Entropy

21 pages, 2867 KiB

Open AccessArticle

A Resource-Efficient Multi-Entropy Fusion Method and Its Application for EEG-Based Emotion Recognition

by Jiawen Li, Guanyuan Feng, Chen Ling, Ximing Ren, Xin Liu, Shuang Zhang, Leijun Wang, Yanmei Chen, Xianxian Zeng and Rongjun Chen

Entropy 2025, 27(1), 96; https://doi.org/10.3390/e27010096 - 20 Jan 2025

Viewed by 553

Abstract

Emotion recognition is an advanced technology for understanding human behavior and psychological states, with extensive applications for mental health monitoring, human–computer interaction, and affective computing. Based on electroencephalography (EEG), the biomedical signals naturally generated by the brain, this work proposes a resource-efficient multi-entropy [...] Read more.

Emotion recognition is an advanced technology for understanding human behavior and psychological states, with extensive applications for mental health monitoring, human–computer interaction, and affective computing. Based on electroencephalography (EEG), the biomedical signals naturally generated by the brain, this work proposes a resource-efficient multi-entropy fusion method for classifying emotional states. First, Discrete Wavelet Transform (DWT) is applied to extract five brain rhythms, i.e., delta, theta, alpha, beta, and gamma, from EEG signals, followed by the acquisition of multi-entropy features, including Spectral Entropy (PSDE), Singular Spectrum Entropy (SSE), Sample Entropy (SE), Fuzzy Entropy (FE), Approximation Entropy (AE), and Permutation Entropy (PE). Then, such entropies are fused into a matrix to represent complex and dynamic characteristics of EEG, denoted as the Brain Rhythm Entropy Matrix (BREM). Next, Dynamic Time Warping (DTW), Mutual Information (MI), the Spearman Correlation Coefficient (SCC), and the Jaccard Similarity Coefficient (JSC) are applied to measure the similarity between the unknown testing BREM data and positive/negative emotional samples for classification. Experiments were conducted using the DEAP dataset, aiming to find a suitable scheme regarding similarity measures, time windows, and input numbers of channel data. The results reveal that DTW yields the best performance in similarity measures with a 5 s window. In addition, the single-channel input mode outperforms the single-region mode. The proposed method achieves 84.62% and 82.48% accuracy in arousal and valence classification tasks, respectively, indicating its effectiveness in reducing data dimensionality and computational complexity while maintaining an accuracy of over 80%. Such performances are remarkable when considering limited data resources as a concern, which opens possibilities for an innovative entropy fusion method that can help to design portable EEG-based emotion-aware devices for daily usage. Full article

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

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20 pages, 1226 KiB

Open AccessArticle

Discontinuous Structural Transitions in Fluids with Competing Interactions

by Ana M. Montero, Santos B. Yuste, Andrés Santos and Mariano López de Haro

Entropy 2025, 27(1), 95; https://doi.org/10.3390/e27010095 - 20 Jan 2025

Viewed by 514

Abstract

This paper explores how competing interactions in the intermolecular potential of fluids affect their structural transitions. This study employs a versatile potential model with a hard core followed by two constant steps, representing wells or shoulders, analyzed in both one-dimensional (1D) and three-dimensional [...] Read more.

This paper explores how competing interactions in the intermolecular potential of fluids affect their structural transitions. This study employs a versatile potential model with a hard core followed by two constant steps, representing wells or shoulders, analyzed in both one-dimensional (1D) and three-dimensional (3D) systems. Comparing these dimensionalities highlights the effect of confinement on structural transitions. Exact results are derived for 1D systems, while the rational function approximation is used for unconfined 3D fluids. Both scenarios confirm that when the steps are repulsive, the wavelength of the oscillatory decay of the total correlation function evolves with temperature either continuously or discontinuously. In the latter case, a discontinuous oscillation crossover line emerges in the temperature–density plane. For an attractive first step and a repulsive second step, a Fisher–Widom line appears. Although the 1D and 3D results share common features, dimensionality introduces differences: these behaviors occur in distinct temperature ranges, require deeper wells, or become attenuated in 3D. Certain features observed in 1D may vanish in 3D. We conclude that fluids with competing interactions exhibit a rich and intricate pattern of structural transitions, demonstrating the significant influence of dimensionality and interaction features. Full article

(This article belongs to the Special Issue Dimensional Crossover in Classical and Quantum Systems)

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23 pages, 2995 KiB

Open AccessArticle

Novel Ensemble Approach with Incremental Information Level and Improved Evidence Theory for Attribute Reduction

by Peng Yu, Yifeng Zheng, Ziwen Liu, Baoya Wei, Wenjie Zhang, Ziqiong Lin and Zhehan Li

Entropy 2025, 27(1), 94; https://doi.org/10.3390/e27010094 - 20 Jan 2025

Viewed by 428

Abstract

With the development of intelligent technology, data in practical applications show exponential growth in quantity and scale. Extracting the most distinguished attributes from complex datasets becomes a crucial problem. The existing attribute reduction approaches focus on the correlation between attributes and labels without [...] Read more.

With the development of intelligent technology, data in practical applications show exponential growth in quantity and scale. Extracting the most distinguished attributes from complex datasets becomes a crucial problem. The existing attribute reduction approaches focus on the correlation between attributes and labels without considering the redundancy. To address the above problem, we propose an ensemble approach based on an incremental information level and improved evidence theory for attribute reduction (IILE). Firstly, the incremental information level reduction measure comprehensively assesses attributes based on reduction capability and redundancy level. Then, an improved evidence theory and approximate reduction methods are employed to fuse multiple reduction results, thereby obtaining an approximately globally optimal and a most representative subset of attributes. Eventually, using different metrics, experimental comparisons are performed on eight datasets to confirm that our proposal achieved better than other methods. The results show that our proposal can obtain more relevant attribute sets by using the incremental information level and improved evidence theory. Full article

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

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5 pages, 166 KiB

Open AccessEditorial

Nonadditive Entropies and Nonextensive Statistical Mechanics

by Ugur Tirnakli

Entropy 2025, 27(1), 93; https://doi.org/10.3390/e27010093 - 20 Jan 2025

Viewed by 393

Abstract

The centennial Boltzmann–Gibbs statistical mechanics [...] Full article

(This article belongs to the Special Issue Nonadditive Entropies and Nonextensive Statistical Mechanics—Dedicated to Professor Constantino Tsallis on the Occasion of His 80th Birthday)

26 pages, 27819 KiB

Open AccessArticle

Spatiotemporal Evolution of the Water System’s Structure and Its Relationship with Urban System Based on Fractal Dimension: A Case Study of the Huaihe River Basin, China

by Hailong Yu, Bin Yu, Xiangmin Zhang, Yong Fan, Sai Wen and Shanshan Jiao

Entropy 2025, 27(1), 92; https://doi.org/10.3390/e27010092 - 20 Jan 2025

Viewed by 468

Abstract

The formation and development of cities are inseparable from a certain scale of water resources. The information contained in the morphological structures of cities and water systems is often overlooked. Exploring the spatiotemporal evolution of water system structures (WSS) and urban system structures [...] Read more.

