Tsallis Entropy Research Articles

2021 Alaska earthquake: entropy approach to its precursors and aftershock regimes

Abstract. We have conducted an entropy analysis in Alaska, a seismic-rich region in a subduction zone that exhibits a nontrivial behavior: the subduction arc alters the seismic activity from the eastern zone to the western zone, demonstrating a decrease in activity along the subduction. We analyze this zone through the Tsallis entropy and the mutability (or dynamic entropy) for the first time. Considering 13 870 seismic events after appropriate filtering, we analyzed a data set for the selected Alaska zone between 2000 and 2023. We have found agreement between the results for the two entropies. We have followed the value of the q parameter of the Tsallis entropy (Sq) finding values between 1.70 and 1.85, in concordance with values found in other seismic regions of the planet. The values of Sq decrease slightly over time but show a broad increase before the major earthquakes. Just opposite to Tsallis entropy, mutability shows a tendency to decrease prior to the major earthquakes. We used the simpler mutability method to further analyze this zone upon dividing the region into four subzones. The results show how mutability can identify the seismic activity in each zone. This study shows how an entropy approach can shed light on understanding the seismicity in subduction zones.

Modified Landauer Principle According to Tsallis Entropy

Modified Landauer Principle According to Tsallis Entropy

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.

Mutual Information Matrix and Global Measure based on Tsallis entropy

Mutual Information Matrix and Global Measure based on Tsallis entropy

Enhancing machine learning multi-class fault detection in electric motors through entropy-based analysis

Abstract In the field of electric motor maintenance, this study introduces a transformative approach by inte-grating entropy-based algorithms with machine learning for enhanced multi-class fault detection. Employing Shannon, Renyi, and Tsallis entropy algorithms on standard fault detection measure-ments, the research significantly advances predictive maintenance strategies through a robust, ear-ly-indication, system-agnostic analysis. Detailed examination is conducted, comparing results de-rived from datasets that include statistical features (excluding entropy) against the proposed entro-py-based datasets, when applied to a Multi-Layer Perceptron Classifier. Optimization of the MLPC and all compared algorithms’ hyperparameters is done using the state-of-the-art Optuna tool to dynamically explore each search space, ensuring that each methodology performs adequately in a timely fashion while allowing for adaptation. The results showcase significant enhancement in clas-sification accuracy of diverse electric motor operational states, facilitating the differentiation be-tween healthy and various levels of fault conditions under assorted load scenarios. Computational analyses reveal favorable results related to execution time and memory overhead, thereby support-ing the practicality in operations constrained by memory resources. Validation of the approach is achieved through laboratory experiments on a purpose-built test bench. Versatility of entropy-based measures through their proposed utilization in diverse fault indications is achieved by a demonstration in the field of mechanical fault detection with a focus on bearing faults through well-respected datasets.

Imprints of Barrow–Tsallis cosmology in primordial gravitational waves

Both the Barrow and Tsallis δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document} entropies are one-parameter generalizations of the black-hole entropy, with the same microcanonical functional form. The ensuing deformation is quantified by a dimensionless parameter Δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta $$\\end{document}, which in the case of Barrow entropy represents the anomalous dimension caused by quantum fluctuations over the horizon surface, while in Tsallis’ case, it describes the deviation of the holographic scaling from extensivity. Here, we utilize the gravity-thermodynamics conjecture with the Barrow–Tsallis entropy to investigate the implications of the related modified Friedmann equations on the spectrum of primordial gravitational waves. We show that, with the experimental sensitivity of the next generation of gravitational wave detectors, such as the Big Bang Observer, it will be possible to discriminate deviations from the Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}CDM model up to Δ≲O(10-3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta \\lesssim \\mathcal {O}(10^{-3})$$\\end{document}. In this limit, Barrow–Tsallis entropy reduces to a logarithmic correction to holographic scaling, which is nearly universally predicted by both entanglement entropy calculations in the UV regime and by several candidate theories of quantum gravity. Hence, our considerations and results are expected to have general validity in the quantum gravity framework.

Information theoretic measures on quantum droplets in ultracold atomic systems

We consider Shannon entropy, Fisher information, Rényi entropy, and Tsallis entropy to study the quantum droplet phase in Bose–Einstein condensates. In the beyond mean-field description, the Gross–Pitaevskii equation with Lee-Huang-Yang correction gives a family of quantum droplets with different chemical potentials. At a larger value of chemical potential, quantum droplet with sharp-top probability density distribution starts to form while it becomes flat top for a smaller value of chemical potential. We show that entropic measures can distinguish the shape change of the probability density distributions and thus can identify the onset of the droplet phase. During the onset of droplet phase, the Shannon entropy decreases gradually with the decrease of chemical potential and attains a minimum in the vicinity where a smooth transition from flat-top to sharp-top QDs occurs. At this stage, the Shannon entropy increases abruptly with the lowering of chemical potential. We observe an opposite trend in the case of Fisher information. These results are found to be consistent with the Rényi and Tsallis entropic measures.

