You seem to have javascript disabled. Please note that many of the page functionalities won't work as expected without javascript enabled.

Search for Articles:

Title / Keyword

Author / Affiliation / Email

Journal

Article Type

Advanced Search

Section

Special Issue

Volume

Issue

Number

Page

Logical OperatorOperator

Search Text

Search Type

Mirror Descent and Exponentiated Gradient Algorithms Using Trace-Form Entropies
Geometry-Based Bounds on the Capacity of Peak-Limited and Band-Limited Signals over the Additive White Gaussian Noise Channel at a High SNR
On Best Erasure Wiretap Codes: Equivocation Matrices and Design Principles
Disentangling Boltzmann Brains, the Time-Asymmetry of Memory, and the Second Law

Journal Description

Entropy

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

Open Access— free for readers, with article processing charges (APC) paid by authors or their institutions.
High Visibility: indexed within Scopus, SCIE (Web of Science), Inspec, PubMed, PMC, Astrophysics Data System, and other databases.
Journal Rank: JCR - Q2 (Physics, Multidisciplinary) / CiteScore - Q1 (Mathematical Physics)
Rapid Publication: manuscripts are peer-reviewed and a first decision is provided to authors approximately 21.5 days after submission; acceptance to publication is undertaken in 2.6 days (median values for papers published in this journal in the second half of 2025).
Recognition of Reviewers: reviewers who provide timely, thorough peer-review reports receive vouchers entitling them to a discount on the APC of their next publication in any MDPI journal, in appreciation of the work done.
Testimonials: See what our editors and authors say about Entropy.
Companion journals for Entropy include: Foundations, Thermo and Complexities.

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

Imprint Information Journal Flyer Open Access ISSN: 1099-4300

Latest Articles

26 pages, 1283 KB

Open AccessArticle

A Propagation Model of Social Hypernetwork Based on Directed Hypergraph

by Lu Yang, Peng-Yue Li, Feng Hu and Zi-Ke Zhang

Entropy 2026, 28(4), 420; https://doi.org/10.3390/e28040420 (registering DOI) - 9 Apr 2026

In the existing research on information propagation modeling in social networks, hypergraphs have been widely applied to characterize the high-order interaction relationships involving multiple nodes. However, most models are still based on the assumption of undirected connections, which leads to certain limitations in [...] Read more.

In the existing research on information propagation modeling in social networks, hypergraphs have been widely applied to characterize the high-order interaction relationships involving multiple nodes. However, most models are still based on the assumption of undirected connections, which leads to certain limitations in depicting the information flow direction and the structural characteristics of propagation chains. To address the above problems, a social hypernetwork propagation model with directional constraints is constructed in this paper by introducing the directed hypergraph structure and combining it with the improved SEIR model. The strength of social relationships is measured by intimacy in the model, and a comprehensive characterization of the information propagation process is achieved by integrating the threshold mechanism of the directed hypergraphs with the attenuation function of information timeliness. In addition, the effectiveness of the proposed model is verified by taking the event of “imposing additional tariffs” as an example, and the evolutionary characteristics of propagation in different network structures, as well as the impacts of user confidence and information timeliness, are analyzed using simulation experiments. The results indicate that the model is applicable to characterizing the information propagation trends and dynamic characteristics in real social networks, and can provide theoretical references and methodological support for the prediction and regulation of network public opinion. Full article

(This article belongs to the Section Complexity)

► Show Figures

Figure 1

32 pages, 6350 KB

Open AccessArticle

Mixed Forecast of Air Quality Index with a Bibranch Parallel Architecture Considering Seasonal Heterogeneity

by Huibin Zeng, Ying Liu, Hongbin Dai, Xue Zhao and Ning Tian

Entropy 2026, 28(4), 419; https://doi.org/10.3390/e28040419 (registering DOI) - 9 Apr 2026

Accurate prediction of the air quality index (AQI) is crucial for understanding urban pollution dynamics and protecting public health. This study proposes a dual-branch fusion framework (CL-XGB-Season) to address seasonal heterogeneity in AQI prediction by integrating temporal dynamic features and static patterns. The [...] Read more.

Accurate prediction of the air quality index (AQI) is crucial for understanding urban pollution dynamics and protecting public health. This study proposes a dual-branch fusion framework (CL-XGB-Season) to address seasonal heterogeneity in AQI prediction by integrating temporal dynamic features and static patterns. The CNN-LSTM branch captures short-term temporal fluctuations, while a seasonally split XGBoost branch fits long-term static patterns via independent submodels for spring, summer, autumn, and winter. SHAP-based interpretability analysis revealed the dominant drivers across different seasons: the “temperature × O₃” interaction feature plays a key role in summer, characterizing the ozone formation mechanism dominated by photochemical reactions under conditions of high temperature and strong solar radiation; whereas the PM_2.5/PM₁₀ ratio is crucial in winter (where pollution is primarily driven by pollutant accumulation). The dual-branch fusion framework was validated using hourly resolution data from Chongqing for the 2020–2025 period. Results indicate that the framework achieved a prediction accuracy of 0.197 root mean square error (nRMSE) and 0.9611 coefficient of determination (R²) on the test set, outperforming eight ablation variants and five baseline models (ARIMA, Transformer, etc.) in comparative experiments. Ablation studies confirm the necessity of dual branches and seasonal modeling, with the full model reducing nRMSE by 19–63% versus single-model variants. This framework maintains stable seasonal performance and provides actionable insights for targeted air quality management. Full article

