Entropy

17 pages, 290 KB

Open AccessArticle

Empirical Nonnegative Finite-Resolution MAR-PID for Continuous Variables

by András Telcs

Entropy 2026, 28(6), 641; https://doi.org/10.3390/e28060641 (registering DOI) - 6 Jun 2026

We develop a finite-resolution empirical framework for applying nonnegative Mages–Anastasiadi–Rohner partial information decomposition (MAR-PID) to continuous and non-binary discrete variables. The variables are represented by recursive quantile binarization. This provides a balanced binary-tree representation at each finite depth. MAR-PID is then applied to [...] Read more.

We develop a finite-resolution empirical framework for applying nonnegative Mages–Anastasiadi–Rohner partial information decomposition (MAR-PID) to continuous and non-binary discrete variables. The variables are represented by recursive quantile binarization. This provides a balanced binary-tree representation at each finite depth. MAR-PID is then applied to binary target components, and the resulting atoms are aggregated back to the original target and source variables. The construction gives nonnegative target-relative information summaries for observed variables up to the chosen empirical resolution. The pipeline consists of conditional channel estimation, bit-level MAR-PID computation, projection of source atoms to the original variables, and aggregation of the resulting information contributions. The obtained quantities are empirical estimates of finite-resolution population quantities. XOR and mixed redundancy–synergy examples show how the representation separates informational mechanisms, which signed interaction-information summaries can conflate. We focus here on the finite-resolution MAR-PID construction and its information-level quantities. Downstream summaries, such as resolution-normalized PID-dimension descriptors and thresholded support-based degrees of freedom, are indicated but left for separate work. Full article

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

12 pages, 345 KB

Open AccessArticle

Delayed-Choice Phenomena in the Projection Evolution Model

by Marek Góźdź, Andrzej Góźdź and Krzysztof Lider

Entropy 2026, 28(6), 640; https://doi.org/10.3390/e28060640 (registering DOI) - 6 Jun 2026

Abstract

In the Schrödinger evolution of a quantum state, time enters as a real parameter representing the coordinate. In a more consistent approach, time should be defined as a quantum observable, with the evolution taking place in a four-dimensional spacetime. This is possible in [...] Read more.

In the Schrödinger evolution of a quantum state, time enters as a real parameter representing the coordinate. In a more consistent approach, time should be defined as a quantum observable, with the evolution taking place in a four-dimensional spacetime. This is possible in the projection evolution model in which the wave function is defined in both space and time. This allows to construct the time operator and discuss the temporal structure of the quantum processes. In this paper, we discuss a photon travelling through a Mach–Zehnder interferometer, focusing the description on the temporal profile of the wave function. We show that in this approach the delayed-choice experiments can be explained by the temporal overlap of the photon and the devices in the interferometer. Full article

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

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21 pages, 22957 KB

Open AccessArticle

Coupled Map Lattice Modeling and Robustness Analysis of Simplicial Complex Networks with Higher-Order Interactions

by Luqian Wang, Jun Yin, Xiujuan Ma and Hongyu Chen

Entropy 2026, 28(6), 639; https://doi.org/10.3390/e28060639 (registering DOI) - 5 Jun 2026

Abstract

Cascading failures in complex networks occur when local node or edge failures propagate to trigger large-scale collapse. Traditional pairwise network models cannot adequately capture group coordination and multi-agent higher-order interactions. Higher-order networks incorporating simplicial structures more accurately represent group and multi-node interactions, providing [...] Read more.

Cascading failures in complex networks occur when local node or edge failures propagate to trigger large-scale collapse. Traditional pairwise network models cannot adequately capture group coordination and multi-agent higher-order interactions. Higher-order networks incorporating simplicial structures more accurately represent group and multi-node interactions, providing a new framework to study cascading failures and network robustness. The paper proposes a higher-order coupled map lattice (CML) model to characterize cascading failures in simplicial complex networks and analyze the influence of higher-order structures on network robustness. Further experiments on fourth-order simplicial networks investigate robustness differences under various topologies and attack strategies. Results indicate that fourth-order simplicial networks are vulnerable to targeted attacks but robust against random failures, regardless of network type. Furthermore in single-order networks, the higher simplex dimensions, the greater robustness. The theoretical perturbation thresholds for third-order networks show a negative correlation between the critical perturbation and the sum of network coupling parameters. These results are validated by analysis of simplices added to ordinary networks, destructive experiments, and empirical networks. This study deepens the understanding of cascading failure mechanisms and robustness in higher-order networks, and provides theoretical guidance for designing resilient networks based on higher-order structures. Full article

