Entropy

22 pages, 2272 KB

Open AccessArticle

Atmospheric Turbulence Channel Modeling and Performance Analysis of a CO-ZP-OFDM Coherent Optical Communication System for UAV Air-to-Ground Scenarios

by Zheming Zhang, Junbin Lou, Yuanjin Lyu, Fanghui Huang, Dawei Wang, Sixu Lu and Yixin He

Entropy 2026, 28(6), 714; https://doi.org/10.3390/e28060714 (registering DOI) - 22 Jun 2026

Abstract

This paper targets the air-to-ground (A2G) data backhaul scenario of UAVs and proposes a communication system based on coherent optical zero-padding orthogonal frequency division multiplexing (CO-ZP-OFDM), which unifies atmospheric turbulence scintillation, pointing errors, and Doppler frequency shift into a composite channel model. The [...] Read more.

This paper targets the air-to-ground (A2G) data backhaul scenario of UAVs and proposes a communication system based on coherent optical zero-padding orthogonal frequency division multiplexing (CO-ZP-OFDM), which unifies atmospheric turbulence scintillation, pointing errors, and Doppler frequency shift into a composite channel model. The system employs the Gamma-Gamma (GG) distribution to describe turbulence-induced intensity fluctuations, a Gaussian beam truncation model to characterize pointing errors, and a dual-pilot method to estimate and compensate the Doppler frequency offset. Furthermore, on a polarization-time-frequency (PTF) three-dimensional orthogonal grid pilot structure, we derive theoretical mean square error (MSE) expressions for the zero-forcing (ZF) and minimum mean square error (MMSE) estimators, and analyze their MSE characteristics under the proposed pilot model. Simulation results show that, under moderate turbulence, the shrinkage factor of the MMSE estimator yields only about

0.4

dB MSE reduction over ZF at

SNR = 10

dB, whereas the full receiver pipeline that combines coherence-bandwidth pilot averaging with the MMSE and maximum ratio combining (MRC) equalizer reduces the empirical MSE by approximately 15 dB. The bit error rate (BER) performance tests indicate that, under turbulence-free conditions with ideal channel estimation, the system can reduce the BER below

10^{- 4}

at an SNR of approximately 12 dB. Under strong turbulence conditions with MMSE channel estimation, the SNR cost required to achieve a BER of

10^{- 3}

is approximately 18 dB, which corresponds to a 3 to 5 dB BER gain over the ZF baseline at the same SNR. Further simulation analysis shows that the average pointing loss is highly sensitive to the angular jitter at the 1 km link distance: an angular jitter of 1 mrad incurs about 18 dB of loss, and a sub-mrad pointing stability (i.e.,

σ_{jit} < 0.062

mrad) is required to keep the average pointing loss below 1 dB. Full article

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

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17 pages, 320 KB

Open AccessArticle

Information Geometry and Asymptotic Theory for SMML Estimators

by Enes Makalic and Daniel F. Schmidt

Entropy 2026, 28(6), 713; https://doi.org/10.3390/e28060713 (registering DOI) - 22 Jun 2026

Abstract

Strict minimum message length (SMML) is an information-theoretic coding principle that represents a continuous statistical model by a finite set of assertions and a partition of the sample space. We show that the SMML objective decomposes into assertion entropy and conditional cross-entropy, balancing [...] Read more.

Strict minimum message length (SMML) is an information-theoretic coding principle that represents a continuous statistical model by a finite set of assertions and a partition of the sample space. We show that the SMML objective decomposes into assertion entropy and conditional cross-entropy, balancing the cost of identifying an assertion against the cost of encoding data under the assigned model. For any fixed partition, the optimal codepoint for each cell is the model distribution that minimises Kullback–Leibler (KL) divergence from the data distribution restricted to that cell. Using the local Fisher–Rao geometry of regular parametric models, we show that, under a high-resolution LAN-scale regime, SMML partitions are asymptotically the pullback, through the maximum-likelihood estimator, of weighted Fisher–Rao Voronoi tessellations in parameter space, with assertion probabilities appearing as additive weights. For regular canonical exponential families, SMML codepoints satisfy a moment-matching condition and admit an interpretation as KL/Bregman centroids, while exact SMML cells are pullbacks of convex polyhedra in sufficient-statistic space. Together, these results show that SMML induces a natural information-geometric quantisation linking entropy-based coding, KL projection, and divergence-based Voronoi geometry. Full article

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

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18 pages, 712 KB

Open AccessHypothesis

Correlation Entropy and Power-Law Kinetics

by Joseph B. Bernstein

Entropy 2026, 28(6), 712; https://doi.org/10.3390/e28060712 (registering DOI) - 21 Jun 2026

Abstract

Power-law kinetics are observed across a wide range of physical, chemical, biological, and engineering systems, yet the thermodynamic origin of the power-law exponent remains incompletely understood. This work proposes a thermodynamic hypothesis in which power-law behavior emerges naturally from correlation-dependent contributions to the [...] Read more.

