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Entropy, Volume 27, Issue 5 (May 2025) – 77 articles

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28 pages, 7273 KiB  
Article
Comparative Study on Flux Solution Methods of Discrete Unified Gas Kinetic Scheme
by Wenqiang Guo
Entropy 2025, 27(5), 528; https://doi.org/10.3390/e27050528 - 15 May 2025
Abstract
In this work, the Simpson method is proposed to calculate the interface flux of a discrete unified gas kinetic scheme (DUGKS) according to the distribution function at the node and the midpoint of the interface, which is noted by Simpson–DUGKS. Moreover, the optimized [...] Read more.
In this work, the Simpson method is proposed to calculate the interface flux of a discrete unified gas kinetic scheme (DUGKS) according to the distribution function at the node and the midpoint of the interface, which is noted by Simpson–DUGKS. Moreover, the optimized DUGKS and Simpson–DUGKS considering the force term are derived. Then, the original DUGKS, optimized DUGKS, and Simpson–DUGKS are compared and analyzed in theory. Finally, the numerical tests are performed under different grid numbers (N). In the steady unidirectional flow (Couette flow and Poiseuille flow), the three methods are stable under different Courant–Friedrichs–Lewy (CFL) numbers, and the calculated L2 errors are the same. In the Taylor–Green vortex flow, the L2 error of the optimized DUGKS is the smallest with respect to the analytical solution of velocity, but the L2 error of the optimized DUGKS is the largest with respect to the analytical solution of density. In the lid-driven cavity flow, the results of the optimized DUGKS deviate more from the reference results in terms of accuracy, especially in the case of a small grid number. In terms of computational efficiency, it should be noted that the computational time of optimized DUGKS increases by about 40% compared with the original DUGKS when CFL = 0.1 and N = 16, and the calculation time of Simpson–DUGKS is reduced by about 59% compared with the original DUGKS when CFL = 0.95 and N = 16. Full article
(This article belongs to the Special Issue Mesoscopic Fluid Mechanics)
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23 pages, 1158 KiB  
Article
Quantum Exact Response Theory Based on the Dissipation Function
by Enrico Greppi and Lamberto Rondoni
Entropy 2025, 27(5), 527; https://doi.org/10.3390/e27050527 - 15 May 2025
Abstract
The exact response theory based on the Dissipation Function applies to general dynamical systems and has yielded excellent results in various applications. In this article, we propose a method to apply it to quantum mechanics. In many quantum systems, it has not yet [...] Read more.
The exact response theory based on the Dissipation Function applies to general dynamical systems and has yielded excellent results in various applications. In this article, we propose a method to apply it to quantum mechanics. In many quantum systems, it has not yet been possible to overcome the perturbative approach, and the most developed theory is the linear one. Extensions of the exact response theory developed in the field of nonequilibrium molecular dynamics could prove useful in quantum mechanics, as perturbations of small systems or far-from-equilibrium states cannot always be taken as small perturbations. Here, we introduce a quantum analogue of the classical Dissipation Function. We then derive a quantum expression for the exact calculation of time-dependent expectation values of observables, in a form analogous to that of the classical theory. We restrict our analysis to finite-dimensional Hilbert spaces, for the sake of simplicity, and we apply our method to specific examples, like qubit systems, for which exact results can be obtained by standard techniques. This way, we prove the consistency of our approach with the existing methods, where they apply. Although not required for open systems, we propose a self-adjoint version of our Dissipation Operator, obtaining a second equivalent expression of response, where the contribution of an anti-self-adjoint operator appears. We conclude by using new formalism to solve the Lindblad equations, obtaining exact results for a specific case of qubit decoherence, and suggesting possible future developments of this work. Full article
(This article belongs to the Special Issue Quantum Nonstationary Systems—Second Edition)
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29 pages, 18881 KiB  
Article
A Novel Entropy-Based Approach for Thermal Image Segmentation Using Multilevel Thresholding
by Thaweesak Trongtirakul, Karen Panetta, Artyom M. Grigoryan and Sos S. Agaian
Entropy 2025, 27(5), 526; https://doi.org/10.3390/e27050526 - 14 May 2025
Abstract
Image segmentation is a fundamental challenge in computer vision, transforming complex image representations into meaningful, analyzable components. While entropy-based multilevel thresholding techniques, including Otsu, Shannon, fuzzy, Tsallis, Renyi, and Kapur approaches, have shown potential in image segmentation, they encounter significant limitations when processing [...] Read more.
Image segmentation is a fundamental challenge in computer vision, transforming complex image representations into meaningful, analyzable components. While entropy-based multilevel thresholding techniques, including Otsu, Shannon, fuzzy, Tsallis, Renyi, and Kapur approaches, have shown potential in image segmentation, they encounter significant limitations when processing thermal images, such as poor spatial resolution, low contrast, lack of color and texture information, and susceptibility to noise and background clutter. This paper introduces a novel adaptive unsupervised entropy algorithm (A-Entropy) to enhance multilevel thresholding for thermal image segmentation. Our key contributions include (i) an image-dependent thermal enhancement technique specifically designed for thermal images to improve visibility and contrast in regions of interest, (ii) a so-called A-Entropy concept for unsupervised thermal image thresholding, and (iii) a comprehensive evaluation using the Benchmarking IR Dataset for Surveillance with Aerial Intelligence (BIRDSAI). Experimental results demonstrate the superiority of our proposal compared to other state-of-the-art methods on the BIRDSAI dataset, which comprises both real and synthetic thermal images with substantial variations in scale, contrast, background clutter, and noise. Comparative analysis indicates improved segmentation accuracy and robustness compared to traditional entropy-based methods. The framework’s versatility suggests promising applications in brain tumor detection, optical character recognition, thermal energy leakage detection, and face recognition. Full article
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15 pages, 1914 KiB  
Article
Nonequilibrium Steady States in Active Systems: A Helmholtz–Hodge Perspective
by Horst-Holger Boltz and Thomas Ihle
Entropy 2025, 27(5), 525; https://doi.org/10.3390/e27050525 - 14 May 2025
Abstract
We revisit the question of the existence of a potential function, the Cole–Hopf transform of the stationary measure, for nonequilibrium steady states, in particular those found in active matter systems. This has been the subject of ongoing research for more than fifty years, [...] Read more.
