Entropy

25 pages, 18810 KB

Open AccessArticle

Application of the Two-Layer Regularized Gated Recurrent Unit (TLR-GRU) Model Enhanced by Sliding Window Features in Water Quality Parameter Prediction

by Xianhe Wang, Meiqi Liu, Ying Li, Adriano Tavares, Weidong Huang and Yanchun Liang

Entropy 2026, 28(2), 186; https://doi.org/10.3390/e28020186 - 6 Feb 2026

Abstract

Water quality monitoring is critical for public health, ecology, and economic sustainability, but traditional methods are limited by temporal-spatial coverage and cost, failing to meet real-time assessment needs. Deep learning for water quality prediction is often hindered by high complexity and noise in [...] Read more.

Water quality monitoring is critical for public health, ecology, and economic sustainability, but traditional methods are limited by temporal-spatial coverage and cost, failing to meet real-time assessment needs. Deep learning for water quality prediction is often hindered by high complexity and noise in raw time series. This study aims to address the high complexity and noise of hydrological time series by proposing a prediction framework integrating sliding window feature enhancement, principal component analysis (PCA), and a two-layer regularized gated recurrent unit (TLR-GRU). The core goal is to achieve high-precision real-time prediction of four key water quality parameters (dissolved oxygen (DO), ammonia nitrogen (

{NH}_{3} - N

), total phosphorus (TP), and total nitrogen (TN)) for aquaculture and irrigation. Sample entropy (SampEn,

m = 2

,

r = 0.2 \times std (X)

), a univariate complexity metric capturing intra-series pattern repetition, quantifies time series regularity, showing sliding windows reduce SampEn by filtering transient noise while retaining ecological patterns. This optimization synergizes with TLR-GRU’s regularization (L2, Dropout) to avoid overfitting. A total of 4970 water quality records (2020–2023, 4 h sampling interval) were collected from a monitoring station in a typical aquaculture-irrigated water body. After dimensionality reduction via PCA, experimental results demonstrate that the TLR-GRU model outperforms six state-of-the-art deep learning models (e.g., TLD-LSTM, WaveNet) on both the base dataset and the sliding window-enhanced dataset. On the latter, DO and TP test set

R^{2}

rise from 0.82 to 0.93 and 0.81 to 0.92, with RMSE decreasing by 49.4% and 55.6%, respectively. This framework supports water resource management, applicable to rivers and lakes beyond aquaculture. Future work will optimize the model and integrate multi-source data. Full article

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

19 pages, 2740 KB

Open AccessArticle

Privacy-Preserving ECC-Based AKA for Resource-Constrained IoT Sensor Networks with Forgotten Password Reset

by Yicheng Yu, Kai Wei, Kun Qi and Wangyu Wu

Entropy 2026, 28(2), 185; https://doi.org/10.3390/e28020185 - 6 Feb 2026

Abstract

Wireless sensor networks (

W S N s

) are extensively used in

I o T

applications. Secure access control and data protection are essential. Nonetheless, the wireless environment has an open nature. The limited resources of sensor devices render [...] Read more.

Wireless sensor networks (

W S N s

) are extensively used in

I o T

applications. Secure access control and data protection are essential. Nonetheless, the wireless environment has an open nature. The limited resources of sensor devices render

W S N s

susceptible to a variety of security attacks, causing significant difficulties in the design phase of efficient authentication and key agreement (

A K A

) protocols. This study proposes a physically unclonable function (

P U F

)-based lightweight and secure

A K A

protocol for

W S N s

based on elliptic curve cryptography (

E C C

). A secure password update scheme is offered, which would allow legitimate users to reset forgotten passwords without re-registration. According to formal security analysis using

B A N

logic and

P r o V e r i f

, the proposed protocol is secure against common attacks. Moreover, from an entropy perspective, the use of dynamic pseudonyms and fresh session randomness increase an adversary’s uncertainty about user identities, thereby limiting identity-related information leakage. Performance evaluation shows that the proposed protocol achieves lower computational and communication overhead than the existing ones, making it suitable for

W S N s

with resource constraints. Full article

(This article belongs to the Special Issue Advances in IoT Security and Privacy)

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25 pages, 4492 KB

Open AccessArticle

Template-Based Catalysis and the Emergence of Collectively Autocatalytic Systems

by Roberto Serra and Marco Villani

Entropy 2026, 28(2), 184; https://doi.org/10.3390/e28020184 - 6 Feb 2026

Abstract

Mathematical and computational models, which have been successfully used in various fields of biology, are particularly relevant in studies on the origin of life, where wet experiments have not yet been able to obtain fully “living” entities from abiotic materials. This paper investigates [...] Read more.

