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Entropy

Entropy is an international and interdisciplinary peer-reviewed open access journal of entropy and information studies, published monthly online by MDPI. The International Society for the Study of Information (IS4SI) and Spanish Society of Biomedical Engineering (SEIB) are affiliated with Entropy and their members receive a discount on the article processing charge.

Open Access— free for readers, with article processing charges (APC) paid by authors or their institutions.
High Visibility: indexed within Scopus, SCIE (Web of Science), Inspec, PubMed, PMC, Astrophysics Data System, and other databases.
Journal Rank: JCR - Q2 (Physics, Multidisciplinary) / CiteScore - Q1 (Mathematical Physics)
Rapid Publication: manuscripts are peer-reviewed and a first decision is provided to authors approximately 21.8 days after submission; acceptance to publication is undertaken in 2.6 days (median values for papers published in this journal in the first half of 2025).
Recognition of Reviewers: reviewers who provide timely, thorough peer-review reports receive vouchers entitling them to a discount on the APC of their next publication in any MDPI journal, in appreciation of the work done.
Testimonials: See what our editors and authors say about Entropy.
Companion journals for Entropy include: Foundations, Thermo and Complexities.

Impact Factor: 2.0 (2024); 5-Year Impact Factor: 2.2 (2024)

Imprint Information Journal Flyer Open Access ISSN: 1099-4300

Latest Articles

20 pages, 26260 KB

Open AccessArticle

AFMNet: A Dual-Domain Collaborative Network with Frequency Prior Guidance for Low-Light Image Enhancement

by Qianqian An and Long Ma

Entropy 2025, 27(12), 1220; https://doi.org/10.3390/e27121220 (registering DOI) - 1 Dec 2025

Low-light image enhancement (LLIE) degradation arises from insufficient illumination, reflectance occlusion, and noise coupling, and it manifests in the frequency domain as suppressed amplitudes with relatively stable phases. To address the fact that pure spatial mappings struggle to balance brightness enhancement and detail [...] Read more.

Low-light image enhancement (LLIE) degradation arises from insufficient illumination, reflectance occlusion, and noise coupling, and it manifests in the frequency domain as suppressed amplitudes with relatively stable phases. To address the fact that pure spatial mappings struggle to balance brightness enhancement and detail fidelity, whereas pure frequency-domain processing lacks semantic modeling, we propose AFMNet—a dual-domain collaborative enhancement network guided by an information-theoretic frequency prior. This prior regularizes global illumination, while spatial branches restore local details. First, a Multi-Scale Amplitude Estimator (MSAE) adaptively generates fine-grained amplitude-modulation maps via multi-scale fusion, encouraging higher output entropy through adaptive spectral-energy redistribution. Next, a Dual-Branch Spectral–Spatial Attention (DBSSA) module—comprising a Frequency-Modulated Attention Block (FMAB) and a Scale-Variable Depth Attention Block (SVDAB)—is employed: FMAB injects the modulation map as a frequency-domain prior into the attention mechanism to conditionally modulate the amplitude of value features while keeping the phase unchanged, thereby helping to preserve structural information in the enhanced output; SVDAB uses multi-scale depthwise-separable convolutions with scale attention to produce adaptively enhanced spatial features. Finally, a Spectral-Gated Feed-Forward Network (SGFFN) applies learnable spectral filters to local features for band-wise selective enhancement. This collaborative design achieves a favorable balance between illumination correction and detail preservation, and AFMNet delivers state-of-the-art performance on multiple low-light enhancement benchmarks. Full article

(This article belongs to the Special Issue Application of Information Theory to Computer Vision and Image Processing, 3rd Edition)

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24 pages, 1378 KB

Open AccessReview

Carnot and Joule Cycles Implied in Generalized Entropy and Exergy Properties as Fundamentals of Extrema Principles

by Pierfrancesco Palazzo

Entropy 2025, 27(12), 1219; https://doi.org/10.3390/e27121219 (registering DOI) - 29 Nov 2025

Thermodynamic laws and principles overarch all domains of physics, chemistry and biology. In this broad perspective, further generalizations with respect to the “state-of-the-art” of current theories are still viable considering all aspects of systems’ states and phenomena. This research aims to discuss the [...] Read more.

