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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

26 pages, 5031 KB

Open AccessArticle

Analysis of Price Dynamic Competition and Stability in Cross-Border E-Commerce Supply Chain Channels Empowered by Blockchain Technology

by Le-Bin Wang, Jian Chai and Lu-Ying Wen

Entropy 2025, 27(10), 1076; https://doi.org/10.3390/e27101076 - 16 Oct 2025

Based on the perspective of multi-stage dynamic competition, this study constructs a discrete dynamic model of price competition between the “direct sales” and “resale” channels in cross-border e-commerce (CBEC) under three blockchain deployment modes. Drawing on nonlinear dynamics theory, the Nash equilibrium of [...] Read more.

Based on the perspective of multi-stage dynamic competition, this study constructs a discrete dynamic model of price competition between the “direct sales” and “resale” channels in cross-border e-commerce (CBEC) under three blockchain deployment modes. Drawing on nonlinear dynamics theory, the Nash equilibrium of the system and its stability conditions are examined. Using numerical simulations, the effects of factors such as the channel price adjustment speed, tariff rate, and commission ratio on the dynamic evolution, entropy, and stability of the system under the empowerment of blockchain technology are investigated. Furthermore, the impact of noise factors on system stability and the corresponding chaos control strategies are further analyzed. This study finds that a single-channel deployment tends to induce asymmetric system responses, whereas dual-channel collaborative deployment helps enhance strategic coordination. An increase in price adjustment speed, tariffs, and commission rates can drive the system’s pricing dynamics from a stable state into chaos, thereby raising its entropy, while the adoption of blockchain technology tends to weaken dynamic stability. Therefore, after deploying blockchain technology, each channel should make its pricing decisions more cautiously. Moderate noise can exert a stabilizing effect, whereas excessive disturbances may cause the system to diverge. Hence, enterprises should carefully assess the magnitude of disturbances and capitalize on the positive effects brought about by moderate fluctuations. In addition, the delayed feedback control method can effectively suppress chaotic fluctuations and enhance system stability, demonstrating strong adaptability across different blockchain deployment modes. Full article

(This article belongs to the Section Multidisciplinary Applications)

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27 pages, 21611 KB

Open AccessArticle

Aggregation in Ill-Conditioned Regression Models: A Comparison with Entropy-Based Methods

by Ana Helena Tavares, Ana Silva, Tiago Freitas, Maria Costa, Pedro Macedo and Rui A. da Costa

Entropy 2025, 27(10), 1075; https://doi.org/10.3390/e27101075 - 16 Oct 2025

Despite the advances on data analysis methodologies in the last decades, most of the traditional regression methods cannot be directly applied to large-scale data. Although aggregation methods are especially designed to deal with large-scale data, their performance may be strongly reduced in ill-conditioned [...] Read more.

Despite the advances on data analysis methodologies in the last decades, most of the traditional regression methods cannot be directly applied to large-scale data. Although aggregation methods are especially designed to deal with large-scale data, their performance may be strongly reduced in ill-conditioned problems (due to collinearity issues). This work compares the performance of a recent approach based on normalized entropy, a concept from information theory and info-metrics, with bagging and magging, two well-established aggregation methods in the literature, providing valuable insights for applications in regression analysis with large-scale data. While the results reveal a similar performance between methods in terms of prediction accuracy, the approach based on normalized entropy largely outperforms the other methods in terms of precision accuracy, even considering a smaller number of groups and observations per group, which represents an important advantage in inference problems with large-scale data. This work also alerts for the risk of using the OLS estimator, particularly under collinearity scenarios, knowing that data scientists frequently use linear models as a simplified view of the reality in big data analysis, and the OLS estimator is routinely used in practice. Beyond the promising findings of the simulation study, our estimation and aggregation strategies show strong potential for real-world applications in fields such as econometrics, genomics, environmental sciences, and machine learning, where data challenges such as noise and ill-conditioning are persistent. Full article

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6 pages, 172 KB

Open AccessEditorial

Advances in Quantum Computation in NISQ Era

by Xu-Dan Xie, Xiaoming Zhang, Balint Koczor and Xiao Yuan

Entropy 2025, 27(10), 1074; https://doi.org/10.3390/e27101074 - 15 Oct 2025

Realizing a universal, fault-tolerant quantum computer remains challenging with current technology [...] Full article

