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Vector Flows That Compute the Capacity of Discrete Memoryless Channels

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 22.3 days after submission; acceptance to publication is undertaken in 2.7 days (median values for papers published in this journal in the second half of 2024).
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: 2.1 (2023); 5-Year Impact Factor: 2.2 (2023)

Imprint Information Journal Flyer Open Access ISSN: 1099-4300

Latest Articles

27 pages, 1571 KiB

Open AccessArticle

Gaussian Versus Mean-Field Models: Contradictory Predictions for the Casimir Force Under Dirichlet–Neumann Boundary Conditions

by Daniel Dantchev, Vassil Vassilev and Joseph Rudnick

Entropy 2025, 27(5), 468; https://doi.org/10.3390/e27050468 - 25 Apr 2025

The mean-field model (MFM) is the workhorse of statistical mechanics: one normally accepts that it yields results which, despite differing numerically from correct ones, are not “very wrong”, in that they resemble the actual behavior of the system as eventually obtained by more [...] Read more.

The mean-field model (MFM) is the workhorse of statistical mechanics: one normally accepts that it yields results which, despite differing numerically from correct ones, are not “very wrong”, in that they resemble the actual behavior of the system as eventually obtained by more advanced treatments. This, for example, turns out to be the case for the Casimir force under, say, Dirichlet–Dirichlet,

(+, +)

and

(+, -)

boundary conditions (BC) for which, according to the general expectations, the MFM is attractive for similar BC or repulsive for dissimilar BC force, with the principally correct position of the maximum strength of the force below or above the critical point

T_{c}

. It turns out, however, that this is not the case with Dirichlet–Neumann (DN) BC. In this case, the mean-field approach leads to an attractive Casimir force. This contradiction with the “boundary condition rule” is cured in the case of the Gaussian model under DN BC. Our results, which are mathematically exact, demonstrate that the Casimir force within the MFM is attractive as a function of temperature T and external magnetic field h, while for the Gaussian model, it is repulsive for

h = 0

and can be, surprisingly, both repulsive and attractive for

h \neq 0

. The treatment of the MFM is based on the exact solution of one non-homogeneous, nonlinear differential equation of second order. The Gaussian model is analyzed in terms of both its continuum and lattice realization. The obtained outcome teaches us that the mean-field results should be accepted with caution in the case of fluctuation-induced forces and ought to be checked against the more precise treatment of fluctuations within the envisaged system. Full article

(This article belongs to the Section Statistical Physics)

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12 pages, 625 KiB

Open AccessArticle

Multiscale Simplicial Complex Entropy Analysis of Heartbeat Dynamics

by Alvaro Zabaleta-Ortega, Carlos Carrizalez-Velazquez, Bibiana Obregón-Quintana and Lev Guzmán-Vargas

Entropy 2025, 27(5), 467; https://doi.org/10.3390/e27050467 - 25 Apr 2025

The present study proposes a multiscale analysis of the simplicial complex approximate entropy (MS-SCAE) applied to cardiac interbeat series. The MS-SCAE method is based on quantifying the changes in the simplicial complex associated with the time series when a coarse-grained procedure is performed. [...] Read more.

The present study proposes a multiscale analysis of the simplicial complex approximate entropy (MS-SCAE) applied to cardiac interbeat series. The MS-SCAE method is based on quantifying the changes in the simplicial complex associated with the time series when a coarse-grained procedure is performed. Our findings are consistent with those of previously reported studies, which indicate that the complexity of healthy interbeat dynamics remains relatively stable over different scales. However, these dynamics undergo changes in the presence of certain cardiac pathologies, such as congestive heart failure and atrial fibrillation. The method we present here allows for effective differentiation between different dynamics and is robust in its ability to characterize both real and simulated sequences. This makes it a suitable candidate for application to a variety of complex signals. Full article

(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications: Fourth Edition)

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30 pages, 521 KiB

Open AccessArticle

Two Types of Temporal Symmetry in the Laws of Nature

by A. Y. Klimenko

Entropy 2025, 27(5), 466; https://doi.org/10.3390/e27050466 - 25 Apr 2025

This work explores the implications of assuming time symmetry and applying bridge-type, time-symmetric temporal boundary conditions to deterministic laws of nature with random components. The analysis, drawing on the works of Kolmogorov and Anderson, leads to two forms of governing equations, referred to [...] Read more.