(This article belongs to the Section Complexity)

17 pages, 2778 KB

Open AccessArticle

Evaluation of Bubble Entropy Using Heart Rate Variability

by Dimitrios Platakis, Roberto Sassi and George Manis

Entropy 2026, 28(6), 638; https://doi.org/10.3390/e28060638 (registering DOI) - 5 Jun 2026

Abstract

Bubble entropy has established its own place in the research community, representing a new and promising definition of entropy. Based on the work required to order a vector in an embedding space of dimension m, Bubble entropy gives a physical interpretation of [...] Read more.

Bubble entropy has established its own place in the research community, representing a new and promising definition of entropy. Based on the work required to order a vector in an embedding space of dimension m, Bubble entropy gives a physical interpretation of what the metric actually computes. In this work, Bubble entropy is evaluated based on its ability to classify RR time series, the time series most commonly considered for entropy-based analysis in the field of biomedical engineering. For this purpose, it is compared with three other definitions of entropy: the most widely used Sample entropy and Approximate entropy, the most relative to Bubble entropy, and also the widely used Permutation entropy. Signals from healthy individuals, in sinus rhythm, are compared with signals from cardiac patients, and machine learning methods are applied to calculate the classification accuracy that each method can achieve. The classifiers chosen are k-Nearest Neighbors, Support Vector Machine, Logistic Regression, and Gaussian Naive Bayes. Feature evaluation methods are also employed to serve as additional measures of effectiveness. Bubble entropy generally manages to achieve better results than Sample entropy, Approximate entropy and Permutation entropy, both in terms of classification accuracy and feature ranking. Full article

(This article belongs to the Special Issue Entropy in Biomedical Engineering, 3rd Edition)

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28 pages, 3181 KB

Open AccessArticle

FedVI: Financial Cross-Domain Federated Learning with Scarce Overlapping Samples via Visual Representation of Heterogeneous Tabular Data and Meta-Optimization

by Kaiqing Yuan and Jiang Wu

Entropy 2026, 28(6), 637; https://doi.org/10.3390/e28060637 - 4 Jun 2026

Abstract

Federated learning offers a promising approach for cross-institutional financial risk control modeling but encounters two key challenges in practice: feature space heterogeneity and low sample overlap rate. Current federated transfer learning methods often rely heavily on sufficient overlapping samples or explicit feature alignment. [...] Read more.

Federated learning offers a promising approach for cross-institutional financial risk control modeling but encounters two key challenges in practice: feature space heterogeneity and low sample overlap rate. Current federated transfer learning methods often rely heavily on sufficient overlapping samples or explicit feature alignment. However, these approaches frequently result in negative transfer when enforced alignment is applied in highly heterogeneous environments. To address this issue, we propose FedVI, a novel federated transfer learning framework that integrates tabular-to-image conversion and meta-learning mechanisms. Moving beyond conventional methods that rely on sample-level alignment, FedVI employs a federated dual-stream feature alignment strategy to securely reconstruct a unified global feature map across institutions. Subsequently, FedVI integrates federated Image Generator for Tabular Data (IGTD) with tabular Transformer technology to convert one-dimensional tabular data into two-dimensional visual-semantic tensors. These tensors effectively fuse spatial topology and semantic information while embedding an independent Mask channel to explicitly retain the true missingness patterns of features. Finally, FedVI adopts the Model-Agnostic Meta-Learning (MAML) architecture to facilitate global parameter optimization. We evaluated FedVI on the real-world Lending Club credit dataset and Home Credit Default Risk datasets under highly heterogeneous federated settings (i.e., heterogeneous feature spaces across three clients and scarce overlapping samples). The results reveal that FedVI achieves competitive performance against advanced baselines such as FedProx, FedRep, and FedKT, particularly in recall and F1-Score. These findings indicate that FedVI can effectively support cross-domain adaptation under heterogeneous federated learning settings. Full article