The formation and development of cities are inseparable from a certain scale of water resources. The information contained in the morphological structures of cities and water systems is often overlooked. Exploring the spatiotemporal evolution of water system structures (WSS) and urban system structures (USS) can reveal the “urban–water” relationship from a new perspective. The Huaihe River Basin (HRB) was selected as the case area, based on the theory of fractal dimensions, grid dimension and multifractal spectrum methods were used to depict the structural evolutionary characteristics of water systems and urban systems from different dimensions. Then, through a comparative analysis of fractal parameters and spectral lines, the characteristics and changing patterns of the “urban-water” relationship in the HRB from 1980 to 2019 were revealed. The results indicate the following: (1) The water system structure in the HRB is complex and exhibits distinct scale characteristics, showing improvement overall and at larger scales while continuously degrading at smaller scales. (2) Both the water system and urban system exhibit increasingly complex spatial development characteristics; however, the USS continues to optimize over time, while the WSS experiences degradation. (3) The development patterns of the water system and urban system are significant differences in the HRB. Urban development mainly relies on outward expansion, whereas the water system is primarily characterized by intensive enhancement. (4) Because of the rapid development of urban areas, water scarcity may occur in densely populated urban areas or larger cities in the future. The research results can serve as a scientific reference for urban planning and water resource management in the HRB. Full article

(This article belongs to the Special Issue Entropy in Landscape Ecology, 4th Edition)

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16 pages, 1260 KiB

Open AccessArticle

Maximizing Free Energy Gain

by Artemy Kolchinsky, Iman Marvian, Can Gokler, Zi-Wen Liu, Peter Shor, Oles Shtanko, Kevin Thompson, David Wolpert and Seth Lloyd

Entropy 2025, 27(1), 91; https://doi.org/10.3390/e27010091 - 20 Jan 2025

Viewed by 1313

Abstract

Maximizing the amount of work harvested from an environment is important for a wide variety of biological and technological processes, from energy-harvesting processes such as photosynthesis to energy storage systems such as fuels and batteries. Here, we consider the maximization of free energy—and [...] Read more.

Maximizing the amount of work harvested from an environment is important for a wide variety of biological and technological processes, from energy-harvesting processes such as photosynthesis to energy storage systems such as fuels and batteries. Here, we consider the maximization of free energy—and by extension, the maximum extractable work—that can be gained by a classical or quantum system that undergoes driving by its environment. We consider how the free energy gain depends on the initial state of the system while also accounting for the cost of preparing the system. We provide simple necessary and sufficient conditions for increasing the gain of free energy by varying the initial state. We also derive simple formulae that relate the free energy gained using the optimal initial state rather than another suboptimal initial state. Finally, we demonstrate that the problem of finding the optimal initial state may have two distinct regimes, one easy and one difficult, depending on the temperatures used for preparation and work extraction. We illustrate our results on a simple model of an information engine. Full article

(This article belongs to the Section Statistical Physics)

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55 pages, 18955 KiB

Open AccessArticle

Structured Dynamics in the Algorithmic Agent

by Giulio Ruffini, Francesca Castaldo and Jakub Vohryzek

Entropy 2025, 27(1), 90; https://doi.org/10.3390/e27010090 - 19 Jan 2025

Viewed by 359

Abstract

In the Kolmogorov Theory of Consciousness, algorithmic agents utilize inferred compressive models to track coarse-grained data produced by simplified world models, capturing regularities that structure subjective experience and guide action planning. Here, we study the dynamical aspects of this framework by examining how [...] Read more.

In the Kolmogorov Theory of Consciousness, algorithmic agents utilize inferred compressive models to track coarse-grained data produced by simplified world models, capturing regularities that structure subjective experience and guide action planning. Here, we study the dynamical aspects of this framework by examining how the requirement of tracking natural data drives the structural and dynamical properties of the agent. We first formalize the notion of a generative model using the language of symmetry from group theory, specifically employing Lie pseudogroups to describe the continuous transformations that characterize invariance in natural data. Then, adopting a generic neural network as a proxy for the agent dynamical system and drawing parallels to Noether’s theorem in physics, we demonstrate that data tracking forces the agent to mirror the symmetry properties of the generative world model. This dual constraint on the agent’s constitutive parameters and dynamical repertoire enforces a hierarchical organization consistent with the manifold hypothesis in the neural network. Our findings bridge perspectives from algorithmic information theory (Kolmogorov complexity, compressive modeling), symmetry (group theory), and dynamics (conservation laws, reduced manifolds), offering insights into the neural correlates of agenthood and structured experience in natural systems, as well as the design of artificial intelligence and computational models of the brain. Full article

(This article belongs to the Special Issue The Mathematics of Structured Experience: Exploring Dynamics, Topology, and Complexity in the Brain)

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27 pages, 1556 KiB

Open AccessArticle

Environmental Performance, Financial Constraints, and Tax Avoidance Practices: Insights from FTSE All-Share Companies

by Probowo Erawan Sastroredjo, Marcel Ausloos and Polina Khrennikova

Entropy 2025, 27(1), 89; https://doi.org/10.3390/e27010089 - 18 Jan 2025

Viewed by 750

Abstract

Through its initiative known as the Climate Change Act (2008), the Government of the United Kingdom encourages corporations to enhance their environmental performance with the significant aim of reducing targeted greenhouse gas emissions by the year 2050. Previous research has predominantly assessed this [...] Read more.