This work explores the implications of assuming time symmetry and applying bridge-type, time-symmetric temporal boundary conditions to deterministic laws of nature with random components. The analysis, drawing on the works of Kolmogorov and Anderson, leads to two forms of governing equations, referred to here as symmetric and antisymmetric. These equations account for the emergence of characteristics associated with conventional thermodynamics, the arrow of time, and a form of antecedent causality. The directional properties of time arise from the mathematical structure of Markov bridges in proximity of the corresponding temporal boundary conditions, without requiring any postulates that impose a preferred direction of time. Full article

(This article belongs to the Special Issue Quantum Mechanics and the Challenge of Time)

18 pages, 898 KiB

Open AccessArticle

The Entropy Analysis Method for Assessing the Efficiency of Workload Distribution Among Medical Institution Personnel

by Oksana Mulesa and Ivanna Dronyuk

Entropy 2025, 27(5), 465; https://doi.org/10.3390/e27050465 - 25 Apr 2025

The aim of this study is to develop a convenient and effective entropy analysis method for assessing the efficiency of workload distribution among medical institution personnel. This research is based on a model for evaluating employee workload in conditional time units—credits—taking into account [...] Read more.

The aim of this study is to develop a convenient and effective entropy analysis method for assessing the efficiency of workload distribution among medical institution personnel. This research is based on a model for evaluating employee workload in conditional time units—credits—taking into account time-and-motion studies and the volume of medical services provided or tasks performed over a given period. The model and method developed by the authors enable the consideration of potential losses of working time and coefficients that determine the percentage of effective working time. The method is based on calculating and analyzing the values of normative and actual workloads of employees. The study introduces such indicators as relative workload, workload distribution entropy, and the entropy of free and excessively worked time credits. During the experimental verification of the developed method for analyzing the activities of a dental clinic, it was demonstrated that the method is both convenient and effective for analyzing the performance of individual employees as well as groups of employees. The results of the method are presented in a convenient and intuitively understandable form. Therefore, this method can serve as an effective tool for identifying internal reserves within the institution and making managerial decisions regarding its further operation. Full article

(This article belongs to the Section Multidisciplinary Applications)

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27 pages, 2723 KiB

Open AccessReview

Phase Stability and Transitions in High-Entropy Alloys: Insights from Lattice Gas Models, Computational Simulations, and Experimental Validation

by Łukasz Łach

Entropy 2025, 27(5), 464; https://doi.org/10.3390/e27050464 - 25 Apr 2025

High-entropy alloys (HEAs) are a novel class of metallic materials composed of five or more principal elements in near-equimolar ratios. This unconventional composition leads to high configurational entropy, which promotes the formation of solid solution phases with enhanced mechanical properties, thermal stability, and [...] Read more.

High-entropy alloys (HEAs) are a novel class of metallic materials composed of five or more principal elements in near-equimolar ratios. This unconventional composition leads to high configurational entropy, which promotes the formation of solid solution phases with enhanced mechanical properties, thermal stability, and corrosion resistance. Phase stability plays a critical role in determining their structural integrity and performance. This study provides a focused review of HEA phase transitions, emphasizing the role of lattice gas models in predicting phase behavior. By integrating statistical mechanics with thermodynamic principles, lattice gas models enable accurate modeling of atomic interactions, phase segregation, and order-disorder transformations. The combination of computational simulations (e.g., Monte Carlo, molecular dynamics) with experimental validation (e.g., XRD, TEM, APT) improves predictive accuracy. Furthermore, advances in data-driven methodologies facilitate high-throughput exploration of HEA compositions, accelerating the discovery of alloys with optimized phase stability and superior mechanical performance. Beyond structural applications, HEAs demonstrate potential in functional domains, such as catalysis, hydrogen storage, and energy technologies. This review brings together theoretical modeling—particularly lattice gas approaches—and experimental validation to form a unified understanding of phase behavior in high-entropy alloys. By highlighting the mechanisms behind phase transitions and their implications for material performance, this work aims to support the design and optimization of HEAs for real-world applications in aerospace, energy systems, and structural materials engineering. Full article

(This article belongs to the Special Issue Statistical Mechanics of Lattice Gases)

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11 pages, 1449 KiB

Open AccessArticle

A Learning Probabilistic Boolean Network Model of a Manufacturing Process with Applications in System Asset Maintenance

by Pedro Juan Rivera Torres, Chen Chen, Sara Rodríguez González and Orestes Llanes Santiago

Entropy 2025, 27(5), 463; https://doi.org/10.3390/e27050463 - 25 Apr 2025

Probabilistic Boolean Networks (PBN) can model the dynamics of complex biological systems, as well as other non-biological systems like manufacturing systems and smart grids. In this proof-of-concept paper, we propose a PBN architecture with a learning process that significantly enhances fault and failure [...] Read more.