Thermodynamic laws and principles overarch all domains of physics, chemistry and biology. In this broad perspective, further generalizations with respect to the “state-of-the-art” of current theories are still viable considering all aspects of systems’ states and phenomena. This research aims to discuss the physical and informational implications of Carnot and Joule cycles and the properties inferred from their definitions, as well as the extrema principles governing non-equilibrium phenomena in complex systems. The approach adopted can be viewed as an analytical variational method focusing cycles’ performances and maxima of properties and parameters along cyclic processes. The dualisms and symmetries characterizing Carnot and Joule cycles imply the inference of the necessity and sufficiency of the stable equilibrium for equality of thermodynamic potentials of any system in any state. The conclusions provide a perspective on complex systems and non-equilibrium processes governed by the extrema principles and the physical and informational properties relating to complexity and self-organization. This treatise also represents a basis and a proposal for further developments looking forward to unifying thermodynamic and informational aspects of extrema principles in the direction of complexity, self-organization, constructal emergence and autopoiesis of non-living and living systems in the frame of a general unifying paradigm. Implications and applications are envisaged in an extended perspective accounting for sustainability, circularity and biotechnologies representing future scenarios of industry and environmental protection. Full article

17 pages, 765 KB

Open AccessArticle

Handwritten Digit Recognition with Flood Simulation and Topological Feature Extraction

by Rafał Brociek, Mariusz Pleszczyński, Jakub Błaszczyk, Maciej Czaicki and Christian Napoli

Entropy 2025, 27(12), 1218; https://doi.org/10.3390/e27121218 (registering DOI) - 29 Nov 2025

This paper introduces a novel approach to handwritten digit recognition based on directional flood simulation and topological feature extraction. While traditional pixel-based methods often struggle with noise, partial occlusion, and limited data, our method leverages the structural integrity of digits by simulating water [...] Read more.

This paper introduces a novel approach to handwritten digit recognition based on directional flood simulation and topological feature extraction. While traditional pixel-based methods often struggle with noise, partial occlusion, and limited data, our method leverages the structural integrity of digits by simulating water flow from image boundaries using a modified breadth-first search (BFS) algorithm. The resulting flooded regions capture stroke directionality, spatial segmentation, and closed-area characteristics, forming a compact and interpretable feature vector. Additional parameters such as inner cavities, perimeter estimation, and normalized stroke density enhance classification robustness. For efficient prediction, we employ the Annoy approximate nearest neighbors algorithm using ensemble-based tree partitioning. The proposed method achieves high accuracy on the MNIST (95.9%) and USPS (93.0%) datasets, demonstrating resilience to rotation, noise, and limited training data. This topology-driven strategy enables accurate digit classification with reduced dimensionality and improved generalization. Full article

(This article belongs to the Special Issue Application of Information Theory to Computer Vision and Image Processing, 3rd Edition)

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24 pages, 2960 KB

Open AccessArticle

Generalized M-Estimation-Based Framework for Robust Guidance Information Extraction

by Jiawei Ren, Xiaoyu Zhang, Shoupeng Li and Panlong Tan

Entropy 2025, 27(12), 1217; https://doi.org/10.3390/e27121217 (registering DOI) - 29 Nov 2025

This study tackles state estimation challenges in guidance information extraction. These challenges arise from non-Gaussian noise. We propose a robust framework to address them. The IMCIF framework effectively handles non-Gaussian noise in seeker measurements. However, noise with unstable and statistically undefined characteristics makes [...] Read more.

This study tackles state estimation challenges in guidance information extraction. These challenges arise from non-Gaussian noise. We propose a robust framework to address them. The IMCIF framework effectively handles non-Gaussian noise in seeker measurements. However, noise with unstable and statistically undefined characteristics makes optimal kernel width selection difficult. This limitation compromises estimation accuracy and may even lead to filter divergence. To resolve this issue, we first linearize the nonlinear model using statistical linear regression and integrate generalized M-estimation with IMCIF. SVD is introduced to enhance numerical stability and mitigate divergence caused by suboptimal kernel width selection. Furthermore, DCS kernel function is employed to address severe non-Gaussian noise induced by large field-of-view operations and target surface reflections. A modified weight function method is proposed to preserve the L2- norm criterion while ensuring estimation accuracy under Gaussian noise. Simulations confirm the algorithm’s precision in Gaussian noise. It also maintains high accuracy under significant non-Gaussian noise, proving robustness. These improvements address both numerical stability and adaptive noise suppression, thereby enhancing system reliability across diverse interference scenarios. This work targets guidance system designers needing real-time algorithms, and filtering researchers interested in robust fusion of M-estimation and information-theoretic learning. Full article

(This article belongs to the Section Multidisciplinary Applications)

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27 pages, 1429 KB

Open AccessArticle

TAN-FGBMLE: Tree-Augmented Naive Bayes Structure Learning Based on Fast Generative Bootstrap Maximum Likelihood Estimation for Continuous-Variable Classification

by Chenghao Wei, Tianyu Zhang, Chen Li, Pukai Wang and Zhiwei Ye

Entropy 2025, 27(12), 1216; https://doi.org/10.3390/e27121216 (registering DOI) - 28 Nov 2025

Tree-Augmented Naive Bayes (TAN) is an interpretable graphical structure model. However, its structure learning for continuous attributes depends on the class-conditional mutual information, which is sensitive to one-dimensional or two-dimensional density estimation. Accurate estimation is challenging under complex distributions such as multi-peak, long-tailed [...] Read more.