Power-law kinetics are observed across a wide range of physical, chemical, biological, and engineering systems, yet the thermodynamic origin of the power-law exponent remains incompletely understood. This work proposes a thermodynamic hypothesis in which power-law behavior emerges naturally from correlation-dependent contributions to the Gibbs free energy. Rather than modifying the classical Boltzmann definition of entropy, a phenomenological Correlation Constant, χ, is introduced to quantify how accumulated microstate evolution influences the accessibility of future states. The resulting correlation entropy contribution produces a free-energy term that modifies the probability of subsequent transitions and leads naturally to power-law kinetic behavior. Positive values of χ correspond to cooperative evolution in which prior evolution promotes future evolution, while negative values correspond to self-limiting behavior in which prior evolution suppresses subsequent evolution. The conventional Arrhenius-Eyring description is recovered as the special case χ = 0. The resulting framework provides a thermodynamic interpretation of the power-law exponent, establishes a connection between entropy, free energy, and kinetic evolution, and offers a unified description applicable to degradation, relaxation, diffusion, fatigue, trapping, and other evolving processes. The present work is intended as a thermodynamic hypothesis motivating further experimental and theoretical investigation of correlation-dependent kinetics. Full article

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

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34 pages, 1792 KB

Open AccessArticle

Research on the Contagion of Systemic Financial Risk Under the Impact of Climate Risks—From the Perspective of Complex Networks and Machine Learning

by Xiao-Li Gong, Xiao-Han Sun and Sergey Aleksandrovich Philin

Entropy 2026, 28(6), 711; https://doi.org/10.3390/e28060711 (registering DOI) - 21 Jun 2026

Abstract

To systematically examine the impact of climate risks on China’s financial system, this study employs the EGARCH-SGED model to precisely fit financial market volatility based on China’s Climate Change News Index. It then combines the LASSO-CoVaR method to measure tail risk spillover effects [...] Read more.

To systematically examine the impact of climate risks on China’s financial system, this study employs the EGARCH-SGED model to precisely fit financial market volatility based on China’s Climate Change News Index. It then combines the LASSO-CoVaR method to measure tail risk spillover effects within China’s financial system under climate risk shocks, constructs a risk contagion network, and innovatively utilizes the RF-AdaBoost model to establish the risk early warning system. Findings reveal that climate risk is a key driver of dynamic correlation evolution within the financial system, with heterogeneous impacts across different markets. Physical climate risk events intensify short-term risk contagion while generating long-term effects; transition risks undergo a dynamic process, initially amplifying uncertainty before enhancing systemic stability over the long term. The RF-AdaBoost model outperforms traditional machine learning models in risk warning, demonstrating outstanding predictive accuracy and generalization capabilities, thereby providing effective intellectual support for climate risk prevention and financial stability management. Full article

(This article belongs to the Section Complexity)

18 pages, 3814 KB

Open AccessArticle

The Statistical-Mechanical Meaning of the Wave Function of Quantum Mechanics

by Alberto Robledo

Entropy 2026, 28(6), 710; https://doi.org/10.3390/e28060710 (registering DOI) - 20 Jun 2026

Abstract

We address the paradoxical transformation of a classical-mechanical particle motion when the space and time scales of observation pass below the uncertainty principle threshold. This is analyzed in the language of classical statistical mechanics, considering specifically many-particle systems inhomogeneous along one spatial direction. [...] Read more.