(This article belongs to the Special Issue Quantum Computing in the NISQ Era)

19 pages, 419 KB

Open AccessArticle

Information-Theoretic Analysis of Selected Water Force Fields: From Molecular Clusters to Bulk Properties

by Rodolfo O. Esquivel, Hazel Vázquez-Hernández and Alexander Pérez de La Luz

Entropy 2025, 27(10), 1073; https://doi.org/10.3390/e27101073 - 15 Oct 2025

We present a comprehensive information-theoretic evaluation of three widely used rigid water models (TIP3P, SPC, and SPC/

ε

) through systematic analysis of water clusters ranging from single molecules to 11-molecule aggregates. Five fundamental descriptors—Shannon entropy, Fisher information, disequilibrium, LMC complexity, and Fisher–Shannon [...] Read more.

We present a comprehensive information-theoretic evaluation of three widely used rigid water models (TIP3P, SPC, and SPC/

ε

) through systematic analysis of water clusters ranging from single molecules to 11-molecule aggregates. Five fundamental descriptors—Shannon entropy, Fisher information, disequilibrium, LMC complexity, and Fisher–Shannon complexity—were calculated in both position and momentum spaces to quantify electronic delocalizability, localization, uniformity, and structural sophistication. Clusters containing 1, 3, 5, 7, 9, and 11 molecules (denoted 1 M, 3 M, 5 M, 7 M, 9 M, and 11 M) were selected to balance computational tractability with representative scaling behavior. Molecular dynamics simulations validated the force fields against experimental bulk properties (density, dielectric constant, self-diffusion coefficient), while statistical analysis using Shapiro–Wilk normality tests and Student’s t-tests ensured robust discrimination between models. Our results reveal distinct scaling behaviors that correlate with experimental accuracy: SPC/

ε

demonstrates superior electronic structure representation with optimal entropy–information balance and enhanced complexity measures, while TIP3P shows excessive localization and reduced complexity that worsen with increasing cluster size. The transferability from clusters to bulk properties is established through systematic convergence of information-theoretic measures toward bulk-like behavior. The methodology establishes information-theoretic analysis as a useful tool for comprehensive force field evaluation. Full article

(This article belongs to the Special Issue The Information-Theoretic Approach in Density Functional Theory and Beyond)

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18 pages, 10279 KB

Open AccessArticle

Hypergraph Representation Learning with Weighted- and Clustering-Biased Random Walks

by Li Liang, Shi-Ming Cai and Shi-Cai Gong

Entropy 2025, 27(10), 1072; https://doi.org/10.3390/e27101072 - 15 Oct 2025

Hypergraphs are powerful tools for modeling complex systems because they naturally encode higher-order interactions. However, most existing hypergraph representation-learning methods still struggle to capture such high-order structures, particularly in heterogeneous hypergraphs, which results in suboptimal performance on structure-sensitive tasks such as node classification. [...] Read more.

Hypergraphs are powerful tools for modeling complex systems because they naturally encode higher-order interactions. However, most existing hypergraph representation-learning methods still struggle to capture such high-order structures, particularly in heterogeneous hypergraphs, which results in suboptimal performance on structure-sensitive tasks such as node classification. This paper presents WCRW-MLP, a new framework that integrates a Weighted- and Clustering-Biased Random Walk (WCRW) with a multi-layer perceptron. WCRW extends second-order random walks by introducing node-pair co-occurrence weights and triadic-closure clustering bias, enabling the walk to favor structurally significant and locally cohesive regions of the hypergraph. The resulting walk sequences are processed with Skip-gram to obtain high-quality structural embeddings, which are then concatenated with node attributes and fed into an MLP for classification. Experiments on several real-world hypergraph benchmarks show that WCRW-MLP consistently surpasses state-of-the-art baselines, validating both the efficacy of the proposed biasing strategy and the overall framework. These results demonstrate that explicitly modeling co-occurrence strength and local clustering is crucial for effective hypergraph embedding. Full article

(This article belongs to the Topic Computational Complex Networks)

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10 pages, 635 KB

Open AccessArticle

Impact of Homophily in Adherence to Anti-Epidemic Measures on the Spread of Infectious Diseases in Social Networks

by Piotr Bentkowski and Tomasz Gubiec

Entropy 2025, 27(10), 1071; https://doi.org/10.3390/e27101071 - 15 Oct 2025

We investigate how homophily in adherence to anti-epidemic measures affects the final size of epidemics in social networks. Using a modified SIR model, we divide agents into two behavioral groups—compliant and non-compliant—and introduce transmission probabilities that depend asymmetrically on the behavior of both [...] Read more.