Tree-Augmented Naive Bayes (TAN) is an interpretable graphical structure model. However, its structure learning for continuous attributes depends on the class-conditional mutual information, which is sensitive to one-dimensional or two-dimensional density estimation. Accurate estimation is challenging under complex distributions such as multi-peak, long-tailed and heteroscedastic cases. To address this issue, we propose a structure learning method for TAN based on Fast Generative Bootstrap Maximum Likelihood Estimation (TAN-FGBMLE). FGBMLE consists of two stages of work. In the first stage, resampling weights and random noise are input into a network generator to rapidly produce candidate parameters, efficiently covering the latent density space without repeated independent optimization. In the second stage, optimal mixture weights are estimated by maximum likelihood estimation, assigning appropriate contributions to each candidate component. This design enables fast and accurate complex density estimation for both single and joint attributes, providing reliable computation of class-conditional mutual information. The TAN structure is then constructed using Prim’s maximum spanning tree algorithm. Experiments show that our estimation method attains higher fitting accuracy and lower runtime compared with traditional nonparametric estimators. By using open-source datasets, the TAN-FGBMLE achieves superior accuracy and recall compared to classic methods, demonstrating good robustness and interpretability. On publicly available real air quality data, it has a high classification result and produces graph structures that more accurately capture dependencies among continuous attributes. Full article

(This article belongs to the Special Issue Bridging Bayesian and Information-Theoretic Approaches in Earth and Environmental System Modeling: Theory and Practical Applications)

14 pages, 1007 KB

Open AccessArticle

Extreme Events and Event Size Fluctuations in Resetting Random Walks on Networks

by Xiaohan Sun, Shaoxiang Zhu and Anlin Li

Entropy 2025, 27(12), 1215; https://doi.org/10.3390/e27121215 (registering DOI) - 28 Nov 2025

Random walks with stochastic resetting, where walkers periodically return to a designated node, have emerged as an important framework for understanding transport processes in complex networks. While resetting is known to optimize search times, its effects on extreme events—defined as exceedances of walker [...] Read more.

Random walks with stochastic resetting, where walkers periodically return to a designated node, have emerged as an important framework for understanding transport processes in complex networks. While resetting is known to optimize search times, its effects on extreme events—defined as exceedances of walker flux above a critical threshold—remain largely unexplored. Such events model critical network phenomena, including traffic congestion, server overloads, and infrastructure failures. In this work, we systematically investigate how stochastic resetting influences both the probability and magnitude of extreme events in complex networks. Through analytical derivation of the stationary occupation probabilities and comprehensive numerical simulations, we demonstrate that resetting significantly reduces the occurrence of extreme events while concentrating event-size fluctuations. Our results reveal a universal suppression effect: increasing the resetting rate

γ

monotonically decreases extreme event probabilities across all nodes, with complete elimination at

γ = 1

. Notably, this suppression is most pronounced for vulnerable low-degree nodes and nodes distant from the resetting node, which experience the largest reduction in both event probability and fluctuation magnitude. These findings provide theoretical foundations for using resetting as a control mechanism to mitigate extreme events in networked systems. Full article

(This article belongs to the Special Issue Transport in Complex Environments)

22 pages, 4609 KB

Open AccessArticle

Statistical CSI-Based Beamspace Transmission for Massive MIMO LEO Satellite Communications

by Qian Dong, Yafei Wang, Nan Hu, Yiming Zhu, Wenjin Wang and Li Chai

Entropy 2025, 27(12), 1214; https://doi.org/10.3390/e27121214 (registering DOI) - 28 Nov 2025

In multibeam low-Earth-orbit (LEO) satellite systems, precoding has emerged as a key technology for mitigating co-channel interference (CCI) and for improving spectral efficiency (SE). However, its practical implementation is challenged by the difficulty of acquiring reliable instantaneous channel state information (iCSI) and by [...] Read more.