Probabilistic Boolean Networks (PBN) can model the dynamics of complex biological systems, as well as other non-biological systems like manufacturing systems and smart grids. In this proof-of-concept paper, we propose a PBN architecture with a learning process that significantly enhances fault and failure prediction in manufacturing systems. This concept was tested using a PBN model of an ultrasound welding process and its machines. Through various experiments, the model successfully learned to maintain a normal operating state. Leveraging the complex properties of PBNs, we utilize them as an adaptive learning tool with positive feedback, demonstrating that these networks may have broader applications than previously recognized. This multi-layered PBN architecture offers substantial improvements in fault detection performance within a positive feedback network structure that shows greater noise tolerance than other methods. Full article

(This article belongs to the Topic AI and Computational Methods for Modelling, Simulations and Optimizing of Advanced Systems: Innovations in Complexity)

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13 pages, 6262 KiB

Open AccessArticle

Spatial, Temporal, and Dynamic Behavior of Different Entropies in Seismic Activity: The February 2023 Earthquakes in Türkiye and Syria

by Denisse Pastén, Eugenio E. Vogel, Gonzalo Saravia and Antonio Posadas

Entropy 2025, 27(5), 462; https://doi.org/10.3390/e27050462 - 25 Apr 2025

Türkiye and Syria were hit by two powerful earthquakes on 6 February 2023. A 7.5 magnitude earthquake, soon followed by a second 7.4 magnitude seism, devastated the area. The present study compares three different entropies using data from 2017 to 2023 (55,823 events) [...] Read more.

Türkiye and Syria were hit by two powerful earthquakes on 6 February 2023. A 7.5 magnitude earthquake, soon followed by a second 7.4 magnitude seism, devastated the area. The present study compares three different entropies using data from 2017 to 2023 (55,823 events) in this region and is the first study to use Shannon entropy, Tsallis entropy, and mutability for analyzing the seismic activity in this region. A couple of years before these large earthquakes, both Shannon entropy and mutability show an overall decrease, potentially indicating upcoming large events; however, the detailed results on mutability offer an advantage, as discussed in this paper. A simultaneous overall increase in Tsallis entropy may also point to some kind of warning of the possible occurrence of large events in the area a couple of years later. The three entropies show how they are presently slowly recovering to previous levels in the affected areas. Longer-term studies combining complementary entropies could help to determine regional seismic risk. Full article

(This article belongs to the Section Non-equilibrium Phenomena)

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45 pages, 6953 KiB

Open AccessReview

A Semantic Generalization of Shannon’s Information Theory and Applications

by Chenguang Lu

Entropy 2025, 27(5), 461; https://doi.org/10.3390/e27050461 - 24 Apr 2025

Does semantic communication require a semantic information theory parallel to Shannon’s information theory, or can Shannon’s work be generalized for semantic communication? This paper advocates for the latter and introduces a semantic generalization of Shannon’s information theory (G theory for short). The core [...] Read more.

Does semantic communication require a semantic information theory parallel to Shannon’s information theory, or can Shannon’s work be generalized for semantic communication? This paper advocates for the latter and introduces a semantic generalization of Shannon’s information theory (G theory for short). The core idea is to replace the distortion constraint with the semantic constraint, achieved by utilizing a set of truth functions as a semantic channel. These truth functions enable the expressions of semantic distortion, semantic information measures, and semantic information loss. Notably, the maximum semantic information criterion is equivalent to the maximum likelihood criterion and similar to the Regularized Least Squares criterion. This paper shows G theory’s applications to daily and electronic semantic communication, machine learning, constraint control, Bayesian confirmation, portfolio theory, and information value. The improvements in machine learning methods involve multi-label learning and classification, maximum mutual information classification, mixture models, and solving latent variables. Furthermore, insights from statistical physics are discussed: Shannon information is similar to free energy; semantic information to free energy in local equilibrium systems; and information efficiency to the efficiency of free energy in performing work. The paper also proposes refining Friston’s minimum free energy principle into the maximum information efficiency principle. Lastly, it compares G theory with other semantic information theories and discusses its limitation in representing the semantics of complex data. Full article