We investigate how homophily in adherence to anti-epidemic measures affects the final size of epidemics in social networks. Using a modified SIR model, we divide agents into two behavioral groups—compliant and non-compliant—and introduce transmission probabilities that depend asymmetrically on the behavior of both the infected and susceptible individuals. We simulate epidemic dynamics on two types of synthetic networks with tunable inter-group connection probability: stochastic block models (SBM) and networks with triadic closure (TC) that better capture local clustering. Our main result reveals a counterintuitive effect: under conditions where compliant infected agents significantly reduce transmission, increasing the separation between groups may lead to a higher fraction of infections in the compliant population. This paradoxical outcome emerges only in networks with clustering (TC), not in SBM, suggesting that local network structure plays a crucial role. These findings highlight that increasing group separation does not always confer protection, especially when behavioral traits amplify within-group transmission. Full article

(This article belongs to the Special Issue Spreading Dynamics in Complex Networks)

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16 pages, 1206 KB

Open AccessArticle

Contrast Analysis on Spin Transport of Multi-Periodic Exotic States in the XXZ Chain

by Shixian Jiang, Jianpeng Liu and Yongqiang Li

Entropy 2025, 27(10), 1070; https://doi.org/10.3390/e27101070 - 15 Oct 2025

Quantum spin transport in integrable systems reveals a rich nonequilibrium phenomena that challenges the conventional hydrodynamic framework. Recent advances in ultracold atom experiments with state preparation and single-site addressing have enabled the understanding of this anomalous behavior. Particularly, the full universality characterization of [...] Read more.

Quantum spin transport in integrable systems reveals a rich nonequilibrium phenomena that challenges the conventional hydrodynamic framework. Recent advances in ultracold atom experiments with state preparation and single-site addressing have enabled the understanding of this anomalous behavior. Particularly, the full universality characterization of exotic initial states, as well as their measurement representation, remain unknown. By employing tensor network and contrast methods, we systematically investigate spin transport in the quantum XXZ spin chain and extract dynamical scaling exponents emerging from two paradigmatic and experimentally attainable initial states, i.e., multi-periodic domain-wall (MPDW) and spin-helix (SH) states. Our results using different values of anisotropic parameters

Δ \in [0, 1.2]

demonstrate the evident impeded transport and the difference between the two states with increasing

Δ

values. Large-scale and consistent simulations confirm the contrast method as a viable scaling extraction approach for exotic states with periodicity within experimentally accessible timescales. Our work establishes a foundation for studying initial memory and the corresponding relations of emergent transport behavior in nonequilibrium quantum systems, opening avenues for the identification of their unique universality classes. Full article

(This article belongs to the Special Issue Emergent Phenomena in Quantum Many-Body Systems)

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21 pages, 512 KB

Open AccessArticle

A Decision Tree Classification Algorithm Based on Two-Term RS-Entropy

by Ruoyue Mao, Xiaoyang Shi and Zhiyan Shi

Entropy 2025, 27(10), 1069; https://doi.org/10.3390/e27101069 - 14 Oct 2025

Classification is an important task in the field of machine learning. Decision tree algorithms are a popular choice for handling classification tasks due to their high accuracy, simple algorithmic process, and good interpretability. Traditional decision tree algorithms, such as ID3, C4.5, and CART, [...] Read more.