We revisit the question of the existence of a potential function, the Cole–Hopf transform of the stationary measure, for nonequilibrium steady states, in particular those found in active matter systems. This has been the subject of ongoing research for more than fifty years, but continues to be relevant. In particular, we want to make a connection to some recent work on the theory of Helmholtz–Hodge decompositions and address the recently suggested notion of typical trajectories in such systems. Full article
(This article belongs to the Collection Foundations of Statistical Mechanics)
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39 pages, 4006 KiB  
Article
Reference Point and Grid Method-Based Evolutionary Algorithm with Entropy for Many-Objective Optimization Problems
by Qi Leng, Bo Shan and Chong Zhou
Entropy 2025, 27(5), 524; https://doi.org/10.3390/e27050524 - 14 May 2025
Abstract
In everyday scenarios, there are many challenges involving multi-objective optimization. As the count of objective functions rises to four or beyond, the problem’s complexity intensifies considerably, often making it challenging for traditional algorithms to arrive at satisfactory solutions. The non-dominated sorting evolutionary reference [...] Read more.
In everyday scenarios, there are many challenges involving multi-objective optimization. As the count of objective functions rises to four or beyond, the problem’s complexity intensifies considerably, often making it challenging for traditional algorithms to arrive at satisfactory solutions. The non-dominated sorting evolutionary reference point-based (NSGA-III) and the grid-based evolutionary algorithms (GrEA) are two prevalent algorithms for many-objective optimization. These two algorithms preserve population diversity by employing reference point and grid mechanisms, respectively. However, they still have limitations when addressing many-objective optimization problems. Due to the uniform distribution of reference points, the reference point-based methods do not obtain good performance on problems with an irregular Pareto front, while grid-based methods do not achieve good results on problems with a regular Pareto front because of the uneven partition of grids. To address the limitations of reference point-based algorithms and grid-based approaches in tackling both regular and irregular problems, a reference point- and grid-based evolutionary algorithm with entropy is proposed for many-objective optimization, denoted as RGEA, which aims to solve both regular and irregular many-objective optimization problems. Entropy is introduced to measure the shape of the Pareto front of a many-objective optimization problem. In RGEA, a parameter α is introduced to determine the interval for calculating the entropy value. By comparing the current entropy value with the maximum entropy value, the reference point-based method or the grid-based method can be determined. In order to verify the performance of the proposed algorithm, a comprehensive experiment was designed on some popular test suites with 3-to-10 objectives. In addition, RGEA was compared against six algorithms without adaptive technology and six algorithms with adaptive technology. A great number of experimental results were obtained showing that RGEA can obtain good results. Full article
(This article belongs to the Topic AI and Computational Methods for Modelling, Simulations and Optimizing of Advanced Systems: Innovations in Complexity, Second Edition)
22 pages, 1850 KiB  
Article
Tail Risk Spillover Between Global Stock Markets Based on Effective Rényi Transfer Entropy and Wavelet Analysis
by Jingjing Jia
Entropy 2025, 27(5), 523; https://doi.org/10.3390/e27050523 - 14 May 2025
Abstract
To examine the spillover of tail-risk information across global stock markets, we select nine major stock markets for the period spanning from June 2014 to May 2024 as the sample data. First, we employ effective Rényi transfer entropy to measure the tail-risk information [...] Read more.
To examine the spillover of tail-risk information across global stock markets, we select nine major stock markets for the period spanning from June 2014 to May 2024 as the sample data. First, we employ effective Rényi transfer entropy to measure the tail-risk information spillover. Second, we construct a Diebold–Yilmaz connectedness table to explore the overall characteristics of tail-risk information spillover across the global stock markets. Third, we integrate wavelet analysis with effective Rényi transfer entropy to assess the multi-scale characteristics of the information spillover. Our findings lead to several key conclusions: (1) US and European stock markets are the primary sources of tail-risk information spillover, while Asian stock markets predominantly act as net information receivers; (2) the intensity of tail-risk information spillover is most pronounced between markets at the medium-high trading frequency, and as trading frequency decreases, information spillover becomes more complex; (3) across all trading frequencies, the US stock market emerges as the most influential, while the Japanese stock market is the most vulnerable. China’s stock market, in contrast, demonstrates relative independence. Full article
(This article belongs to the Special Issue Complexity in Financial Networks)
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18 pages, 286 KiB  
Article
The Physics and Metaphysics of Social Powers: Bridging Cognitive Processing and Social Dynamics, a New Perspective on Power Through Active Inference
by Mahault Albarracin, Sonia de Jager and David Hyland
Entropy 2025, 27(5), 522; https://doi.org/10.3390/e27050522 - 14 May 2025
Abstract
Power operates across multiple scales, from physical action to complex social dynamics, and is constrained by fundamental principles. In the social realm, power is shaped by interactions and cognitive capacity: socially-facilitated empowerment enhances an agent’s information-processing ability, either by delegating tasks or leveraging [...] Read more.
Power operates across multiple scales, from physical action to complex social dynamics, and is constrained by fundamental principles. In the social realm, power is shaped by interactions and cognitive capacity: socially-facilitated empowerment enhances an agent’s information-processing ability, either by delegating tasks or leveraging collective resources. This computational advantage expands access to policies and buffers against vulnerabilities, amplifying an individual’s or group’s influence. In AIF, social power emerges from the capacity to attract attention and process information effectively. Our semantic habitat—narratives, ideologies, representations, etc.—functions through attentional scripts that coordinate social behavior. Shared scripts shape power dynamics by structuring collective attention. Speculative scripts serve as cognitive tools for low-risk learning, allowing agents to explore counterfactuals and refine predictive models. However, dominant scripts can reinforce misinformation, echo chambers, and power imbalances by directing collective attention toward self-reinforcing policies. We argue that power through scripts stems not only from associations with influential agents but also from the ability to efficiently process information, creating a feedback loop of increasing influence. This reframes power beyond traditional material and cultural dimensions, towards an informational and computational paradigm—what we term possibilistic power, i.e., the capacity to explore and shape future trajectories. Understanding these mechanisms has critical implications for political organization and technological foresight. Full article
(This article belongs to the Special Issue Active Inference in Cognitive Neuroscience)
22 pages, 287 KiB  
Article
Two-Step Estimation Procedure for Parametric Copula-Based Regression Models for Semi-Competing Risks Data
by Qingmin Zhang, Bowen Duan, Małgorzata Wojtyś and Yinghui Wei
Entropy 2025, 27(5), 521; https://doi.org/10.3390/e27050521 - 13 May 2025
Abstract
Non-terminal and terminal events in semi-competing risks data are typically associated and may be influenced by covariates. We employed regression modeling for semi-competing risks data under a copula-based framework to evaluate the effects of covariates on the two events and the association between [...] Read more.