(This article belongs to the Special Issue Advancing Privacy-Preserving Federated Learning: Innovative Frameworks and Protocols)

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28 pages, 1030 KB

Open AccessArticle

Integrating Consensus-Based Group Decision Making into the Graph Model for Conflict Resolution in Complex Conflict Environments

by Hengjie Zhang, Xiaoying Lu and Fang Wang

Entropy 2026, 28(6), 636; https://doi.org/10.3390/e28060636 - 4 Jun 2026

Abstract

Complex conflict environments often involve composite conflicting parties composed of multiple individuals with divergent assessments, posing significant challenges for preference elicitation and conflict analysis in the Graph Model for Conflict Resolution (GMCR). Existing option prioritization methods in GMCR rely on ordinal rankings of [...] Read more.

Complex conflict environments often involve composite conflicting parties composed of multiple individuals with divergent assessments, posing significant challenges for preference elicitation and conflict analysis in the Graph Model for Conflict Resolution (GMCR). Existing option prioritization methods in GMCR rely on ordinal rankings of statement importance and binary evaluations of state support, which limits their ability to capture nuanced and heterogeneous preferences. Moreover, current GMCR frameworks lack a systematic mechanism to reconcile divergent individual assessments and construct collective preferences. To address these gaps, this study integrates consensus-based group decision making into the GMCR framework through flexible assessments. Specifically, pairwise comparisons are employed to represent the relative importance of statements, while continuous values in [⁰,¹] are used to characterize the degree of statement support for states. To reconcile heterogeneous assessments, a minimum adjustment-based consensus reaching process is developed. Furthermore, grounded in inequity aversion theory from behavioral economics, fairness concern is incorporated to model individuals’ sensitivity to adjustment differences between themselves and others. Based on this mechanism, two minimum adjustment consensus models with fairness concern are proposed. The resulting consensus-based preferences are subsequently integrated into GMCR stability analysis to identify equilibrium solutions. A supply chain carbon reduction case is used to illustrate the implementation process and applicability of the proposed framework. Full article

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

44 pages, 11374 KB

Open AccessArticle

Cyber Defense Effectiveness Evaluation for ICS Under Uncertainty: A Dynamic Bayesian Network Approach with Information Entropy

by Rongbao Kang, Zhiyong Zhang, Xiao Zhang, Jianfeng Chen, Ruoyu Xu, Yongdong Zhang and Zhihong Rao

Entropy 2026, 28(6), 635; https://doi.org/10.3390/e28060635 - 4 Jun 2026

Abstract

Proactive defense planning in Industrial Control Systems (ICS) is critically constrained by two intertwined types of uncertainty: epistemic uncertainty, arising from the defender’s limited observability of the system state and incomplete knowledge of attacker strategies, and aleatoric uncertainty, stemming from the stochastic nature [...] Read more.

Proactive defense planning in Industrial Control Systems (ICS) is critically constrained by two intertwined types of uncertainty: epistemic uncertainty, arising from the defender’s limited observability of the system state and incomplete knowledge of attacker strategies, and aleatoric uncertainty, stemming from the stochastic nature of state transitions and the propagation of disturbances through inter-device dependencies. These factors significantly complicate the quantitative assessment of defense strategies before deployment. To address this challenge, this study proposes a Dynamic Bayesian Network (DBN)-based framework that explicitly models four sources of uncertainty. Within this framework, the expectation of the effectiveness differential is coupled with its information entropy to jointly quantify expected performance and prediction uncertainty. A casestudy on a typical substation automation system, complemented by systematic ablation experiments, demonstrates that the framework can effectively distinguish the relative effectiveness of defense strategies. The framework maintains robust assessment results under up to 15% noise in Conditional Probability Tables (CPTs). The ablation experiments further quantify the individual contributions of observability, dependency propagation, and attacker strategy to prediction uncertainty, and reveal a non-trivial coupling between epistemic and aleatoric uncertainty. This research provides theoretical and methodological support for resilience-oriented cyber defense planning in ICS. Full article