Classification is an important task in the field of machine learning. Decision tree algorithms are a popular choice for handling classification tasks due to their high accuracy, simple algorithmic process, and good interpretability. Traditional decision tree algorithms, such as ID3, C4.5, and CART, differ primarily in their criteria for splitting trees. Shannon entropy, Gini index, and mean squared error are all examples of measures that can be used as splitting criteria. However, their performance varies on different datasets, making it difficult to determine the optimal splitting criterion. As a result, the algorithms lack flexibility. In this paper, we introduce the concept of generalized entropy from information theory, which unifies many splitting criteria under one free parameter, as the split criterion for decision trees. We propose a new decision tree algorithm called RSE (RS-Entropy decision tree). Additionally, we improve upon a two-term information measure method by incorporating penalty terms and coefficients into the split criterion, leading to a new decision tree algorithm called RSEIM (RS-Entropy Information Method). In theory, the improved algorithms RSE and RSEIM are more flexible due to the presence of multiple free parameters. In experiments conducted on several datasets, using genetic algorithms to optimize the parameters, our proposed RSE and RSEIM methods significantly outperform traditional decision tree methods in terms of classification accuracy without increasing the complexity of the resulting trees. Full article

(This article belongs to the Section Multidisciplinary Applications)

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11 pages, 275 KB

Open AccessArticle

Relativistic Limits on the Discretization and Temporal Resolution of a Quantum Clock

by Tommaso Favalli

Entropy 2025, 27(10), 1068; https://doi.org/10.3390/e27101068 - 14 Oct 2025

We provide a brief discussion regarding relativistic limits on the discretization and temporal resolution of time values in a quantum clock. Our clock is characterized by a time observable chosen to be the complement of a bounded and discrete Hamiltonian that can have [...] Read more.

We provide a brief discussion regarding relativistic limits on the discretization and temporal resolution of time values in a quantum clock. Our clock is characterized by a time observable chosen to be the complement of a bounded and discrete Hamiltonian that can have an equally spaced or a generic spectrum. In the first case, the time observable can be described by a Hermitian operator, and we find a limit in the discretization for the time eigenvalues. Nevertheless, in both cases, the time observable can be described by a POVM, and, by increasing the number of time states, we show how the bound on the minimum time quantum can be reduced and identify the conditions under which the clock values can be treated as continuous. Finally, we find a limit for the temporal resolution of our time observable when the clock is used (together with light signals) in a relativistic framework for the measurement of spacetime distances. Full article

(This article belongs to the Special Issue Time in Quantum Mechanics)

15 pages, 10451 KB

Open AccessArticle

Noise Robustness of Transcript-Based Estimators for Properties of Interactions

by Manuel Adams and Klaus Lehnertz

Entropy 2025, 27(10), 1067; https://doi.org/10.3390/e27101067 - 14 Oct 2025

We investigate the robustness of transcript-based estimators for properties of interactions against various types of noise, ranging from colored noise to isospectral noise. We observe that all estimators are sensitive to symmetric and asymmetric contamination at signal-to-noise ratios that are orders of magnitude [...] Read more.

We investigate the robustness of transcript-based estimators for properties of interactions against various types of noise, ranging from colored noise to isospectral noise. We observe that all estimators are sensitive to symmetric and asymmetric contamination at signal-to-noise ratios that are orders of magnitude higher than those typically encountered in real-world applications. While different coupling regimes can still be distinguished and characterized sufficiently well, the strong impact of noise on the estimator for the direction of interaction can lead to severe misinterpretations of the underlying coupling structure. Full article

(This article belongs to the Special Issue Ordinal Patterns-Based Tools and Their Applications)

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16 pages, 1101 KB

Open AccessArticle

Analysis of Complex Network Attack and Defense Game Strategies Under Uncertain Value Criterion

by Chaoqi Fu and Zhuoying Shi

Entropy 2025, 27(10), 1066; https://doi.org/10.3390/e27101066 - 14 Oct 2025

The study of attack–defense game decision making in critical infrastructure systems confronting intelligent adversaries, grounded in complex network theory, has emerged as a prominent topic in the field of network security. Most existing research centers on game-theoretic analysis under conditions of complete information [...] Read more.