Non-terminal and terminal events in semi-competing risks data are typically associated and may be influenced by covariates. We employed regression modeling for semi-competing risks data under a copula-based framework to evaluate the effects of covariates on the two events and the association between them. Due to the complexity of the copula structure, we propose a new method that integrates a novel two-step algorithm with the Bound Optimization by Quadratic Approximation (BOBYQA) method. This approach effectively mitigates the influence of initial values and demonstrates greater robustness. The simulations validate the performance of the proposed method. We further applied our proposed method to the Amsterdam Cohort Study (ACS) real data, where some improvements could be found. Full article
23 pages, 3783 KiB  
Article
Design of Covert Communication Waveform Based on Phase Randomization
by Wenjie Zhou, Zhenyong Wang, Jun Shi and Qing Guo
Entropy 2025, 27(5), 520; https://doi.org/10.3390/e27050520 - 13 May 2025
Viewed by 101
Abstract
Covert wireless communication is designed to securely transmit hidden information between two devices. Its primary objective is to conceal the existence of transmitted data, rendering communication signals difficult for unauthorized parties to detect, intercept, or decipher during transmission. In this paper, we propose [...] Read more.
Covert wireless communication is designed to securely transmit hidden information between two devices. Its primary objective is to conceal the existence of transmitted data, rendering communication signals difficult for unauthorized parties to detect, intercept, or decipher during transmission. In this paper, we propose a Noise-like Multi-Carrier Random Phase Communication System (NRPCS) to enhance covert wireless communication by significantly complicating the detection and interception of transmitted signals. The proposed system utilizes bipolar modulation and Cyclic Code Shift Keying (CCSK) modulation, complemented by a random sequence generation mechanism, to increase the randomness and complexity of the transmitted signals. A mathematical model of the NRPCS waveform is formulated, and detailed analyses of the system’s time-domain basis functions, correlation properties, and power spectral characteristics are conducted to substantiate its noise-like behavior. Simulation results indicate that, compared to traditional fixed-frequency transmission methods, NRPCS substantially improves both the Low Probability of Detection (LPD) and the Low Probability of Interception (LPI). Further research results demonstrate that unauthorized eavesdroppers are unable to effectively demodulate signals without knowledge of the employed modulation scheme, thus significantly enhancing the overall security of communication. Full article
(This article belongs to the Special Issue Wireless Communications: Signal Processing Perspectives)
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2 pages, 342 KiB  
Correction
Correction: Klemann et al. Quantifying the Resilience of a Healthcare System: Entropy and Network Science Perspectives. Entropy 2024, 26, 21
by Désirée Klemann, Windi Winasti, Fleur Tournois, Helen Mertens and Frits van Merode
Entropy 2025, 27(5), 519; https://doi.org/10.3390/e27050519 - 13 May 2025
Viewed by 50
Abstract
In the original publication [...] Full article
(This article belongs to the Section Multidisciplinary Applications)
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46 pages, 510 KiB  
Article
Towards Nonlinearity: The p-Regularity Theory
by Ewa Bednarczuk, Olga Brezhneva, Krzysztof Leśniewski, Agnieszka Prusińska and Alexey A. Tret’yakov
Entropy 2025, 27(5), 518; https://doi.org/10.3390/e27050518 - 12 May 2025
Viewed by 82
Abstract
We present recent advances in the analysis of nonlinear problems involving singular (degenerate) operators. The results are obtained within the framework of p-regularity theory, which has been successfully developed over the past four decades. We illustrate the theory with applications to degenerate [...] Read more.
We present recent advances in the analysis of nonlinear problems involving singular (degenerate) operators. The results are obtained within the framework of p-regularity theory, which has been successfully developed over the past four decades. We illustrate the theory with applications to degenerate problems in various areas of mathematics, including optimization and differential equations. In particular, we address the problem of describing the tangent cone to the solution set of nonlinear equations in singular cases. The structure of p-factor operators is used to propose optimality conditions and to construct novel numerical methods for solving degenerate nonlinear equations and optimization problems. The numerical methods presented in this paper represent the first approaches targeting solutions to degenerate problems such as the Van der Pol differential equation, boundary-value problems with small parameters, and partial differential equations where Poincaré’s method of small parameters fails. Additionally, these methods may be extended to nonlinear degenerate dynamical systems and other related problems. Full article
(This article belongs to the Section Complexity)
18 pages, 356 KiB  
Article
Entropy of the Quantum–Classical Interface: A Potential Metric for Security
by Sarah Chehade, Joel A. Dawson, Stacy Prowell and Ali Passian
Entropy 2025, 27(5), 517; https://doi.org/10.3390/e27050517 - 12 May 2025
Viewed by 188
Abstract
Hybrid quantum–classical systems are emerging as key platforms in quantum computing, sensing, and communication technologies, but the quantum–classical interface (QCI)—the boundary enabling these systems—introduces unique and largely unexplored security vulnerabilities. This position paper proposes using entropy-based metrics to monitor and enhance security, specifically [...] Read more.