We address the paradoxical transformation of a classical-mechanical particle motion when the space and time scales of observation pass below the uncertainty principle threshold. This is analyzed in the language of classical statistical mechanics, considering specifically many-particle systems inhomogeneous along one spatial direction. We employ the density functional formalism in its square-gradient form and find: (i) The macroscopic solution is analogous to the classical trajectory of a particle under a potential of force given by (minus) the free energy density. Whereas, (ii) fluctuations around the solution in (i) are equal to the quantum-mechanical wave functions of a particle under a potential given by the curvature of the free energy density. We illustrate this situation with three textbook examples: A particle in a box, the harmonic oscillator, and the hydrogen atom. We show that their time-independent Schrödinger equation wave functions describe, respectively, the fluctuations of a fluid interface, of critical point fluctuations, and of a confined ideal gas. At large scales, sharp probability distributions make fluctuations irrelevant; the vanishing of the first variation yields the macroscopically observable statistical-mechanical non-uniformity, equivalent to the classical particle trajectory. But at sufficiently small scales, with necessarily very few particles, distributions appear much wider, fluctuations dominate, and one obtains the Schrödinger equation (for the microscopic potential). Full article

(This article belongs to the Special Issue Quantum Ontology: Theory and Applications)

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15 pages, 464 KB

Open AccessArticle

A Fault Diagnosis Method for Transmission Networks Based on Multi-Source Information Fusion

by Shifu Gu, Xiaotian Chen, Tao Wang, Quanlin Leng and Chunyu Zhou

Entropy 2026, 28(6), 709; https://doi.org/10.3390/e28060709 (registering DOI) - 20 Jun 2026

Abstract

In order to solve the miscalculation problem caused by the distortion and loss of fault information caused by the traditional transmission grid fault diagnosis method due to the severe meteorological environment, a transmission grid fault diagnosis method based on multi-source information fusion is [...] Read more.

In order to solve the miscalculation problem caused by the distortion and loss of fault information caused by the traditional transmission grid fault diagnosis method due to the severe meteorological environment, a transmission grid fault diagnosis method based on multi-source information fusion is proposed. Firstly, the pulse fault degree, amplitude fault degree and meteorological fault degree are obtained by analyzing the switching, electrical and meteorological information from multiple sources using the binary reasoning spiking neural P systems, Hilbert–Huang transform and meteorological fusion methods, respectively. Then, the fault diagnosis results are obtained by fusing the various fault degrees using the analytic hierarchy process. Finally, simulation experiments are conducted on the standard IEEE39-bus system built by PSCAD simulation software, and the results verify the feasibility and effectiveness of the proposed diagnosis method in this paper. Full article

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

29 pages, 18277 KB

Open AccessArticle

Task Graph Generation for Heterogeneous UAV Swarms in Partially Observable Adversarial Environments

by Wenxin Li and Yongxin Feng

Entropy 2026, 28(6), 708; https://doi.org/10.3390/e28060708 (registering DOI) - 18 Jun 2026

Abstract

In partially observable adversarial environments, heterogeneous unmanned aerial vehicle (UAV) swarms must generate tasks online from noisy observations while respecting platform capabilities, consumable resources, and structural dependencies among tasks. This paper proposes a task graph generation method that converts local observations, target beliefs, [...] Read more.

In partially observable adversarial environments, heterogeneous unmanned aerial vehicle (UAV) swarms must generate tasks online from noisy observations while respecting platform capabilities, consumable resources, and structural dependencies among tasks. This paper proposes a task graph generation method that converts local observations, target beliefs, and UAV resource states into executable task graphs with explicit resource semantics and inter-task relations. The method first constructs a sufficiently expressive candidate task graph in the belief and resource spaces. An offline search teacher then evaluates future trajectory particles, resource feasibility, and structural interaction values to produce supervision for node selection, marginal task value, and relation prediction. A relation-biased graph attention network learns to generate task graphs online, and a task manager further performs task filtering, dependency repair, conflict completion, and resource checking. Simulation results under complex observation pressure and unseen adversarial strategies show that the proposed method consistently improves structural generation quality and execution feasibility. Compared with Graphormer, it improves the task-graph utility, task-edge F1-score, and executable-graph ratio by 5.83%, 5.41%, and 2.68%, respectively, while reducing the infeasible-task ratio by 35.14%. These results indicate that combining an offline search teacher with resource-constrained graph modeling provides an effective front-end task organization mechanism for heterogeneous UAV swarm planning. Full article

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13 pages, 1560 KB

Open AccessArticle

Dual-Channel Voice Communication System Based on One-Way Quantum Secure Direct Communication—Classical Optical Communication Hybrid Mode

by Xiuwei Chen, Dong Pan and Jianxing Guo

Entropy 2026, 28(6), 707; https://doi.org/10.3390/e28060707 (registering DOI) - 18 Jun 2026

Abstract

Quantum secure direct communication, as an important branch of quantum communication, possesses strict information-theoretic security and can achieve secure communication in channel environments with noise interference and eavesdropping threats. As voice communication is the most fundamental and widespread communication method in daily life, [...] Read more.