In multibeam low-Earth-orbit (LEO) satellite systems, precoding has emerged as a key technology for mitigating co-channel interference (CCI) and for improving spectral efficiency (SE). However, its practical implementation is challenged by the difficulty of acquiring reliable instantaneous channel state information (iCSI) and by the high computational complexity induced by large-scale antenna arrays, making it incompatible with fixed codebook-based beamforming schemes commonly adopted in operational systems. In this analysis, we propose a beamspace transmission framework leveraging statistical CSI (sCSI) and achieves reduced computational complexity compared with antenna-domain precoding designs. Specifically, we first propose a low-complexity beam selection algorithm that selects a small subset of beams for each user terminal (UT) from a fixed beamforming codebook, using only the UTs’ two-dimensional (2D) angular information. To suppress CCI among beams, we then derive a beamspace weighted minimum mean square error (WMMSE) precoding scheme based on the equivalent beamspace channel matrix. The derivation employs an sCSI-based WMMSE (sWMMSE) formulation derived from an upper bound approximation of the ergodic sum rate, which provides a tighter estimate than the expected mean square error (MSE)-based lower bound approximation. Simulation results demonstrate that the proposed sCSI-based beamspace transmission scheme achieves a favorable trade-off between performance and computational complexity. Full article

(This article belongs to the Topic Advances in Sixth Generation and Beyond (6G&B))

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29 pages, 561 KB

Open AccessArticle

Exploring Universal Domain Adaptation with CLIP Models: A Calibration Method

by Bin Deng

Entropy 2025, 27(12), 1213; https://doi.org/10.3390/e27121213 (registering DOI) - 28 Nov 2025

CLIP models have shown their impressive learning and transfer capabilities in a wide range of visual tasks. It is, however, interesting that these foundation models have not been fully explored for Universal Domain Adaptation (UniDA). In this paper, we make comprehensive empirical studies [...] Read more.

CLIP models have shown their impressive learning and transfer capabilities in a wide range of visual tasks. It is, however, interesting that these foundation models have not been fully explored for Universal Domain Adaptation (UniDA). In this paper, we make comprehensive empirical studies of state-of-the-art UniDA methods using these foundation models. We first demonstrate that although the foundation models greatly improve the performance of the baseline method (which trains the models on the source data alone), existing UniDA methods struggle to improve over the baseline. This suggests that new research efforts are necessary for UniDA using these foundation models. Finally, we observe that calibration of CLIP models plays a key role in UniDA. To this end, we propose a very simple calibration method via automatic temperature scaling, which significantly enhances the baseline’s out-of-class detection capability. We show that a single learned temperature outperforms previous approaches in most benchmark tasks when adapting from CLIP models, excelling in evaluation metrics including H-score and a newly proposed Universal Classification Rate (UCR) metric. We hope that our investigation and the proposed simple framework can serve as a strong baseline to facilitate future studies in this field. Full article

(This article belongs to the Special Issue Entropy in Machine Learning Applications, 2nd Edition)

26 pages, 2619 KB

Open AccessArticle

Pseudo-Sample Generation and Self-Supervised Framework for Infrared Dim and Small Target Detection

by Jinxin Guo, Weida Zhan, Dehua Huo, Depeng Zhu, Yu Chen, Yichun Jiang and Xiaoyu Xu

Entropy 2025, 27(12), 1212; https://doi.org/10.3390/e27121212 (registering DOI) - 28 Nov 2025

Infrared dim and small target detection is crucial for long-range sensing. However, its deep representation learning is severely constrained by the scarcity of accurately annotated real data, and related research remains underdeveloped. Existing data generation methods based on patch synthesis or geometric transformations [...] Read more.

Infrared dim and small target detection is crucial for long-range sensing. However, its deep representation learning is severely constrained by the scarcity of accurately annotated real data, and related research remains underdeveloped. Existing data generation methods based on patch synthesis or geometric transformations fail to incorporate the physical degradation mechanisms of infrared imaging systems and reasonable environmental constraints, leading to significant discrepancies between synthetic data and real-world scenarios. To address this issue, this paper proposes a novel pseudo-sample generation paradigm based on physics-informed degradation modeling and high-order constraints. First, we construct an infrared image degradation model that decouples the degradation processes of targets and backgrounds at the signal level, achieving accurate modeling of real infrared imaging while ensuring the reliability of the degradation process through information fidelity optimization. Second, an online grid-based high-order constraint strategy is designed, which synergistically integrates global semantic, local structural, and grayscale constraints based on statistical distribution consistency to generate a high-fidelity infrared simulation dataset. Finally, we build a complete self-supervised detection framework incorporating classical neural networks, customized loss functions, and two-dimensional information evaluation metrics. Extensive experiments demonstrate that the synthetic data generated by our method significantly outperforms existing simulated datasets on authenticity metrics. It also effectively enhances the generalization performance of various detectors in real-world scenarios, achieving detection accuracy superior to baseline models trained on traditional simulated data. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