(This article belongs to the Section Multidisciplinary Applications)

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25 pages, 957 KB

Open AccessArticle

ODE-BPA: A Novel Basic Probability Assignment Generation Method Based on OTSU and Deng Entropy

by Xinghua Zhou, Luyuan Chen, Enze Mao and Yutong He

Entropy 2026, 28(6), 634; https://doi.org/10.3390/e28060634 - 3 Jun 2026

Abstract

Constructing basic probability assignments (BPAs) objectively and adaptively remains an important issue in Dempster–Shafer theory. To reduce the uncertainty caused by rigid nested focal element structures, this paper proposes ODE-BPA, a BPA generation method based on OTSU and Deng entropy. In ODE-BPA, hypothesis [...] Read more.

Constructing basic probability assignments (BPAs) objectively and adaptively remains an important issue in Dempster–Shafer theory. To reduce the uncertainty caused by rigid nested focal element structures, this paper proposes ODE-BPA, a BPA generation method based on OTSU and Deng entropy. In ODE-BPA, hypothesis support values are normalized and sorted into an ordered sequence. OTSU is then used to construct two non-overlapping focal elements, while Deng entropy regulates the confidence degree for mass allocation. Experiments are conducted on twenty-five benchmark datasets, including mapping-structure comparison, ablation analysis, comparisons with existing BPA generation methods and classical machine learning classifiers, as well as the robustness evaluation under noisy and conflicting evidence. Rigorous statistical analysis demonstrates that ODE-BPA achieves competitive accuracy and ranking among existing BPA generation methods; exhibits performance comparable to that of SVM, naive Bayes, and decision tree under standard settings; and alleviates the influence of local noise and conflicting evidence in certain cases. Full article

(This article belongs to the Special Issue Recent Progress in Uncertainty Measures)

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32 pages, 405 KB

Open AccessArticle

Entropy Gap as a Measure of Epistemic Caution in Credal Sets Generated from Data

by María Isabel A. Benítez, Carlos J. Mantas and Joaquín Abellán

Entropy 2026, 28(6), 633; https://doi.org/10.3390/e28060633 - 3 Jun 2026

Abstract

Imprecise probability models generated from data represent epistemic uncertainty by replacing the precise empirical distribution with a set of compatible probability distributions. When this set is described by reachable probability intervals, the induced bounds are tight, so the represented imprecision is not inflated [...] Read more.

Imprecise probability models generated from data represent epistemic uncertainty by replacing the precise empirical distribution with a set of compatible probability distributions. When this set is described by reachable probability intervals, the induced bounds are tight, so the represented imprecision is not inflated by unattainable interval limits. This paper studies the informational effect of this replacement through the epistemic entropy gap, defined as the difference between the maximum entropy over the induced credal set and the Shannon entropy of the empirical distribution. The gap is a differential quantity: it measures the additional uncertainty introduced by the imprecise model beyond the observed frequencies. We analyze it for three reachable interval models generated from multinomial data: the Imprecise Dirichlet Model, the

ϵ

-contamination model and the approximated Non-Parametric Predictive Inference model. The analysis covers its main properties, its asymptotic behavior and its role in entropy equivalent calibration of model parameters. The results show that the entropy gap offers a common informational scale for comparing how different imprecise models represent the same empirical evidence, and helps interpret the degree of caution associated with limited data reliability and with empirical distributions that may otherwise lead to overconfident uncertainty assessments. Full article

(This article belongs to the Section Multidisciplinary Applications)

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19 pages, 3160 KB

Open AccessArticle

Insights of Photocatalytic Properties of Fe/TiO₂ Bio-Based Particles: Experimental and Modeling Design Toward Methyl Orange Photodegradation

by Aleksandar Jovanović, Amil Aligayev, Mladen Bugarčić, Dimitrije Anđić, Ulkar Samadova, Jelena Dimitrijević, Miroslav Sokić and Qing Huang

Entropy 2026, 28(6), 632; https://doi.org/10.3390/e28060632 - 3 Jun 2026

Abstract

This study investigates the electronic and photocatalytic properties of greenly fabricated rutile-phase titanium dioxide (bTiO₂) modified with iron vanadate (Fe/bTiO₂/VO₄) and vanadium-substituted goethite (Fe/bTiO₂/VOOH) by detailed experimental assay and density functional theory (DFT) calculations. Our [...] Read more.