Mathematical and computational models, which have been successfully used in various fields of biology, are particularly relevant in studies on the origin of life, where wet experiments have not yet been able to obtain fully “living” entities from abiotic materials. This paper investigates mathematical and computational models of interacting polymers in prebiotic environments to understand how molecular replication and protocell reproduction could emerge. This study builds on the Binary Polymer Model (K-BPM), in which polymers are represented as binary strings that undergo catalyzed condensation and cleavage reactions, by introducing a biologically relevant variant (C-BPM), where catalytic activity depends on polymer structure. The model is analyzed with respect to the formation of autocatalytic networks, formalized as Reflexive Autocatalytic Food-generated (RAF) sets, embedded in a protocell in order to simulate their dynamics. The results show clear differences between K-BPM and C-BPM models. They also show that the existence of a RAF does not guarantee the survival of a population of protocells, although it may be possible when only a subset of the existing species partakes in the RAF, thus suggesting that small autocatalytic networks may have preceded the larger networks found in modern life. Full article

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18 pages, 1035 KB

Open AccessArticle

Narrative Divergence and Disinformation: An Entropic Model for Assessing the Informative Utility of Public Information Sources

by José Ignacio Peláez, Gustavo Fabian Vaccaro and Felix Infante León

Entropy 2026, 28(2), 183; https://doi.org/10.3390/e28020183 - 6 Feb 2026

Abstract

In today’s information ecosystem, disinformation threatens civic autonomy and the stability of public discourse. Beyond the intentional spread of false information, it often appears as narrative divergence among sources interpreting shared events, generating fragmentation and measurable losses in structural coherence. This study examines [...] Read more.

In today’s information ecosystem, disinformation threatens civic autonomy and the stability of public discourse. Beyond the intentional spread of false information, it often appears as narrative divergence among sources interpreting shared events, generating fragmentation and measurable losses in structural coherence. This study examines disinformation within an entropic structural framework, defining it as narrative disorder and epistemic incoherence in information systems. The approach moves beyond fact-checking by treating narrative structure and informational order as quantifiable attributes of public communication. We present the QVP-RI (Relational Information Valuation) operator, a computational model that quantifies narrative divergence through informational entropy and normalized structural divergence, without issuing truth assessments. Implemented through state-of-the-art NLP pipelines and entropic analysis, the operator maps narrative structure and epistemic order across plural media environments. Unlike accuracy-driven approaches, it evaluates narrative coherence and informational utility (IU) as complementary indicators of epistemic value. Experimental validation with 500 participants confirms the robustness of the structural–entropic model and identifies high divergence regions, revealing communication vulnerabilities and showing how narrative disorder enables disinformation dynamics. The QVP-RI operator thus offers a computationally grounded tool for analyzing disinformation as narrative divergence and for strengthening epistemic order in open information systems. Full article

(This article belongs to the Special Issue Complexity of Social Networks)

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28 pages, 11769 KB

Open AccessArticle

Entropy-Guided Regime Switching for Railway Passenger Flow Forecasting: An Adaptive EA-ARIMA-Informer Framework

by Silun Tan, Xinghua Shan, Zhengzheng Wei, Shuo Zhao and Jinfei Wu

Entropy 2026, 28(2), 182; https://doi.org/10.3390/e28020182 - 5 Feb 2026

Abstract

Railway passenger flow forecasting plays a critical role in operational efficiency and resource allocation for transportation systems. However, existing deep learning approaches suffer significant performance degradation when facing rare but high-impact events, primarily due to sample scarcity and their inability to distinguish between [...] Read more.