Through its initiative known as the Climate Change Act (2008), the Government of the United Kingdom encourages corporations to enhance their environmental performance with the significant aim of reducing targeted greenhouse gas emissions by the year 2050. Previous research has predominantly assessed this encouragement favourably, suggesting that improved environmental performance bolsters governmental efforts to protect the environment and fosters commendable corporate governance practices among companies. Studies indicate that organisations exhibiting strong corporate social responsibility (CSR), environmental, social, and governance (ESG) criteria, or high levels of environmental performance often engage in lower occurrences of tax avoidance. However, our findings suggest that an increase in environmental performance may paradoxically lead to a rise in tax avoidance activities. Using a sample of 567 firms listed on the FTSE All Share from 2014 to 2022, our study finds that firms associated with higher environmental performance are more likely to avoid taxation. The study further documents that the effect is more pronounced for firms facing financial constraints. Entropy balancing, propensity score matching analysis, the instrumental variable method, and the Heckman test are employed in our study to address potential endogeneity concerns. Collectively, the findings of our study suggest that better environmental performance helps explain the variation in firms’ tax avoidance practices. Full article

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

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27 pages, 3978 KiB

Open AccessArticle

Dissipation Alters Modes of Information Encoding in Small Quantum Reservoirs near Criticality

by Krai Cheamsawat and Thiparat Chotibut

Entropy 2025, 27(1), 88; https://doi.org/10.3390/e27010088 - 18 Jan 2025

Viewed by 626

Abstract

Quantum reservoir computing (QRC) has emerged as a promising paradigm for harnessing near-term quantum devices to tackle temporal machine learning tasks. Yet, identifying the mechanisms that underlie enhanced performance remains challenging, particularly in many-body open systems where nonlinear interactions and dissipation intertwine in [...] Read more.

Quantum reservoir computing (QRC) has emerged as a promising paradigm for harnessing near-term quantum devices to tackle temporal machine learning tasks. Yet, identifying the mechanisms that underlie enhanced performance remains challenging, particularly in many-body open systems where nonlinear interactions and dissipation intertwine in complex ways. Here, we investigate a minimal model of a driven-dissipative quantum reservoir described by two coupled Kerr-nonlinear oscillators, an experimentally realizable platform that features controllable coupling, intrinsic nonlinearity, and tunable photon loss. Using Partial Information Decomposition (PID), we examine how different dynamical regimes encode input drive signals in terms of redundancy (information shared by each oscillator) and synergy (information accessible only through their joint observation). Our key results show that, near a critical point marking a dynamical bifurcation, the system transitions from predominantly redundant to synergistic encoding. We further demonstrate that synergy amplifies short-term responsiveness, thereby enhancing immediate memory retention, whereas strong dissipation leads to more redundant encoding that supports long-term memory retention. These findings elucidate how the interplay of instability and dissipation shapes information processing in small quantum systems, providing a fine-grained, information-theoretic perspective for analyzing and designing QRC platforms. Full article

(This article belongs to the Special Issue Quantum Computing in the NISQ Era)

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18 pages, 277 KiB

Open AccessArticle

Fitting Copulas with Maximal Entropy

by Milan Bubák and Mirko Navara

Entropy 2025, 27(1), 87; https://doi.org/10.3390/e27010087 - 18 Jan 2025

Viewed by 333

Abstract

We deal with two-dimensional copulas from the perspective of their differential entropy. We formulate a problem of finding a copula with maximum differential entropy when some copula values are given. As expected, the solution is a copula with a piecewise constant density (a [...] Read more.

We deal with two-dimensional copulas from the perspective of their differential entropy. We formulate a problem of finding a copula with maximum differential entropy when some copula values are given. As expected, the solution is a copula with a piecewise constant density (a checkerboard copula). This allows us to simplify the optimization of the continuous objective function, the differential entropy, to an optimization of finitely many density values. We present several ideas to simplify this problem. It has a feasible numerical solution. We also present several instances that admit closed-form solutions. Full article

(This article belongs to the Special Issue Quantum Probability and Randomness V)

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14 pages, 369 KiB

Open AccessArticle

Statistical Mechanics of Directed Networks

by Marián Boguñá and M. Ángeles Serrano

Entropy 2025, 27(1), 86; https://doi.org/10.3390/e27010086 - 18 Jan 2025

Viewed by 411

Abstract

Directed networks are essential for representing complex systems, capturing the asymmetry of interactions in fields such as neuroscience, transportation, and social networks. Directionality reveals how influence, information, or resources flow within a network, fundamentally shaping the behavior of dynamical processes and distinguishing directed [...] Read more.

Directed networks are essential for representing complex systems, capturing the asymmetry of interactions in fields such as neuroscience, transportation, and social networks. Directionality reveals how influence, information, or resources flow within a network, fundamentally shaping the behavior of dynamical processes and distinguishing directed networks from their undirected counterparts. Robust null models are crucial for identifying meaningful patterns in these representations, yet designing models that preserve key features remains a significant challenge. One such critical feature is reciprocity, which reflects the balance of bidirectional interactions in directed networks and provides insights into the underlying structural and dynamical principles that shape their connectivity. This paper introduces a statistical mechanics framework for directed networks, modeling them as ensembles of interacting fermions. By controlling the reciprocity and other network properties, our formalism offers a principled approach to analyzing directed network structures and dynamics, introducing new perspectives and models and analytical tools for empirical studies. Full article

(This article belongs to the Special Issue 180th Anniversary of Ludwig Boltzmann)

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22 pages, 444 KiB

Open AccessFeature PaperArticle

Some New Constructions of q-ary Codes for Correcting a Burst of at Most t Deletions

by Wentu Song, Kui Cai and Tony Q. S. Quek

Entropy 2025, 27(1), 85; https://doi.org/10.3390/e27010085 - 18 Jan 2025

Viewed by 343

Abstract

In this paper, we construct q-ary codes for correcting a burst of at most t deletions, where

t, q \geq 2

are arbitrarily fixed positive integers. We consider two scenarios of error correction: the classical error correcting codes, which recover each [...] Read more.