The study of attack–defense game decision making in critical infrastructure systems confronting intelligent adversaries, grounded in complex network theory, has emerged as a prominent topic in the field of network security. Most existing research centers on game-theoretic analysis under conditions of complete information and assumes that the attacker and defender share congruent criteria for evaluating target values. However, in reality, asymmetric value perception may lead to different evaluation criteria for both the offensive and defensive sides. This paper examines the game problem wherein the attacker and defender possess distinct target value evaluation criteria. The research findings reveal that both the attacker and defender have their own “advantage ranges” for value assessment, and topological heterogeneity is the reason for this phenomenon. Within their respective advantage ranges, the attacker or defender can adopt clear-cut strategies to secure optimal benefits—without needing to consider their opponents’ decisions. Outside these ranges, we explore how the attacker can leverage small-sample detection outcomes to probabilistically infer defenders’ strategies, and we further analyze the attackers’ preference strategy selections under varying acceptable security thresholds and penalty coefficients. The research results deliver more practical solutions for games involving uncertain value criteria. Full article

(This article belongs to the Section Complexity)

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17 pages, 1117 KB

Open AccessArticle

High-Efficiency Lossy Source Coding Based on Multi-Layer Perceptron Neural Network

by Yuhang Wang, Weihua Chen, Linjing Song, Zhiping Xu, Dan Song and Lin Wang

Entropy 2025, 27(10), 1065; https://doi.org/10.3390/e27101065 - 14 Oct 2025

With the rapid growth of data volume in sensor networks, lossy source coding systems achieve high–efficiency data compression with low distortion under limited transmission bandwidth. However, conventional compression algorithms rely on a two–stage framework with high computational complexity and frequently struggle to balance [...] Read more.

With the rapid growth of data volume in sensor networks, lossy source coding systems achieve high–efficiency data compression with low distortion under limited transmission bandwidth. However, conventional compression algorithms rely on a two–stage framework with high computational complexity and frequently struggle to balance compression performance with generalization ability. To address these issues, an end–to–end lossy compression method is proposed in this paper. The approach integrates an enhanced belief propagation algorithm with a multi–layer perceptron neural network, aiming to introduce a novel joint optimization architecture described as “encoding–structured encoding–decoding”. In addition, a quantization module incorporating random perturbation and the straight–through estimator is designed to address the non–differentiability in the quantization process. Simulation results demonstrate that the proposed system significantly improves compression performance while offering superior generalization and reconstruction quality. Furthermore, the designed neural architecture is both simple and efficient, reducing system complexity and enhancing feasibility for practical deployment. Full article

(This article belongs to the Special Issue Next-Generation Channel Coding: Theory and Applications)

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18 pages, 1960 KB

Open AccessArticle

CasDacGCN: A Dynamic Attention-Calibrated Graph Convolutional Network for Information Popularity Prediction

by Bofeng Zhang, Yanlin Zhu, Zhirong Zhang, Kaili Liao, Sen Niu, Bingchun Li and Haiyan Li

Entropy 2025, 27(10), 1064; https://doi.org/10.3390/e27101064 - 14 Oct 2025

Information popularity prediction is a critical problem in social network analysis. With the increasing prevalence of social platforms, accurate prediction of the diffusion process has become increasingly important. Existing methods mainly rely on graph neural networks to model structural relationships, but they are [...] Read more.

Information popularity prediction is a critical problem in social network analysis. With the increasing prevalence of social platforms, accurate prediction of the diffusion process has become increasingly important. Existing methods mainly rely on graph neural networks to model structural relationships, but they are often insufficient in capturing the complex interplay between temporal evolution and local cascade structures, especially in real-world scenarios involving sparse or rapidly changing cascades. To address this issue, we propose the Cascading Dynamic attention-calibrated Graph Convolutional Network, named CasDacGCN. It enhances prediction performance through spatiotemporal feature fusion and adaptive representation learning. The model integrates snapshot-level local encoding, global temporal modeling, cross-attention mechanisms, and a hypernetwork-based sample-wise calibration strategy, enabling flexible modeling of multi-scale diffusion patterns. Results from experiments demonstrate that the proposed model consistently surpasses existing approaches on two real-world datasets, validating its effectiveness in popularity prediction tasks. Full article

(This article belongs to the Special Issue Entropy-Centric Intelligent Computation with Graph: In Pursuit of Advanced Computational Theories, Methods, and Applications)

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24 pages, 9099 KB

Open AccessArticle

Dynamic MAML with Efficient Multi-Scale Attention for Cross-Load Few-Shot Bearing Fault Diagnosis

by Qinglei Zhang, Yifan Zhang, Jiyun Qin, Jianguo Duan and Ying Zhou

Entropy 2025, 27(10), 1063; https://doi.org/10.3390/e27101063 - 14 Oct 2025

Accurate bearing fault diagnosis under various operational conditions presents significant challenges, mainly due to the limited availability of labeled data and the domain mismatches across different operating environments. In this study, an adaptive meta-learning framework (AdaMETA) is proposed, which combines dynamic task-aware model-independent [...] Read more.