(This article belongs to the Special Issue Semantic Information Theory)

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16 pages, 553 KiB

Open AccessArticle

Improving Phrase Segmentation in Symbolic Folk Music: A Hybrid Model with Local Context and Global Structure Awareness

by Xin Guan, Zhilin Dong, Hui Liu and Qiang Li

Entropy 2025, 27(5), 460; https://doi.org/10.3390/e27050460 - 24 Apr 2025

The segmentation of symbolic music phrases is crucial for music information retrieval and structural analysis. However, existing BiLSTM-CRF methods mainly rely on local semantics, making it difficult to capture long-range dependencies, leading to inaccurate phrase boundary recognition across measures or themes. Traditional Transformer [...] Read more.

The segmentation of symbolic music phrases is crucial for music information retrieval and structural analysis. However, existing BiLSTM-CRF methods mainly rely on local semantics, making it difficult to capture long-range dependencies, leading to inaccurate phrase boundary recognition across measures or themes. Traditional Transformer models use static embeddings, limiting their adaptability to different musical styles, structures, and melodic evolutions. Moreover, multi-head self-attention struggles with local context modeling, causing the loss of short-term information (e.g., pitch variation, melodic integrity, and rhythm stability), which may result in over-segmentation or merging errors. To address these issues, we propose a segmentation method integrating local context enhancement and global structure awareness. This method overcomes traditional models’ limitations in long-range dependency modeling, improves phrase boundary recognition, and adapts to diverse musical styles and melodies. Specifically, dynamic note embeddings enhance contextual awareness across segments, while an improved attention mechanism strengthens both global semantics and local context modeling. Combining these strategies ensures reasonable phrase boundaries and prevents unnecessary segmentation or merging. The experimental results show that our method outperforms the state-of-the-art methods for symbolic music phrase segmentation, with phrase boundaries better aligned to musical structures. Full article

(This article belongs to the Section Multidisciplinary Applications)

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28 pages, 5922 KiB

Open AccessArticle

Thoughtseeds: A Hierarchical and Agentic Framework for Investigating Thought Dynamics in Meditative States

by Prakash Chandra Kavi, Gorka Zamora-López, Daniel Ari Friedman and Gustavo Patow

Entropy 2025, 27(5), 459; https://doi.org/10.3390/e27050459 - 24 Apr 2025

The Thoughtseeds Framework introduces a novel computational approach to modeling thought dynamics in meditative states, conceptualizing thoughtseeds as dynamic attentional agents that integrate information. This hierarchical model, structured as nested Markov blankets, comprises three interconnected levels: (i) knowledge domains as information repositories, (ii) [...] Read more.

The Thoughtseeds Framework introduces a novel computational approach to modeling thought dynamics in meditative states, conceptualizing thoughtseeds as dynamic attentional agents that integrate information. This hierarchical model, structured as nested Markov blankets, comprises three interconnected levels: (i) knowledge domains as information repositories, (ii) the Thoughtseed Network where thoughtseeds compete, and (iii) meta-cognition regulating awareness. It simulates focused-attention Vipassana meditation via rule-based training informed by empirical neuroscience research on attentional stability and neural dynamics. Four states—breath_control, mind_wandering, meta_awareness, and redirect_breath—emerge organically from thoughtseed interactions, demonstrating self-organizing dynamics. Results indicate that experts sustain control dominance to reinforce focused attention, while novices exhibit frequent, prolonged mind_wandering episodes, reflecting beginner instability. Integrating Global Workspace Theory and the Intrinsic Ignition Framework, the model elucidates how thoughtseeds shape a unitary meditative experience through meta-awareness, balancing epistemic and pragmatic affordances via active inference. Synthesizing computational modeling with phenomenological insights, it provides an embodied perspective on cognitive state emergence and transitions, offering testable predictions about meditation skill development. The framework yields insights into attention regulation, meta-cognitive awareness, and meditation state emergence, establishing a versatile foundation for future research into diverse meditation practices (e.g., Open Monitoring, Non-Dual Awareness), cognitive development across the lifespan, and clinical applications in mindfulness-based interventions for attention disorders, advancing our understanding of the nature of mind and thought. Full article

(This article belongs to the Special Issue Integrated Information Theory and Consciousness II)