Quantum secure direct communication, as an important branch of quantum communication, possesses strict information-theoretic security and can achieve secure communication in channel environments with noise interference and eavesdropping threats. As voice communication is the most fundamental and widespread communication method in daily life, guaranteeing its security and efficiency has become an important research topic in current communication technology. One-way quantum secure direct communication technology can build an efficient and reliable security barrier for voice communication services, effectively preventing the leakage of private information in voice communication. This paper proposes a duplex voice communication scheme based on one-way quantum secure direct communication. By adopting a method combining multi-task parallel processing and stream processing, the communication rate and transmission delay performance of the system are significantly improved. Relying on quantum secure direct communication technology and the one-time-key encryption channel within the system, duplex voice communication is achieved securely. The real-time temperature drift compensation algorithm is introduced to ensure the long-term stable operation of the system. At the same time, through the real-time temperature drift prediction mechanism, the strategy selection during the call process is optimized to ensure the quality of the voice communication. To verify the feasibility and performance of this scheme, a one-way quantum secure direct communication duplex voice communication system was built in the laboratory environment, and comprehensive performance indicator tests were conducted. The test results show that the constructed one-way quantum secure direct communication system can fully meet the performance requirements of duplex voice communication. The realization of this system successfully achieves the goal of secure and efficient quantum voice communication, laying an important technical foundation for further expanding the practical application scenarios of quantum communication technology and promoting the industrialization development of quantum communication. Full article

(This article belongs to the Special Issue New Advances in Quantum Communication and Networks, 2nd Edition)

20 pages, 373 KB

Open AccessArticle

Forward-Secure Linearly Homomorphic Signature Scheme in the Standard Model and Its Application

by Linlin Wang and Zuling Chang

Entropy 2026, 28(6), 706; https://doi.org/10.3390/e28060706 (registering DOI) - 18 Jun 2026

Abstract

Linearly homomorphic signatures (LHSs) are widely used in scenarios such as network coding and the Internet of Things, but their security faces the serious threat of key leakage. To address this issue, this paper introduces a forward secure mechanism into LHSs, aiming to [...] Read more.

Linearly homomorphic signatures (LHSs) are widely used in scenarios such as network coding and the Internet of Things, but their security faces the serious threat of key leakage. To address this issue, this paper introduces a forward secure mechanism into LHSs, aiming to construct a linearly homomorphic signature (LHS) scheme that can resist the risk of key leakage. By combining the binary tree minimal cover set mechanism with lattice-based extension algorithms, we construct an LHS scheme that supports time-period key updates. We prove its forward secure unforgeability under the standard model (SM) by reducing it to the Short Integer Solution (SIS) problem. To the best of our knowledge, this scheme is the first provably secure lattice-based forward secure linearly homomorphic signature (FSLHS) scheme in the SM, filling a theoretical gap in existing research. Furthermore, we apply this scheme to a smart grid data acquisition system and verify its practicality through concrete performance analysis. Full article

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

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37 pages, 5828 KB

Open AccessArticle

Geodesic Execution Slippage: A Statistical Physics Framework for Cryptocurrency Liquidity Risk

by Ntebogang Dinah Moroke and Lebotsa Daniel Metsileng

Entropy 2026, 28(6), 705; https://doi.org/10.3390/e28060705 (registering DOI) - 18 Jun 2026

Abstract

Standard cryptocurrency transaction cost models assume flat geometry and assign execution cost as a proportional fee. This paper proposes GEODEX, a framework that models execution slippage as the geodesic arc length on the Fisher information manifold of a Markov-switching GARCH maximum-entropy model, augmented [...] Read more.