This study investigates the electronic and photocatalytic properties of greenly fabricated rutile-phase titanium dioxide (bTiO₂) modified with iron vanadate (Fe/bTiO₂/VO₄) and vanadium-substituted goethite (Fe/bTiO₂/VOOH) by detailed experimental assay and density functional theory (DFT) calculations. Our analysis of the density of states (DOS), band structure, and work function reveals that both dopant systems significantly modify the electronic structure of pure rutile bTiO₂. The dye methyl orange (MO) was used as the model pollutant. During photodegradation tests, parameters such as the reaction time, solid-to-liquid ratio, initial concentrations of the photocatalyst and dye, as well as distance of the lamp from the reactor and pH were varied. Degradation kinetics follows the equation of the pseudo-first order law for both photocatalysts (k_VO4 = 0.058 min⁻¹ and k_VOOH = 0.065 min⁻¹), while degradation efficiencies of 92% and 99% were observed after 120 min at pH 3, respectively. Specifically, the DOS analysis highlights the contribution of Fe 3d and V 3d orbitals, which create new electronic states within the bandgap, facilitating charge transfer. These insights provide a strong foundation for the rational design of novel, highly efficient Fe/bTiO₂-based photocatalysts for the degradation of organic pollutants in water. Full article

(This article belongs to the Special Issue Unraveling Water–Nanomaterial Interactions)

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15 pages, 388 KB

Open AccessReview

Entropy Is Not Extensive

by Chris Jeynes and Michael C. Parker

Entropy 2026, 28(6), 631; https://doi.org/10.3390/e28060631 - 3 Jun 2026

Abstract

The Gibbs Paradox (concerning the entropy of mixing and entropic extensivity) was explored in depth by Edwin Jaynes (1992). We take up Jaynes’ treatment, considering the special cases for which entropy is (approximately) extensive, and the general case in which it is not. [...] Read more.

The Gibbs Paradox (concerning the entropy of mixing and entropic extensivity) was explored in depth by Edwin Jaynes (1992). We take up Jaynes’ treatment, considering the special cases for which entropy is (approximately) extensive, and the general case in which it is not. We also explore the Holographic Principle which (strictly speaking) excludes the extensivity of entropy. The formalism of Quantitative Geometrical Thermodynamics shows that, being isomorphic to energy, it is entropy production (not entropy) that is extensive. As a corollary, Shannon information is also not extensive, although information production is extensive. Full article

(This article belongs to the Section Thermodynamics)

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19 pages, 2496 KB

Open AccessArticle

SCKM: Symmetric Co-Skew Moment for User Selection in Federated Learning

by Liangyan Li, Yangyi Liu, Yimo Ning, Stefano Rini and Jun Chen

Entropy 2026, 28(6), 630; https://doi.org/10.3390/e28060630 - 2 Jun 2026

Abstract

We introduce the symmetric co-skewness moment (SCKM)—a third-order informational dissimilarity metric that consistently outperforms state-of-the-art client-selection heuristics in federated learning (FL) under heterogeneous data. Unlike similarity-driven schemes, SCKM minimizes redundancy by favoring clients with complementary gradients, delivering faster and more stable convergence even [...] Read more.