(This article belongs to the Special Issue Uncertainty-Aware Feature Learning and Anomaly Detection for Unlabeled Complex Data)

► Show Figures

Figure 1

19 pages, 2474 KB

Open AccessArticle

Power Laws in Empirical Eigenvalue Spectra

by Benyuan Liu, Yung-Ying Chen, M. Shane Li, Vanessa Thomasin Morgan, Eslam Abdelaleem and Audrey Sederberg

Entropy 2026, 28(4), 418; https://doi.org/10.3390/e28040418 - 9 Apr 2026

The critical brain hypothesis proposes that neural systems operate near a phase transition to optimize information processing. A key method for investigating this hypothesis is the phenomenological renormalization group (pRG), which looks for scale-invariant features across levels of coarse-graining. One such feature is [...] Read more.

The critical brain hypothesis proposes that neural systems operate near a phase transition to optimize information processing. A key method for investigating this hypothesis is the phenomenological renormalization group (pRG), which looks for scale-invariant features across levels of coarse-graining. One such feature is the power-law scaling of eigenvalues of covariance matrices of coarse-grained variables. However, the estimation of this scaling exponent,

μ

, often relies on linear regression over arbitrarily selected ranges of the plot of eigenvalues versus rank. This heuristic “eyeballing” introduces uncontrolled bias and complicates the interpretation of observed scaling relationships. In order to obtain a more robust estimation of

μ

, we do not fit the standard eigenvalue-vs-rank relationship, but rather the density of eigenvalues, for which standard protocols exist for fitting power laws to empirical data distributions. We demonstrate this approach using a synthetic model that replicates the scaling signatures of neural data while providing control over the system’s exponents as well as neural data obtained from publicly available Neuropixels recordings. We also establish standards for the minimal data required to quantify power-law behavior in a pRG eigenvalue analysis. Our approach contributes a tool for understanding the fundamental limitations imposed by spatial and temporal constraints of experimental datasets, which is required to rigorously assess the neural criticality hypothesis. Full article

(This article belongs to the Special Issue Information-Theoretic Methods in Computational Neuroscience)

► Show Figures

Figure 1

39 pages, 468 KB

Open AccessArticle

4+1 Gravitation in the SHP Formalism

by Martin Land

Entropy 2026, 28(4), 417; https://doi.org/10.3390/e28040417 - 8 Apr 2026

The Stueckelberg–Horwitz–Piron (SHP) formalism describes particles and fields traced out as spacetime events functionally dependent on an external evolution parameter

τ

. This approach addresses a number of difficulties associated with the problem of time. In SHP general relativity, the state of the [...] Read more.

The Stueckelberg–Horwitz–Piron (SHP) formalism describes particles and fields traced out as spacetime events functionally dependent on an external evolution parameter

τ

. This approach addresses a number of difficulties associated with the problem of time. In SHP general relativity, the state of the unconstrained phase space variables

{x^{μ} (τ), p_{ν} (τ)}

specifies a 4D block spacetime

M (τ)

that evolves to an infinitesimally close 4D block spacetime

M (τ + δ τ)

under a scalar Hamiltonian. As the configuration of matter and energy evolves with

τ

it induces changes in the spacetime metric

γ_{μ ν} (x, τ)

, leading to

τ

-dependent geodesic equations for the phase space variables. The 4+1 approach in gravitation generalizes the 3+1 formalism of Arnowitt, Deser, and Misner (ADM) to construct

τ

-dependent Einstein field equations, a canonical Hamiltonian formalism, and an initial value problem for

γ_{μ ν} (x, τ)

. To conform to known gravitational phenomenology, we must respect the 5D symmetries associated with the free fields—the geometrical constructs relevant to

M (τ)

as an embedded hypersurface—and the O(3,1) symmetries of 4D matter. The 4+1 formalism has been discussed in a series of publications. The goal of this paper is to provide a systematic review of the subject, make a few corrections and some significant additions, and present the theory in a concise and orderly fashion. Full article

(This article belongs to the Special Issue Hamiltonian Dynamics in Fundamental Physics)

20 pages, 388 KB

Open AccessArticle

Koopman–von Neumann and Weyl–Wigner Phase-Space Formulation of Inviscid Euler Flows

by Sandor M. Molnar and Joseph R. Godfrey

Entropy 2026, 28(4), 416; https://doi.org/10.3390/e28040416 - 7 Apr 2026

We develop a unified Koopman–von Neumann (KvN) operator and Weyl–Wigner phase-space framework for inviscid ideal (barotropic) Euler flows. Our approach reformulates the nonlinear fluid dynamics as a linear KvN evolution on an enlarged field phase space, thereby enabling us to apply tools developed [...] Read more.