Railway passenger flow forecasting plays a critical role in operational efficiency and resource allocation for transportation systems. However, existing deep learning approaches suffer significant performance degradation when facing rare but high-impact events, primarily due to sample scarcity and their inability to distinguish between routine patterns and disruption regimes. To address these challenges, this study introduces EA-ARIMA-Informer, an adaptive forecasting framework that integrates entropy-augmented ARIMA with Informer through an entropy-guided regime-switching mechanism. The passenger flow series is characterized through a multi-dimensional entropy space comprising four complementary measures: Sample Entropy quantifies local regularity and predictability, Permutation Entropy captures the complexity of ordinal dynamics, Transfer Entropy measures causal information flow from external events (holidays, weather) to passenger demand, and the Conditional Entropy Growth Factor (CEGF)—a novel metric introduced herein—detects regime transitions by tracking the rate of uncertainty change between consecutive time windows. These entropy indicators serve dual roles as feature inputs for representation learning and as state identifiers for segmenting the time series into stable and fluctuating regimes with distinct predictability properties. An adaptive dual-path architecture is then designed accordingly: EA-ARIMA handles low-entropy stable regimes where linear seasonality dominates, while EA-Informer processes high-entropy fluctuating regimes requiring nonlinear residual modeling, with CEGF-guided gating dynamically controlling component weights. Unlike conventional black-box gating mechanisms, this entropy-based switching provides physically interpretable signals that explain when and why different model components dominate the forecast. The framework is validated on a large-scale dataset covering nearly 300 Chinese cities over three years (2017–2019), encompassing normal operations, holiday peaks, and extreme weather disruptions. Experimental results demonstrate that EA-ARIMA-Informer achieves a MAPE of 4.39% for large-scale cities and 7.82% for data-scarce small cities (Tier-3), substantially outperforming standalone ARIMA, XGBoost, and Informer, which yield 15.95%, 13.75%, and 12.87%, respectively, for Tier-3 cities. Ablation studies confirm that both entropy-based feature augmentation and CEGF-guided regime switching contribute significantly to these performance gains, establishing a new paradigm for interpretable and adaptive forecasting in complex transportation systems. Full article

(This article belongs to the Section Multidisciplinary Applications)

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23 pages, 3517 KB

Open AccessArticle

Finite-Size Thermodynamics of the Two-Dimensional Dipolar Q-Clock Model

by Michel Aguilera, Francisco J. Peña, Eugenio E. Vogel and Patricio Vargas

Entropy 2026, 28(2), 181; https://doi.org/10.3390/e28020181 - 5 Feb 2026

Abstract

We present a fully controlled thermodynamic study of the two-dimensional dipolar Q-state clock model on small square lattices with free boundaries, combining exhaustive state enumeration with noise-free evaluation of canonical observables. We resolve the complete energy spectra and degeneracies [...] Read more.

We present a fully controlled thermodynamic study of the two-dimensional dipolar Q-state clock model on small square lattices with free boundaries, combining exhaustive state enumeration with noise-free evaluation of canonical observables. We resolve the complete energy spectra and degeneracies

{E_{n}, c_{n}}

for the Ising case (

Q = 2

) on lattices of size

L = 3, 4, 5

, and for clock symmetries

Q = 4, 6, 8

on a

3 \times 3

lattice, tracking how the competition between exchange and long-range dipolar interactions reorganizes the low-energy manifold as the ratio

α = D / J

is varied. Beyond a finite-size characterization, we identify several qualitatively new thermodynamic signatures induced solely by dipolar anisotropy. First, we demonstrate that ground-state level crossings generated by long-range interactions appear as exact zeros of the specific heat in the limit

C (T \to 0, α)

, establishing an unambiguous correspondence between microscopic spectral rearrangements and macroscopic caloric response. Second, we show that the shape of the associated Schottky-like anomalies encodes detailed information about the degeneracy structure of the competing low-energy states: odd lattices (

L = 3, 5

) display strongly asymmetric peaks due to unbalanced multiplicities, whereas the even lattice (

L = 4

) exhibits three critical values of

α

accompanied by nearly symmetric anomalies, reflecting paired degeneracies and revealing lattice parity as a key organizing principle. Third, we uncover a symmetry-driven crossover with increasing Q: while the

Q = 2

and

Q = 4

models retain sharp dipolar-induced critical points and pronounced low-temperature structure, for

Q \geq 6

, the energy landscape becomes sufficiently smooth to suppress ground-state crossings altogether, yielding purely thermal specific-heat maxima. Altogether, our results provide a unified, size- and symmetry-resolved picture of how long-range anisotropy, lattice parity, and discrete rotational symmetry shape the thermodynamics of mesoscopic magnetic systems. We show that dipolar interactions alone are sufficient to generate nontrivial critical-like caloric behavior in clusters as small as

3 \times 3

, establishing exact finite-size benchmarks directly relevant for van der Waals nanomagnets, artificial spin-ice arrays, and dipolar-coupled nanomagnetic structures. Full article