35 pages, 11904 KB

Open AccessArticle

AdaptPest-Net: A Task-Adaptive Network with Graph–Mamba Fusion for Multi-Scale Agricultural Pest Recognition

by Jixiang Zou, Wenzhong Yang, Chuanxiang Li and Zhishan Feng

Entropy 2025, 27(12), 1211; https://doi.org/10.3390/e27121211 (registering DOI) - 28 Nov 2025

Accurate pest classification is critical for precision agriculture, yet existing deep learning methods face challenges including computational inefficiency from uniform sample processing and inadequate modeling of complex feature relationships. This paper proposes AdaptPest-Net, a task-adaptive architecture with three key innovations: (1) Sample-Difficulty-Aware Dynamic [...] Read more.

Accurate pest classification is critical for precision agriculture, yet existing deep learning methods face challenges including computational inefficiency from uniform sample processing and inadequate modeling of complex feature relationships. This paper proposes AdaptPest-Net, a task-adaptive architecture with three key innovations: (1) Sample-Difficulty-Aware Dynamic Routing (SDADR) employs Gaussian-gated path selection to adaptively route samples through shallow, medium, or deep networks based on predicted classification difficulty, improving accuracy by matching network capacity to sample complexity; (2) Graph Convolution-Mamba Fusion (GCMF) synergistically combines a 3-layer GCN with adaptive adjacency for explicit spatial structural modeling and a selective Mamba state-space model (T = 8, input-dependent parameters) for temporal feature dynamics, capturing complementary topological relationships and long-range dependencies through parallel dual-branch extraction; (3) Bidirectional Cross-Modal Attention enables deep integration between spatial and temporal modalities through mutual enhancement with batch-level knowledge transfer and adaptive gate fusion, where graph topology directly guides Mamba’s feature evolution. Comprehensive experiments on IP102 and D0 demonstrate that AdaptPest-Net achieves 78.4% accuracy on IP102, representing a significant improvement over existing methods, while D0 experiments validate strong cross-dataset generalization capability with 99.85% accuracy. Full article

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23 pages, 1977 KB

Open AccessArticle

Non-Parametric Goodness-of-Fit Tests Using Tsallis Entropy Measures

by Mehmet Siddik Cadirci

Entropy 2025, 27(12), 1210; https://doi.org/10.3390/e27121210 (registering DOI) - 28 Nov 2025

We develop goodness-of-fit (GOF) procedures rooted in Tsallis entropy, with a particular emphasis on multivariate exponential-power (generalized Gaussian) and q-Gaussian models. The GOF statistic compares a closed-form Tsallis entropy under the null with a non-parametric k-nearest-neighbor (k-NN) estimator. We [...] Read more.

We develop goodness-of-fit (GOF) procedures rooted in Tsallis entropy, with a particular emphasis on multivariate exponential-power (generalized Gaussian) and q-Gaussian models. The GOF statistic compares a closed-form Tsallis entropy under the null with a non-parametric k-nearest-neighbor (k-NN) estimator. We establish consistency and mean-square convergence of the estimator under mild regularity and tail assumptions, discuss an asymptotic normality regime as

q \to 1

, and calibrate critical values by parametric bootstrap/permutation. Extensive Monte Carlo experiments report empirical size, power, and runtime. These are reported across dimensions, k, and q. An applied example illustrates practical calibration and sensitivity, which are essential for accurate measurement. Full article

(This article belongs to the Special Issue Statistical Planning, Inference, and Decision Making in High-Dimensional Data Analysis)

14 pages, 482 KB

Open AccessArticle

Wealth Distribution Under Power Trading Frequencies and Transitions of Agents

by Rongmei Sun, Shaoyong Lai and Xia Zhou

Entropy 2025, 27(12), 1209; https://doi.org/10.3390/e27121209 (registering DOI) - 28 Nov 2025

We construct a kinetic model to investigate transactions among two populations from different countries. In our model, power collision kernels and a transfer operator are introduced into the Boltzmann equation. Through the method of continuous trading limits, we derive the Fokker–Planck equations and [...] Read more.