We introduce the symmetric co-skewness moment (SCKM)—a third-order informational dissimilarity metric that consistently outperforms state-of-the-art client-selection heuristics in federated learning (FL) under heterogeneous data. Unlike similarity-driven schemes, SCKM minimizes redundancy by favoring clients with complementary gradients, delivering faster and more stable convergence even at high heterogeneity levels. Operating on highly compressed 0.5% gradient summaries, our framework provides two operating modes for different deployment scales: (i) SCKM-Select directly ranks and schedules a small candidate pool, whereas (ii) SCKM-Cluster adds a fast, elbow-guided clustering step to scalably choose from thousands of users. We evaluate both variants on a VGG-16 model across multiple non-IID partition schemes and initializations, observing consistent gains over leading cosine-similarity, loss-sketch, and max-diversity baselines—without increasing the communication budget. Full article

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

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15 pages, 633 KB

Open AccessArticle

Extended Divergence on a Foliation by Continuous-Type Escort Distributions

by Keiko Uohashi

Entropy 2026, 28(6), 629; https://doi.org/10.3390/e28060629 - 2 Jun 2026

Abstract

From an information geometric perspective, this study considers a natural foliation of dualistic structures associated with escort distributions of exponential families. We propose an extended divergence on this foliation by continuous-type escort distributions. Specifically, we investigate the foliation formed by escort distributions to [...] Read more.

From an information geometric perspective, this study considers a natural foliation of dualistic structures associated with escort distributions of exponential families. We propose an extended divergence on this foliation by continuous-type escort distributions. Specifically, we investigate the foliation formed by escort distributions to analyze the transition of q-parameters, rather than relying on a fixed parameter. Within this foliation, distinct q-parameters and their corresponding dualistic

α

-parameters were defined on each leaf. Finally, we present a decomposition of the extended divergence on this foliation, providing an analog to the method previously established for discrete escort distributions. Full article

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

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48 pages, 3703 KB

Open AccessReview

Early Warning Signals in Ecological Time-Series

by Roberto Alvarez-Martinez and Pedro Miramontes

Entropy 2026, 28(6), 628; https://doi.org/10.3390/e28060628 - 2 Jun 2026

Abstract

Ecosystems can undergo abrupt, often irreversible transitions between alternative states—phenomena termed critical transitions or regime shifts—with profound consequences for biodiversity, ecosystem services, and human well-being. Early warning signals (EWSs) derived from time-series analysis offer the prospect of anticipating such transitions before they occur, [...] Read more.

Ecosystems can undergo abrupt, often irreversible transitions between alternative states—phenomena termed critical transitions or regime shifts—with profound consequences for biodiversity, ecosystem services, and human well-being. Early warning signals (EWSs) derived from time-series analysis offer the prospect of anticipating such transitions before they occur, potentially enabling preventive management intervention. This review provides a comprehensive synthesis of EWS methods for ecological systems, encompassing theoretical foundations, statistical indicators, empirical applications, and emerging methodological frontiers. We examine the dynamical basis of EWS in critical slowing down theory, wherein systems approaching bifurcation points exhibit characteristic statistical signatures including rising autocorrelation, increasing variance, and spectral reddening. We present a systematic overview of proposed indicators discuss moving-window frameworks for their computation, and critically evaluate preprocessing requirements and sensitivity to analytical choices. Empirical applications across major ecosystem types—including lakes, coral reefs, grasslands, forests, and marine fisheries—reveal both successes and limitations, with EWS performance depending critically on data quality, transition mechanism, and system-specific dynamics. We address recent advances including machine learning approaches, non-equilibrium thermodynamic indicators, multivariate extensions, and the important distinction between bifurcation-induced, noise-induced, and rate-induced tipping. We conclude with recommendations for specialists, emphasizing the integration of EWS within broader monitoring frameworks, systematic sensitivity analysis, and the interpretation of indicators as probabilistic assessments of changing resilience rather than deterministic predictions of imminent collapse. Full article

(This article belongs to the Section Multidisciplinary Applications)

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23 pages, 323 KB

Open AccessArticle

Towards a Tensor Product Structure-Grounded Mereology

by Matías Pasqualini and Sebastian Fortin

Entropy 2026, 28(6), 627; https://doi.org/10.3390/e28060627 - 2 Jun 2026

Abstract

This paper aims to lay some groundwork for a systematic framework for quantum mereology based on the tensor product structures (TPSs) of Hilbert space. While previous work has suggested that such a TPS-based mereology might conform to classical extensional mereology, we demonstrate that [...] Read more.

