Nonextensive Entropy Research Articles

Non-extensive entropy and power-law inflation: Implications for observations

This study explores the interaction between non-extensive entropic FLRW cosmology and the power-law inflationary model, with a focus on the overlap between the scalar spectral index “[Formula: see text]” and the tensor-to-scalar ratio “r”. Based on a conjecture that non-extensive entropy alters the energy–momentum content of the cosmic fluid, the analysis examines how these overlaps shift with different model parameters and compares the findings to those from Bekenstein–Hawking (BH) entropic cosmology. The study highlights the impact of Tsallis, Rényi, and Sharma–Mittal entropies, uncovering a significant correlation between “[Formula: see text]” and “r” that suggests a deeper connection in power-law inflationary dynamics. The results demonstrate that non-extensive entropies not only enable viable inflation with a graceful exit but also address limitations inherent in the standard BH entropic framework, emphasizing the importance of precise parameter estimation. Specifically, Tsallis entropy allows for power-law inflation with [Formula: see text] to [Formula: see text] in alignment with Planck 2018 data. Moreover, the [Formula: see text] parameter in Rényi and Sharma–Mittal entropy models must be extremely small ([Formula: see text] in Planck mass units) to achieve successful power-law inflation with an e-folding number around 55–65, suggesting a unified thermodynamic perspective in cosmological studies.

Look Beyond Additivity and Extensivity of Entropy for Black Hole and Cosmological Horizons.

We present a comparative analysis of the plethora of nonextensive and/or nonadditive entropies which go beyond the standard Boltzmann-Gibbs formulation. After defining the basic notions of additivity, extensivity, and composability, we discuss the properties of these entropies and their mutual relations, if they exist. The results are presented in two informative tables that are of strong interest to the gravity and cosmology community in the context of the recently intensively explored horizon entropies for black hole and cosmological models. Gravitational systems admit long-range interactions, which usually lead to a break of the standard additivity rule for thermodynamic systems composed of subsystems in Boltzmann-Gibbs thermodynamics. The features of additivity, extensivity, and composability are listed systematically. A brief discussion on the validity of the notion of equilibrium temperature for nonextensive systems is also presented.

Multiplicity dependence of the pT -spectra for identified particles and its relationship with partonic entropy

We investigate the multiplicity dependence of the transverse momentum pT spectra of hadrons produced in high-energy collisions. We propose that the partonic distribution be parametrized by its nonextensive entropy and the parton saturation scale Qs(x). These two variables can be identified from the produced charged hadron distributions and provide important information on the gluon dynamics at the moment of interaction. From this perspective, we interpret data from different ALICE multiplicity classes at s=13 TeV and s=5.02 TeV. A multiplicity-dependent scaling function is presented and the dependence of the interaction area on multiplicity is also investigated. Published by the American Physical Society 2024

On the foundations of the maximum entropy principle using Fenchel duality for Shannon and Tsallis entropies

In this work, we address two main objectives. The first one is to provide a rigorous foundation to the maximum entropy principle in statistical physics, by making use of the Fenchel-Rockafellar duality. The second objective is to discuss the well-foundedness of the so-called escort distributions in the context of non-extensive entropy maximization. The duality treatment of maximum entropy confirms the non-rigorous results obtained via the usual variational calculus, however, the use of escort distributions yields undefined behavior when used consistently, and only leads to the desired results when used in an ad-hoc manner.

New solutions for the generalized q-deformed wave equation with q-translation symmetry

Abstract In this work, we explore the generalized discrete wave equation, which utilizes a specific irregular space interval. The introduction of this irregular space interval is motivated by its connection to the q-addition, a mathematical operation that arises in the nonextensive entropy theory. By taking the continuous limit, we obtain the wave equation with q-deformation, which captures the effects of the q-addition. To solve the generalized q-deformed wave equation, we investigate three different methods: the separation method, the reduced differential transform method, and the finite difference method. These methods offer distinct approaches for finding solutions to the equation. By comparing the results obtained from each method, we can evaluate their effectiveness and identify their respective strengths and limitations in solving the generalized q-deformed wave equation. The solutions obtained from this newly defined equation have potential applications in modeling physical systems with violated symmetries. The inclusion of the q-deformation allows for a more comprehensive description of such systems, which may exhibit nonextensive behavior or possess irregularities in their spatial intervals. By incorporating these features into the wave equation, we can improve our understanding and modeling capabilities of complex physical phenomena.