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19 pages, 1050 KiB

Open AccessArticle

Density Distribution of Strongly Quantum Degenerate Fermi Systems Simulated by Fictitious Identical Particle Thermodynamics

by Bo Yang, Hongsheng Yu, Shujuan Liu and Fengzheng Zhu

Entropy 2025, 27(5), 458; https://doi.org/10.3390/e27050458 - 24 Apr 2025

The exchange antisymmetry of identical fermions leads to an exponential computational bottleneck in ab initio simulations, known as the fermion sign problem. The thermodynamic approach of fictitious identical particles (Y. Xiong and H. Xiong, J. Chem. Phys. 157, 094112 (2022)) provides an efficient [...] Read more.

The exchange antisymmetry of identical fermions leads to an exponential computational bottleneck in ab initio simulations, known as the fermion sign problem. The thermodynamic approach of fictitious identical particles (Y. Xiong and H. Xiong, J. Chem. Phys. 157, 094112 (2022)) provides an efficient and accurate means to simulate some fermionic systems by overcoming the fermion sign problem. This method has been significantly promoted and used by National Ignition Facilities for the ab initio simulations and is believed to have wide application prospects in warm dense quantum matter (T. Dornheim et al., arXiv: 2402.19113 (2023)). By utilizing the fictitious identical particles in the bosonic regime and constant energy extrapolation method (Y. Xiong and H. Xiong, Phys. Rev. E 107, 055308 (2023); T. Morresi and G. Garberoglio, Phys. Rev. B 111, 014521 (2025)), there are promising results in simulating the energy of strongly quantum degenerate fermionic systems. The previous works mainly concern the energy of Fermi systems or only consider situations of weak quantum degeneracy. In this study, we extend the concept of the constant energy extrapolation method and demonstrate the potential of the constant density extrapolation method to accurately simulate the density distribution of fermionic systems in strongly quantum degenerate conditions. Furthermore, based on the energy derived from the constant energy extrapolation method, we present simulation results for the entropy of fermions. Full article

(This article belongs to the Section Statistical Physics)

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15 pages, 4569 KiB

Open AccessArticle

Improved EEG-Based Emotion Classification via Stockwell Entropy and CSP Integration

by Yuan Lu and Jingying Chen

Entropy 2025, 27(5), 457; https://doi.org/10.3390/e27050457 - 24 Apr 2025

Traditional entropy-based learning methods primarily extract the relevant entropy measures directly from EEG signals using sliding time windows. This study applies differential entropy to a time-frequency domain that is decomposed by Stockwell transform, proposing a novel EEG emotion recognition method combining Stockwell entropy [...] Read more.

Traditional entropy-based learning methods primarily extract the relevant entropy measures directly from EEG signals using sliding time windows. This study applies differential entropy to a time-frequency domain that is decomposed by Stockwell transform, proposing a novel EEG emotion recognition method combining Stockwell entropy and a common spatial pattern (CSP). The results demonstrate that Stockwell entropy effectively captures the entropy features of high-frequency signals, and CSP-transformed Stockwell entropy features show superior discriminative capability for different emotional states. The experimental results indicate that the proposed method achieves excellent classification performance in the Gamma band (30–46 Hz) for emotion recognition. The combined approach yields high classification accuracy for binary tasks (“positive vs. neutral”, “negative vs. neutral”, and “positive vs. negative”) and maintains satisfactory performance in the three-class task (“positive vs. negative vs. neutral”). Full article

(This article belongs to the Special Issue Assessing Complexity in Physiological Systems through Biomedical Signals Analysis II)

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13 pages, 540 KiB

Open AccessArticle

Transmit Power Optimization for Simultaneous Wireless Information and Power Transfer-Assisted IoT Networks with Integrated Sensing and Communication and Nonlinear Energy Harvesting Model

by Chengrui Zhou, Xinru Wang, Yanfei Dou and Xiaomin Chen

Entropy 2025, 27(5), 456; https://doi.org/10.3390/e27050456 - 24 Apr 2025

Integrated sensing and communication (ISAC) can improve the energy harvesting (EH) efficiency of simultaneous wireless information and power transfer (SWIPT)-assisted IoT networks by enabling precise energy harvest. However, the transmit power is increased in the hybrid system due to the fact that the [...] Read more.