This paper aims to lay some groundwork for a systematic framework for quantum mereology based on the tensor product structures (TPSs) of Hilbert space. While previous work has suggested that such a TPS-based mereology might conform to classical extensional mereology, we demonstrate that the space of all possible tensor product structures for a given Hilbert space lacks the lattice-theoretic structure characteristic of classical partitions. Specifically, we show that this space admits no canonical meet operation, thus violating the global validity of a natural extension of the weak supplementation principle. These structural features suggest that quantum mereology, even when based on TPSs, should exhibit certain non-extensional behavior: what counts as a “part” is decomposition-relative in a stronger sense, with different decompositions allowed to be mutually incompatible. The resulting picture challenges extensional interpretations of quantum composition and underscores the need for a formally richer, genuinely non-classical mereology. Full article

(This article belongs to the Special Issue Quantum Mereologies and Quantum Inspired Set Theories and Logics)

25 pages, 3303 KB

Open AccessArticle

Phases and Dynamics of an Impurity Immersed in One-Dimensional Quantum Droplets

by Dimitrios Diplaris, Ilias A. Englezos, Friethjof Theel, Peter Schmelcher and Simeon I. Mistakidis

Entropy 2026, 28(6), 626; https://doi.org/10.3390/e28060626 - 2 Jun 2026

Abstract

We explore the ground-state properties of a single impurity immersed in a one-dimensional quantum droplet medium formed by a two-component Bose mixture. Relying on ab initio simulations, we demonstrate that tuning the impurity–droplet interactions allows to controllably reshape the droplets’ density profiles and [...] Read more.

We explore the ground-state properties of a single impurity immersed in a one-dimensional quantum droplet medium formed by a two-component Bose mixture. Relying on ab initio simulations, we demonstrate that tuning the impurity–droplet interactions allows to controllably reshape the droplets’ density profiles and associated correlation patterns. For attractive impurity-medium couplings, the impurity becomes localized within the droplet, which exhibits a density hump at the vicinity of the impurity, while repulsive interactions facilitate phase separation. Comparing our many-body results with the appropriate extended Gross–Pitaevskii description, we find adequate agreement for the droplet density profiles, with the effective field approach systematically overestimating impurity localization. Following a release of the external trap, we unveil that the sign and magnitude of the interactions between the impurity and the droplet hosts dictate the response of the three-component setting, which experiences expansion unless strongly attractive intercomponent couplings are present. These results corroborate the role and presence of correlations in impurity–droplet mixtures and inspire future investigations on impurity physics for probing droplet configurations. Full article

(This article belongs to the Special Issue Quantum Dynamics in Hybrid Systems)

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18 pages, 634 KB

Open AccessArticle

Curiosity-Driven Exploration with Information Bottleneck Representations and Matrix-Based Mutual Information

by Zhaoxu Meng and Yong Cui

Entropy 2026, 28(6), 625; https://doi.org/10.3390/e28060625 - 2 Jun 2026

Abstract

Curiosity empowers humans to ask questions about the world and explore it without relying on extrinsic, encouraging rewards such as money. To investigate how this mechanism drives exploration, we implement a curiosity-based approach and test it in a reinforcement learning environment. We define [...] Read more.

Curiosity empowers humans to ask questions about the world and explore it without relying on extrinsic, encouraging rewards such as money. To investigate how this mechanism drives exploration, we implement a curiosity-based approach and test it in a reinforcement learning environment. We define curiosity using a hybrid intrinsic signal based on prediction error and the rarity of state–action pairs. To address the curse of dimensionality in raw pixel inputs, we adopt the Information Bottleneck (IB) principle to learn low-dimensional representations that are both compact and predictive. We introduce two formulations for computing mutual information—one based on entropy decomposition and the other on matrix-based Rényi entropy—and compare their effectiveness. Experiments on Acrobot show substantially improved exploration efficiency over Intrinsic Curiosity Module (ICM), Random Network Distillation (RND), and a k-NN novelty baseline, while results on MountainCar indicate that the proposed method is not uniformly superior in low-dimensional environments. These findings suggest that IB-shaped representations and matrix-based information objectives are most beneficial when observations are high-dimensional or dynamics are structurally complex. Full article