In this paper, we construct q-ary codes for correcting a burst of at most t deletions, where

t, q \geq 2

are arbitrarily fixed positive integers. We consider two scenarios of error correction: the classical error correcting codes, which recover each codeword from one read (channel output), and the reconstruction codes, which allow to recover each codeword from multiple channel reads. For the first scenario, our construction has redundancy

\log n + 8 \log \log n + o (\log \log n)

bits, encoding complexity

O (q^{7 t} n {(\log n)}^{3})

and decoding complexity

O (n \log n)

. For the reconstruction scenario, our construction can recover the codewords with two reads and has redundancy

8 \log \log n + o (\log \log n)

bits. The encoding complexity of this construction is

O (q^{7 t} n {(\log n)}^{3})

, and decoding complexity is

O (q^{9 t} {(n \log n)}^{3})

. Both of our constructions have lower redundancy than the best known existing works. We also give explicit encoding functions for both constructions that are simpler than previous works. Full article

(This article belongs to the Special Issue Coding Theory and Its Applications)

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24 pages, 14297 KiB

Open AccessArticle

Image Encryption Method Based on Three-Dimensional Chaotic Systems and V-Shaped Scrambling

by Lei Wang, Wenjun Song, Jiali Di, Xuncai Zhang and Chengye Zou

Entropy 2025, 27(1), 84; https://doi.org/10.3390/e27010084 - 17 Jan 2025

Viewed by 398

Abstract

With the increasing importance of securing images during network transmission, this paper introduces a novel image encryption algorithm that integrates a 3D chaotic system with V-shaped scrambling techniques. The proposed method begins by constructing a unique 3D chaotic system to generate chaotic sequences [...] Read more.

With the increasing importance of securing images during network transmission, this paper introduces a novel image encryption algorithm that integrates a 3D chaotic system with V-shaped scrambling techniques. The proposed method begins by constructing a unique 3D chaotic system to generate chaotic sequences for encryption. These sequences determine a random starting point for V-shaped scrambling, which facilitates the transformation of image pixels into quaternary numbers. Subsequently, four innovative bit-level scrambling strategies are employed to enhance encryption strength. To further improve randomness, DNA encoding is applied to both the image and chaotic sequences, with chaotic sequences directing crossover and DNA operations. Ciphertext feedback is then utilized to propagate changes across the image, ensuring increased complexity and security. Extensive simulation experiments validate the algorithm’s robust encryption performance for grayscale images, yielding uniformly distributed histograms, near-zero correlation values, and an information entropy value of 7.9975, approaching the ideal threshold. The algorithm also features a large key space, providing robust protection against brute force attacks while effectively resisting statistical, differential, noise, and cropping attacks. These results affirm the algorithm’s reliability and security for image communication and transmission. Full article

(This article belongs to the Special Issue Image Encryption and Privacy Protection Based on Chaotic Systems—Second Edition)

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27 pages, 1692 KiB

Open AccessArticle

Optimizing Hydrogen Production in the Co-Gasification Process: Comparison of Explainable Regression Models Using Shapley Additive Explanations

by Thavavel Vaiyapuri

Entropy 2025, 27(1), 83; https://doi.org/10.3390/e27010083 - 17 Jan 2025

Viewed by 478

Abstract

The co-gasification of biomass and plastic waste offers a promising solution for producing hydrogen-rich syngas, addressing the rising demand for cleaner energy. However, optimizing this complex process to maximize hydrogen yield remains challenging, particularly when balancing diverse feedstocks and improving process efficiency. While [...] Read more.

The co-gasification of biomass and plastic waste offers a promising solution for producing hydrogen-rich syngas, addressing the rising demand for cleaner energy. However, optimizing this complex process to maximize hydrogen yield remains challenging, particularly when balancing diverse feedstocks and improving process efficiency. While machine learning (ML) has shown significant potential in simulating and optimizing such processes, there is no clear consensus on the most effective regression models for co-gasification, especially with limited experimental data. Additionally, the interpretability of these models is a key concern. This study aims to bridge these gaps through two primary objectives: (1) modeling the co-gasification process using seven different ML algorithms, and (2) developing a framework for evaluating model interpretability, ultimately identifying the most suitable model for process optimization. A comprehensive set of experiments was conducted across three key dimensions, generalization ability, predictive accuracy, and interpretability, to thoroughly assess the models. Support Vector Regression (SVR) exhibited superior performance, achieving the highest coefficient of determination (

R^{2}

) of 0.86. SVR outperformed other models in capturing non-linear dependencies and demonstrated effective overfitting mitigation. This study further highlights the limitations of other ML models, emphasizing the importance of regularization and hyperparameter tuning in improving model stability. By integrating Shapley Additive Explanations (SHAP) into model evaluation, this work is the first to provide detailed insights into feature importance and demonstrate the operational feasibility of ML models for industrial-scale hydrogen production in the co-gasification process. The findings contribute to the development of a robust framework for optimizing co-gasification, supporting the advancement of sustainable energy technologies and the reduction of greenhouse gas (GHG) emissions. 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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21 pages, 1162 KiB

Open AccessArticle

Forecasting Stock Market Indices Using Integration of Encoder, Decoder, and Attention Mechanism

by Tien Thanh Thach

Entropy 2025, 27(1), 82; https://doi.org/10.3390/e27010082 - 17 Jan 2025

Viewed by 585

Abstract

Accurate forecasting of stock market indices is crucial for investors, financial analysts, and policymakers. The integration of encoder and decoder architectures, coupled with an attention mechanism, has emerged as a powerful approach to enhance prediction accuracy. This paper presents a novel framework that [...] Read more.