We develop a unified Koopman–von Neumann (KvN) operator and Weyl–Wigner phase-space framework for inviscid ideal (barotropic) Euler flows. Our approach reformulates the nonlinear fluid dynamics as a linear KvN evolution on an enlarged field phase space, thereby enabling us to apply tools developed for quantum mechanics (Weyl quantization, Moyal ⋆-products, and Wigner functionals) to a classical fluid. We construct the appropriate KvN generator (including the required Jacobian term for unitarity) and derive the evolution equation for the corresponding Wigner functional. This framework clarifies when the classical Liouville (Vlasov) description is exact—namely, in quadratic or linear regimes where the Moyal bracket reduces to the Poisson bracket—and when higher-order quantum-like corrections become significant in fully nonlinear regimes. As an analytic example, we obtain a closed-form Wigner solution for a one-dimensional Burgers flow (pressureless Euler) and verify, term by term, that it reproduces the expected Liouville transport (with distributional contributions at the shock). We also compare the phase-space approach with a kinetic (Vlasov–monokinetic) formulation and outline the extension of the framework to three-dimensional flows using a Clebsch variable representation. Full article

(This article belongs to the Section Multidisciplinary Applications)

► Show Figures

Figure 1

13 pages, 2717 KB

Open AccessArticle

Confinement-Tunable Spatial Distribution of Physisorbed Hydrogen in Defective Carbon Nanotube Bundles

by Shuming Yang, Kun Qiu, Gang Sun and Huaze Shen

Entropy 2026, 28(4), 415; https://doi.org/10.3390/e28040415 - 7 Apr 2026

Spatial confinement strongly affects matter by altering structural stability, relaxation times, and equilibrium properties. Interest in hydrogen storage within carbon nanotube bundles has grown because it addresses practical energy needs while revealing rich confined-fluid physics. Understanding how geometry and defects influence hydrogen structure [...] Read more.

Spatial confinement strongly affects matter by altering structural stability, relaxation times, and equilibrium properties. Interest in hydrogen storage within carbon nanotube bundles has grown because it addresses practical energy needs while revealing rich confined-fluid physics. Understanding how geometry and defects influence hydrogen structure and dynamics is essential to the development of effective storage materials. Here, we investigate how confinement in single-walled carbon nanotube (SWCNT) bundles with vacancies alters the spatial distribution and phase behavior of physisorbed hydrogen. At low temperature, hydrogen forms solid-like, cylindrical layered structures both inside and outside the tubes. Raising the temperature broadens these layers and produces a liquid-like arrangement within the confined regions. This confined solid-to-liquid crossover controls storage capacity and release behavior and can be tuned by temperature, confinement dimensions, and vacancy defects. Full article

(This article belongs to the Special Issue Structure, Dynamics, Phase Behavior and Applications of the Complex Disordered Systems)

► Show Figures

Figure 1

24 pages, 5257 KB

Open AccessArticle

Research on Colorization Algorithm for γ-Photon Flow Field Images Using the SECN Model

by Hui Xiao, Liying Hou, Jiantang Liu and Shengjun Huang

Entropy 2026, 28(4), 414; https://doi.org/10.3390/e28040414 - 4 Apr 2026

γ-photon tomography, which leverages the high penetration and electrical neutrality of high-energy γ-photons, offers a promising non-contact approach for industrial flow field monitoring. However, γ-photon flow-field images are inherently grayscale and exhibit probabilistic statistical imaging characteristics, leading to color banding artifacts when processed [...] Read more.

γ-photon tomography, which leverages the high penetration and electrical neutrality of high-energy γ-photons, offers a promising non-contact approach for industrial flow field monitoring. However, γ-photon flow-field images are inherently grayscale and exhibit probabilistic statistical imaging characteristics, leading to color banding artifacts when processed by mainstream colorization algorithms like DeOldify, which compromise structural continuity and visual consistency. To address this issue, this paper proposes a Structure Enhancement Colorization Network (SECN) model for γ-photon flow-field image colorization. A U-Net + GAN framework is employed, with ResNet101 as the generator backbone. It integrates structure-aware enhancement and multi-scale attention modules, while the discriminator incorporates enhanced blocks for improved boundary and texture discrimination. By adaptively fusing global–local features across channel and spatial dimensions, the SECN model effectively suppresses color banding artifacts and enhances structural consistency. To validate the effectiveness of the proposed algorithm, two CFD-simulated γ-photon flow-field image colorization scenarios—namely a large-scale vortex wake and a horizontal wake—are used as evaluation targets. In terms of image quality metrics, the proposed colorization algorithm achieves PSNR, SSIM, FID, and MAE values of 32.5831, 0.8612, 17.8514, and 0.0191, respectively, corresponding to improvements over DeOldify of 4.54%, 2.82%, 5.18%, and 11.16%. When considering information entropy, the proposed colorization algorithm achieves an average entropy value of 4.0257, marking a 4.44% increase compared to DeOldify’s 3.8543, demonstrating superior information preservation and reduced uncertainty in reconstructing complex probabilistic structures. Furthermore, from the perspective of parameter inversion, the temperature inversion MAPE is 7.60%, which is a significant reduction of 18.42% compared to that of DeOldify. Full article