Hybrid quantum–classical systems are emerging as key platforms in quantum computing, sensing, and communication technologies, but the quantum–classical interface (QCI)—the boundary enabling these systems—introduces unique and largely unexplored security vulnerabilities. This position paper proposes using entropy-based metrics to monitor and enhance security, specifically at the QCI. We present a theoretical security outline that leverages well-established information-theoretic entropy measures, such as Shannon entropy, von Neumann entropy, and quantum relative entropy, to detect anomalous behaviors and potential breaches at the QCI. By linking entropy fluctuations to scenarios of practical relevance—including quantum key distribution, quantum sensing, and hybrid control systems—we promote the potential value and applicability of entropy-based security monitoring. While explicitly acknowledging practical limitations and theoretical assumptions, we argue that entropy-based metrics provide a complementary approach to existing security methods, inviting further empirical studies and theoretical refinements that can strengthen future quantum technologies. Full article
(This article belongs to the Section Quantum Information)
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21 pages, 7300 KiB  
Article
Public Opinion Propagation Prediction Model Based on Dynamic Time-Weighted Rényi Entropy and Graph Neural Network
by Qiujuan Tong, Xiaolong Xu, Jianke Zhang and Huawei Xu
Entropy 2025, 27(5), 516; https://doi.org/10.3390/e27050516 - 12 May 2025
Viewed by 148
Abstract
Current methods for public opinion propagation prediction struggle to jointly model temporal dynamics, structural complexity, and dynamic node influence in evolving social networks. To overcome these limitations, this paper proposes a public opinion dissemination prediction model based on the integration of dynamic time-weighted [...] Read more.
Current methods for public opinion propagation prediction struggle to jointly model temporal dynamics, structural complexity, and dynamic node influence in evolving social networks. To overcome these limitations, this paper proposes a public opinion dissemination prediction model based on the integration of dynamic time-weighted Rényi entropy (DTWRE) and graph neural networks. By incorporating a time-weighted mechanism, the model devises two tiers of Rényi entropy metrics—local node entropy and global time-step entropy—to effectively quantify the uncertainty and complexity of network topology at different time points. Simultaneously, by integrating DTWRE features with high-dimensional node embeddings generated by Node2Vec and utilizing GraphSAGE to construct a spatiotemporal fusion modeling framework, the model achieves precise prediction of link formation and key node identification in public opinion dissemination. The model was validated on multiple public opinion datasets, and the results indicate that, compared to baseline methods, it exhibits significant advantages in several evaluation metrics such as AUC, thereby fully demonstrating the effectiveness of the dynamic time-weighted mechanism in capturing the temporal evolution of public opinion dissemination and the dynamic changes in network structure. Full article
(This article belongs to the Special Issue Information-Theoretic Approaches for Machine Learning and AI)
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18 pages, 417 KiB  
Article
Comparing Singlet Testing Schemes
by George Cowperthwaite and Adrian Kent
Entropy 2025, 27(5), 515; https://doi.org/10.3390/e27050515 - 11 May 2025
Viewed by 93
Abstract
We compare schemes for testing whether two parties share a two-qubit singlet state. The first, standard, scheme tests Braunstein–Caves (or CHSH) inequalities, comparing the correlations of local measurements drawn from a fixed finite set against the quantum predictions for a singlet. The second, [...] Read more.
We compare schemes for testing whether two parties share a two-qubit singlet state. The first, standard, scheme tests Braunstein–Caves (or CHSH) inequalities, comparing the correlations of local measurements drawn from a fixed finite set against the quantum predictions for a singlet. The second, alternative, scheme tests the correlations of local measurements, drawn randomly from the set of those that are θ-separated on the Bloch sphere, against the quantum predictions. We formulate each scheme as a hypothesis test and then evaluate the test power in a number of adversarial scenarios involving an eavesdropper altering or replacing the singlet qubits. We find the ‘random measurement’ test to be superior in most natural scenarios. Full article
(This article belongs to the Special Issue Editorial Board Members' Collection Series on Quantum Entanglement)
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25 pages, 461 KiB  
Article
A Deflationary Account of Information in Terms of Probability
by Riccardo Manzotti
Entropy 2025, 27(5), 514; https://doi.org/10.3390/e27050514 - 11 May 2025
Viewed by 259
Abstract
In this paper, I argue that information is nothing more than an abstract object; therefore, it does not exist fundamentally. It is neither a concrete physical entity nor a form of “stuff” that “flows” through communication channels or that is “carried” by vehicles [...] Read more.
In this paper, I argue that information is nothing more than an abstract object; therefore, it does not exist fundamentally. It is neither a concrete physical entity nor a form of “stuff” that “flows” through communication channels or that is “carried” by vehicles or that is stored in memories, messages, books, or brains—these are misleading metaphors. To support this thesis, I adopt three different approaches. First, I present a series of concrete cases that challenge our commonsensical belief that information is a real entity. Second, I apply Eleaticism (the principle that entities lacking causal efficacy do not exist). Finally, I provide a mathematical derivation showing that information reduces to probability and is therefore unnecessary both ontologically and epistemically. In conclusion, I maintain that information is a causally redundant epistemic construct that does not exist fundamentally, regardless of its remarkable epistemic convenience. What, then, is information? It is merely a very efficient way of describing reality—a manner of speaking, nothing more. Full article
(This article belongs to the Special Issue Integrated Information Theory and Consciousness II)
20 pages, 24982 KiB  
Article
A Novel One-Dimensional Chaotic System for Image Encryption in Network Transmission Through Base64 Encoding
by Linqing Huang, Qingye Huang, Han Chen, Shuting Cai, Xiaoming Xiong and Jian Yang
Entropy 2025, 27(5), 513; https://doi.org/10.3390/e27050513 - 10 May 2025
Viewed by 160
Abstract
Continuous advancements in digital image transmission technology within network environments have heightened the necessity for secure, convenient, and well-suited image encryption systems. Base64 encoding possesses the ability to convert raw data into printable ASCII characters, facilitating excellent stability transmission across various communication protocols. [...] Read more.