Standard cryptocurrency transaction cost models assume flat geometry and assign execution cost as a proportional fee. This paper proposes GEODEX, a framework that models execution slippage as the geodesic arc length on the Fisher information manifold of a Markov-switching GARCH maximum-entropy model, augmented by a joint curvature–topological fragmentation alarm. The Curvature-Fragmentation Law (Proposition 2) is an analytically derived heuristic. Its empirical validity is confirmed across four crisis episodes. Ablation confirms that each geometric component contributes uniquely: removing the geodesic increases mean squared prediction error by 2.9%, removing topological data analysis by 2.1%, and removing curvature by 1.5%. On five cryptocurrency markets (BTC, ETH, XRP, LTC, and BCH), over 2253 daily observations, the framework achieves competitive prediction error and is the only single-signal model retained in the Model Confidence Set at

α = 0.10

against eight benchmarks. A joint curvature–topological alarm fires a median of two days before price-based circuit breaker thresholds across four crisis episodes, including the Terra collapse (May 2022) and FTX bankruptcy (November 2022). Online inference requires under one second; full offline calibration requires approximately 28 h. The framework requires no additional data beyond the upstream estimation pipeline and supports SDG 10 (Reduced Inequalities) and SDG 16 (Strong Institutions) by enabling accessible geometric liquidity intelligence for regulators and smaller market participants. Full article

(This article belongs to the Special Issue Geometric Perspectives in Emergent Phenomena: From Phase Transitions to Machine Learning)

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16 pages, 327 KB

Open AccessArticle

Toward a Tripartite Taxonomy of Entropy in Physics

by Antoine Druilhe

Entropy 2026, 28(6), 704; https://doi.org/10.3390/e28060704 - 18 Jun 2026

Abstract

The term “entropy” denotes several mathematically distinct quantities across modern physics, including thermodynamic, statistical, quantum-informational, and geometric notions that are often conflated in foundational discussions. We propose an operational distinction among three such quantities: a geometric capacity entropy

S_{capacity}

proportional to a [...] Read more.

The term “entropy” denotes several mathematically distinct quantities across modern physics, including thermodynamic, statistical, quantum-informational, and geometric notions that are often conflated in foundational discussions. We propose an operational distinction among three such quantities: a geometric capacity entropy

S_{capacity}

proportional to a region’s bounding area, a microscopic content entropy

S_{content}

given by the fine-grained von Neumann entropy of the reduced state, and a thermodynamic entropy

S_{thermo}

corresponding to the observer-relative ignorance that remains after accessible information is accounted for. We argue that keeping these quantities distinct is not merely terminological: within this framework, the second law of thermodynamics can be formulated as a typical consequence of unitary dynamics combined with bounded observational access, rather than as an independent postulate. The distinction also clarifies which entropy enters established results such as the Bekenstein–Hawking entropy of black holes and the Clausius relation in Jacobson’s thermodynamic derivation of Einstein’s equations. The proposed framework is conceptual and does not modify established physical theories; it is intended as a useful clarification for informational approaches to physics. Full article

(This article belongs to the Section Quantum Information)

23 pages, 2980 KB

Open AccessArticle

Grouped Feature Representation and Gated Multilayer Perceptron for Event-Level Football Pass Outcome Prediction

by Yijuan Yuan, Shaosong Wang, Yonghong Deng and Zhibin Li

Entropy 2026, 28(6), 703; https://doi.org/10.3390/e28060703 - 17 Jun 2026

Abstract

Accurate prediction of football pass outcomes is important for tactical analysis, decision evaluation, and skill-oriented feedback in student football training and physical education. However, event-level pass outcome prediction remains challenging because pass success is jointly influenced by spatial context, defensive pressure, receiver-related cues, [...] Read more.

Accurate prediction of football pass outcomes is important for tactical analysis, decision evaluation, and skill-oriented feedback in student football training and physical education. However, event-level pass outcome prediction remains challenging because pass success is jointly influenced by spatial context, defensive pressure, receiver-related cues, and historical coordination between players. To address this issue, this study proposes an information-guided multilayer perceptron (IGMLP) based on grouped feature representation and gated feature fusion using structured event data. In the proposed framework, input variables are organized into interpretable semantic feature groups, including contextual features, pressure-aware features, historical coordination features, and receiver-related features. These groups are encoded through separate branches and adaptively fused by a group-level gating mechanism for nonlinear pass outcome modeling. Unlike conventional gated neural architectures that usually apply generic gates to hidden units, channels, or sequential states, the proposed gated design operates at the semantic feature-group level and adaptively weights football-specific information sources according to their relevance to each pass event. Using the StatsBomb open-event dataset, both prediction and recognition paths were constructed, and the proposed model was compared with standard multilayer perceptron (MLP), residual neural network (ResNet), boosting tree (BT), convolutional neural network (CNN), and long short-term memory network (LSTM). In the prediction path, IGMLP achieved an Accuracy of 0.9184, Precision of 0.9295, Recall of 0.9837, F1-score of 0.9558, and AUC of 0.9325. In the recognition path, IGMLP achieved an Accuracy of 0.9808, Precision of 0.9882, Recall of 0.9902, F1-score of 0.9893, and AUC of 0.9925. These results indicate that semantic feature grouping and gated feature fusion are effective for event-level football pass outcome prediction. Full article