(This article belongs to the Section Thermodynamics)

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23 pages, 2302 KB

Open AccessArticle

Learnable Feature Disentanglement with Temporal-Complemented Motion Enhancement for Micro-Expression Recognition

by Yu Qian, Shucheng Huang and Kai Qu

Entropy 2026, 28(2), 180; https://doi.org/10.3390/e28020180 - 4 Feb 2026

Abstract

Micro-expressions (MEs) are involuntary facial movements that reveal genuine emotions, holding significant value in fields like deception detection and psychological diagnosis. However, micro-expression recognition (MER) is fundamentally challenged by the entanglement of subtle emotional motions with identity-specific features. Traditional methods, such as those [...] Read more.

Micro-expressions (MEs) are involuntary facial movements that reveal genuine emotions, holding significant value in fields like deception detection and psychological diagnosis. However, micro-expression recognition (MER) is fundamentally challenged by the entanglement of subtle emotional motions with identity-specific features. Traditional methods, such as those based on Robust Principal Component Analysis (RPCA), attempt to separate identity and motion components through fixed preprocessing and coarse decomposition. However, these methods can inadvertently remove subtle emotional cues and are disconnected from subsequent module training, limiting the discriminative power of features. Inspired by the Bruce–Young model of facial cognition, which suggests that facial identity and expression are processed via independent neural routes, we recognize the need for a more dynamic, learnable disentanglement paradigm for MER. We propose LFD-TCMEN, a novel network that introduces an end-to-end learnable feature disentanglement framework. The network is synergistically optimized by a multi-task objective unifying orthogonality, reconstruction, consistency, cycle, identity, and classification losses. Specifically, the Disentangle Representation Learning (DRL) module adaptively isolates pure motion patterns from subject-specific appearance, overcoming the limitations of static preprocessing, while the Temporal-Complemented Motion Enhancement (TCME) module integrates purified motion representations—highlighting subtle facial muscle activations—with optical flow dynamics to comprehensively model the spatiotemporal evolution of MEs. Extensive experiments on CAS(ME)³ and DFME benchmarks demonstrate that our method achieves state-of-the-art cross-subject performance, validating the efficacy of the proposed learnable disentanglement and synergistic optimization. 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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20 pages, 2137 KB

Open AccessArticle

A Partitioned Finite Difference Method for Heat Transfer with Moving Line and Plane Heat Sources

by Jun Li and Yingjun Jiang

Entropy 2026, 28(2), 179; https://doi.org/10.3390/e28020179 - 4 Feb 2026

Abstract

This study proposes an efficient numerical scheme for simulating heat transfer governed by the diffusion equation with moving singular sources. The work addresses two-dimensional problems with line sources and three-dimensional problems with plane sources, which are prevalent in irreversible thermodynamic processes. Developed within [...] Read more.

This study proposes an efficient numerical scheme for simulating heat transfer governed by the diffusion equation with moving singular sources. The work addresses two-dimensional problems with line sources and three-dimensional problems with plane sources, which are prevalent in irreversible thermodynamic processes. Developed within a finite difference framework, the method employs a partitioned discretization strategy to accurately resolve the solution singularity near the heat source—a region critical for precise local entropy production analysis. In the immediate vicinity of the source, we analytically derive and incorporate the solution’s “jump” conditions to construct specialized finite difference approximations. Away from the source, standard second-order-accurate schemes are applied. This hybrid approach yields a globally second-order convergent spatial discretization. The resulting sparse system is efficient for large-scale simulation of dissipative systems. The accuracy and efficacy of the proposed method are demonstrated through numerical examples, providing a reliable tool for the detailed study of energy distribution in non-equilibrium thermal processes. Full article

(This article belongs to the Section Thermodynamics)

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11 pages, 334 KB

Open AccessArticle

Entropy in Exact 2D Navier–Stokes and 3D Burgers Gas Flows

by Philip Broadbridge

Entropy 2026, 28(2), 178; https://doi.org/10.3390/e28020178 - 3 Feb 2026

Abstract

Two exact solutions are constructed for viscous compressible gas dynamics in two and three dimensions. The first is a steady vortex, with explicit solutions for the full Navier–Stokes system of velocity, density, temperature and pressure. In contrast, the second is a time-dependent radial [...] Read more.