(This article belongs to the Special Issue The Information Bottleneck Method: Theory and Applications)

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43 pages, 24379 KB

Open AccessArticle

An Adaptive Refined Composite Multiscale Differential Symbolic Entropy Rooted in LSC-SAO and Its Application in Fault Diagnosis

by Min Mao, Jingzong Yang, Chao Zhou, Chengjiang Zhou and Xuefeng Li

Entropy 2026, 28(6), 624; https://doi.org/10.3390/e28060624 - 1 Jun 2026

Abstract

Accurate fault diagnosis of rotating machinery is critical for ensuring the reliability of the energy, industrial, and transportation sectors. However, conventional methods face significant challenges, including the susceptibility of the Snow Ablation Optimizer (SAO) to local optima, the instability of Multiscale Differential Symbolic [...] Read more.

Accurate fault diagnosis of rotating machinery is critical for ensuring the reliability of the energy, industrial, and transportation sectors. However, conventional methods face significant challenges, including the susceptibility of the Snow Ablation Optimizer (SAO) to local optima, the instability of Multiscale Differential Symbolic Entropy (MDSE) with short time series, and the non-adaptability of Support Vector Machine parameters. To address these issues, this study proposes a parameter-adaptive fault diagnosis framework integrating an improved SAO with Adaptive Refined Composite Multiscale Differential Symbolic Entropy (Adaptive-RCMDSE). First, the Logistic Sine Cosine strategy (LSC) is introduced to enhance SAO’s global search capability, forming the LSC-SAO algorithm. Subsequently, an Adaptive-RCMDSE method is developed wherein LSC-SAO optimizes the control parameter to significantly improve feature stability for short time series. Furthermore, an Adaptive Support Vector Machine (Adaptive-SVM) model is constructed, employing LSC-SAO to automatically tune the penalty factor and kernel parameters for precise fault identification. Finally, validation is performed on gearbox, ball bearing, and axle box bearing datasets. Results indicate that the proposed method achieves superior diagnostic performance, with average accuracies of 99.70%, 99.29%, and 99.28%, respectively, outperforming existing methods. This work provides an effective and robust solution for intelligent health monitoring of rotating machinery. Full article

(This article belongs to the Section Multidisciplinary Applications)

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19 pages, 638 KB

Open AccessArticle

Symptom-Based Contextual Models of Cognition and Judgment for Resolving Biased Decision Making Under Uncertainty

by Gueorgui Petkov

Entropy 2026, 28(6), 623; https://doi.org/10.3390/e28060623 - 1 Jun 2026

Abstract

This article presents a contextual entropic model for understanding, explaining, and resolving cognitive biases associated with thinking under uncertainty. This article applies a unique performance evaluation of teamwork method, enabling a qualitative and quantitative assessment of symptom-based context and ensuring a clear and [...] Read more.

This article presents a contextual entropic model for understanding, explaining, and resolving cognitive biases associated with thinking under uncertainty. This article applies a unique performance evaluation of teamwork method, enabling a qualitative and quantitative assessment of symptom-based context and ensuring a clear and rational interpretation of decision making in ambiguous and risky situations. This human reliability assessment method also contributes to a deeper understanding of the iterative and complementary nature of cognition and decision making. The main idea is the quantum-like understanding of the dual image of a symptom as a wave and a bit of information in thought processes. Thus, each context alternative is identified by a unique combination of three-valued states of the socio-technical system at discrete points in time—recognized, unrecognized, and unrecognizable. Judgment is a wave-like process driven by the interfering sum of the amplitudes of symptom recognition shifts. By modeling stepwise cognitive processes by adding and subtracting symptoms or stimuli, we can estimate and compare the likelihood ratios between biased judgments. This article presents three canonical filters used to improve cognitive theories and decision-making methods—Ellsberg’s two-color, three-color, and four-color paradoxes—to demonstrate the power of the symptom-based context model. Full article

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