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

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29 pages, 1562 KB

Open AccessArticle

ICU Delirium as a Failure of Predictive Synchronization: A Two-Agent Active Inference Model

by Luca M. Possati

Entropy 2026, 28(6), 702; https://doi.org/10.3390/e28060702 - 17 Jun 2026

Abstract

This paper presents a computational model of delirium in the Intensive Care Unit (ICU), in which delirium is defined as the endpoint of a self-reinforcing cycle of predictive failure between two bidirectionally coupled agents: the patient and the ICU room environment. Drawing on [...] Read more.

This paper presents a computational model of delirium in the Intensive Care Unit (ICU), in which delirium is defined as the endpoint of a self-reinforcing cycle of predictive failure between two bidirectionally coupled agents: the patient and the ICU room environment. Drawing on the active inference framework and the free energy principle, the paper proposes that delirium is not a property of the patient in isolation but a relational phenomenon that emerges when the environment persistently fails to predict the patient’s internal state. This failure triggers a causal feedback mechanism in which desynchronization pressure progressively sharpens the patient’s prior beliefs—implementing precision rigidity in the correct active inference sense: not a brain overwhelmed by noise but a brain locked into a state that incoming observations can no longer update. The model is implemented as a two-agent POMDP in which both agents maintain generative models and continuously attempt to predict each other’s states. The room agent (R)—understood as the environment-side sensing–inference–actuation loop, whether instantiated by clinical staff or by an automated monitoring system—infers the patient (P)’s latent parameters

(θ_{cog}, θ_{emo})

over time and builds a progressively personalized generative model of the patient. Synchronization is operationalized via two commensurable directional surprisal metrics:

S_{R \to P} = - ln Q_{R} (s^{*})

, the room’s surprisal at the patient’s true state, and

S_{P \to R} = - ln P (o_{R} ∣ Q_{P})

, the patient’s surprisal at the room’s observations. A systematic ablation study across four model variants shows that room inference is the architectural component necessary to reproduce the synchronization–delirium relationship: when the room infers, the association between synchronization and declared delirium is strong and stable, whereas a non-inferring room collapses to ceiling delirium rates and a weak association.

θ

learning and the prior-sharpening feedback do not increase the strength of this association; instead they shape the phenotypic gradient, reducing ceiling effects in vulnerable phenotypes and amplifying the separation between them. The model is presented as a computational hypothesis generator rather than a calibrated clinical predictor, and its implications for ICU design are discussed. Full article

(This article belongs to the Section Multidisciplinary Applications)

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13 pages, 296 KB

Open AccessArticle

Explicit Geodesic Projection Distance on the Statistical Manifold of Multivariate Elliptical Distributions

by Xiangbing Chen, Yingying Wang and Jihong Xiao

Entropy 2026, 28(6), 701; https://doi.org/10.3390/e28060701 - 17 Jun 2026

Abstract

Geodesic projection distances on statistical manifolds have been widely applied across various research fields. Nevertheless, the existing closed-form solution is only available for the Gaussian manifold. Multivariate elliptical distributions (MEDs) include the Gaussian, generalized Gaussian, Student’s t distribution, contaminated Gaussian, and other heavy-tailed [...] Read more.