Continuous advancements in digital image transmission technology within network environments have heightened the necessity for secure, convenient, and well-suited image encryption systems. Base64 encoding possesses the ability to convert raw data into printable ASCII characters, facilitating excellent stability transmission across various communication protocols. In this paper, base64 encoding is first used in image encryption to pursue high application value in network transmission. First, a novel one-dimensional discrete chaotic system (1D-LSCM) with complex chaotic behavior is introduced and extensively tested. Second, a new multi-image encryption algorithm based on the proposed 1D-LSCM and base64 encoding is presented. Technically, three original grayscale images are constructed as a color image and encoded in base64 characters. To purse high plaintext sensitivity, the original image is input to the SHA-256 hash function and its output is used to influence the generated keystream employed in the permutation and diffusion process. After scramble and diffusion operations, the base64 ciphertext is obtained. Finally, test results derived from comprehensive tests prove that our proposed algorithm has remarkable security and encryption efficiency. Full article
(This article belongs to the Section Multidisciplinary Applications)
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22 pages, 6099 KiB  
Article
Fault Diagnosis of Planetary Gearbox Based on Hierarchical Refined Composite Multiscale Fuzzy Entropy and Optimized LSSVM
by Xin Xia and Xiaolu Wang
Entropy 2025, 27(5), 512; https://doi.org/10.3390/e27050512 - 10 May 2025
Viewed by 144
Abstract
Efficient extraction and classification of fault features remain critical challenges in planetary gearbox fault diagnosis. A fault diagnosis framework is proposed that integrates hierarchical refined composite multiscale fuzzy entropy (HRCMFE) for feature extraction and a gray wolf optimization (GWO)-optimized least squares support vector [...] Read more.
Efficient extraction and classification of fault features remain critical challenges in planetary gearbox fault diagnosis. A fault diagnosis framework is proposed that integrates hierarchical refined composite multiscale fuzzy entropy (HRCMFE) for feature extraction and a gray wolf optimization (GWO)-optimized least squares support vector machine (LSSVM) for classification. Firstly, the HRCMFE is developed for feature extraction, which combines the segmentation advantage of hierarchical entropy (HE) and the computational stability advantage of refined composite multiscale fuzzy entropy (RCMFE). Secondly, the hyperparameters of LSSVM are optimized by GWO using a proposed fitness function. Finally, fault diagnosis of the planetary gearbox is achieved by the optimized LSSVM using the HRCMFE-extracted features. Simulation and experimental study results indicate that the proposed method demonstrates superior effectiveness in both feature discriminability and diagnosis accuracy. Full article
(This article belongs to the Special Issue Entropy-Based Fault Diagnosis: From Theory to Applications)
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18 pages, 729 KiB  
Article
Characterization of the Performance of an XXZ Three-Spin Quantum Battery
by Suman Chand, Dario Ferraro and Niccolò Traverso Ziani
Entropy 2025, 27(5), 511; https://doi.org/10.3390/e27050511 - 10 May 2025
Viewed by 264
Abstract
Quantum batteries represent a new and promising technological application of quantum mechanics, offering the potential for enhanced energy storage and fast charging. In this work, we study a quantum battery composed of three two-level systems with XXZ coupling operating under open boundary conditions. [...] Read more.
Quantum batteries represent a new and promising technological application of quantum mechanics, offering the potential for enhanced energy storage and fast charging. In this work, we study a quantum battery composed of three two-level systems with XXZ coupling operating under open boundary conditions. We investigate the role played by ferromagnetic and antiferromagnetic initial configurations on the charging dynamics of the battery. Two charging mechanisms are explored: static charging, where the battery interacts with a constant classical external field, and harmonic charging, where the field oscillates periodically over time. Our results demonstrate that static charging can be more efficient in the ferromagnetic case, achieving maximum energy due to complete population inversion between the ground and excited states. In contrast, harmonic charging excels in the antiferromagnetic case. By analyzing the stored energy and the average charging power in these two regimes, we highlight the impact of anisotropy on the performance of quantum batteries. Our findings provide valuable insights for optimizing quantum battery performance based on the system’s initial state and coupling configuration, paving the way for the study of more efficient quantum devices for energy storage. Full article
(This article belongs to the Special Issue Non-Equilibrium Quantum Many-Body Dynamics)
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16 pages, 276 KiB  
Article
Underlying Geometric Flow in Hamiltonian Evolution
by Gil Elgressy and Lawrence Horwitz
Entropy 2025, 27(5), 510; https://doi.org/10.3390/e27050510 - 9 May 2025
Viewed by 88
Abstract
In this paper, an underlying perturbed Ricci flow construction is made within the metric operator space, originating from the Heisenberg dynamical equations, to formulate a method which appears to provide a new geometric approach for the geometric formulation of the quantum mechanical dynamics. [...] Read more.
In this paper, an underlying perturbed Ricci flow construction is made within the metric operator space, originating from the Heisenberg dynamical equations, to formulate a method which appears to provide a new geometric approach for the geometric formulation of the quantum mechanical dynamics. A quantum mechanical notion of stability and local instability is introduced within the quantum mechanical theory, based on the quantum mechanical dynamical equations governing the evolution of the tensor metric operator. The stability analysis is conducted in the topology of little Ho¨lder spaces of metrics which the tensor metric operator acts on. Finally, a theorem is introduced in an attempt to characterize the stability properties of the quantum mechanical system such that it brings the quantum mechanical dynamics into the analysis. Full article
(This article belongs to the Special Issue Unstable Hamiltonian Systems and Scattering Theory)
57 pages, 10943 KiB  
Review
Jean-Marie Souriau’s Symplectic Foliation Model of Sadi Carnot’s Thermodynamics
by Frédéric Barbaresco
Entropy 2025, 27(5), 509; https://doi.org/10.3390/e27050509 - 9 May 2025
Viewed by 133
Abstract
The explanation of thermodynamics through geometric models was initiated by seminal figures such as Carnot, Gibbs, Duhem, Reeb, and Carathéodory. Only recently, however, has the symplectic foliation model, introduced within the domain of geometric statistical mechanics, provided a geometric definition of entropy as [...] Read more.