Analyzing inflationary parameters and swampland conjectures of modified cosmology according to various entropies

This paper investigates the inflationary phase of the FRW universe within the context of generalized Chaplygin gas. We explore modified cosmological scenarios incorporating two entropy corrections, namely Tsallis and Barrow. These proposed entropies lead to modifications in the Friedmann equations. By incorporating the generalized Chaplygin gas into these equations, we determine various parameters including the slow roll parameters, number of e-folds, curvature perturbation, tensor-to-scalar ratio, and spectral index. We observe variations in the values of the tensor-to-scalar ratio r and the scalar spectral index ns with respect to the Tsallis and Barrow parameters. Additionally, we plot the r−ns plane. For the Tsallis entropy, we obtain r≤0.012, r≤0.027, and r≤0.008, with ns lying between (0.961,0.964), (0.957,0.966), and (0.956,0.964), respectively. In Barrow cosmology, we find r≤0.0037, r≤0.12, and r≤0.010, while ns falls within (0.92,0.97), (0.855,0.995), and (0.960,0.966) for the chaotic, SFI, and exponential potentials, respectively. These graphical analyses demonstrate the compatibility of our models with the latest Planck data. Moreover, by considering these entropy corrections, we determine the bounds of the Swampland de Sitter conjecture for generalized Chaplygin gas. In all cases, this bound remains less than unity, satisfying the required condition for the Swampland de Sitter conjecture.

Exploring Tsallis thermodynamics for boundary conformal field theories in gauge/gravity duality

Inspired by AdS/CFT interpretation Karch and Robinson (2015), we analyze thermodynamics for thermal boundary conformal field theories (CFT) that are dual to four-dimensional charged anti-de Sitter (AdS) black holes embedded in 11-dimensional M-theory-inspired models with AdS4×S7 space–time, employing the framework of Tsallis entropy. The latter is recognized as a nonadditive extension of the Boltzmann–Gibbs entropy that can satisfy the thermodynamic extensivity for black holes. Specifically, we consider AdS black holes in AdS4×S7 spacetime, interpreting the number of M2-branes as a thermodynamic variable. Using the Tsallis entropy enables us to highlight peculiar thermodynamic features, resorting to essential tools such as the chemical potential and Gibbs energy while examining phase transitions along iso-charge partitions. Then, we leveraged a class of geometrothermodynamic formalisms, including Weinhold, Ruppeiner, Quevedo I, and II metrics. Quevedo’s formulations provide richer information about phase transitions than the first two methods. Our study sheds new light on AdS black holes in a thermodynamically proper context, deepening our understanding of the role of non-extensivity in the critical behavior of such complex systems.

Worst-case distortion riskmetrics and weighted entropy with partial information

In this paper, we discuss the worst-case distortion riskmetrics for general distributions when only partial information (mean and variance) is known. This result is applicable to a general class of distortion risk measures and variability measures. Furthermore, we also consider the worst-case weighted entropy for general distributions when only partial information is available. Specifically, we provide some applications for entropies, weighted entropies and risk measures. The commonly used entropies include Gini functional, cumulative residual entropy, tail-Gini functional, cumulative Tsallis past entropy, extended Gini coefficient, among others. The risk measures contain some premium principles and shortfalls based on entropy. The shortfalls include the Gini shortfall, extended Gini shortfall, shortfall of cumulative residual entropy and shortfall of cumulative residual Tsallis entropy with order α.

Bitcoin, Fintech, Energy Consumption, and Environmental Pollution Nexus: Chaotic Dynamics with Threshold Effects in Tail Dependence, Contagion, and Causality