Two-parameter deformed quantum mechanics based on Fibonacci calculus and Debye crystal model of two-parameter deformed quantum statistics

Starting on the basis of Fibonacci calculus and Fibonacci oscillator algebra, we introduce the main properties to develop a new formalism for the two-parameter (q1,q2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({q}_{1},{q}_{2})$$\\end{document}-deformed quantum mechanics, where q1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${q}_{1}$$\\end{document} and q2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${q}_{2}$$\\end{document} are real positive independent deformation parameters. As applications of such a two-parameter deformed formalism, we investigate the behavior of a quantum particle in some different physical phenomena covering the free particle and the inverse-harmonic potential case. The effect of two deformation parameters on the wave functions for these applications is studied. Another application is carried out onto the quantum statistics of lattice oscillations through a model of the (q1,q2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({q}_{1},{q}_{2})$$\\end{document}-deformed phonon gas, and it is shown that the high- and low-temperature behavior of the model specific heat differs notably from the classical theories for the interval 0<(q1,q2)<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<({q}_{1},{q}_{2})<\\infty $$\\end{document}. We also construct a two-parameter deformed non-extensive entropy based on some elements of the Fibonacci calculus and discuss its possible connection with the Tsallis entropy in non-extensive statistical mechanics. Finally, other possible application areas of the present two-parameter (q1,q2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({q}_{1},{q}_{2})$$\\end{document}-deformed construction on quantum mechanics are discussed.

Nonextensive entropies impact onto thermodynamics and phase structure of Kerr–Newman black holes

Taking the nonextensive Tsallis and Rényi entropies into account, we explore thermodynamic properties and phase transitions of the Kerr–Newman black holes (KNBH) in the microcanonical and canonical ensembles. We also compare our results with those obtained by attributing the Bekenstein–Hawking entropy bound to the mentioned black holes. Our analysis indicates that, similarly to the standard Boltzmann picture, isolated KNBH in the microcanonical approach are stable against axisymmetric perturbations in both Tsallis and Rényi models. On the other hand, in considering the case when the black holes are enveloped by a bath of thermal radiation in the canonical treatment, the KNBH based on the Tsallis and Rényi entropies can be stable for some values of the entropy parameters, in contrast to the traditional Boltzmann framework. For the case of Rényi entropy, we find that a Hawking-Page transition and a first order small black hole/large black hole transition can occur in a similar fashion as in rotating black holes in an anti-de Sitter space. Finally, we employ the Ruppeneir geometrothermodynamic technique to provide a new perspective on studying the nature of interactions between black hole microstructures, revealing a non-trivial impact of nonextensive entropies.

Dynamic convolution layer based optimization techniques for object classification and semantic segmentation

&lt;p&gt;Providing meaningful classification for each pixel in an image is a primary goal of computer vision, and the tasks of object classification and semantic segmentation are among the field’s greatest challenges. To improve object classification, this study presents a novel method that combines semantic segmentation with dynamic convolution layer-based optimization techniques. In the proposed method, a Refined Convolution Neural Network (R-CNN) is used, which uses non-extensive entropy to dynamically increase the size of its convolutional layers. The Common Objects in Context (COCO) dataset is used to assess the performance of the model. The model performs exceptionally well at different Intersections over Union (IoU) cutoffs, with average precision values of 40.1, 61.9, and 45.4, respectively, for Average Precision (AP), AP&lt;sub&gt;50&lt;/sub&gt;, and AP&lt;sub&gt;75&lt;/sub&gt;. These results demonstrate the model’s efficiency in discriminating between various image contents. Additionally, the model predicts an image’s outcome on average in just 0.901 s. The model has been proven to be superior through various performance evaluation parameters, showing an average mean precision of 91.78%. This study demonstrates the power of combining dynamic convolution layers with semantic segmentation to improve object classification accuracy, a key component in the development of computer vision applications.&lt;/p&gt;