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

► Show Figures

Figure 1

29 pages, 10333 KB

Open AccessArticle

Chaotic Characteristics Analysis of a Strongly Dissipative Nonlinearly Coupled Chaotic System and Its Application in DNA-Encoded RGB Image Encryption

by Zhixin Yu, Zean Tian, Biao Wang, Wei Wang, Ning Pan, Yang Wang, Qian Fang, Xin Zuo, Luxue Yu, Yuxin Jiang, Long Tian and Feiyan Yan

Entropy 2026, 28(4), 413; https://doi.org/10.3390/e28040413 - 4 Apr 2026

This paper proposes a novel four-dimensional strongly dissipative nonlinearly coupled hyperchaotic system, investigates its dynamical characteristics, and demonstrates its applicability through Deoxyribonucleic Acid (DNA)-encoded RGB image encryption. First, a four-dimensional nonlinearly coupled hyperchaotic system with strong dissipativity is constructed. Nonlinear dynamics analysis methods, [...] Read more.

This paper proposes a novel four-dimensional strongly dissipative nonlinearly coupled hyperchaotic system, investigates its dynamical characteristics, and demonstrates its applicability through Deoxyribonucleic Acid (DNA)-encoded RGB image encryption. First, a four-dimensional nonlinearly coupled hyperchaotic system with strong dissipativity is constructed. Nonlinear dynamics analysis methods, including phase trajectory diagrams, Lyapunov exponent spectra, and bifurcation diagrams, are employed to thoroughly reveal the system’s complex dynamical evolution mechanisms. The analysis indicates that the system not only possesses a wide range of chaotic parameters but also exhibits rich phenomena of multiple coexisting attractors, demonstrating a high degree of multistability. This characteristic offers potential advantages for image encryption, as it increases the diversity of dynamical behaviors and enhances sensitivity to initial conditions. The physical realizability of the chaotic behavior is further verified through an analog circuit implementation. Consequently, the system supports the design of encryption algorithms with larger key spaces, stronger resistance to phase space reconstruction, and improved pseudo-randomness, making it particularly suitable for applications with extremely high security requirements. Subsequently, leveraging the highly random chaotic sequences generated by this system, combined with various DNA coding rules and operations, the RGB image components are scrambled and diffused for encryption. Security analysis demonstrates that the algorithm effectively passes examinations across multiple dimensions, including histogram analysis, information entropy, adjacent pixel correlation, Number of Pixel Change Rate (NPCR), Unified Average Changing Intensity (UACI), and The Peak Signal-to-noise Ratio (PSNR). It achieves favorable encryption results, significantly enhances image resistance against attacks, and provides a reliable technical solution for the secure transmission of remote sensing and military images. Full article

(This article belongs to the Special Issue Nonlinear Dynamics of Complex Systems)

► Show Figures

Figure 1

23 pages, 1645 KB

Open AccessArticle

Secure Cooperative Communications in 6G Networks: A Constrained Hierarchical Reinforcement Learning Framework with Hybrid Action Space

by Xiaosi Tian, Zulin Wang and Yuanhan Ni

Entropy 2026, 28(4), 412; https://doi.org/10.3390/e28040412 - 4 Apr 2026

With the rapid evolution toward 6G networks, ensuring robust physical layer security (PLS) in highly dynamic and heterogeneous wireless environments has become a key challenge. Traditional security methods often struggle to adapt to time-varying channels, especially in the absence of perfect channel state [...] Read more.

With the rapid evolution toward 6G networks, ensuring robust physical layer security (PLS) in highly dynamic and heterogeneous wireless environments has become a key challenge. Traditional security methods often struggle to adapt to time-varying channels, especially in the absence of perfect channel state information. Furthermore, the dynamic nature of node selection and power allocation in heterogeneous networks creates a complex hybrid action space operating across multiple timescales, significantly complicating the design of efficient and adaptive security strategies. To address this, this paper proposes a novel constrained hierarchical reinforcement learning (CHRL) framework for secure cooperative communications in next-generation wireless systems. The framework is designed to optimize secrecy performance within a hybrid action space comprising both discrete node selection and continuous power allocation, operating at different timescales. By hierarchically decoupling the joint optimization problem, the upper layer performs risk-aware node selection to maximize long-term secrecy capacity (SC) while guaranteeing a stable and secure link. At the lower layer, we develop a constrained MiniMax Multi-objective Deep Deterministic Policy Gradient (M3DDPG) algorithm that optimizes power allocation considering worst-case conditions. Lagrange multipliers are integrated to enforce a strictly positive SC constraint throughout transmission, effectively preventing security outages. Simulation results under time-varying Rayleigh fading channels demonstrate that the proposed CHRL framework outperforms existing HRL methods, achieving up to 17% improvement in SC while strictly maintaining security constraints. These results validate the effectiveness of the proposed approach for enhancing PLS in next-generation cooperative wireless networks. Full article

(This article belongs to the Special Issue Physical Layer Security for Next-Generation Wireless Networks: Theory, Technologies, and Applications)

► Show Figures

Figure 1

22 pages, 10553 KB

Open AccessArticle

Reconstruction of Multiplex Networks with Correlated Layers

by Valerio Gemmetto and Diego Garlaschelli

Entropy 2026, 28(4), 411; https://doi.org/10.3390/e28040411 - 4 Apr 2026

In many situations, the complete microscopic structure of a network is empirically inaccessible and has to be inferred from aggregate information using some probabilistic model. While several network reconstruction methods have been developed in the case of single-layer networks where nodes can be [...] Read more.