Two exact solutions are constructed for viscous compressible gas dynamics in two and three dimensions. The first is a steady vortex, with explicit solutions for the full Navier–Stokes system of velocity, density, temperature and pressure. In contrast, the second is a time-dependent radial solution to the 3D vector Burgers’ equation, with a constant injection rate from a spherical interior surface. That solution is shock-like at low Reynolds numbers. In both cases, expressions are given for the local density of entropy production. Full article

(This article belongs to the Special Issue Insight into Entropy)

22 pages, 512 KB

Open AccessArticle

Dynamical Casimir Effect Under the Action of Gravitational Waves

by Gustavo de Oliveira, Thiago Henrique Moreira and Lucas Chibebe Céleri

Entropy 2026, 28(2), 177; https://doi.org/10.3390/e28020177 - 3 Feb 2026

Abstract

Several nontrivial phenomena emerge when a quantum field is subjected to dynamical perturbations, with prominent examples including the Hawking and Unruh effects, as well as the dynamical Casimir effect. In this work, we compute the number of particles produced via the dynamical Casimir [...] Read more.

Several nontrivial phenomena emerge when a quantum field is subjected to dynamical perturbations, with prominent examples including the Hawking and Unruh effects, as well as the dynamical Casimir effect. In this work, we compute the number of particles produced via the dynamical Casimir effect in an ideal cavity, where one of the mirrors is allowed to move under the influence of a gravitational wave. Assuming an oscillatory mirror motion and a plane gravitational wave, we identify the resonance conditions that lead to an exponential increase in the number of created particles through parametric amplification. Full article

(This article belongs to the Special Issue Quantum Nonstationary Systems—Second Edition)

22 pages, 437 KB

Open AccessArticle

Adjusted Kolmogorov Complexity of Binary Words with Empirical Entropy Normalization

by Brani Vidakovic

Entropy 2026, 28(2), 176; https://doi.org/10.3390/e28020176 - 3 Feb 2026

Abstract

Kolmogorov complexity of a finite binary word reflects both algorithmic structure and the empirical distribution of symbols appearing in the word. Words with symbol frequencies far from one half belong to smaller combinatorial classes and therefore appear less complex under the standard definition. [...] Read more.

Kolmogorov complexity of a finite binary word reflects both algorithmic structure and the empirical distribution of symbols appearing in the word. Words with symbol frequencies far from one half belong to smaller combinatorial classes and therefore appear less complex under the standard definition. In this paper, an entropy-normalized complexity measure is introduced that divides the Kolmogorov complexity of a word by the empirical entropy of its observed distribution of zeros and ones. This adjustment isolates intrinsic descriptive complexity from the purely combinatorial effect of symbol imbalance. For Martin–Löf random sequences under constructive exchangeable measures, the adjusted complexity grows linearly and converges to one. A pathological construction shows that regularity of the underlying measure is essential. The proposed framework connects Kolmogorov complexity, empirical entropy, and randomness in a natural manner and suggests applications in randomness testing and in the analysis of structured binary data. Full article

(This article belongs to the Special Issue Entropy: Exploring Complexity and Information in Science)

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20 pages, 597 KB

Open AccessArticle

BeamNet: Unsupervised Beamforming for ISAC Systems Under Imperfect CSI

by Helitha Nimnaka, Samiru Gayan, Ruhui Zhang, Hazer Inaltekin and H. Vincent Poor

Entropy 2026, 28(2), 175; https://doi.org/10.3390/e28020175 - 3 Feb 2026

Abstract

Integrated sensing and communication (ISAC) is expected to be a key enabler for future wireless networks, improving spectral and hardware efficiency by jointly performing radar sensing and wireless communication within a unified framework. This paper proposes BeamNet, an unsupervised deep learning framework [...] Read more.