The explanation of thermodynamics through geometric models was initiated by seminal figures such as Carnot, Gibbs, Duhem, Reeb, and Carathéodory. Only recently, however, has the symplectic foliation model, introduced within the domain of geometric statistical mechanics, provided a geometric definition of entropy as an invariant Casimir function on symplectic leaves—specifically, the coadjoint orbits of the Lie group acting on the system, where these orbits are interpreted as level sets of entropy. We present a symplectic foliation interpretation of thermodynamics, based on Jean-Marie Souriau’s Lie group thermodynamics. This model offers a Lie algebra cohomological characterization of entropy, viewed as an invariant Casimir function in the coadjoint representation. The dual space of the Lie algebra is foliated into coadjoint orbits, which are identified with the level sets of entropy. Within the framework of thermodynamics, dynamics on symplectic leaves—described by the Poisson bracket—are associated with non-dissipative phenomena. Conversely, on the transversal Riemannian foliation (defined by the level sets of energy), the dynamics, characterized by the metric flow bracket, induce entropy production as transitions occur from one symplectic leaf to another. Full article
(This article belongs to the Special Issue 200 Years Anniversary of “Sadi Carnot, Réflexions Sur La Puissance Motrice Du Feu”; Bachelier: Paris, France, 1824)
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16 pages, 3758 KiB  
Article
In-Plane Gradient Magnetic Field-Induced Topological Defects in Rotating Spin-1 Bose–Einstein Condensates with SU(3) Spin-Orbit Coupling
by Hui Yang, Peng-Yu Li and Bo Yu
Entropy 2025, 27(5), 508; https://doi.org/10.3390/e27050508 - 9 May 2025
Viewed by 158
Abstract
We study the topological defects and spin structures of rotating SU(3) spin–orbit-coupled spin F=1 Bose–Einstein condensates (BECs) in an in-plane quadrupole field with ferromagnetic spin interaction, and the BECs is confined by a harmonic trap. Without rotation, as the quadrupole field [...] Read more.
We study the topological defects and spin structures of rotating SU(3) spin–orbit-coupled spin F=1 Bose–Einstein condensates (BECs) in an in-plane quadrupole field with ferromagnetic spin interaction, and the BECs is confined by a harmonic trap. Without rotation, as the quadrupole field strength is increased, the spin F=1 BECs with SU(3) spin–orbit coupling (SOC) evolves from the initial Thomas–Fermi phase into the stripe phase; then, it enters a vortex–antivortex cluster state and eventually a polar-core vortex state. In the absence of rotation with the given quadrupole field, the enhancing SU(3) SOC strength can cause a phase transition from a central Mermin–Ho vortex to a vortex–antivortex cluster, subsequently converting to a bending vortex–antivortex chain. In addition, when considering rotation, it is found that this system generates the following five typical quantum phases: a three-vortex-chain cluster structure with mutual angles of approximately 2π3, a tree-fork-like vortex chain cluster, a rotationally symmetric vortex necklace, a diagonal vortex chain cluster, and a density hole vortex cluster. Particularly, the system exhibits unusual topological structures and spin textures, such as a bending half-skyrmion–half-antiskyrmion (meron–antimeron) chain, three half-skyrmion (meron) chains with mutual angles of an approximately 2π3, slightly curved diagonal half-skyrmion (meron) cluster lattice, a skyrmion–half-skyrmion (skyrmion-meron) necklace, and a tree-fork-like half-skyrmion (meron) chain cluster lattice. Full article
(This article belongs to the Section Statistical Physics)
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22 pages, 7901 KiB  
Article
Integrating Human Mobility Models with Epidemic Modeling: A Framework for Generating Synthetic Temporal Contact Networks
by Diaoulé Diallo, Jurij Schoenfeld, René Schmieding, Sascha Korf, Martin J. Kühn and Tobias Hecking
Entropy 2025, 27(5), 507; https://doi.org/10.3390/e27050507 - 8 May 2025
Viewed by 220
Abstract
High-resolution temporal contact networks are useful ingredients for realistic epidemic simulations. Existing solutions typically rely either on empirical studies that capture fine-grained interactions via Bluetooth or wearable sensors in confined settings or on large-scale simulation frameworks that model entire populations using generalized assumptions. [...] Read more.
High-resolution temporal contact networks are useful ingredients for realistic epidemic simulations. Existing solutions typically rely either on empirical studies that capture fine-grained interactions via Bluetooth or wearable sensors in confined settings or on large-scale simulation frameworks that model entire populations using generalized assumptions. However, for most realistic modeling of epidemic spread and the evaluation of countermeasures, there is a critical need for highly resolved, temporal contact networks that encompass multiple venues without sacrificing the intricate dynamics of real-world contacts. This paper presents an integrated approach for generating such networks by coupling Bayesian-optimized human mobility models (HuMMs) with a state-of-the-art epidemic simulation framework. Our primary contributions are twofold: First, we embed empirically calibrated HuMMs into an epidemic simulation environment to create a parameterizable, adaptive engine for producing synthetic, high-resolution, population-wide temporal contact network data. Second, we demonstrate through empirical evaluations that our generated networks exhibit realistic interaction structures and infection dynamics. In particular, our experiments reveal that while variations in population size do not affect the underlying network properties—a crucial feature for scalability—altering location capacities naturally influences local connectivity and epidemic outcomes. Additionally, sub-graph analyses confirm that different venue types display distinct network characteristics consistent with their real-world contact patterns. Overall, this integrated framework provides a scalable and empirically grounded method for epidemic simulation, offering a powerful tool for generating and simulating contact networks. Full article
(This article belongs to the Special Issue Spreading Dynamics in Complex Networks)
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21 pages, 3009 KiB  
Article
Karatsuba Algorithm Revisited for 2D Convolution Computation Optimization
by Qi Wang, Jianghan Zhu, Can He, Shihang Wang, Xingbo Wang, Yuan Ren and Terry Tao Ye
Entropy 2025, 27(5), 506; https://doi.org/10.3390/e27050506 - 8 May 2025
Viewed by 147
Abstract
Convolution plays a significant role in many scientific and technological computations, such as artificial intelligence and signal processing. Convolutional computations consist of many dot-product operations (multiplication–accumulation, or MAC), for which the Winograd algorithm is currently the most widely used method to reduce the [...] Read more.