Accurate bearing fault diagnosis under various operational conditions presents significant challenges, mainly due to the limited availability of labeled data and the domain mismatches across different operating environments. In this study, an adaptive meta-learning framework (AdaMETA) is proposed, which combines dynamic task-aware model-independent meta-learning (DT-MAML) with efficient multi-scale attention (EMA) modules to enhance the model’s ability to generalize and improve diagnostic performance in small-sample bearing fault diagnosis across different load scenarios. Specifically, a hierarchical encoder equipped with C-EMA is introduced to effectively capture multi-scale fault features from vibration signals, greatly improving feature extraction under constrained data conditions. Furthermore, DT-MAML dynamically adjusts the inner-loop learning rate based on task complexity, promoting efficient adaptation to diverse tasks and mitigating domain bias. Comprehensive experimental evaluations on the CWRU bearing dataset, conducted under carefully designed cross-domain scenarios, demonstrate that AdaMETA achieves superior accuracy (up to 99.26%) and robustness compared to traditional meta-learning and classical diagnostic methods. Additional ablation studies and noise interference experiments further validate the substantial contribution of the EMA module and the dynamic learning rate components. Full article

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24 pages, 7771 KB

Open AccessArticle

Cross-Domain OTFS Detection via Delay–Doppler Decoupling: Reduced-Complexity Design and Performance Analysis

by Mengmeng Liu, Shuangyang Li, Baoming Bai and Giuseppe Caire

Entropy 2025, 27(10), 1062; https://doi.org/10.3390/e27101062 - 13 Oct 2025

In this paper, a reduced-complexity cross-domain iterative detection for orthogonal time frequency space (OTFS) modulation is proposed that exploits channel properties in both time and delay–Doppler domains. Specifically, we first show that in the time-domain effective channel, the path delay only introduces interference [...] Read more.

In this paper, a reduced-complexity cross-domain iterative detection for orthogonal time frequency space (OTFS) modulation is proposed that exploits channel properties in both time and delay–Doppler domains. Specifically, we first show that in the time-domain effective channel, the path delay only introduces interference among samples in adjacent time slots, while the Doppler becomes a phase term that does not affect the channel sparsity. This investigation indicates that the effects of delay and Doppler can be decoupled and treated separately. This “band-limited” matrix structure further motivates us to apply a reduced-size linear minimum mean square error (LMMSE) filter to eliminate the effect of delay in the time domain, while exploiting the cross-domain iteration for minimizing the effect of Doppler by noticing that the time and Doppler are a Fourier dual pair. Furthermore, we apply eigenvalue decomposition to the reduced-size LMMSE estimator, which makes the computational complexity independent of the number of cross-domain iterations, thus significantly reducing the computational complexity. The bias evolution and variance evolution are derived to evaluate the average MSE performance of the proposed scheme, which shows that the proposed estimators suffer from only negligible estimation bias in both time and DD domains. Particularly, the state (MSE) evolution is compared with bounds to verify the effectiveness of the proposed scheme. Simulation results demonstrate that the proposed scheme achieves almost the same error performance as the optimal detection, but only requires a reduced complexity. Full article

(This article belongs to the Special Issue Delay-Doppler Domain Communications for Future Wireless Networks 2nd Edition)

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24 pages, 6365 KB

Open AccessArticle

Synergizing High-Quality Tourism Development and Digital Economy: A Coupling Coordination Analysis in Chinese Prefecture-Level Cities

by Yuyan Luo, Yue Wang, Ziqi Pan, Huilin Li, Bin Lai and Yong Qin

Entropy 2025, 27(10), 1061; https://doi.org/10.3390/e27101061 - 12 Oct 2025

The rapid development of the digital economy (DE) provides a new driving force for high-quality tourism development (HQTD). How to coordinate HQTD and DE is an urgent issue to be resolved. In this study, the coupling coordination degree (CCD) between HQTD and DE [...] Read more.