We construct a kinetic model to investigate transactions among two populations from different countries. In our model, power collision kernels and a transfer operator are introduced into the Boltzmann equation. Through the method of continuous trading limits, we derive the Fokker–Planck equations and find their entropy steady-state solutions for two special cases. Numerical experiments are provided to illustrate our results under different parameters. The conclusion shows that the increase of agents’ trading willingness and the decrease of agents’ transaction rate (the increase of saving rate) narrow the gap in agents’ wealth. If there is a significant disparity in the trading willingness between agents from the two countries, the overall wealth distribution exhibits a bimodal distribution pattern. Full article

(This article belongs to the Section Statistical Physics)

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24 pages, 3897 KB

Open AccessArticle

Argon Ion Treatment of Multi-Material Layered Surface-Electrode Traps for Noise Mitigation

by Deviprasath Palani, Florian Hasse, Philip Kiefer, Frederick Böckling, Daniel L. Stick, Dustin Hite, Ulrich Warring and Tobias Schaetz

Entropy 2025, 27(12), 1208; https://doi.org/10.3390/e27121208 (registering DOI) - 28 Nov 2025

Electric-field noise near ion-trap electrodes limits motional coherence and represents a key obstacle to scaling trapped-ion quantum systems. Here, we investigate how in situ Ar⁺ sputtering modifies motional heating and dephasing in multi-material surface-electrode traps. Trapped ions serve as local probes of [...] Read more.

Electric-field noise near ion-trap electrodes limits motional coherence and represents a key obstacle to scaling trapped-ion quantum systems. Here, we investigate how in situ Ar⁺ sputtering modifies motional heating and dephasing in multi-material surface-electrode traps. Trapped ions serve as local probes of electric-field fluctuations before and after controlled sputtering cycles. The data reveal a non-monotonic dependence of both the dephasing rate and the electric-field noise on the extent of Ar⁺ sputtering, with coherence initially improving while heating rates increase, followed by a reversal at longer exposures. This behavior highlights the intricate balance between beneficial surface cleaning and detrimental structural modification, driven by changes in surface morphology, redeposition of sputtered material, and diffusion on the surface, underscoring the complex interplay between surface composition and motional stability in multi-material electrode systems. Post-treatment scanning electron microscopy and energy-dispersive X-ray spectroscopy confirm significant modification of the multilayer structure. Technical noise was independently verified to be well below the observed levels. These findings indicate that in situ sputtering modifies surface properties in ways that can either mitigate or enhance electric-field noise, underscoring the need for precise control of material interfaces in next-generation ion-trap architectures. Full article

(This article belongs to the Special Issue Quantum Computing with Trapped Ions)

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18 pages, 35381 KB

Open AccessArticle

Stabilized Radial Basis Function Finite Difference Schemes with Mass Conservation for the Cahn–Hilliard Equation on Surfaces

by Jinwei Qiao, Yuanyang Qiao and Yinnian He

Entropy 2025, 27(12), 1206; https://doi.org/10.3390/e27121206 - 28 Nov 2025

It is well known that the Cahn–Hilliard equation satisfies the energy dissipation law and the mass conservation property. Recently, the radial basis function–finite difference (RBF–FD) approach and its numerous variants have garnered significant attention for the numerical solution of surface-related problems, owing to [...] Read more.

It is well known that the Cahn–Hilliard equation satisfies the energy dissipation law and the mass conservation property. Recently, the radial basis function–finite difference (RBF–FD) approach and its numerous variants have garnered significant attention for the numerical solution of surface-related problems, owing to their intrinsic advantage in handling complex geometries. However, existing RBF–FD schemes generally fail to preserve mass conservation when solving the Cahn–Hilliard equation on smooth closed surfaces. In this paper, based on an

L^{2}

projection method, two numerically efficient RBF–FD schemes are proposed to achieve mass-conservative numerical solutions, which are demonstrated to preserve the mass conservation law under relatively mild time-step constraints. Spatial discretization is performed using the RBF–FD method, while based on the convex splitting method and a linear stabilization technique, the first-order backward Euler formula (BDF1) and the second-order Crank–Nicolson (CN) scheme are employed for temporal integration. Extensive numerical experiments not only validate the performance of the proposed numerical schemes but also demonstrate their ability to utilize mild time steps for long-term phase-separation simulations. Full article

(This article belongs to the Section Thermodynamics)

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18 pages, 2576 KB

Open AccessArticle

Physical Origins of Memory Effects in a Non-Markovian Quantum Evolution

by Shao-Cheng Hou, Yu-Han Zhou, Xing-Yuan Zhang and Xue-Xi Yi

Entropy 2025, 27(12), 1207; https://doi.org/10.3390/e27121207 - 27 Nov 2025

We quantitatively investigate the physical origins of the non-Markovianity measure proposed in our previous work, which can be directly interpreted as memory effects, i.e., the dependence of a quantum system’s future evolution on its history. Using the properties of the trace norm and [...] Read more.