Extensive entropy: the case of zero entropy defect

This paper shows that the Rényi and Boltzmann-Gibbs (BG) extensive entropies share the same functional relationship with the nonextensive entropy associated with kappa distributions, which coincides with the well-known Havrda/Charvát/Daróczy/Tsallis (HCDT) entropy. We find that while the relationship between kappa/HCDT and Rényi entropies is merely a mathematical identity between their entropic statistical definitions, the relationship between kappa/HCDT and BG entropies is based on their thermodynamic connection. The latter connects the entropy between a system characterized by correlations among their all constituents (kappa/HCDT entropy) and the entropy of the same system but with no correlations among their constituents (BG entropy). The origin of this relationship, and its connection with thermodynamics, is examined using the concept of entropy defect, that is, the decrease in a system’s entropy caused by the presence of long-range correlations among its constituents; in the limiting case of zero correlations, the entropy defect vanishes and the entropy becomes extensive and expressed by the BG formulation.

Modified cosmology from quantum deformed entropy

Jalalzadeh (2022), established that the thermodynamical entropy of a quantum-deformed black hole with horizon area A can be written as Sq=πsinA8GN/sinπ2N, where N=Lq2/LP2, LP being the Planck length and Lq denoting, generically, the q-deformed cosmic event horizon distance Lq. Motivated by this, we now extend the framework constructed in Jalalzadeh (2022) towards the Friedmann and Raychaudhuri equations describing spatially homogeneous and isotropic universe dynamics. Our procedure in this paper involves a twofold assumption. On the one hand, we take the entropy associated with the apparent horizon of the Robertson–Walker universe in the form of the aforementioned expression. On the other hand, we assume that the unified first law of thermodynamics, dE=TdS+WdV, holds on the apparent horizon. Subsequently, we find a novel modified cosmological scenario characterized by quantum-deformed (q-deformed) Friedmann and Raychaudhuri equations containing additional components that generate an effective dark energy sector. Our results indicate an effective dark energy component, which can explain the Universe’s late-time acceleration. Moreover, the Universe follows the standard thermal history, with a transition redshift from deceleration to acceleration at ztran=0.5. More precisely, according to our model, at a redshift of z=0.377, the effective dark energy dominates with a de Sitter universe in the long run. We include the evolution of luminosity distance, μ, the Hubble parameter, H(z), and the deceleration parameter, q(z), versus redshift. Finally, we have conducted a comparative analysis of our proposed model with others involving non-extensive entropies.

Nonextensive realizations in interacting ion channels: Implications for mechano-electrical transducer mechanisms

We propose a theoretical model to investigate the thermodynamics of single and coupled two-state ion channels, associated with mechanoelectrical transduction (MET) and hair cell biophysics. The modeling was based on the Tsallis nonextensive statistical mechanics. The choice for a nonextensive framework in modeling ion channels is encouraged on the fact that we take into account the presence of interactions or long-range correlations in the dynamics of single and coupled ion channels. However, the basic assumptions that support Boltzmann–Gibbs statistics, traditionally used to model ion channel dynamics, state that the system is formed by independent or weakly interacting elements. Despite being well studied in many biological systems, the literature has not yet addressed the study of both entropy and mutual information related to isolated or physically interacting pairs of MET channels. Inspired by hair cell biophysics, we show how the presence of nonextensivity, or subadditivity and superadditivity modulates the nonextensive entropy and mutual information as functions of stereocilia displacements. We also observe that the magnitude of the interaction between the two channels, given by a nonextensive parameter, influences the amplitude of the nonextensive joint entropy and mutual information as functions of the hair cell displacements. Finally, we show how nonextensivity regulates the current versus displacement curve for a single and a pair of interacting two-state channels. The present findings shed light on the thermodynamic process involved in the molecular mechanisms of the auditory system.

Nonequilibrium distributions from the Fokker-Planck equation: Kappa distributions and Tsallis entropy.

Nonequilibrium systems in chemistry and physics are generally modeled with the Boltzmann, Fokker-Planck, and Master equations. There has been a considerable interest in the nonequilibrium distributions of electrons and ions in space physics in different environments as well as in other systems. An often-used empirical model to characterize these distributions, especially in space physics, is the Kappa distribution. There have been numerous efforts to provide a theoretical basis for the Kappa distribution that include the Fokker-Planck equationwith specific drift and diffusion coefficients. Alternatively, the maximization of the Tsallis nonextensive entropy provides the desired Kappa distribution. This paper examines three families of Fokker-Planck equationsthat provide a steady-state Kappa distribution as well as a myriad of other nonequilibrium distributions. The relationship of these works with analogous studies of distributions with asymptotic high-energy tails is also considered. It is clear that the many different nonequilibrium distribution functions that can occur cannot all be rationalized with Gibbs-Boltzmann statistical mechanics, which uniquely gives equilibrium distributions, or with the Tsallis nonextensive entropy, which gives uniquely the Kappa distribution. The current research is directed towards an improved understanding of the origin of nonequilibrium distributions in several specific systems.