The rapid development of the digital economy (DE) provides a new driving force for high-quality tourism development (HQTD). How to coordinate HQTD and DE is an urgent issue to be resolved. In this study, the coupling coordination degree (CCD) between HQTD and DE in Chinese prefecture-level cities is analysed using the CCD model, and the factors driving CCD are identified by Shapley additive explanations (SHAP). The results show that (1) Chinese city-level HQTD and DE show a rising trend from 2010 to 2019. The national average rises from 0.1807 and 0.2434 in 2010 to 0.2318 and 0.4113 in 2019, respectively, with HQTD’s development lagging noticeably behind DE. (2) CCD exhibits marked inter-regional disparities and intra-regional clustering. The northwest region has the lowest values, with many cities’ CCD below 0.5, indicating an imbalanced status. In 2019, all cities in the eastern region are in a balanced status, with Shanghai exceeding 0.8. (3) Total social retail sales per capita and percentage of tertiary sector are the key drivers of CCD; economic development and urbanisation rate exhibit a non-linear relationship with CCD. The CCD in developed cities in the east and north is driven by consumption, whereas the northwest region is primarily influenced by factors related to labour capital. Based on these conclusions, some policy implications are provided for the synergistic development of HQTD and DE. Full article

(This article belongs to the Section Multidisciplinary Applications)

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23 pages, 7050 KB

Open AccessArticle

Secure and Efficient Lattice-Based Ring Signcryption Scheme for BCCL

by Yang Zhang, Pengxiao Duan, Chaoyang Li, Haseeb Ahmad and Hua Zhang

Entropy 2025, 27(10), 1060; https://doi.org/10.3390/e27101060 - 12 Oct 2025

Blockchain-based cold chain logistics (BCCL) systems establish a new logistics data-sharing mechanism with blockchain technology, which destroys the traditional data island problem and promotes cross-institutional data interoperability. However, security vulnerabilities, risks of data loss, exposure of private information, and particularly the emergence of [...] Read more.

Blockchain-based cold chain logistics (BCCL) systems establish a new logistics data-sharing mechanism with blockchain technology, which destroys the traditional data island problem and promotes cross-institutional data interoperability. However, security vulnerabilities, risks of data loss, exposure of private information, and particularly the emergence of quantum-based attacks pose heightened threats to the existing BCCL framework. This paper first introduces a transaction privacy preserving (TPP) model for BCCLS that aggregates the blockchain and ring signcryption scheme together to strengthen the security of the data exchange process. Then, a lattice-based ring signcryption (LRSC) scheme is proposed. This LRSC utilizes the lattice assumption to enhance resistance against quantum attacks while employing ring mechanisms to safeguard the anonymity and privacy of the actual signer. It also executes signature and encryption algorithms simultaneously to improve algorithm execution efficiency. Moreover, the formal security proof results show that this LRSC can capture the signer’s confidentiality and unforgeability. Experimental findings indicate that the LRSC scheme achieves higher efficiency compared with comparable approaches. The proposed TPP model and LRSC scheme effectively facilitate cross-institutional logistics data exchange and enhance the utilization of logistics information via the BCCL system. Full article

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18 pages, 357 KB

Open AccessArticle

Exact ODE Framework for Classical and Quantum Corrections for the Lennard-Jones Second Virial Coefficient

by Zhe Zhao, Alfredo González-Calderón, Jorge Adrián Perera-Burgos, Antonio Estrada, Horacio Hernández-Anguiano, Celia Martínez-Lázaro and Yanmei Li

Entropy 2025, 27(10), 1059; https://doi.org/10.3390/e27101059 - 11 Oct 2025

The second virial coefficient (SVC) of the Lennard-Jones fluid is a cornerstone of molecular theory, yet its calculation has traditionally relied on the complex integration of the pair potential. This work introduces a fundamentally different approach by reformulating the problem in terms of [...] Read more.