In many situations, the complete microscopic structure of a network is empirically inaccessible and has to be inferred from aggregate information using some probabilistic model. While several network reconstruction methods have been developed in the case of single-layer networks where nodes can be connected only by one type of link, the problem is still largely unexplored in the case of multiplex networks where several interdependent layers, each representing a distinct mode of connection, coexist. Even the most advanced network reconstruction techniques, when applied to each layer separately, may fail in replicating the inter-layer dependencies embodying the essence of multiplex networks. Here we develop a methodology to reconstruct a class of correlated multiplexes which includes, as a specific example that we study in detail, the multiplex network of international trade in different products. Our method starts from virtually any reconstruction model that successfully reproduces a set of desired marginal properties of each layer separately, i.e., node strengths and/or node degrees. It then introduces the minimal dependency structure required to replicate an additional set of higher-order properties, namely the portion of each node’s degree and each node’s strength that is shared and/or reciprocated across pairs of layers. These properties are found to provide empirically robust measures of inter-layer coupling, allowing for an accurate reconstruction of the world trade multiplex network. Our method allows for joint multi-layer connection probabilities to be reliably reconstructed from marginal ones, effectively bridging the gap between single-layer information and global multiplex properties. Full article

(This article belongs to the Section Multidisciplinary Applications)

► Show Figures

Figure 1

15 pages, 1517 KB

Open AccessArticle

Intrinsic Cause–Effect Power: The Tradeoff Between Differentiation and Specification

by William G. P. Mayner, William Marshall and Giulio Tononi

Entropy 2026, 28(4), 410; https://doi.org/10.3390/e28040410 - 4 Apr 2026

Integrated information theory (IIT) starts from the existence of consciousness and characterizes its essential properties: every experience is intrinsic, specific, unitary, definite, and structured. IIT then formulates existence and its essential properties operationally in terms of cause–effect power of a substrate of units. [...] Read more.

Integrated information theory (IIT) starts from the existence of consciousness and characterizes its essential properties: every experience is intrinsic, specific, unitary, definite, and structured. IIT then formulates existence and its essential properties operationally in terms of cause–effect power of a substrate of units. Here, we address IIT’s operational requirements for existence by considering that, to have cause–effect power, to have it intrinsically, and to have it specifically, substrate units in their actual state must both (i) ensure the intrinsic availability of a repertoire of cause–effect states, and (ii) increase the probability of a specific cause–effect state. We showed previously that requirement (ii) can be assessed by the intrinsic difference of a state’s probability from maximal differentiation. Here, we show that requirement (i) can be assessed by the intrinsic difference from maximal specification. These points and their consequences for integrated information are illustrated using simple systems of micro units. When applied to macro units and systems of macro units such as neural systems, a tradeoff between differentiation and specification is a necessary condition for intrinsic existence—and therefore, according to IIT, for consciousness. Full article

(This article belongs to the Collection Advances in Integrated Information Theory)

► Show Figures

Figure 1

20 pages, 893 KB

Open AccessArticle

Step-Wise Dual Dynamic DPSGD: Enhancing Performance on Imbalanced Medical Datasets with Differential Privacy

by Xiaobo Huang and Fang Xie

Entropy 2026, 28(4), 409; https://doi.org/10.3390/e28040409 - 4 Apr 2026

The application of differential privacy in deep learning often leads to significant performance degradation on class-imbalanced medical datasets. Methods such as adding noise to gradients for differential privacy are effective on large datasets, like MNIST and CIFAR-100, but perform poorly on small, imbalanced [...] Read more.

The application of differential privacy in deep learning often leads to significant performance degradation on class-imbalanced medical datasets. Methods such as adding noise to gradients for differential privacy are effective on large datasets, like MNIST and CIFAR-100, but perform poorly on small, imbalanced medical datasets, like HAM10000 and ISIC2019. This is because the imbalanced distribution causes the gradients from the few-shot classes to be clipped, resulting in the loss of crucial information, while the majority classes dominate the learning process. This leads the model to fall into suboptimal solutions early. To address this issue, we propose SDD-DPSGD, which uses a step-wise dynamic exponential scheduling mechanism for noise and clipping thresholds to preserve gradient information. By allocating more privacy budget and employing higher clipping thresholds during the initial training phases, the model can avoid suboptimal solutions and improve its performance. Experiments show that SDD-DPSGD outperforms comparable algorithms on the HAM10000 dataset, and the ISIC2019 dataset. Full article

(This article belongs to the Special Issue Information-Theoretic Security and Privacy)

► Show Figures

Figure 1

34 pages, 7929 KB

Open AccessReview

Interior Microstates and Black Hole Entropy

by Martin Sasieta

Entropy 2026, 28(4), 408; https://doi.org/10.3390/e28040408 - 3 Apr 2026

Semiclassical gravity admits a vast set of candidate black hole interior states, raising the question of which of these correspond to independent quantum microstates that account for black hole entropy. In this review, we survey several explicit constructions of black hole interior microstates [...] Read more.