Geodesic projection distances on statistical manifolds have been widely applied across various research fields. Nevertheless, the existing closed-form solution is only available for the Gaussian manifold. Multivariate elliptical distributions (MEDs) include the Gaussian, generalized Gaussian, Student’s t distribution, contaminated Gaussian, and other heavy-tailed models. Despite their widespread practical applicability, explicit geodesic projection solutions for MED manifolds remain unexplored. This paper derives the explicit geodesic projection distance from an arbitrary point on the MED manifold equipped with the Fisher metric onto its commonly used submanifold with a fixed mean vector. The core theoretical contribution lies in a novel symmetry-exploitation technique proposed to tackle the highly nonlinear and complex geodesic equations, and this methodological framework is readily extendable to other statistical manifolds. The derived results substantially advance the state-of-the-art by generalizing existing theories from the Gaussian manifold to the broader family of MEDs. Full article

(This article belongs to the Special Issue Methods from Differential Topology and Differential Geometry in Information Geometry)

20 pages, 2114 KB

Open AccessArticle

A Study on a Method for Detecting Surface Defects in Optical Modules Based on Information Entropy Feature Extraction

by Longbing Yang, Quan Xu, Min Liao, Kang Sun, Rujie Xiang, Yanbin Duan and Haonan Xu

Entropy 2026, 28(6), 700; https://doi.org/10.3390/e28060700 - 17 Jun 2026

Abstract

Optical modules serve as the core transmission interfaces for artificial intelligence computing networks and digital communications. In recent years, demand for these modules has experienced explosive growth. During mass production, the requirements for the accuracy of surface defect detection and noise resistance have [...] Read more.

Optical modules serve as the core transmission interfaces for artificial intelligence computing networks and digital communications. In recent years, demand for these modules has experienced explosive growth. During mass production, the requirements for the accuracy of surface defect detection and noise resistance have continued to rise. Existing POL detection models are susceptible to environmental noise interference; effective defect information is easily overwhelmed by noise entropy, and these models exhibit a high false negative rate for low-contrast and minute defects. This paper proposes a traditional image processing detection scheme that incorporates information entropy constraints. All experimental samples were collected from actual industrial mass production lines. The core process includes: noise suppression during the calibration stage using an entropy-weighted Hough transform; Canny edge detection combined with local entropy filtering for contour localization; and defect fusion recognition based on Hu similarity matching and entropy difference verification. Experimental results show that, compared to traditional POL methods, the proposed approach (WOMC) achieves an average improvement of 35.77% in image clarity and approximately a 2.25-fold increase in detection rate under Gaussian and salt-and-pepper noise conditions. According to statistical analysis of the experiments, this method achieved an accuracy of 96.67%, a recall rate of 97.32%, and a false positive rate of 3.31% in defect detection. In addition, the comprehensive performance score of this detection model reached 96.99%. Moreover, it does not require the deployment of deep-learning models, has a low computing power cost, and is suitable for the detection requirements of large-scale mass production. Full article

(This article belongs to the Special Issue Information Theoretic Learning with Its Applications)

21 pages, 1855 KB

Open AccessArticle

Automated Working Alliance Assessment in Psychological Counseling Using Gemini and XGBoost

by Yuexi Li, Ningtao Sun, Zhuoxi Mai, Dalin Li, Guifang Fu and Xueling Yang

Entropy 2026, 28(6), 699; https://doi.org/10.3390/e28060699 - 17 Jun 2026

Abstract

Session dialogue assessment based on machine learning is gradually becoming an effective solution for therapeutic alliance measurement which is an important factor for successful psychotherapy. However, most existing models assume clean and pre-structured dialogue transcripts, whereas real-world counseling documentation often contains heterogeneous case [...] Read more.

Session dialogue assessment based on machine learning is gradually becoming an effective solution for therapeutic alliance measurement which is an important factor for successful psychotherapy. However, most existing models assume clean and pre-structured dialogue transcripts, whereas real-world counseling documentation often contains heterogeneous case reports. This gap limits the applicability of current automated assessment models in realistic documentation scenarios. In this work, we propose a framework for automated working alliance assessment from complex, multilingual reports. First, language-specific BERT models are fine-tuned to process case reports across different languages, enabling accurate speaker role delineation and dialogue structuring. Second, Gemini-2.5-Flash is leveraged to annotate the dialogues with working alliance ratings. Third, a hybrid feature representation strategy is then developed to jointly capture linguistic style and semantic content from the counseling dialogues. Furthermore, an entropy-based mutual information analysis is conducted to identify the most informative linguistic features. Finally, the extracted hybrid features serve as inputs to XGBoost for alliance assessment. In experiments, the proposed framework shows better performance in the comparison with SOTA methods and generalization ability. Full article

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

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19 pages, 45330 KB

Open AccessArticle

Complexity Entropy Analysis of Grid Chaotic System: Image Encryption and DSP Implementation

by Gang Hu, Baolin Kang and Xiaolin Ye

Entropy 2026, 28(6), 698; https://doi.org/10.3390/e28060698 - 16 Jun 2026

Abstract

In this research, based on Adomian decomposition method (ADM), we construct true fractional-order differential equations. Due to the boosting function brought by the sine function, the system can output infinite coexistence attractors on y–z planes. In particular, this grid effect becomes [...] Read more.