Accurate forecasting of stock market indices is crucial for investors, financial analysts, and policymakers. The integration of encoder and decoder architectures, coupled with an attention mechanism, has emerged as a powerful approach to enhance prediction accuracy. This paper presents a novel framework that leverages these components to capture complex temporal dependencies and patterns within stock price data. The encoder effectively transforms an input sequence into a dense representation, which the decoder then uses to reconstruct future values. The attention mechanism provides an additional layer of sophistication, allowing the model to selectively focus on relevant parts of the input sequence for making predictions. Furthermore, Bayesian optimization is employed to fine-tune hyperparameters, further improving forecast precision. Our results demonstrate a significant improvement in forecast precision over traditional recurrent neural networks. This indicates the potential of our integrated approach to effectively handle the complex patterns and dependencies in stock price data. Full article

(This article belongs to the Collection Advances in Applied Statistical Mechanics)

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17 pages, 821 KiB

Open AccessArticle

Measuring the Risk Spillover Effect of RCEP Stock Markets: Evidence from the TVP-VAR Model and Transfer Entropy

by Yijiang Zou, Qinghua Chen, Jihui Han and Mingzhong Xiao

Entropy 2025, 27(1), 81; https://doi.org/10.3390/e27010081 - 17 Jan 2025

Viewed by 429

Abstract

This paper selects daily stock market trading data of RCEP member countries from 3 December 2007 to 9 December 2024 and employs the Time-Varying Parameter Vector Autoregression (TVP-VAR) model and transfer entropy to measure the time-varying volatility spillover effects among the stock markets [...] Read more.

This paper selects daily stock market trading data of RCEP member countries from 3 December 2007 to 9 December 2024 and employs the Time-Varying Parameter Vector Autoregression (TVP-VAR) model and transfer entropy to measure the time-varying volatility spillover effects among the stock markets of the sampled countries. The results indicate that the signing of the RCEP has strengthened the interconnectedness of member countries’ stock markets, with an overall upward trend in volatility spillover effects, which become even more pronounced during periods of financial turbulence. Within the structure of RCEP member stock markets, China is identified as a net risk receiver, while countries like Japan and South Korea act as net risk spillover contributors. This highlights the current “fragility” of China’s stock market, making it susceptible to risk shocks from the stock markets of economically developed RCEP member countries. This analysis suggests that significant changes in bidirectional risk spillover relationships between China’s stock market and those of other RCEP members coincided with the signing and implementation of the RCEP agreement. Full article

(This article belongs to the Special Issue Risk Spillover and Transfer Entropy in Complex Financial Networks)

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17 pages, 887 KiB

Open AccessArticle

Bidimensional Increment Entropy for Texture Analysis: Theoretical Validation and Application to Colon Cancer Images

by Muqaddas Abid, Muhammad Suzuri Hitam, Rozniza Ali, Hamed Azami and Anne Humeau-Heurtier

Entropy 2025, 27(1), 80; https://doi.org/10.3390/e27010080 - 17 Jan 2025

Viewed by 433

Abstract

Entropy algorithms are widely applied in signal analysis to quantify the irregularity of data. In the realm of two-dimensional data, their two-dimensional forms play a crucial role in analyzing images. Previous works have demonstrated the effectiveness of one-dimensional increment entropy in detecting abrupt [...] Read more.

Entropy algorithms are widely applied in signal analysis to quantify the irregularity of data. In the realm of two-dimensional data, their two-dimensional forms play a crucial role in analyzing images. Previous works have demonstrated the effectiveness of one-dimensional increment entropy in detecting abrupt changes in signals. Leveraging these advantages, we introduce a novel concept, two-dimensional increment entropy (IncrEn2D), tailored for analyzing image textures. In our proposed method, increments are translated into two-letter words, encoding both the size (magnitude) and direction (sign) of the increments calculated from an image. We validate the effectiveness of this new entropy measure by applying it to MIX_2D(p) processes and synthetic textures. Experimental validation spans diverse datasets, including the Kylberg dataset for real textures and medical images featuring colon cancer characteristics. To further validate our results, we employ a support vector machine model, utilizing multiscale entropy values as feature inputs. A comparative analysis with well-known bidimensional sample entropy (SampEn_2D) and bidimensional dispersion entropy (DispEn_2D) reveals that IncrEn_2D achieves an average classification accuracy surpassing that of other methods. In summary, IncrEn_2D emerges as an innovative and potent tool for image analysis and texture characterization, offering superior performance compared to existing bidimensional entropy measures. Full article

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

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18 pages, 504 KiB

Open AccessArticle

Multi-Condition Remaining Useful Life Prediction Based on Mixture of Encoders

by Yang Liu, Bihe Xu and Yangli-ao Geng

Entropy 2025, 27(1), 79; https://doi.org/10.3390/e27010079 - 17 Jan 2025

Viewed by 451

Abstract

Accurate Remaining Useful Life (RUL) prediction is vital for effective prognostics in and the health management of industrial equipment, particularly under varying operational conditions. Existing approaches to multi-condition RUL prediction often treat each working condition independently, failing to effectively exploit cross-condition knowledge. To [...] Read more.

Accurate Remaining Useful Life (RUL) prediction is vital for effective prognostics in and the health management of industrial equipment, particularly under varying operational conditions. Existing approaches to multi-condition RUL prediction often treat each working condition independently, failing to effectively exploit cross-condition knowledge. To address this limitation, this paper introduces MoEFormer, a novel framework that combines a Mixture of Encoders (MoE) with a Transformer-based architecture to achieve precise multi-condition RUL prediction. The core innovation lies in the MoE architecture, where each encoder is designed to specialize in feature extraction for a specific operational condition. These features are then dynamically integrated through a gated mixture module, enabling the model to effectively leverage cross-condition knowledge. A Transformer layer is subsequently employed to capture temporal dependencies within the input sequence, followed by a fully connected layer to produce the final prediction. Additionally, we provide a theoretical performance guarantee for MoEFormer by deriving a lower bound for its error rate. Extensive experiments on the widely used C-MAPSS dataset demonstrate that MoEFormer outperforms several state-of-the-art methods for multi-condition RUL prediction. Full article

(This article belongs to the Section Multidisciplinary Applications)

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17 pages, 1273 KiB

Open AccessArticle

Anomalous Behavior of the Non-Hermitian Topological System with an Asymmetric Coupling Impurity

by Junjie Wang, Fude Li and Weijun Cheng

Entropy 2025, 27(1), 78; https://doi.org/10.3390/e27010078 - 17 Jan 2025

Viewed by 494

Abstract

A notable feature of systems with non-Hermitian skin effects is the sensitivity to boundary conditions. In this work, we introduce one type of boundary condition provided by a coupling impurity. We consider a system where a two-level system as an impurity couples to [...] Read more.