Semiclassical gravity admits a vast set of candidate black hole interior states, raising the question of which of these correspond to independent quantum microstates that account for black hole entropy. In this review, we survey several explicit constructions of black hole interior microstates in AdS₂ holography and AdS/CFT and assess whether they furnish bases of the black hole Hilbert space. We further highlight the settings in which non-perturbative effects in the gravitational path integral, captured by spacetime wormholes, resolve the resulting overcounting and reproduce the black hole entropy from state counting. Full article

(This article belongs to the Special Issue Coarse and Fine-Grained Aspects of Gravitational Entropy)

► Show Figures

Figure 1

19 pages, 333 KB

Open AccessArticle

Quantum Codes from Galois Hulls of Constacyclic Codes over a Finite Non-Chain Ring

by Enbin Zhang, Bo Kong and Xiying Zheng

Entropy 2026, 28(4), 407; https://doi.org/10.3390/e28040407 - 3 Apr 2026

We study the algebraic structures of constacyclic codes over a new finite non-chain ring

R

and their l-Galois dual codes based on a new Gray map

ϕ

and determine the dimensions of the l-Galois hull of constacyclic codes over

R

. [...] Read more.

We study the algebraic structures of constacyclic codes over a new finite non-chain ring

R

and their l-Galois dual codes based on a new Gray map

ϕ

and determine the dimensions of the l-Galois hull of constacyclic codes over

R

. Furthermore, we propose a quantum construction X of the Galois inner product to construct quantum codes by combining the Calderbank–Shor–Steane (CSS) construction, Hermitian construction and construction X. As an application, by calculating the dimensions of the l-Galois hull of constacyclic codes over

R

, we obtain some new quantum maximum distance separable (QMDS) codes and new quantum error-correcting codes (QECCs) that are better than the best known ones. Full article

(This article belongs to the Section Quantum Information)

15 pages, 1217 KB

Open AccessArticle

Detecting Phase Transitions from Data Using Generative Learning

by Xiyu Zhou, Yan Mi and Pan Zhang

Entropy 2026, 28(4), 406; https://doi.org/10.3390/e28040406 - 3 Apr 2026

Identifying phase transitions in complex many-body systems traditionally necessitates the definition of specific order parameters, a task often requiring prior knowledge of the statistical model and the symmetry-breaking mechanism. In this work, we propose a framework for detecting phase transitions directly from raw [...] Read more.

Identifying phase transitions in complex many-body systems traditionally necessitates the definition of specific order parameters, a task often requiring prior knowledge of the statistical model and the symmetry-breaking mechanism. In this work, we propose a framework for detecting phase transitions directly from raw (experimental) data without requiring knowledge of the underlying model Hamiltonian, parameters, or pre-defined labels. Inspired by generative modeling in machine learning, our method utilizes autoregressive networks to estimate the normalized probability distribution of the system from raw configuration data. We then quantify the intrinsic sensitivity of this learned distribution to control parameters (such as temperature) to construct a robust indicator of phase transitions. This indicator is based on the expectation of the change in absolute logarithmic probability, derived entirely from the raw data. Our approach is purely data-driven: it takes raw data across varying control parameters as input and outputs the most likely estimate of the phase transition point. To validate our approach, we conduct extensive numerical experiments on the 2D Ising model on both triangular and square lattices, and on the Sherrington–Kirkpatrick (SK) model utilizing raw data generated via Markov Chain Monte Carlo and Tensor Network methods. The results demonstrate that our generative approach accurately identifies phase transitions using only raw data. Our framework provides a general tool for exploring critical phenomena in model systems, with the potential to be extended to realistic experimental data where theoretical descriptions remain incomplete. Full article

(This article belongs to the Special Issue Applications of Information-Theoretic Concepts for Generative AI Systems)

► Show Figures

Figure 1

5 pages, 3652 KB

Open AccessEditorial

Causal Graphical Models and Their Applications

by Luis Enrique Sucar and David Danks

Entropy 2026, 28(4), 405; https://doi.org/10.3390/e28040405 - 3 Apr 2026

The study of causality [...] Full article

(This article belongs to the Special Issue Causal Graphical Models and Their Applications)

10 pages, 250 KB

Open AccessBrief Report

Antiunitary Symmetry in Non-Hermitian Dissipative Dynamics and Neutron Scattering

by László Deák

Entropy 2026, 28(4), 404; https://doi.org/10.3390/e28040404 - 3 Apr 2026

Symmetry transformations are defined by operators in quantum mechanics that preserve the modulus of the scalar product between Hilbert space vectors. According to Wigner’s theorem, any such transformation is represented by either a unitary linear operator or an antiunitary (isometric conjugate-linear) operator. Although [...] Read more.

