave as |T−Tcq|−ρ in which Tcq denotes the condensation temperature and ρ is the critical exponent. We show that the determined critical exponents are independent of nonextensive parameter. So, the scaling properties near the transition temperature indicates that the universality class of nonextensive systems is the same as extensive ones.