Integrated sensing and communication (ISAC) can improve the energy harvesting (EH) efficiency of simultaneous wireless information and power transfer (SWIPT)-assisted IoT networks by enabling precise energy harvest. However, the transmit power is increased in the hybrid system due to the fact that the sensing signals are required to be transferred in addition to the communication data. This paper aims to tackle this issue by formulating an optimization problem to minimize the transmit power of the base station (BS) under a nonlinear EH model, considering the coexistence of power-splitting users (PSUs) and time-switching users (TSUs), as well as the beamforming vector associated with PSUs and TSUs. A two-layer algorithm based on semi-definite relaxation is proposed to tackle the complexity issue of the non-convex optimization problem. The global optimality is theoretically analyzed, and the impact of each parameter on system performance is also discussed. Numerical results indicate that TSUs are more prone to saturation compared to PSUs under identical EH requirements. The minimal required transmit power under the nonlinear EH model is much lower than that under the linear EH model. Moreover, it is observed that the number of TSUs is the primary limiting factor for the minimization of transmit power, which can be effectively mitigated by the proposed algorithm. Full article

(This article belongs to the Special Issue Integrated Sensing and Communication (ISAC) in 6G)

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19 pages, 5288 KiB

Open AccessArticle

Multi-Particle-Collision Simulation of Heat Transfer in Low-Dimensional Fluids

by Rongxiang Luo and Stefano Lepri

Entropy 2025, 27(5), 455; https://doi.org/10.3390/e27050455 - 24 Apr 2025

The simulation of the transport properties of confined, low-dimensional fluids can be performed efficiently by means of multi-particle collision (MPC) dynamics with suitable thermal-wall boundary conditions. We illustrate the effectiveness of the method by studying the dimensionality effects and size-dependence of thermal conduction, [...] Read more.

The simulation of the transport properties of confined, low-dimensional fluids can be performed efficiently by means of multi-particle collision (MPC) dynamics with suitable thermal-wall boundary conditions. We illustrate the effectiveness of the method by studying the dimensionality effects and size-dependence of thermal conduction, since these properties are of crucial importance for understanding heat transfer at the micro–nanoscale. We provide a sound numerical evidence that the simple MPC fluid displays the features previously predicted from hydrodynamics of lattice systems: (1) in 1D, the thermal conductivity

κ

diverges with the system size L as

κ \sim L^{1 / 3}

and its total heat current autocorrelation function

C (t)

decays with the time t as

C (t) \sim t^{- 2 / 3}

; (2) in 2D,

κ

diverges with L as

κ \sim \ln (L)

and its

C (t)

decays with t as

C (t) \sim t^{- 1}

; (3) in 3D, its

κ

is independent with L and its

C (t)

decays with t as

C (t) \sim t^{- 3 / 2}

. For weak interaction (the nearly integrable case) in 1D and 2D, there exists an intermediate regime of sizes where kinetic effects dominate and transport is diffusive before crossing over to the expected anomalous regime. The crossover can be studied by decomposing the heat current in two contributions, which allows for a very accurate test of the predictions. In addition, we also show that, upon increasing the aspect ratio of the system, there exists a dimensional crossover from 2D or 3D dimensional behavior to the 1D one. Finally, we show that an applied magnetic field renders the transport normal, indicating that pseudomomentum conservation is not sufficient for the anomalous heat conduction behavior to occur. Full article

(This article belongs to the Special Issue Advances in Molecular Dynamics Simulations: Understanding Statistical Physics and Complex Systems Across Scales)

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17 pages, 400 KiB

Open AccessArticle

Efficient Circuit Implementations of Continuous-Time Quantum Walks for Quantum Search

by Renato Portugal and Jalil Khatibi Moqadam

Entropy 2025, 27(5), 454; https://doi.org/10.3390/e27050454 - 23 Apr 2025

Quantum walks are a powerful framework for simulating complex quantum systems and designing quantum algorithms, particularly for spatial search on graphs, where the goal is to find a marked vertex efficiently. In this work, we present efficient quantum circuits that implement the evolution [...] Read more.