The study investigates the nonlinear contagion, tail dependence, and Granger causality relations with TAR-TR-GARCH–copula causality methods for daily Bitcoin, Fintech, energy consumption, and CO2 emissions in addition to examining these series for entropy, long-range dependence, fractionality, complexity, chaos, and nonlinearity with a dataset spanning from 25 June 2012 to 22 June 2024. Empirical results from Shannon, Rényi, and Tsallis entropy measures; Kolmogorov–Sinai complexity; Hurst–Mandelbrot and Lo’s R/S tests; and Phillips’ and Geweke and Porter-Hudak’s fractionality tests confirm the presence of entropy, complexity, fractionality, and long-range dependence. Further, the largest Lyapunov exponents and Hurst exponents confirm chaos across all series. The BDS test confirms nonlinearity, and ARCH-type heteroskedasticity test results support the basis for the use of novel TAR-TR-GARCH–copula causality. The model estimation results indicate moderate to strong levels of positive and asymmetric tail dependence and contagion under distinct regimes. The novel method captures nonlinear causality dynamics from Bitcoin and Fintech to energy consumption and CO2 emissions as well as causality from energy consumption to CO2 emissions and bidirectional feedback between Bitcoin and Fintech. These findings underscore the need to take the chaotic and complex dynamics seriously in policy and decision formulation and the necessity of eco-friendly technologies for Bitcoin and Fintech.

A generalised entropic measure of steering using Tsallis entropies and the relationship with existent steering criteria

Abstract EPR steering is an asymmetric quantum correlation, as such, it is an important resource for quantum information protocols, like those used in quantum key distribution that are of tremendous relevance nowadays. The security of the aforementioned protocols can be shown by using entropic uncertainty measures, which are based on quantum information quantities, among which the entropies are found. In this work, we propose an entropic measure of steering, starting from the detection of steering given in a previous entropic uncertainty relation-based criterion, as well as using a generalised entropy, the Tsallis entropies. We define the entropic measure of steering in terms of an indicator, a quantity depending on the entropies of the measurements of the system, which assigns a value to the steering content of quantum states, while also comprising the asymmetry property of steering. We investigate the properties of the proposed quantity as a measure of steering and its asymmetry and show how it can be used for specific examples. For instance, a class of noisy two-qubit states that includes the Werner states is used to demonstrate the potential of such a proposal. Furthermore, a comparison is also made with the detection of steering in these states using existing steering criteria and the amount of steering a geometric measure for the Werner state gives.

The Application of Tsallis Entropy Based Self-Adaptive Algorithm for Multi-Threshold Image Segmentation.

Tsallis entropy has been widely used in image thresholding because of its non-extensive properties. The non-extensive parameter q contained in this entropy plays an important role in various adaptive algorithms and has been successfully applied in bi-level image thresholding. In this paper, the relationships between parameter q and pixels' long-range correlations have been further studied within multi-threshold image segmentation. It is found that the pixels' correlations are remarkable and stable for images generated by a known physical principle, such as infrared images, medical CT images, and color satellite remote sensing images. The corresponding non-extensive parameter q can be evaluated by using the self-adaptive Tsallis entropy algorithm. The results of this algorithm are compared with those of the Shannon entropy algorithm and the original Tsallis entropy algorithm in terms of quantitative image quality evaluation metrics PSNR (Peak Signal-to-Noise Ratio) and SSIM (Structural Similarity). Furthermore, we observed that for image series with the same background, the q values determined by the adaptive algorithm are consistently kept in a narrow range. Therefore, similar or identical scenes during imaging would produce similar strength of long-range correlations, which provides potential applications for unsupervised image processing.

Stability of the logarithmic Sobolev inequality and uncertainty principle for the Tsallis entropy

We consider the stability of the functional inequalities concerning the entropy functional. For the Boltzmann–Shannon entropy, the logarithmic Sobolev inequality holds as a lower bound of the entropy by the Fisher information, and the Heisenberg uncertainty principle follows from combining it with the Shannon inequality. The optimizer for these inequalities is the Gauss function, which is a fundamental solution to the heat equation. In the fields of statistical mechanics and information theory, the Tsallis entropy is known as a one-parameter extension of the Boltzmann–Shannon entropy, and the Wasserstein gradient flow of it corresponds to the quasilinear diffusion equation. We consider the improvement and stability of the optimizer for the logarithmic Sobolev inequality related to the Tsallis entropy. Furthermore, we show the stability results of the uncertainty principle concerning the Tsallis entropy.

A Proposed Novel Approach in Estimation of Unbiased Community Rarity Using the Locked Matrix and Tsallis Entropy

A Proposed Novel Approach in Estimation of Unbiased Community Rarity Using the Locked Matrix and Tsallis Entropy

Lagrangian formulation of the Tsallis entropy

We investigate the gravitational origin of the Tsallis entropy, characterized by the nonadditive index δ. Utilizing Wald's formalism within the framework of f(R) modified theories of gravity, we evaluate the entropy on the black hole horizon for constant curvature solutions to the spherically symmetric vacuum field equations. In so doing, we demonstrate that the Tsallis entropy can be effectively derived from a generalization of the Lagrangian L∝R1+ϵ, where ϵ=δ−1 quantifies small deviations from general relativity. In conclusion, we examine the physical implications of our findings in light of cosmological observations and elaborate on the possibility of solving the thermodynamic instability of Schwarzschild black holes.