Integrated sensing and communication (ISAC) is expected to be a key enabler for future wireless networks, improving spectral and hardware efficiency by jointly performing radar sensing and wireless communication within a unified framework. This paper proposes BeamNet, an unsupervised deep learning framework for transmit beamforming in dual-function radar-communication systems operating over general fading with imperfect channel state information (CSI). BeamNet maps noisy estimates of the communication and sensing channels to a transmit beamforming vector and is trained end-to-end by maximizing a weighted sum of the communication rate (CR) and sensing rate (SR), thereby learning the CR–SR Pareto frontier without beamforming labels or embedded optimization solvers. Using Rayleigh fading with perfect CSI, we first show that BeamNet reproduces the analytical Pareto-optimal beamforming solutions. We then use BeamNet to characterize, for Nakagami-m and Rician fading, the CR–SR trade-off across a range of fading parameters, and to assess robustness under distribution mismatch between training and test channels. Finally, under imperfect CSI, we demonstrate that BeamNet yields CR–SR trade-offs that are consistently sandwiched between the perfect-CSI and mismatched analytical baselines, outperforming the closed-form beamformer applied to imperfect CSI and recovering part of the performance loss caused by channel estimation errors. These results indicate that unsupervised learning offers a flexible and robust approach to ISAC beamforming in fading environments with imperfect channel knowledge. Full article

(This article belongs to the Special Issue Joint Sensing, Communication, and Computation)

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26 pages, 6232 KB

Open AccessArticle

MFE-YOLO: A Multi-Scale Feature Enhanced Network for PCB Defect Detection with Cross-Group Attention and FIoU Loss

by Ruohai Di, Hao Fan, Hanxiao Feng, Zhigang Lv, Lei Shu, Rui Xie and Ruoyu Qian

Entropy 2026, 28(2), 174; https://doi.org/10.3390/e28020174 - 2 Feb 2026

Abstract

The detection of defects in Printed Circuit Boards (PCBs) is a critical yet challenging task in industrial quality control, characterized by the prevalence of small targets and complex backgrounds. While deep learning models like YOLOv5 have shown promise, they often lack the ability [...] Read more.

The detection of defects in Printed Circuit Boards (PCBs) is a critical yet challenging task in industrial quality control, characterized by the prevalence of small targets and complex backgrounds. While deep learning models like YOLOv5 have shown promise, they often lack the ability to quantify predictive uncertainty, leading to overconfident errors in challenging scenarios—a major source of false alarms and reduced reliability in automated manufacturing inspection lines. From a Bayesian perspective, this overconfidence signifies a failure in probabilistic calibration, which is crucial for trustworthy automated inspection. To address this, we propose MFE-YOLO, a Bayesian-enhanced detection framework built upon YOLOv5 that systematically integrates uncertainty-aware mechanisms to improve both accuracy and operational reliability in real-world settings. First, we construct a multi-background PCB defect dataset with diverse substrate colors and shapes, enhancing the model’s ability to generalize beyond the single-background bias of existing data. Second, we integrate the Convolutional Block Attention Module (CBAM), reinterpreted through a Bayesian lens as a feature-wise uncertainty weighting mechanism, to suppress background interference and amplify salient defect features. Third, we propose a novel FIoU loss function, redesigned within a probabilistic framework to improve bounding box regression accuracy and implicitly capture localization uncertainty, particularly for small defects. Extensive experiments demonstrate that MFE-YOLO achieves state-of-the-art performance, with mAP@0.5 and mAP@0.5:0.95 values of 93.9% and 59.6%, respectively, outperforming existing detectors, including YOLOv8 and EfficientDet. More importantly, the proposed framework yields better-calibrated confidence scores, significantly reducing false alarms and enabling more reliable human-in-the-loop verification. This work provides a deployable, uncertainty-aware solution for high-throughput PCB inspection, advancing toward trustworthy and efficient quality control in modern manufacturing environments. Full article

(This article belongs to the Special Issue Bayesian Networks and Causal Discovery)

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48 pages, 1830 KB

Open AccessArticle

An Information–Theoretic Model of Abduction for Detecting Hallucinations in Explanations

by Boris Galitsky

Entropy 2026, 28(2), 173; https://doi.org/10.3390/e28020173 - 2 Feb 2026

Abstract

We present an Information–Theoretic Model of Abduction for Detecting Hallucinations in Generative Models, a neuro-symbolic framework that combines entropy-based inference with abductive reasoning to identify unsupported or contradictory content in large language model outputs. Our approach treats hallucination detection as a dual optimization [...] Read more.