A notable feature of systems with non-Hermitian skin effects is the sensitivity to boundary conditions. In this work, we introduce one type of boundary condition provided by a coupling impurity. We consider a system where a two-level system as an impurity couples to a nonreciprocal Su–Schrieffer–Heeger chain under periodic boundary conditions at two points with asymmetric couplings. We first study the spectrum of the system and find that asymmetric couplings lead to topological phase transitions. Meanwhile, a striking feature is that the coupling impurity can act as an effective boundary, and asymmetric couplings can also induce a flexibly adjusted zero mode. It is localized at one of the two effective boundaries or both of them by tuning coupling strengths. Moreover, we uncover three types of localization behaviors of eigenstates for this non-Hermitian impurity system with on-site disorder. These results corroborate the potential for control of a class of non-Hermitian systems with coupling impurities. Full article

(This article belongs to the Special Issue Entropy: From Atoms to Complex Systems)

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17 pages, 1432 KiB

Open AccessArticle

Distinguishing Ideal and Non-Ideal Chemical Systems Based on Kinetic Behavior

by Gregory Yablonsky and Vladislav Fedotov

Entropy 2025, 27(1), 77; https://doi.org/10.3390/e27010077 - 16 Jan 2025

Viewed by 484

Abstract

This paper focuses on differentiating between ideal and non-ideal chemical systems based on their kinetic behavior within a closed isothermal chemical environment. Non-ideality is examined using the non-ideal Marcelin–de Donde model. The analysis primarily addresses ‘soft’ non-ideality, where the equilibrium composition for a [...] Read more.

This paper focuses on differentiating between ideal and non-ideal chemical systems based on their kinetic behavior within a closed isothermal chemical environment. Non-ideality is examined using the non-ideal Marcelin–de Donde model. The analysis primarily addresses ‘soft’ non-ideality, where the equilibrium composition for a reversible non-ideal chemical system is identical to the corresponding composition for the ideal chemical system. Our approach in distinguishing the ideal and non-ideal systems is based on the properties of the special event, i.e., event, the time of which is well-defined. For the single-step first-order reaction in the ideal system, this event is the half-time-decay point, or the intersection point. For the two consecutive reversible reactions in the ideal system, A ↔ B ↔ C, this event is the extremum obtained within the conservatively perturbed equilibrium (CPE) procedure. For the non-ideal correspondent models, the times of chosen events significantly depend on the initial concentrations. The obtained difference in the behavior of the times of these events (intersection point and CPE-extremum point) between the ideal and non-ideal systems is proposed as the kinetic fingerprint for distinguishing these systems. Full article

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

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14 pages, 2466 KiB

Open AccessArticle

Statistical Complexity Analysis of Sleep Stages

by Cristina D. Duarte, Marianela Pacheco, Francisco R. Iaconis, Osvaldo A. Rosso, Gustavo Gasaneo and Claudio A. Delrieux

Entropy 2025, 27(1), 76; https://doi.org/10.3390/e27010076 - 16 Jan 2025

Viewed by 446

Abstract

Studying sleep stages is crucial for understanding sleep architecture, which can help identify various health conditions, including insomnia, sleep apnea, and neurodegenerative diseases, allowing for better diagnosis and treatment interventions. In this paper, we explore the effectiveness of generalized weighted permutation entropy (GWPE) [...] Read more.

Studying sleep stages is crucial for understanding sleep architecture, which can help identify various health conditions, including insomnia, sleep apnea, and neurodegenerative diseases, allowing for better diagnosis and treatment interventions. In this paper, we explore the effectiveness of generalized weighted permutation entropy (GWPE) in distinguishing between different sleep stages from EEG signals. Using classification algorithms, we evaluate feature sets derived from both standard permutation entropy (PE) and GWPE to determine which set performs better in classifying sleep stages, demonstrating that GWPE significantly enhances sleep stage differentiation, particularly in identifying the transition between N1 and REM sleep. The results highlight the potential of GWPE as a valuable tool for understanding sleep neurophysiology and improving the diagnosis of sleep disorders. 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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15 pages, 2287 KiB

Open AccessArticle

Transport Numbers and Electroosmosis in Cation-Exchange Membranes with Aqueous Electrolyte Solutions of HCl, LiCl, NaCl, KCl, MgCl₂, CaCl₂ and NH₄Cl

by Simon B. B. Solberg, Zelalem B. Deress, Marte H. Hvamstad and Odne S. Burheim

Entropy 2025, 27(1), 75; https://doi.org/10.3390/e27010075 - 15 Jan 2025

Viewed by 535

Abstract

Electroosmosis reduces the available energy from ion transport arising due to concentration gradients across ion-exchange membranes. This work builds on previous efforts to describe the electroosmosis, the permselectivity and the apparent transport number of a membrane, and we show new measurements of concentration [...] Read more.

Electroosmosis reduces the available energy from ion transport arising due to concentration gradients across ion-exchange membranes. This work builds on previous efforts to describe the electroosmosis, the permselectivity and the apparent transport number of a membrane, and we show new measurements of concentration cells with the Selemion CMVN cation-exchange membrane and single-salt solutions of HCl, LiCl, NaCl, MgCl₂, CaCl₂ and NH₄Cl. Ionic transport numbers and electroosmotic water transport relative to the membrane are efficiently obtained from a relatively new permselectivity analysis method. We find that the membrane can be described as perfectly selective towards the migration of the cation, and that

{Cl}^{-}

does not contribute to the net electric current. For the investigated salts, we obtained water transference coefficients,

t_{w}

, of 1.1 ± 0.8 for HCl, 9.2 ± 0.8 for LiCl, 4.9 ± 0.2 for NaCl, 3.7 ± 0.4 for KCl, 8.5 ± 0.5 for MgCl₂, 6.2 ± 0.6 for CaCl₂ and 3.8 ± 0.5 for NH₄Cl. However, as the test compartment concentrations of LiCl, MgCl₂ and CaCl₂ increased past 3.5, 1.3 and 1.4 mol kg⁻¹, respectively, the water transference coefficients appeared to decrease. The presented methods are generally useful for characterising concentration polarisation phenomena in electrochemistry, and may aid in the design of more efficient electrochemical cells. Full article

(This article belongs to the Special Issue The Entropy Production—as Cornerstone in Applied Nonequilibrium Thermodynamics—Dedicated to Professor Signe Kjelstrup on the Occasion of Her 75th Birthday)

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6 pages, 181 KiB

Open AccessArticle

The Gibbs Fundamental Relation as a Tool for Relativity

by Friedrich Herrmann and Michael Pohlig

Entropy 2025, 27(1), 74; https://doi.org/10.3390/e27010074 - 15 Jan 2025

Viewed by 434

Abstract

When relativistic physics is lectured on, interest is focused on the behavior of mechanical and electromagnetic quantities during a reference frame change. However, not only mechanical and electromagnetic quantities transform during a reference frame change; thermodynamic and chemical quantities do too. We will [...] Read more.