In this research, based on Adomian decomposition method (ADM), we construct true fractional-order differential equations. Due to the boosting function brought by the sine function, the system can output infinite coexistence attractors on y–z planes. In particular, this grid effect becomes increasingly obvious as the fractional order increases. Based on this boosting grid idea, in combination with the fractal dynamics, we construct some fractal patterns, e.g., Koch snows. These fractal diagrams all present grid fractal shapes. And then, we design a grid image encryption algorithm. This algorithm is proven to have higher security. The combination of chaos and fractals explores a new research direction. It provides new ideas for research in related fields. Full article

(This article belongs to the Section Complexity)

34 pages, 4045 KB

Open AccessArticle

Quantum-Tunnelling Oscillators for Cognitive Modelling and Neural Computation: Foundations, Machine-Vision Realisation and Applications

by Ivan S. Maksymov

Entropy 2026, 28(6), 697; https://doi.org/10.3390/e28060697 - 16 Jun 2026

Abstract

I present a quantum-tunnelling oscillator model as a universal dynamical engine for two paradigmatic problems in quantum cognition theory—optical illusion perception and group decision making—where individuals are treated as quantum-mechanical agents whose choices shift through context-dependent transitions rather than simple probabilities. I show [...] Read more.

I present a quantum-tunnelling oscillator model as a universal dynamical engine for two paradigmatic problems in quantum cognition theory—optical illusion perception and group decision making—where individuals are treated as quantum-mechanical agents whose choices shift through context-dependent transitions rather than simple probabilities. I show that, when networked together, these units form a quantum-cognitive neural system that reproduces familiar collective and perceptual phenomena while naturally accommodating counterintuitive processes that challenge classical models. Bridging ideas from quantum cognition theory and neural networks, this approach offers a compact, physically grounded way to describe how real individuals and groups think, perceive and decide. Full article

(This article belongs to the Special Issue Dynamic Models of Group Decision Making)

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30 pages, 887 KB

Open AccessArticle

Topology-Oblivious Random-Walk Key Relaying in Quantum Key Distribution Networks

by Krišjānis Petručeņa, Sergejs Kozlovičs, Juris Vīksna, Elīna Kalniņa, Reinis Isaks, Edgars Celms, Lelde Lāce and Edgars Rencis

Entropy 2026, 28(6), 696; https://doi.org/10.3390/e28060696 - 16 Jun 2026

Abstract

Quantum key distribution (QKD) networks require relaying when distant key management entities share no direct quantum link. Most relay strategies, however, rely on centralized control or globally maintained routing state. This paper asks whether useful security and efficiency can still be obtained with [...] Read more.

Quantum key distribution (QKD) networks require relaying when distant key management entities share no direct quantum link. Most relay strategies, however, rely on centralized control or globally maintained routing state. This paper asks whether useful security and efficiency can still be obtained with topology-oblivious stochastic forwarding. It studies the security-overhead trade-off in a model in which fragmented key material is relayed via random-walk variants and reconstructed under privacy amplification. The analysis asks whether strictly local forwarding can retain useful information-theoretic security (ITS). Evaluation on the GÉANT topology, representing a European academic backbone network, shows clear differences between random-walk variants. The proposed highest-score-neighbor local path-diversification heuristic reduces the probability that relayed key material passes through a compromised node. The evaluation also shows that scouting-based loop erasure significantly shortens sampled routes and improves throughput in the model. Against one- to three-node cartels, random flow protects slightly more source–target pairs than a static disjoint-multipath method on the evaluated topologies. These findings position topology-oblivious stochastic forwarding as a simpler decentralized design for QKD relaying without centralized orchestration or gossip protocols. Full article

(This article belongs to the Special Issue Classical and Quantum Networks: Theory, Modeling and Optimization, 2nd Edition)

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