We quantitatively investigate the physical origins of the non-Markovianity measure proposed in our previous work, which can be directly interpreted as memory effects, i.e., the dependence of a quantum system’s future evolution on its history. Using the properties of the trace norm and the trace distance, we find that the strength of memory effects in an evolution is upper (lower) bounded by the sum (difference) of two quantities. One originates from (bounded by) the change of environment state caused by the system, the other from (bounded by) the correlations between the system and the environment. The simulation results for the Jaynes–Cummings model show that the two origins may contribute to the memory effects in different manners, depending on the initial states of the environment and the system. Full article

(This article belongs to the Special Issue Non-Markovian Open Quantum Systems)

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21 pages, 1144 KB

Open AccessArticle

Distributed Data Classification with Coalition-Based Decision Trees and Decision Template Fusion

by Katarzyna Kusztal and Małgorzata Przybyła-Kasperek

Entropy 2025, 27(12), 1205; https://doi.org/10.3390/e27121205 - 27 Nov 2025

In distributed data environments, classification tasks are challenged by inconsistencies across independently maintained sources. These environments are inherently characterized by high informational uncertainty. Our framework addresses this challenge through a structured process designed for the reduction of entropy in the overall decision-making process. [...] Read more.

In distributed data environments, classification tasks are challenged by inconsistencies across independently maintained sources. These environments are inherently characterized by high informational uncertainty. Our framework addresses this challenge through a structured process designed for the reduction of entropy in the overall decision-making process. This paper proposes a novel framework that integrates conflict analysis, coalition formation, decision tree induction, and decision template fusion to address these challenges. The method begins by identifying compatible data sources using Pawlak’s conflict model, forming coalitions that aggregate complementary information. Each coalition trains a decision tree classifier, and the final decision is derived through decision templates that fuse probabilistic outputs from all models. The proposed approach is compared with a variant that does not use coalitions, where each local source is modeled independently. Additionally, the framework extends previous work based on decision rules by introducing decision trees, which offer greater modeling flexibility while preserving interpretability. Experimental results on benchmark datasets from the UCI repository demonstrate that the proposed method consistently outperforms both the non-coalition variant and the rule-based version, particularly under moderate data dispersion. The key contributions of this work include the integration of coalition-based modeling with decision trees, the use of decision templates for interpretable fusion, and the demonstration of improved classification performance across diverse scenarios. Full article

(This article belongs to the Special Issue Entropy Method for Decision Making with Uncertainty)

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15 pages, 1474 KB

Open AccessArticle

Entropy, Periodicity and the Probability of Primality

by Grenville J. Croll

Entropy 2025, 27(12), 1204; https://doi.org/10.3390/e27121204 - 27 Nov 2025

The distribution of prime numbers has long been viewed as a balance between order and randomness. In this work, we investigate the relationship between entropy, periodicity, and primality through the computational framework of the binary derivative. We prove that periodic numbers are composite [...] Read more.

The distribution of prime numbers has long been viewed as a balance between order and randomness. In this work, we investigate the relationship between entropy, periodicity, and primality through the computational framework of the binary derivative. We prove that periodic numbers are composite in all bases except for a single trivial case and establish a set of twelve theorems governing the behavior of primes and composites in terms of binary periodicity. Building upon these results, we introduce a novel scale-invariant entropic measure of primality, denoted p(s′), which provides an exact and unconditional entropic probability of primality derived solely from the periodic structure of a binary number and its binary derivatives. We show that p(s′) is quadratic, statistically well-defined, and strongly correlated with our earlier BiEntropy measure of binary disorder. Empirical analyses across several numerical ranges demonstrate that the variance in prime density relative to quadratic expectation is small, binormal, and constrained by the central limit theorem. These findings reveal a deep connection between entropy and the randomness of the primes, offering new insights into the entropic structure of number theory, with implications for the Riemann Hypothesis, special classes of primes, and computational applications in cryptography. Full article

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34 pages, 1452 KB

Open AccessArticle

A Masi-Entropy Image Thresholding Based on Long-Range Correlation

by Perfilino Eugênio Ferreira Júnior, Vinícius Moreira Mello, Enzo P. Silva Ribeiro and Gilson Antonio Giraldi

Entropy 2025, 27(12), 1203; https://doi.org/10.3390/e27121203 - 27 Nov 2025

Entropy-based image thresholding is one of the most widely used segmentation techniques in image processing. The Tsallis and Masi entropies are information measures that can capture long-range interactions in various physical systems. On the other hand, Shannon entropy is more appropriate for short-range [...] Read more.