Quantum walks are a powerful framework for simulating complex quantum systems and designing quantum algorithms, particularly for spatial search on graphs, where the goal is to find a marked vertex efficiently. In this work, we present efficient quantum circuits that implement the evolution operator of continuous-time quantum-walk-based search algorithms for three graph families: complete graphs, complete bipartite graphs, and hypercubes. For complete and complete bipartite graphs, our circuits exactly implement the evolution operator. For hypercubes, we propose an approximate implementation that closely matches the exact evolution operator as the number of vertices increases. Our Qiskit simulations demonstrate that even for low-dimensional hypercubes, the algorithm effectively identifies the marked vertex. Furthermore, the approximate implementation developed for hypercubes can be extended to a broad class of graphs, enabling efficient quantum search in scenarios where exact implementations are impractical. Full article

(This article belongs to the Special Issue Quantum Walks for Quantum Technologies)

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45 pages, 1892 KiB

Open AccessReview

Integrating Dynamical Systems Modeling with Spatiotemporal scRNA-Seq Data Analysis

by Zhenyi Zhang, Yuhao Sun, Qiangwei Peng, Tiejun Li and Peijie Zhou

Entropy 2025, 27(5), 453; https://doi.org/10.3390/e27050453 - 22 Apr 2025

Understanding the dynamic nature of biological systems is fundamental to deciphering cellular behavior, developmental processes, and disease progression. Single-cell RNA sequencing (scRNA-seq) has provided static snapshots of gene expression, offering valuable insights into cellular states at a single time point. Recent advancements in [...] Read more.

Understanding the dynamic nature of biological systems is fundamental to deciphering cellular behavior, developmental processes, and disease progression. Single-cell RNA sequencing (scRNA-seq) has provided static snapshots of gene expression, offering valuable insights into cellular states at a single time point. Recent advancements in temporally resolved scRNA-seq, spatial transcriptomics (ST), and time-series spatial transcriptomics (temporal-ST) have further revolutionized our ability to study the spatiotemporal dynamics of individual cells. These technologies, when combined with computational frameworks such as Markov chains, stochastic differential equations (SDEs), and generative models like optimal transport and Schrödinger bridges, enable the reconstruction of dynamic cellular trajectories and cell fate decisions. This review discusses how these dynamical system approaches offer new opportunities to model and infer cellular dynamics from a systematic perspective. Full article

(This article belongs to the Special Issue Complexity, Information and Quantitative Modelling in Single Cell Multiomics)

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21 pages, 6529 KiB

Open AccessArticle

The Supervised Information Bottleneck

by Nir Z. Weingarten, Zohar Yakhini, Moshe Butman and Ronit Bustin

Entropy 2025, 27(5), 452; https://doi.org/10.3390/e27050452 - 22 Apr 2025

The Information Bottleneck (IB) framework offers a theoretically optimal approach to data modeling, although it is often intractable. Recent efforts have optimized supervised deep neural networks (DNNs) using a variational upper bound on the IB objective, leading to enhanced robustness to adversarial attacks. [...] Read more.

The Information Bottleneck (IB) framework offers a theoretically optimal approach to data modeling, although it is often intractable. Recent efforts have optimized supervised deep neural networks (DNNs) using a variational upper bound on the IB objective, leading to enhanced robustness to adversarial attacks. In these studies, supervision assumes a dual role: sometimes as a presumably constant and observed random variable and at other times as its variational approximation. This work proposes an extension to the IB framework and, consequent to the derivation of its variational bound, that resolves this duality. Applying the resulting bound as an objective for supervised DNNs induces empirical improvements and provides an information-theoretic motivation for decoder regularization. Full article

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

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18 pages, 3845 KiB

Open AccessArticle

Mutual Information Neural-Estimation-Driven Constellation Shaping Design and Performance Analysis

by Xiuli Ji, Qian Wang, Liping Qian and Pooi-Yuen Kam

Entropy 2025, 27(4), 451; https://doi.org/10.3390/e27040451 - 21 Apr 2025

The choice of constellations largely affects the performance of both wireless and optical communications. To address increasing capacity requirements, constellation shaping, especially for high-order modulations, is imperative in high-speed coherent communication systems. This paper, thus, proposes novel mutual information neural estimation (MINE)-based geometric, [...] Read more.