Generalized uncertainty principle impact on nonextensive black hole thermodynamics

The effect of the generalized uncertainty principle (GUP) on nonextensive thermodynamics applied to black holes, as well as the sparsity of the radiation at different temperatures associated with each nonextensive entropy, is investigated. We examine the Rényi, Tsallis–Cirto, Kaniadakis, Sharma Mittal, and Barrow entropies, temperatures, and heat capacities and show that, in each case, due to GUP corrections, the temperature and entropy have finite values, implying that the final state of the black hole is a remnant at the end of the evaporation process and that the sparsity of the radiation for massless bosons at each temperature depends on the mass of the black hole. We also find that GUP reduces the value of the sparsity profile for each case as compared to the sparsity parameter at Hawking temperature, which is always constant throughout the evaporation.

Image thresholding through nonextensive entropies and long-range correlation

Image thresholding through nonextensive entropies and long-range correlation

Time evolution of electron distributions to bimodal steady states for electrons dilutely dispersed in theinert gases Ar, Kr, and Xe with deep Ramsauer Townsend minima in themomentum transfer cross section

The current paper considers the thermalization of an ensemble of electrons under the influence of an external electric field and dilutely dispersed in one of the inert gas moderators, Argon, Krypton or Xenon for which the electron momentum transfer cross sections have deep Ramsauer-Townsend minima. As a consequence, the steady state electron distribution functions are bimodal over a small range of external electric field strengths. The current work is directed towards the time evolution of the electron distribution function determined from the numerical solution of the Fokker-Planck equation. The kinetic theory of electrons dilutely dispersed in a heat bath of atoms at temperature T b has a very long history. The solution of the Fokker-Planck equation can be expressed as a sum of exponentials of the form where λ n are the eigenvalues of the Fokker-Planck operator. Alternatively, a finite difference algorithm is used to solve the time dependent Fokker-Planck equation to give the time dependent electron energy distribution function. We demonstrate the evolution of the initial Maxwellian into a nonequilibrium bimodal distribution which cannot be rationalized with either the Gibbs-Boltzmann entropy or the Tsallis nonextensive entropy. Instead, the time dependent approach of an initial Maxwellian to the bimodal distribution is described in terms of the Kullback-Leibler entropy. We also demonstrate the inapplicability of the Boltzmann entropy nor the Tsallis entropy for a model system with a power law momentum transfer cross section of the form, σ(x) = σ 0/x p , where is the reduced speed. This model with p = 2 is also employed to demonstrate a steady-state Kappa distribution which features prominently in space physics and other fields. For p > 2, we show distribution functions that increase without bound analogous to runaway electrons. The steady nonequilibrium distributions are interpreted as solutions of a Pearson ordinary differential equation.

Some Non-Obvious Consequences of Non-Extensiveness of Entropy

Non-additive (or non-extensive) entropies have long been intensively studied and used in various fields of scientific research. This was due to the desire to describe the commonly observed quasi-power rather than the exponential nature of various distributions of the variables of interest when considered in the full available space of their variability. In this work we will concentrate on the example of high energy multiparticle production processes and will limit ourselves to only one form of non-extensive entropy, namely the Tsallis entropy. We will discuss some points not yet fully clarified and present some non-obvious consequences of non-extensiveness of entropy when applied to production processes.