Symmetry transformations are defined by operators in quantum mechanics that preserve the modulus of the scalar product between Hilbert space vectors. According to Wigner’s theorem, any such transformation is represented by either a unitary linear operator or an antiunitary (isometric conjugate-linear) operator. Although antiunitary symmetries—most notably time reversal and charge conjugation—are encountered less frequently than unitary ones, they are fundamental to the description of non-conservative and reversible systems. The most frequently treated antiunitary operators are the involutive ones, called conjugations. Any antiunitary operator can be written as a product of a conjugation and a unitary operator. Considering general scattering problems defined by a scattering potential and separating conjugation from the symmetry operator, one can find the role of complex symmetric (in other words self-transpose) unitary operators in physical problems. This approach provides a robust framework for analyzing the role of non-Hermitian symmetries in wave scattering and dissipative dynamics. To demonstrate the practical applicability of these theoretical concepts, we analyze the case of polarized neutron reflectometry (PNR). We show that the scattering potential in PNR, comprising nuclear and magnetic terms, can satisfy the condition of being unitarily equivalent to its transpose, thereby guaranteeing reciprocity under specific orientations of the magnetic field. Full article

(This article belongs to the Special Issue Dissipative Physical Dynamics)

19 pages, 12990 KB

Open AccessArticle

Multistructural and Multiscale Instability Characterization of Gas–Liquid Two-Phase Flow with MRA-CMESSE

by Qing-Ming Sun, Qing-Chao Yu and Di Ba

Entropy 2026, 28(4), 403; https://doi.org/10.3390/e28040403 - 2 Apr 2026

Characterizing instability in gas–liquid flows is difficult because flow dynamics interact across multiple scales. In this work, we develop an integrated framework that combines multi-resolution analysis with composite multiscale equiprobable symbolic sample entropy (MRA-CMESSE). This combination enables us to examine flow instability from [...] Read more.

Characterizing instability in gas–liquid flows is difficult because flow dynamics interact across multiple scales. In this work, we develop an integrated framework that combines multi-resolution analysis with composite multiscale equiprobable symbolic sample entropy (MRA-CMESSE). This combination enables us to examine flow instability from a multistructural and multiscale perspective. A comprehensive evaluation across four distinct metrics shows that our method is more robust to changes in data length than multiscale sample entropy and composite multiscale sample entropy approaches. Furthermore, MRA-CMESSE is applied to analyze differential pressure time series from vertical air–water two-phase flow, providing a quantitative characterization of the instability of three flow patterns. Among these, bubble flow is the most unstable, with energy spread out and high complexity at small scales; slug flow is the most stable, with its energy focused at larger scales with low complexity, and churn flow falls in between. A central finding is that as superficial gas velocity increases, energy and complexity shift to the meso-scale and micro-scale. This quantitative analysis identifies increased agitation at the meso-scale and micro-scale as the primary driver of enhanced overall flow instability. This framework offers a new quantitative basis for analyzing gas–liquid two-phase flows and strengthens the physical foundation for the monitoring and control of related industrial systems. Full article

(This article belongs to the Section Multidisciplinary Applications)

► Show Figures

Figure 1

21 pages, 1058 KB

Open AccessArticle

Sequential Change Detection with Local Differential Privacy

by Lixing Zhang, Xuran Liu, Ruizhi Zhang and Liyan Xie

Entropy 2026, 28(4), 402; https://doi.org/10.3390/e28040402 - 2 Apr 2026

Sequential change detection is a fundamental problem in statistics and signal processing, with the CUSUM procedure widely used to achieve minimax detection delay under a prescribed false alarm rate when pre- and post-change distributions are fully known. However, in many practical settings, raw [...] Read more.

Sequential change detection is a fundamental problem in statistics and signal processing, with the CUSUM procedure widely used to achieve minimax detection delay under a prescribed false alarm rate when pre- and post-change distributions are fully known. However, in many practical settings, raw observations cannot be shared with a trusted central curator, and privacy must be enforced at the data source, which prevents the computation of exact CUSUM statistics. We therefore introduce a local differentially private (DP) variant called LDP-CUSUM, which first applies a local DP mechanism to transform the raw data into privatized observations and then applies a CUSUM procedure to detect the change. We derive closed-form bounds on the average run length to false alarm and on the worst-case average detection delay, explicitly characterizing the tradeoff among privacy level, false alarm rate, and detection efficiency. Numerical simulations and a real-data case study were conducted to demonstrate the detection efficiency of our proposed LDP-CUSUM across various scenarios. Full article

(This article belongs to the Special Issue Information-Theoretic Security and Privacy)

► Show Figures

Figure 1

28 pages, 13989 KB

Open AccessArticle

Stable Boundaries of Opinion Dynamics in Heterogeneous Spatial Complex Networks

by Mats Bierwirth and Johannes Lengler

Entropy 2026, 28(4), 401; https://doi.org/10.3390/e28040401 - 2 Apr 2026

We investigate majority-vote opinion dynamics on Geometric Inhomogeneous Random Graphs (GIRGs), a powerful model for spatial complex networks. In contrast to classic coarsening dynamics, where a single opinion typically achieves global consensus, our simulations reveal that sufficiently large, localized opinion domains do not [...] Read more.