When relativistic physics is lectured on, interest is focused on the behavior of mechanical and electromagnetic quantities during a reference frame change. However, not only mechanical and electromagnetic quantities transform during a reference frame change; thermodynamic and chemical quantities do too. We will study the transformations of temperature and chemical potential, show how to obtain the corresponding transformation equations with little effort, and exploit the fact that the energy conjugate extensive quantities, namely entropy and amount of substance, are Lorentz-invariant. Full article

(This article belongs to the Special Issue 200 Years Anniversary of “Sadi Carnot, Réflexions Sur La Puissance Motrice Du Feu”; Bachelier: Paris, France, 1824)

23 pages, 5323 KiB

Open AccessArticle

Entropies in Electric Circuits

by Angel Cuadras, Victoria J. Ovejas and Herminio Martínez-García

Entropy 2025, 27(1), 73; https://doi.org/10.3390/e27010073 - 15 Jan 2025

Viewed by 404

Abstract

The present study examines the relationship between thermal and configurational entropy in two resistors in parallel and in series. The objective is to introduce entropy in electric circuit analysis by considering the impact of system geometry on energy conversion in the circuit. Thermal [...] Read more.

The present study examines the relationship between thermal and configurational entropy in two resistors in parallel and in series. The objective is to introduce entropy in electric circuit analysis by considering the impact of system geometry on energy conversion in the circuit. Thermal entropy is derived from thermodynamics, whereas configurational entropy is derived from network modelling. It is observed that the relationship between thermal entropy and configurational entropy varies depending on the configuration of the resistors. In parallel resistors, thermal entropy decreases with configurational entropy, while in series resistors, the opposite is true. The implications of the maximum power transfer theorem and constructal law are discussed. The entropy generation for resistors at different temperatures was evaluated, and it was found that the consideration of resistor configurational entropy change was necessary for consistency. Furthermore, for the sake of generalization, a similar behaviour was observed in time-dependent circuits, either for resistor–capacitor circuits or circuits involving degradation. Full article

(This article belongs to the Section Multidisciplinary Applications)

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24 pages, 18984 KiB

Open AccessArticle

Maximum-Power Stirling-like Heat Engine with a Harmonically Confined Brownian Particle

by Irene Prieto-Rodríguez, Antonio Prados and Carlos A. Plata

Entropy 2025, 27(1), 72; https://doi.org/10.3390/e27010072 - 15 Jan 2025

Viewed by 491

Abstract

Heat engines transform thermal energy into useful work, operating in a cyclic manner. For centuries, they have played a key role in industrial and technological development. Historically, only gases and liquids have been used as working substances, but the technical advances achieved in [...] Read more.

Heat engines transform thermal energy into useful work, operating in a cyclic manner. For centuries, they have played a key role in industrial and technological development. Historically, only gases and liquids have been used as working substances, but the technical advances achieved in recent decades allow for expanding the experimental possibilities and designing engines operating with a single particle. In this case, the system of interest cannot be addressed at a macroscopic level and their study is framed in the field of stochastic thermodynamics. In the present work, we study mesoscopic heat engines built with a Brownian particle submitted to harmonic confinement and immersed in a fluid acting as a thermal bath. We design a Stirling-like heat engine, composed of two isothermal and two isochoric branches, by controlling both the stiffness of the harmonic trap and the temperature of the bath. Specifically, we focus on the irreversible, non-quasi-static case—whose finite duration enables the engine to deliver a non-zero output power. This is a crucial aspect, which enables the optimisation of the thermodynamic cycle by maximising the delivered power—thereby addressing a key goal at the practical level. The optimal driving protocols are obtained by using both variational calculus and optimal control theory tools. Furthermore, we numerically explore the dependence of the maximum output power and the corresponding efficiency on the system parameters. Full article

(This article belongs to the Special Issue Control of Driven Stochastic Systems: From Shortcuts to Optimality)

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22 pages, 8302 KiB

Open AccessArticle

Refraction of the Two-Photon Multimode Field via a Three-Level Atom

by Trever Harborth and Yuri Rostovtsev

Entropy 2025, 27(1), 71; https://doi.org/10.3390/e27010071 - 15 Jan 2025

Viewed by 681

Abstract

Classically, the refractive index of a medium is due to a response on said medium from an electromagnetic field. It has been shown that a single two-level atom interacting with a single photon undergoes dispersion. The following extends that analyses to a three-level [...] Read more.

Classically, the refractive index of a medium is due to a response on said medium from an electromagnetic field. It has been shown that a single two-level atom interacting with a single photon undergoes dispersion. The following extends that analyses to a three-level system interacting with two photons. Analysis of the system is completed both numerically for all photonic field modes, and analytically for an adiabatic solution of a single field mode. The findings are not only interesting for understanding additional physical phenomena due to the increased complexity of a three-level, two-photon system, but are also necessary for advancing applications such as quantum communications, quantum computation, and quantum information. Full article

(This article belongs to the Special Issue Entropy, Quantum Information and Entanglement)

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16 pages, 1633 KiB

Open AccessArticle

Advancing Rice Grain Impurity Segmentation with an Enhanced SegFormer and Multi-Scale Feature Integration

by Xiulin Qiu, Hongzhi Yao, Qinghua Liu, Hongrui Liu, Haozhi Zhang and Mengdi Zhao

Entropy 2025, 27(1), 70; https://doi.org/10.3390/e27010070 - 15 Jan 2025

Viewed by 469

Abstract

During the rice harvesting process, severe occlusion and adhesion exist among multiple targets, such as rice, straw, and leaves, making it difficult to accurately distinguish between rice grains and impurities. To address the current challenges, a lightweight semantic segmentation algorithm for impurities based [...] Read more.