The second virial coefficient (SVC) of the Lennard-Jones fluid is a cornerstone of molecular theory, yet its calculation has traditionally relied on the complex integration of the pair potential. This work introduces a fundamentally different approach by reformulating the problem in terms of ordinary differential equations (ODEs). For the classical component of the SVC, we generalize the confluent hypergeometric and Weber–Hermite equations. For the first quantum correction, we present entirely new ODEs and their corresponding exact-analytical solutions. The most striking result of this framework is the discovery that these ODEs can be transformed into Schrödinger-like equations. The classical term corresponds to a harmonic oscillator, while the quantum correction includes additional inverse-power potential terms. This formulation not only provides a versatile method for expressing the virial coefficient through a linear combination of functions (including Kummer, Weber, and Whittaker functions) but also reveals a profound and previously unknown mathematical structure underlying a classical thermodynamic property. Full article

(This article belongs to the Collection Foundations of Statistical Mechanics)

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16 pages, 2423 KB

Open AccessArticle

Exploring Ohm’s Law: The Randomness of Determinism

by Angel Cuadras, Marina Cuadras-Alba and Gaia Cuadras-Alba

Entropy 2025, 27(10), 1058; https://doi.org/10.3390/e27101058 - 11 Oct 2025

Ohm’s law has become ubiquitous in numerous scientific and technical disciplines. Generally, the subject is introduced to students in secondary school as fundamental technical knowledge. The present study proposes a visual model to facilitate the comprehension of Ohm’s law in electron transport in [...] Read more.

Ohm’s law has become ubiquitous in numerous scientific and technical disciplines. Generally, the subject is introduced to students in secondary school as fundamental technical knowledge. The present study proposes a visual model to facilitate the comprehension of Ohm’s law in electron transport in solids to pre-university and university students. The objective is to facilitate students’ comprehension of the correlation between electron movement in solids, as depicted by a current, and the energy of the system, which is introduced by the electric field and the material’s structure. The approach’s originality lies in its novel strategy for describing electron trajectory randomization. This enables the establishment of a relationship between the material’s structure and its resistivity. Moreover, the description of electron transport and scattering processes is presented regarding different types of entropy. It shows that electrons follow the maximum trajectory entropy and that thermal entropy has a quadratic relationship with configurational entropy. The determinism of Ohm’s law is inferred from statistical entropy. Full article

(This article belongs to the Section Multidisciplinary Applications)

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17 pages, 2165 KB

Open AccessArticle

Seizure Type Classification Based on Hybrid Feature Engineering and Mutual Information Analysis Using Electroencephalogram

by Yao Miao

Entropy 2025, 27(10), 1057; https://doi.org/10.3390/e27101057 - 11 Oct 2025

Epilepsy has diverse seizure types that challenge diagnosis and treatment, requiring automated and accurate classification to improve patient outcomes. Traditional electroencephalogram (EEG)-based diagnosis relies on manual interpretation, which is subjective and inefficient, particularly for multi-class differentiation in imbalanced datasets. This study aims to [...] Read more.

Epilepsy has diverse seizure types that challenge diagnosis and treatment, requiring automated and accurate classification to improve patient outcomes. Traditional electroencephalogram (EEG)-based diagnosis relies on manual interpretation, which is subjective and inefficient, particularly for multi-class differentiation in imbalanced datasets. This study aims to develop a hybrid framework for automated multi-class seizure type classification using segment-wise EEG processing and multi-band feature engineering to enhance precision and address data challenges. EEG signals from the TUSZ dataset were segmented into 1-s windows with 0.5-s overlaps, followed by the extraction of multi-band features, including statistical measures, sample entropy, wavelet energies, Hurst exponent, and Hjorth parameters. The mutual information (MI) approach was employed to select the optimal features, and seven machine learning models (SVM, KNN, DT, RF, XGBoost, CatBoost, LightGBM) were evaluated via 10-fold stratified cross-validation with a class balancing strategy. The results showed the following: (1) XGBoost achieved the highest performance (accuracy: 0.8710, F1 score: 0.8721, AUC: 0.9797), with

γ

-band features dominating importance. (2) Confusion matrices indicated robust discrimination but noted overlaps in focal subtypes. This framework advances seizure type classification by integrating multi-band features and the MI method, which offers a scalable and interpretable tool for supporting clinical epilepsy diagnostics. Full article

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

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