Unified formalism for the law of emergence from the first law of thermodynamics

We derive a unified expansion law for our Universe from the first law of thermodynamics on the apparent horizon, where entropic evolution depicts the emergence of cosmic space. The derivation advances a general form for degrees of freedom on the surface and bulk, which provides a natural generalization for the expansion law proposed by Padmanabhan. The general expression for the surface degrees of freedom differs from the natural expectation, ${N}_{\mathrm{sur}}=4S$ in the emergent gravity paradigm for general theories of gravity. The derivation also provides justification for the selection of Gibbons-Hawking temperature in the original expansion law and for the use of areal volume in the nonflat Friedmann-Robertson-Walker universe. Since the unified expansion law exclusively depends on the form of entropy, the method is applicable to obtain the expansion law in any gravity theory without any additional ad hoc assumptions. From the general expansion law, we have obtained the expansion law corresponding to different theories of gravity like ($n+1$) Einstein, Gauss-Bonnet, Lovelock, and Ho\ifmmode \check{r}\else \v{r}\fi{}ava-Lifshitz. We also obtained the expansion law for nonextensive entropy like Tsallis entropy from the unified expansion law.

Equilibrium temperature for black holes with nonextensive entropy

Hawking temperature has been widely utilized in the literature as the temperature that corresponds to various nonextensive entropies. In this study, we analyze the compatibility of the Hawking temperature with the nonextensive entropies. We demonstrate that, for every nonextensive entropy, one may define an effective temperature (which we call equilibrium temperature) by utilizing the equilibrium condition, and that there is always an additive equilibrium entropy associated with this effective temperature. Except for Bekenstein entropy, we show that Hawking temperature is thermodynamically inconsistent with other nonextensive entropies. We focus on the equilibrium requirement for the Tsallis–Cirto black hole entropy and demonstrate that the Bekenstein–Hawking entropy is the related equilibrium entropy, and the Hawking temperature is the associated equilibrium temperature for the Tsallis–Cirto black hole entropy.

Response Characteristics of Weak Current Stimulated from Coal under an Impact Load and Its Generation Mechanism

Understanding the response law and mechanism of weak currents stimulated from coal under an impact load is significant for the prediction of coal bumps in deep coal mines. In this paper, the system for the weak current measurement of coal under an impact load is established and the response characteristics of weak currents induced by the deformation of coal under an impact load are investigated. Physical models are established to describe the process of charge transfer and explain the generation mechanism of those currents. The results show that a transient current is stimulated from the coal sample when an impact load is applied, and then, the current decays slowly, tending to be a stable value that is slightly greater than the background current. The weak current flows from the loaded volume to the unloaded volume of the coal and increases with the impact velocity in a negative exponential form. Analysis of weak currents using non-extensive entropy shows that the attenuation of the weak current obeys non-extensive statistical mechanics and the non-extensive parameter q is greater than 2. The carriers are mainly electrons, of which the distribution obeys the tip effect that electrons tend to enrich towards the tip of a crack. The generation mechanism of those weak currents induced by coal deformation is the instantaneous movement of electrons under a density difference caused by the tip effect. Research results can provide a new perspective to understand the electric phenomena of coal under an impact load as well as a new method for coal bump prediction.

Investigating Dynamical Complexity and Fractal Characteristics of Bitcoin/US Dollar and Euro/US Dollar Exchange Rates around the COVID-19 Outbreak.

This article investigates the dynamical complexity and fractal characteristics changes of the Bitcoin/US dollar (BTC/USD) and Euro/US dollar (EUR/USD) returns in the period before and after the outbreak of the COVID-19 pandemic. More specifically, we applied the asymmetric multifractal detrended fluctuation analysis (A-MF-DFA) method to investigate the temporal evolution of the asymmetric multifractal spectrum parameters. In addition, we examined the temporal evolution of Fuzzy entropy, non-extensive Tsallis entropy, Shannon entropy, and Fisher information. Our research was motivated to contribute to the comprehension of the pandemic's impact and the possible changes it caused in two currencies that play a key role in the modern financial system. Our results revealed that for the overall trend both before and after the outbreak of the pandemic, the BTC/USD returns exhibited persistent behavior while the EUR/USD returns exhibited anti-persistent behavior. Additionally, after the outbreak of COVID-19, there was an increase in the degree of multifractality, a dominance of large fluctuations, as well as a sharp decrease of the complexity (i.e., increase of the order and information content and decrease of randomness) of both BTC/USD and EUR/USD returns. The World Health Organization (WHO) announcement, in which COVID-19 was declared a global pandemic, appears to have had a significant impact on the sudden change in complexity. Our findings can help both investors and risk managers, as well as policymakers, to formulate a comprehensive response to the occurrence of such external events.