During the rice harvesting process, severe occlusion and adhesion exist among multiple targets, such as rice, straw, and leaves, making it difficult to accurately distinguish between rice grains and impurities. To address the current challenges, a lightweight semantic segmentation algorithm for impurities based on an improved SegFormer network is proposed. To make full use of the extracted features, the decoder was redesigned. First, the Feature Pyramid Network (FPN) was introduced to optimize the structure, selectively fusing the high-level semantic features and low-level texture features generated by the encoder. Secondly, a Part Large Kernel Attention (Part-LKA) module was designed and introduced after feature fusion to help the model focus on key regions, simplifying the model and accelerating computation. Finally, to compensate for the lack of spatial interaction capabilities, Bottleneck Recursive Gated Convolution (B-

g^{n}

Conv) was introduced to achieve effective segmentation of rice grains and impurities. Compared with the original model, the improved model’s pixel accuracy (PA) and F1 score increased by 1.6% and 3.1%, respectively. This provides a valuable algorithmic reference for designing a real-time impurity rate monitoring system for rice combine harvesters. Full article

(This article belongs to the Special Issue Application of Information Theory to Computer Vision and Image Processing II)

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26 pages, 348 KiB

Open AccessArticle

Reporting Standards for Bayesian Network Modelling

by Martine J. Barons, Anca M. Hanea, Steven Mascaro and Owen Woodberry

Entropy 2025, 27(1), 69; https://doi.org/10.3390/e27010069 - 15 Jan 2025

Viewed by 537

Abstract

Reproducibility is a key measure of the veracity of a modelling result or finding. In other research areas, notably in medicine, reproducibility is supported by mandating the inclusion of an agreed set of details into every research publication, facilitating systematic reviews, transparency and [...] Read more.

Reproducibility is a key measure of the veracity of a modelling result or finding. In other research areas, notably in medicine, reproducibility is supported by mandating the inclusion of an agreed set of details into every research publication, facilitating systematic reviews, transparency and reproducibility. Governments and international organisations are increasingly turning to modelling approaches in the development and decision-making for policy and have begun asking questions about accountability in model-based decision making. The ethical issues of relying on modelling that is biased, poorly constructed, constrained by heroic assumptions and not reproducible are multiplied when such models are used to underpin decisions impacting human and planetary well-being. Bayesian Network modelling is used in policy development and decision support across a wide range of domains. In light of the recent trend for governments and other organisations to demand accountability and transparency, we have compiled and tested a reporting checklist for Bayesian Network modelling which will bring the desirable level of transparency and reproducibility to enable models to support decision making and allow the robust comparison and combination of models. The use of this checklist would support the ethical use of Bayesian network modelling for impactful decision making and research. Full article

(This article belongs to the Special Issue Bayesian Network Modelling in Data Sparse Environments)

12 pages, 834 KiB

Open AccessArticle

A Post-Processing Method for Quantum Random Number Generator Based on Zero-Phase Component Analysis Whitening

by Longju Liu, Jie Yang, Mei Wu, Jinlu Liu, Wei Huang, Yang Li and Bingjie Xu

Entropy 2025, 27(1), 68; https://doi.org/10.3390/e27010068 - 14 Jan 2025

Viewed by 548

Abstract

Quantum Random Number Generators (QRNGs) have been theoretically proven to be able to generate completely unpredictable random sequences, and have important applications in many fields. However, the practical implementation of QRNG is always susceptible to the unwanted classical noise or device imperfections, which [...] Read more.

Quantum Random Number Generators (QRNGs) have been theoretically proven to be able to generate completely unpredictable random sequences, and have important applications in many fields. However, the practical implementation of QRNG is always susceptible to the unwanted classical noise or device imperfections, which inevitably diminishes the quality of the generated random bits. It is necessary to perform the post-processing to extract the true quantum randomness contained in raw data generated by the entropy source of QRNG. In this work, a novel post-processing method for QRNG based on Zero-phase Component Analysis (ZCA) whitening is proposed and experimentally verified through both time and spectral domain analysis, which can effectively reduce auto-correlations and flatten the spectrum of the raw data, and enhance the random number generation rate of QRNG. Furthermore, the randomness extraction is performed after ZCA whitening, after which the final random bits can pass the NIST test. Full article

(This article belongs to the Special Issue Network Information Theory and Its Applications)

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17 pages, 4366 KiB

Open AccessArticle

Shannon Entropy Computations in Navier–Stokes Flow Problems Using the Stochastic Finite Volume Method

by Marcin Kamiński and Rafał Leszek Ossowski

Entropy 2025, 27(1), 67; https://doi.org/10.3390/e27010067 - 14 Jan 2025

Viewed by 469

Abstract

The main aim of this study is to achieve the numerical solution for the Navier–Stokes equations for incompressible, non-turbulent, and subsonic fluid flows with some Gaussian physical uncertainties. The higher-order stochastic finite volume method (SFVM), implemented according to the iterative generalized stochastic perturbation [...] Read more.

The main aim of this study is to achieve the numerical solution for the Navier–Stokes equations for incompressible, non-turbulent, and subsonic fluid flows with some Gaussian physical uncertainties. The higher-order stochastic finite volume method (SFVM), implemented according to the iterative generalized stochastic perturbation technique and the Monte Carlo scheme, are engaged for this purpose. It is implemented with the aid of the polynomial bases for the pressure–velocity–temperature (PVT) solutions, for which the weighted least squares method (WLSM) algorithm is applicable. The deterministic problem is solved using the freeware OpenFVM, the computer algebra software MAPLE 2019 is employed for the LSM local fittings, and the resulting probabilistic quantities are computed. The first two probabilistic moments, as well as the Shannon entropy spatial distributions, are determined with this apparatus and visualized in the FEPlot software. This approach is validated using the 2D heat conduction benchmark test and then applied for the probabilistic version of the 3D coupled lid-driven cavity flow analysis. Such an implementation of the SFVM is applied to model the 2D lid-driven cavity flow problem for statistically homogeneous fluid with limited uncertainty in its viscosity and heat conductivity. Further numerical extension of this technique is seen in an application of the artificial neural networks, where polynomial approximation may be replaced automatically by some optimal, and not necessarily polynomial, bases. Full article

(This article belongs to the Section Multidisciplinary Applications)

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