Convolution plays a significant role in many scientific and technological computations, such as artificial intelligence and signal processing. Convolutional computations consist of many dot-product operations (multiplication–accumulation, or MAC), for which the Winograd algorithm is currently the most widely used method to reduce the number of MACs. The Karatsuba algorithm, since its introduction in the 1960s, has been traditionally used as a fast arithmetic method to perform multiplication between large-bit-width operands. It had not been exploited to accelerate 2D convolution computations before. In this paper, we revisited the Karatsuba algorithm and exploited it to reduce the number of MACs in 2D convolutions. The matrices are first segmented into tiles in a divide-and-conquer method, and the resulting submatrices are overlapped to construct the final output matrix. Our analysis and benchmarks have shown that for convolution operations of the same dimensions, the Karatsuba algorithm requires the same number of multiplications but fewer additions as compared with the Winograd algorithm. A pseudocode implementation is also provided to demonstrate the complexity reduction in Karatsuba-based convolution. FPGA implementation of Karatsuba-based convolution also achieves 33.6% LUTs (Look -up Tables) reduction compared with Winograd-based implementation. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
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18 pages, 5837 KiB  
Article
Quantitative Assessment of the Trigger Effect of Proton Flux on Seismicity
by Alexey Lyubushin and Eugeny Rodionov
Entropy 2025, 27(5), 505; https://doi.org/10.3390/e27050505 - 8 May 2025
Viewed by 210
Abstract
An estimate of the trigger effect of the proton flux on seismicity was obtained. The proton flux time series with a time step of 5 min, 2000–2024, was analyzed. In each time interval of 5 days, statistics of the proton flux time series [...] Read more.
An estimate of the trigger effect of the proton flux on seismicity was obtained. The proton flux time series with a time step of 5 min, 2000–2024, was analyzed. In each time interval of 5 days, statistics of the proton flux time series were calculated: mean values, logarithm of kurtosis, spectral slope, singularities spectrum support width, wavelet-based entropy, and the Donoho–Johnston wavelet-based index. For each of the used statistics, time points of local extrema were found, and for each pair of time sequences of proton flux statistics and earthquakes with a magnitude of at least 6.5 in sliding time windows, the “advance measures” of each time sequence relative to the other were estimated using a model of the intensity of interacting point processes. The difference between the “direct” measure of the advance of time points of local extrema of proton flux statistics relative to the time moments of earthquakes and the “inverse” measure of the advance was calculated. The maximum proportion of the intensity of seismic events for which the proton flux was a trigger was estimated as 0.28 for using the points of the local minima of the singularities spectrum support width. Full article
(This article belongs to the Special Issue Time Series Analysis in Earthquake Complex Networks)
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19 pages, 3480 KiB  
Article
Forensic Support for Abraham et al.’s BB Protocol
by Qidi You, Hongjian Yang, Xiyong Zhang, Xiaotong Jiang, Kaiwen Guo and Kexin Hu
Entropy 2025, 27(5), 504; https://doi.org/10.3390/e27050504 - 8 May 2025
Viewed by 192
Abstract
The consensus protocol is a fundamental building block in distributed computing and has been widely used in blockchain systems in recent years. Paxos, introduced by Lamport, stands out as one of the most widely adopted consensus protocols and has found application in renowned [...] Read more.
The consensus protocol is a fundamental building block in distributed computing and has been widely used in blockchain systems in recent years. Paxos, introduced by Lamport, stands out as one of the most widely adopted consensus protocols and has found application in renowned distributed systems, including Google’s Spanner system. Abraham et al. analyzed the FaB Paxos protocol, a Byzantine version of Paxos. They abstracted the single-shot FaB Paxos into a Byzantine broadcast protocol and further gave an enhanced protocol known as Abraham et al.’s BB. Abraham et al.’s BB protocol achieved optimal two-round message interaction under good conditions, satisfying the optimal fault tolerance threshold of n=5t−1 where n represents total number of nodes in the system and t denotes the tolerable number of Byzantine nodes. This paper delves into scenarios wherein the actual number of Byzantine nodes surpasses the fault tolerance threshold during the operation of Abraham et al.’s BB protocol. To address this, we propose a forensic protocol designed to offer forensic support in cases of agreement violations. The forensic protocol aims to label Byzantine nodes through irrefutable evidence. We analyze the forensic protocol, elucidating the number of Byzantine nodes that the forensic protocol can label under different circumstances, along with the corresponding number of required messages. Additionally, we present an impossibility result, indicating that forensic support for Abraham et al.’s BB is impossible when the number of Byzantine nodes exceeds 2t−2. Full article
(This article belongs to the Special Issue Information-Theoretic Cryptography and Security)
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20 pages, 2304 KiB  
Article
Resilient Topology Reconfiguration for Industrial Internet of Things: A Feature-Driven Approach Against Heterogeneous Attacks
by Tianyu Wang, Dong Li, Bowen Zhang, Xianda Liu and Wenli Shang
Entropy 2025, 27(5), 503; https://doi.org/10.3390/e27050503 - 7 May 2025
Viewed by 139
Abstract
This paper proposes a feature-driven topology reconfiguration framework to enhance the resilience of Industrial Internet of Things (IIoT) systems against heterogeneous attacks. By dynamically partitioning IIoT into subnetworks based on localized attack features and reconstructing each subnetwork with tailored topologies, our framework significantly [...] Read more.
This paper proposes a feature-driven topology reconfiguration framework to enhance the resilience of Industrial Internet of Things (IIoT) systems against heterogeneous attacks. By dynamically partitioning IIoT into subnetworks based on localized attack features and reconstructing each subnetwork with tailored topologies, our framework significantly improves connectivity and communication efficiency. Evaluations on a real-world dataset (Tech-Routers-RF) characterizing IIoT topologies with 2113 nodes show that under diverse attack scenarios, connectivity and communication efficiency improve by more than 70% and 50%, respectively. Leveraging information entropy to quantify the trade-off between structural diversity and connection predictability, our work bridges adaptive network design with real-world attack dynamics, offering a scalable solution for securing large-scale IIoT deployments. Full article
(This article belongs to the Special Issue Spreading Dynamics in Complex Networks)
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19 pages, 3857 KiB  
Article
2024 ‘Key Reflections’ on Sadi Carnot’s 1824 ‘Réflexions’ and 200 Year Legacy
by Milivoje M. Kostic
Entropy 2025, 27(5), 502; https://doi.org/10.3390/e27050502 - 7 May 2025
Viewed by 141
Abstract
This author is not a philosopher nor a historian of science, but an engineering thermodynamicist. In that regard, and in addition to various philosophical “why and how” treatises and existing historical analyses, the physical and logical “what it is” [...] Read more.