Entropy-based image thresholding is one of the most widely used segmentation techniques in image processing. The Tsallis and Masi entropies are information measures that can capture long-range interactions in various physical systems. On the other hand, Shannon entropy is more appropriate for short-range correlations. In this paper, we have improved a thresholding technique based on Tsallis and Shannon formulas by using Masi entropy. Specifically, we replace the Tsallis information measure with Masi’s one, obtaining better results than the original methodology. As the proposed method depends on an entropic parameter, we designed a thresholding algorithm that incorporates a simulated annealing procedure for parameter optimization. Then, we compared our results with thresholding methods that use just Masi (or Tsallis), or a combination of them, Shannon, Sine, and Hill entropies. The comparison is enriched with a kernel version of a support vector machine, as well as a discussion of our proposal in relation to deep learning approaches. Quantitative measures of segmentation accuracy demonstrated the superior performance of our method in infrared, nondestructive testing (NDT), as well as RGB images from the BSDS500 dataset. Full article

(This article belongs to the Section Signal and Data Analysis)

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37 pages, 4917 KB

Open AccessArticle

Transformer and Pre-Transformer Model-Based Sentiment Prediction with Various Embeddings: A Case Study on Amazon Reviews

by Ismail Duru and Ayşe Saliha Sunar

Entropy 2025, 27(12), 1202; https://doi.org/10.3390/e27121202 - 27 Nov 2025

Sentiment analysis is essential for understanding consumer opinions, yet selecting the optimal models and embedding methods remains challenging, especially when handling ambiguous expressions, slang, or mismatched sentiment–rating pairs. This study provides a comprehensive comparative evaluation of sentiment classification models across three paradigms: traditional [...] Read more.

Sentiment analysis is essential for understanding consumer opinions, yet selecting the optimal models and embedding methods remains challenging, especially when handling ambiguous expressions, slang, or mismatched sentiment–rating pairs. This study provides a comprehensive comparative evaluation of sentiment classification models across three paradigms: traditional machine learning, pre-transformer deep learning, and transformer-based models. Using the Amazon Magazine Subscriptions 2023 dataset, we evaluate a range of embedding techniques, including static embeddings (GloVe, FastText) and contextual transformer embeddings (BERT, DistilBERT, etc.). To capture predictive confidence and model uncertainty, we include categorical cross-entropy as a key evaluation metric alongside accuracy, precision, recall, and F1-score. In addition to detailed quantitative comparisons, we conduct a systematic qualitative analysis of misclassified samples to reveal model-specific patterns of uncertainty. Our findings show that FastText consistently outperforms GloVe in both traditional and LSTM-based models, particularly in recall, due to its subword-level semantic richness. Transformer-based models demonstrate superior contextual understanding and achieve the highest accuracy (92%) and lowest cross-entropy loss (0.25) with DistilBERT, indicating well-calibrated predictions. To validate the generalisability of our results, we replicated our experiments on the Amazon Gift Card Reviews dataset, where similar trends were observed. We also adopt a resource-aware approach by reducing the dataset size from 25 K to 20 K to reflect real-world hardware constraints. This study contributes to both sentiment analysis and sustainable AI by offering a scalable, entropy-aware evaluation framework that supports informed, context-sensitive model selection for practical applications. Full article

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30 pages, 1203 KB

Open AccessArticle

Quantum AI in Speech Emotion Recognition

by Michael Norval and Zenghui Wang

Entropy 2025, 27(12), 1201; https://doi.org/10.3390/e27121201 - 26 Nov 2025

We evaluate a hybrid quantum–classical pipeline for speech emotion recognition (SER) on a custom Afrikaans corpus using MFCC-based spectral features with pitch and energy variants, explicitly comparing three quantum approaches—a variational quantum classifier (VQC), a quantum support vector machine (QSVM), and a Quantum [...] Read more.

We evaluate a hybrid quantum–classical pipeline for speech emotion recognition (SER) on a custom Afrikaans corpus using MFCC-based spectral features with pitch and energy variants, explicitly comparing three quantum approaches—a variational quantum classifier (VQC), a quantum support vector machine (QSVM), and a Quantum Approximate Optimisation Algorithm (QAOA)-based classifier—against a CNN–LSTM (CLSTM) baseline. We detail the classical-to-quantum data encoding (angle embedding with bounded rotations and an explicit feature-to-qubit map) and report test accuracy, weighted precision, recall, and F1. Under ideal analytic simulation, the quantum models reach 41–43% test accuracy; under a realistic 1% NISQ noise model (100–1000 shots) this degrades to 34–40%, versus 73.9% for the CLSTM baseline. Despite the markedly lower empirical accuracy—expected in the NISQ era—we provide an end-to-end, noise-aware hybrid SER benchmark and discuss the asymptotic advantages of quantum subroutines (Chebyshev-based quantum singular value transformation, quantum walks, and block encoding) that become relevant only in the fault-tolerant regime. Full article

(This article belongs to the Special Issue The Future of Quantum Machine Learning and Quantum AI, 2nd Edition)

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