We present an Information–Theoretic Model of Abduction for Detecting Hallucinations in Generative Models, a neuro-symbolic framework that combines entropy-based inference with abductive reasoning to identify unsupported or contradictory content in large language model outputs. Our approach treats hallucination detection as a dual optimization problem: minimizing the information gain between source-conditioned and response-conditioned belief distributions, while simultaneously selecting the minimal abductive hypothesis capable of explaining discourse-salient claims. By incorporating discourse structure through RST-derived EDU weighting, the model distinguishes legitimate abductive elaborations from claims that cannot be justified under any computationally plausible hypothesis. Experimental evaluation across medical, factual QA, and multi-hop reasoning datasets demonstrates that the proposed method outperforms state-of-the-art neural and symbolic baselines in both accuracy and interpretability. Qualitative analysis further shows that the framework successfully exposes plausible-sounding but abductively unsupported model errors, including real hallucinations generated by GPT-5.1. Together, these results indicate that integrating Information–Theoretic divergence and abductive explanation provides a principled and effective foundation for robust hallucination detection in generative systems. Full article

(This article belongs to the Special Issue Information Theory in Artificial Intelligence)

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22 pages, 1100 KB

Open AccessArticle

Statistical Distribution and Entropy of Multi-Scale Returns: A Coarse-Grained Analysis and Evidence for a New Stylized Fact

by Alejandro Raúl Hernández-Montoya

Entropy 2026, 28(2), 172; https://doi.org/10.3390/e28020172 - 2 Feb 2026

Abstract

Financial time series often show periods during which market index values or asset prices increase or decrease monotonically. These events are known as price runs, uninterrupted trends, or simply runs. By identifying such runs in the daily DJIA and IPC indices from 2 [...] Read more.

Financial time series often show periods during which market index values or asset prices increase or decrease monotonically. These events are known as price runs, uninterrupted trends, or simply runs. By identifying such runs in the daily DJIA and IPC indices from 2 January 1990 to 17 October 2025, we construct their associated returns to obtain a non-arbitrary sample of multi-scale returns, which we call trend returns (TReturns). The timescale of each multi-scale return is determined by the exponentially distributed duration of its corresponding run. We empirically show that the distribution of these coarse-grained returns exhibits distinctive statistical properties: the central region displays an exponential decay, likely resulting from the exponential distribution of trend durations, while the tails follow a power-law decay. This combination of exponential central behavior and asymptotic power-law decay has also been observed in other complex systems, and our findings provide additional evidence of its natural emergence. We also explore the informational properties of multi-scale returns using three measures: Shannon entropy, permutation entropy, and compression-based complexity. We find that Shannon entropy increases with coarse-graining, indicating a wider range of values; permutation entropy drops sharply, revealing underlying temporal patterns; and compression ratios improve, reflecting suppressed randomness. Overall, these findings suggest that constructing TReturns filters out microscopic noise, reveals structured temporal patterns, and provides a complementary and clear view of market behavior. Full article

(This article belongs to the Special Issue Entropy, Econophysics, and Complexity)

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9 pages, 255 KB

Open AccessArticle

Quasi-Power Law Ensembles: Nonextensive Statistics or Superstatistics

by Maciej Rybczyński, Grzegorz Wilk and Zbigniew Włodarczyk

Entropy 2026, 28(2), 171; https://doi.org/10.3390/e28020171 - 2 Feb 2026

Abstract

In phenomenological studies of multiparticle production, transverse-momentum spectra measured in experiments frequently display an approximately power-law falloff, for which the Tsallis-type functional form is commonly employed as an effective parametrization. Within this framework, the emergence of such spectra is interpreted as a manifestation [...] Read more.

In phenomenological studies of multiparticle production, transverse-momentum spectra measured in experiments frequently display an approximately power-law falloff, for which the Tsallis-type functional form is commonly employed as an effective parametrization. Within this framework, the emergence of such spectra is interpreted as a manifestation of nonextensive statistical behavior. An analogous power-law structure, however, can be reproduced without explicitly postulating Tsallis statistics by assuming the presence of intrinsic fluctuations of the local temperature (T) in the hadronizing medium; in that case, the observed deviations from a purely exponential spectrum are encapsulated by the nonextensivity index (q). We show that temperature fluctuation mechanisms capable of generating Tsallis-like power-law distributions in multiparticle production necessarily induce nontrivial inter-particle correlations among the emitted hadrons. Building on this observation, we outline a strategy to discriminate fluctuations realized on an event-by-event basis from those arising predominantly through event-to-event variability. Such a separation may be particularly pertinent for the characterization of high-multiplicity (high-density) final states produced at the Large Hadron Collider. Full article

(This article belongs to the Special Issue Complexity in High-Energy Physics: A Nonadditive Entropic Perspective)

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14 pages, 457 KB

Open AccessReview

Density Functional Theory and Information-Theoretic Diagnostics of Quantum Phase Transitions

by E. Romera and Á. Nagy

Entropy 2026, 28(2), 170; https://doi.org/10.3390/e28020170 - 1 Feb 2026

Abstract

Within density functional theory (DFT), where the density is the fundamental variable, quantum phase transitions (QPTs) can be formulated through a Hamiltonian

\hat{H} = {\hat{H}}_{0} + \sum_{i} ξ_{i} {\hat{A}}_{i}

, such that the control parameters [...] Read more.