We investigate majority-vote opinion dynamics on Geometric Inhomogeneous Random Graphs (GIRGs), a powerful model for spatial complex networks. In contrast to classic coarsening dynamics, where a single opinion typically achieves global consensus, our simulations reveal that sufficiently large, localized opinion domains do not disappear. Instead, they stabilize, leading to a persistent coexistence of competing opinions. To understand the mechanism behind this arrested coarsening, we develop and analyze a tractable mean-field model of the interface between two opinion domains. Our main theoretical result rigorously establishes the existence of a stable, non-trivial limiting distribution for the interface profile in a mean-field analysis. This demonstrates that the boundary between opinions is stationary, providing a mathematical explanation for how complex network geometry can support robust opinion diversity in social systems. Full article

(This article belongs to the Special Issue Advances in Complex Networks and Their Applications, from COMPLEX NETWORKS 2025)

► Show Figures

Figure 1

More Articles...

Submit to Entropy Review for Entropy

Journal Menu

Journal Browser

► Journal Browser

Highly Accessed Articles

View More...

Latest Books

More Books and Reprints...

E-Mail Alert

News

4 March 2026
MDPI’s 2025 Best Paper Awards—Award-Winning Papers Announced

22 January 2026
“Do Not Be Afraid of New Things”: Prof. Michele Parrinello on Scientific Curiosity and the Importance of Fundamental Research

9 April 2026
World Quantum Day, 14 April 2026

More News & Announcements...

Topics

Propose a Topic

Topic in AI, Buildings, Computers, Drones, Entropy, Symmetry

Applications of Machine Learning in Large-Scale Optimization and High-Dimensional Learning Topic Editors: Jeng-Shyang Pan, Junzo Watada, Vaclav Snasel, Pei Hu
Deadline: 30 April 2026

Topic in Algorithms, Energies, Entropy, MAKE, Materials, Laboratories, Complexities

AI and Computational Methods for Modelling, Simulations and Optimizing of Advanced Systems: Innovations in Complexity, 2nd Edition Topic Editors: Jaroslaw Krzywanski, Marcin Sosnowski, Karolina Grabowska, Dorian Skrobek, Anna Zylka, Agnieszka Kijo-Kleczkowska, Bashar Shboul, Tomasz Czakiert
Deadline: 30 June 2026

Topic in Atmosphere, Earth, Encyclopedia, Entropy, Fractal Fract, MAKE, Meteorology

Revisiting Butterfly Effect, Multiscale Dynamics, and Predictability Using Ai-Enhanced Modeling Framework (AEMF) and Chaos Theory Topic Editors: Bo-Wen Shen, Roger A. Pielke Sr., Xubin Zeng
Deadline: 31 July 2026

Topic in Applied Sciences, Computers, Energies, Entropy, Mathematics

Numerical Methods and Computer Simulations in Energy Analysis, 3rd Edition Topic Editors: Marcin Kamiński, Mateus Mendes
Deadline: 31 August 2026

More Topics

Conferences

Propose a Conference Collaboration

1–3 July 2026 Entropy 2026: Exploring Complexity and Information in Science

30 October–3 November 2026 The 1st International Conference on Modern Mathematical Physics (ICMMP 2026)

10–13 May 2026 34th European Symposium on Applied Thermodynamics - ESAT 2026

More Conferences...

Special Issues

Propose a Special Issue

Special Issue in Entropy

Thermodynamic and Transport Properties of Plasmas Guest Editors: Gianpiero Colonna, Annarita Laricchiuta, Elena Kustova
Deadline: 15 April 2026

Special Issue in Entropy

Nonextensive Statistical Mechanics in Astrophysics Guest Editor: Jiulin Du
Deadline: 15 April 2026

Special Issue in Entropy

Advances in Probabilistic Machine Learning Guest Editors: Martin Trapp, Pierre Alquier
Deadline: 16 April 2026

Special Issue in Entropy

Entropy in Quantum Systems and Quantum Field Theory (QFT): Gravity, 3rd Edition Guest Editor: Ignazio Licata
Deadline: 17 April 2026

More Special Issues

Topical Collections

Topical Collection in Entropy

Advances in Applied Statistical Mechanics Collection Editor: Antonio M. Scarfone

Topical Collection in Entropy

Wavelets, Fractals and Information Theory Collection Editor: Carlo Cattani

Topical Collection in Entropy

Feature Papers in Information Theory Collection Editors: Raúl Alcaraz, Luca Faes, Leandro Pardo, Boris Ryabko

Topical Collection in Entropy

Disorder and Biological Physics Collection Editors: Hong Qian, Felix Ritort, Lamberto Rondoni

More Topical Collections

Back to TopTop