Electrical brain networks before and after transcranial pulsed shockwave stimulation in Alzheimer's patients.

Alzheimer's disease (AD) is a neurodegenerative disorder that dramatically affects cognitive abilities and represents the most common cause of dementia. Currently, pharmacological interventions represent the main treatment to deal with the symptoms of AD; however, alternative approaches are readily sought. Transcranial pulse stimulation (TPS) is an emerging non-invasive neuromodulation technique that uses short, repetitive shockwaves with the potential to provide a wide range of vascular, metabolic, and neurotrophic changes and that has recently been shown to improve cognitive abilities in AD. This exploratory study aims to gain insight into the neurophysiological effect of one session of TPS in AD as reflected in electroencephalographic measures, e.g., spectral power, coherence, Tsallis entropy (TE), and cross-frequency coupling (cfc). We document changes in power (frontal and occipital), coherence (frontal, occipital and temporal), and TE (temporal and frontal) as well as changes in cfc (parietal-frontal, parietal-temporal, frontal-temporal). Our results emphasize the role of electroencephalographic measures as prospective markers for the neurophysiological effect of TPS.

An overview of the development and applications of information entropy

With information entropy gradually taking the lead in modern information theory development, it begins to hold greater influence over multiple research areas as well as technology innovation. This paper aims to clarify peoples confusion with the development of entropy theory and provide a brief overview of the origin of entropy theory, including the original Shannons proposal, variants such as relative entropy and conditional entropy, and entropy concepts proposed by other scientists, such as Rnyi Tsallis entropy. The paper also includes the current application of entropy, studies hotspots, and predicts future entropy development trends. This research paper is able to add more coherence and consistency to information entropys development, helping more people to better understand the concept of entropy and its derivation. At the same time, with hotspots of entropy fields of study, this paper hopes to attract more people to devote themselves to studying entropy-related fields, and boost technological development.

Polygon relations and subadditivity of entropic measures for discrete and continuous multipartite entanglement

In a recent work by us Ge et al [Phys. Rev. A 110, L010402 (2024)], we have derived a series of polygon relations of bipartite entanglement measures that is useful to reveal entanglement properties of discrete, continuous, and even hybrid multipartite quantum systems. In this work, with the information-theoretical measures of Rényi and Tsallis entropies, we study the relationship between the polygon relation and the subadditivity of entropy. In particular, the entropy-polygon relations are derived for pure multi-qubit states and then generalized to multi-mode Gaussian states, by utilizing the known results from the quantum marginal problem. Then the equivalence between the polygon relation and subadditivity is established, in the sense that for all discrete or continuous multipartite states, the polygon relation holds if and only if the underlying entropy is subadditive. As a byproduct, the subadditivity of Rényi and Tsallis entropies is proven for all bipartite Gaussian states. Finally, the difference between polygon relations and monogamy relations is clarified, and generalizations of our results are discussed. Our work provides a better understanding of the rich structure of multipartite states, and hence is expected to be helpful for the study of multipartite entanglement.

Testing for Markovian character of transfer of fluctuations in solar wind turbulence on kinetic scales.

We apply statistical analysis to search for processes responsible for turbulence in physical systems. In our previous studies, we have shown that solar wind turbulence in the inertial range of large magnetohydrodynamic scales exhibits Markov properties. We have recently extended this approach on much smaller kinetic scales. Here we are testing for the Markovian character of stochastic processes in a kinetic regime based on magnetic field and velocity fluctuations in the solar wind, measured onboard the Magnetospheric Multiscale (MMS) mission: behind the bow shock, inside the magnetosheath, and near the magnetopause. We have verified that the Chapman-Kolmogorov necessary conditions for Markov processes is satisfied for local transfer of energy between the magnetic and velocity fields also on kinetic scales. We have confirmed that for magnetic fluctuations, the first Kramers-Moyal coefficient is linear, while the second term is quadratic, corresponding to drift and diffusion processes in the resulting Fokker-Planck equation. It means that magnetic self-similar turbulence is described by generalized Ornstein-Uhlenbeck processes. We show that for the magnetic case, the Fokker-Planck equationleads to the probability density functions of the kappa distributions, which exhibit global universal scale invariance with a linear scaling and lack of intermittency. On the contrary, for velocity fluctuations, higher order Kramers-Moyal coefficients should be taken into account and hence scale invariance is not observed. However, the nonextensity parameter in Tsallis entropy provides a robust measure of the departure of the system from equilibrium. The obtained results are important for a better understanding of the physical mechanism governing turbulent systems in space and laboratory.