The choice of constellations largely affects the performance of both wireless and optical communications. To address increasing capacity requirements, constellation shaping, especially for high-order modulations, is imperative in high-speed coherent communication systems. This paper, thus, proposes novel mutual information neural estimation (MINE)-based geometric, probabilistic, and joint constellation shaping schemes, i.e., the MINE-GCS, MINE-PCS, and MINE-JCS, to maximize mutual information (MI) via emerging deep learning (DL) techniques. Innovatively, we first introduce the MINE module to effectively estimate and maximize MI through backpropagation, without clear knowledge of the channel state information. Then, we train encoder and probability generator networks with different signal-to-noise ratios to optimize the distribution locations and probabilities of the points, respectively. Note that MINE transforms the precise MI calculation problem into a parameter optimization problem. Our MINE-based schemes only optimize the transmitter end, and avoid the computational and structural complexity in traditional shaping. All the designs were verified through simulations as having superior performance for MI, among which the MINE-JCS undoubtedly performed the best for additive white Gaussian noise, compared to the unshaped QAMs and even the end-to-end training and other DL-based joint shaping schemes. For example, the low-order 8-ary MINE-GCS could achieve an MI gain of about 0.1 bits/symbol compared to the unshaped Star-8QAM. It is worth emphasizing that our proposed schemes achieve a balance between implementation complexity and MI performance, and they are expected to be applied in various practical scenarios with different noise and fading levels in the future. Full article

(This article belongs to the Special Issue Advances in Modern Channel Coding)

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16 pages, 3473 KiB

Open AccessArticle

Information Theory Quantifiers in Cryptocurrency Time Series Analysis

by Micaela Suriano, Leonidas Facundo Caram, Cesar Caiafa, Hernán Daniel Merlino and Osvaldo Anibal Rosso

Entropy 2025, 27(4), 450; https://doi.org/10.3390/e27040450 - 21 Apr 2025

This paper investigates the temporal evolution of cryptocurrency time series using information measures such as complexity, entropy, and Fisher information. The main objective is to differentiate between various levels of randomness and chaos. The methodology was applied to 176 daily closing price time [...] Read more.

This paper investigates the temporal evolution of cryptocurrency time series using information measures such as complexity, entropy, and Fisher information. The main objective is to differentiate between various levels of randomness and chaos. The methodology was applied to 176 daily closing price time series of different cryptocurrencies, from October 2015 to October 2024, with more than 30 days of data and not completely null. Complexity–entropy causality plane (CECP) analysis reveals that daily cryptocurrency series with lengths of two years or less exhibit chaotic behavior, while those longer than two years display stochastic behavior. Most longer series resemble colored noise, with the parameter k varying between 0 and 2. Additionally, Natural Language Processing (NLP) analysis identified the most relevant terms in each white paper, facilitating a clustering method that resulted in four distinct clusters. However, no significant characteristics were found across these clusters in terms of the dynamics of the time series. This finding challenges the assumption that project narratives dictate market behavior. For this reason, investment recommendations should prioritize real-time informational metrics over whitepaper content. Full article

(This article belongs to the Special Issue Nonlinear Dynamics and Applications—In Honor of Professor Osvaldo Anibal Rosso's 70th Birthday)

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17 pages, 885 KiB

Open AccessReview

Maximum Entropy Production Principle of Thermodynamics for the Birth and Evolution of Life

by Yasuji Sawada, Yasukazu Daigaku and Kenji Toma

Entropy 2025, 27(4), 449; https://doi.org/10.3390/e27040449 - 21 Apr 2025

Research on the birth and evolution of life are reviewed with reference to the maximum entropy production principle (MEPP). It has been shown that this principle is essential for consistent understanding of the birth and evolution of life. First, a recent work for [...] Read more.

Research on the birth and evolution of life are reviewed with reference to the maximum entropy production principle (MEPP). It has been shown that this principle is essential for consistent understanding of the birth and evolution of life. First, a recent work for the birth of a self-replicative system as pre-RNA life is reviewed in relation to the MEPP. A critical condition of polymer concentration in a local system is reported by a dynamical system approach, above which, an exponential increase of entropy production is guaranteed. Secondly, research works of early stage of evolutions are reviewed; experimental research for the numbers of cells necessary for forming a multi-cellular organization, and numerical research of differentiation of a model system and its relation with MEPP. It is suggested by this review article that the late stage of evolution is characterized by formation of society and external entropy production. A hypothesis on the general route of evolution is discussed from the birth to the present life which follows the MEPP. Some examples of life which happened to face poor thermodynamic condition are presented with thermodynamic discussion. It is observed through this review that MEPP is consistently useful for thermodynamic understanding of birth and evolution of life, subject to a thermodynamic condition far from equilibrium. Full article

(This article belongs to the Special Issue Thermodynamics of Dissipative Structures and Related Emergent Phenomena)

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