Within density functional theory (DFT), where the density is the fundamental variable, quantum phase transitions (QPTs) can be formulated through a Hamiltonian

\hat{H} = {\hat{H}}_{0} + \sum_{i} ξ_{i} {\hat{A}}_{i}

, such that the control parameters

{ξ_{i}}

are in bijective correspondence (in the nondegenerate case) with the “densities”

a_{i} = ⟨ {\hat{A}}_{i} ⟩

, and the functional

Q ({a_{i}})

acts as the Legendre transform of the energy; this structure even permits the use of Rényi entropy (for a given order) as an alternative control parameter, while degeneracy can be handled via a subspace density. On this foundation, information-theoretic measures provide sensitive diagnostics of criticality: fidelity and its susceptibility

χ

, Fisher information, relative Rényi entropy, and the Kullback–Leibler divergence are locally linked by

R^{q} \approx q I_{K L} \approx 2 q χ {(δ λ)}^{2}

, revealing their proportionality in the small-parameter-shift regime. Applied to the Dicke model, numerical analyses show that fidelity exhibits pronounced curvature or divergence near

λ_{c} = \sqrt{ω ω_{0}} / 2

and that the response sharpens with increasing j, corroborating that these information measures capture QPTs with precision within the DFT framework. Full article

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

14 pages, 695 KB

Open AccessArticle

The Calhoun Experiment Study by Means of Agent-Based Modeling

by Tomasz M. Gwizdałła and Jakub Duś

Entropy 2026, 28(2), 169; https://doi.org/10.3390/e28020169 - 1 Feb 2026

Abstract

More than 50 years ago John D. Calhoun conducted a series of experiments devoted to studying the social behavior of mice. The longest of them lasted more than four years and led to the creation of the concept of the so-called “behavioral sink”. [...] Read more.

More than 50 years ago John D. Calhoun conducted a series of experiments devoted to studying the social behavior of mice. The longest of them lasted more than four years and led to the creation of the concept of the so-called “behavioral sink”. The population of mice, which had all possible resources at their disposal but were located in a limited room, disintegrated after an initial phase of strong development. In our paper we are going to reproduce the effects of this experiment. The crucial problem in every simulation approach is to determine the set of the most important parameters which influence the global as well as local effects of the simulated process. In the studied case we have the problem that a lot of important information is missing. The author of the original work focused rather on social mechanisms, often omitting key numerical data related to the course of the experiment. In this paper we try to propose a certain set of parameters. By using them we can reproduce Calhoun’s results qualitatively and, with some deviation, quantitatively. Full article

(This article belongs to the Section Complexity)

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17 pages, 3006 KB

Open AccessArticle

A New Quantum Video Processing Algorithm Based on the NEQR Technique

by Adrian Prodan, Alexandru-Gabriel Tudorache and Vasile Manta

Entropy 2026, 28(2), 168; https://doi.org/10.3390/e28020168 - 1 Feb 2026

Abstract

The main goal of this paper is to present a new way of processing a video file using a combination of multiple quantum methods. The design is built upon the novel enhanced quantum representation technique, NEQR, which is then expanded using ideas such [...] Read more.

The main goal of this paper is to present a new way of processing a video file using a combination of multiple quantum methods. The design is built upon the novel enhanced quantum representation technique, NEQR, which is then expanded using ideas such as image segmentation, implemented with the help of one or multiple comparators, binarization and cycle shift. This approach allows us to process all frames in parallel according to the desired parameters—one or more thresholds. A demonstration circuit for the proposed design, using a couple of frames, that sums together all the concepts is implemented using the Python programming language and Qiskit open-source framework, made available by IBM. The circuits are analyzed in the experimental section, using the Simulator component and configured using the noise properties of real devices, where we present different relevant metrics obtained by processing the simulation results. Full article

(This article belongs to the Special Issue Quantum Entanglement and Quantum Algorithms)

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