This author is not a philosopher nor a historian of science, but an engineering thermodynamicist. In that regard, and in addition to various philosophical “why and how” treatises and existing historical analyses, the physical and logical “what it is” reflections, as sequential Key Points, where a key Sadi Carnot reasoning infers the next one, along with novel contributions and original generalizations, are presented. We need to keep in mind that in Sadi Carnot’s time (early 1800s), steam engines were inefficient (below 5%, so the heat in and out was comparable within experimental uncertainty, as if caloric were conserved), the conservation of caloric flourished (might be a fortunate misconception leading to the critical analogy with the waterwheel), and many critical thermal concepts, including the conservation of energy (The First Law), were not even established. If Clausius and Kelvin earned the title “Fathers of thermodynamics”, then Sadi Carnot was ‘the ingenious’ “Forefather of thermodynamics-to-become”. Full article
(This article belongs to the Section Thermodynamics)
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12 pages, 898 KiB  
Article
Network and Phase Symmetries Reveal That Amplitude Dynamics Stabilize Decoupled Oscillator Clusters
by Jeffrey Emenheiser, Anastasiya Salova, Jordan Snyder, James P. Crutchfield and Raissa M. D’Souza
Entropy 2025, 27(5), 501; https://doi.org/10.3390/e27050501 - 7 May 2025
Viewed by 80
Abstract
Oscillator networks display intricate synchronization patterns. Determining their stability typically requires incorporating the symmetries of the network coupling. Going beyond analyses that appeal only to a network’s automorphism group, we explore synchronization patterns that emerge from the phase-shift invariance of the dynamical equations [...] Read more.
Oscillator networks display intricate synchronization patterns. Determining their stability typically requires incorporating the symmetries of the network coupling. Going beyond analyses that appeal only to a network’s automorphism group, we explore synchronization patterns that emerge from the phase-shift invariance of the dynamical equations and symmetries in the nodes. We show that these nonstructural symmetries simplify stability calculations. We analyze a ring-network of phase–amplitude oscillators that exhibits a “decoupled” state in which physically-coupled nodes appear to act independently due to emergent cancellations in the equations of dynamical evolution. We establish that this state can be linearly stable for a ring of phase–amplitude oscillators, but not for a ring of phase-only oscillators that otherwise require explicit long-range, nonpairwise, or nonphase coupling. In short, amplitude–phase interactions are key to stable synchronization at a distance. Full article
(This article belongs to the Section Statistical Physics)
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26 pages, 8765 KiB  
Article
Precision in Brief: The Bayesian Hurst–Kolmogorov Method for the Assessment of Long-Range Temporal Correlations in Short Behavioral Time Series
by Madhur Mangalam and Aaron D. Likens
Entropy 2025, 27(5), 500; https://doi.org/10.3390/e27050500 - 6 May 2025
Viewed by 259
Abstract
Various fields within biological and psychological inquiry recognize the significance of exploring long-range temporal correlations to study phenomena. However, these fields face challenges during this transition, primarily stemming from the impracticality of acquiring the considerably longer time series demanded by canonical methods. The [...] Read more.
Various fields within biological and psychological inquiry recognize the significance of exploring long-range temporal correlations to study phenomena. However, these fields face challenges during this transition, primarily stemming from the impracticality of acquiring the considerably longer time series demanded by canonical methods. The Bayesian Hurst–Kolmogorov (HK) method estimates the Hurst exponents of time series—quantifying the strength of long-range temporal correlations or “fractality”—more accurately than the canonical detrended fluctuation analysis (DFA), especially when the time series is short. Therefore, the systematic application of the HK method has been encouraged to assess the strength of long-range temporal correlations in empirical time series in behavioral sciences. However, the Bayesian foundation of the HK method fuels reservations about its performance when artifacts corrupt time series. Here, we compare the HK method’s and DFA’s performance in estimating the Hurst exponents of synthetic long-range correlated time series in the presence of additive white Gaussian noise, fractional Gaussian noise, short-range correlations, and various periodic and non-periodic trends. These artifacts can affect the accuracy and variability of the Hurst exponent and, therefore, the interpretation and generalizability of behavioral research findings. We show that the HK method outperforms DFA in most contexts—while both processes break down for anti-persistent time series, the HK method continues to provide reasonably accurate H values for persistent time series as short as N=64 samples. Not only can the HK method detect long-range temporal correlations accurately, show minimal dispersion around the central tendency, and not be affected by the time series length, but it is also more immune to artifacts than DFA. This information becomes particularly valuable in favor of choosing the HK method over DFA, especially when acquiring a longer time series proves challenging due to methodological constraints, such as in studies involving psychological phenomena that rely on self-reports. Moreover, it holds significance when the researcher foreknows that the empirical time series may be susceptible to contamination from these processes. Full article
(This article belongs to the Section Complexity)
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10 pages, 8363 KiB  
Article
Improved Reconstruction of Chaotic Signals from Ordinal Networks
by Antonio Politi and Leonardo Ricci
Entropy 2025, 27(5), 499; https://doi.org/10.3390/e27050499 - 6 May 2025
Viewed by 177
Abstract
Permutation entropy is customarily implemented to quantify the intrinsic indeterminacy of complex time series, under the assumption that determinism manifests itself by lowering the (permutation) entropy of the resulting symbolic sequence. We expect this to be roughly true, but, in general, it is [...] Read more.
Permutation entropy is customarily implemented to quantify the intrinsic indeterminacy of complex time series, under the assumption that determinism manifests itself by lowering the (permutation) entropy of the resulting symbolic sequence. We expect this to be roughly true, but, in general, it is not clear to what extent a given ordinal pattern indeed provides a faithful reconstruction of the original signal. Here, we address this question by attempting the reconstruction of the original time series by invoking an ergodic Markov approximation of the symbolic dynamics, thereby inverting the encoding procedure. Using the Hénon map as a testbed, we show that a meaningful reconstruction can also be made in the presence of a small observational noise. Full article
(This article belongs to the Special Issue Ordinal Patterns-Based Tools and Their Applications)
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