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Entropy, Volume 27, Issue 8 (August 2025) – 95 articles

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33 pages, 2080 KiB  
Article
Latent Class Analysis with Arbitrary-Distribution Responses
by Huan Qing and Xiaofei Xu
Entropy 2025, 27(8), 866; https://doi.org/10.3390/e27080866 - 14 Aug 2025
Abstract
The latent class model has been proposed as a powerful tool in understanding human behavior for various fields such as social, psychological, behavioral, and biological sciences. However, one important limitation of the latent class model is that it is primarily applied to data [...] Read more.
The latent class model has been proposed as a powerful tool in understanding human behavior for various fields such as social, psychological, behavioral, and biological sciences. However, one important limitation of the latent class model is that it is primarily applied to data with binary responses or categorical responses, making it fail to model real-world data with continuous or negative responses. In many applications, ignoring the weights throws out a lot of potentially valuable information contained in the weights. To address this limitation, we propose a novel generative model, the arbitrary-distribution latent class model (adLCM). Our model enables the generation of data’s response matrix from an arbitrary distribution with a latent class structure. When compared to the latent class model, our adLCM is both more realistic and general. To our knowledge, our adLCM is the first model for latent class analysis with any real-valued responses, including continuous, negative, and signed values, thereby extending the classical latent class model beyond its traditional limitation to binary or categorical outcomes. We investigate the identifiability of the model and propose an efficient algorithm for estimating the latent classes and other model parameters. We show that the proposed algorithm enjoys consistent estimation. The performance of our algorithm is evaluated using both computer-generated data and real-world personality test data. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
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20 pages, 402 KiB  
Article
Variations on the Expectation Due to Changes in the Probability Measure
by Samir M. Perlaza and Gaetan Bisson
Entropy 2025, 27(8), 865; https://doi.org/10.3390/e27080865 - 14 Aug 2025
Abstract
In this paper, closed-form expressions for the variation of the expectation of a given function due to changes in the probability measure (probability distribution drifts) are presented. These expressions unveil interesting connections with Gibbs probability measures, information projections, Pythagorean identities for relative entropy, [...] Read more.
In this paper, closed-form expressions for the variation of the expectation of a given function due to changes in the probability measure (probability distribution drifts) are presented. These expressions unveil interesting connections with Gibbs probability measures, information projections, Pythagorean identities for relative entropy, mutual information, and lautum information. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
23 pages, 374 KiB  
Article
Empirical Lossless Compression Bound of a Data Sequence
by Lei M. Li
Entropy 2025, 27(8), 864; https://doi.org/10.3390/e27080864 - 14 Aug 2025
Abstract
We consider the lossless compression bound of any individual data sequence. Conceptually, its Kolmogorov complexity is such a bound yet uncomputable. According to Shannon’s source coding theorem, the average compression bound is nH, where n is the number of words and [...] Read more.
We consider the lossless compression bound of any individual data sequence. Conceptually, its Kolmogorov complexity is such a bound yet uncomputable. According to Shannon’s source coding theorem, the average compression bound is nH, where n is the number of words and H is the entropy of an oracle probability distribution characterizing the data source. The quantity nH(θ^n) obtained by plugging in the maximum likelihood estimate is an underestimate of the bound. Shtarkov showed that the normalized maximum likelihood (NML) distribution is optimal in a minimax sense for any parametric family. Fitting a data sequence—without any a priori distributional assumption—by a relevant exponential family, we apply the local asymptotic normality to show that the NML code length is nH(θ^n)+d2logn2π+log∫Θ|I(θ)|1/2dθ+o(1), where d is dictionary size, |I(θ)| is the determinant of the Fisher information matrix, and Θ is the parameter space. We demonstrate that sequentially predicting the optimal code length for the next word via a Bayesian mechanism leads to the mixture code whose length is given by nH(θ^n)+d2logn2π+log|I(θ^n)|1/2w(θ^n)+o(1), where w(θ) is a prior. The asymptotics apply to not only discrete symbols but also continuous data if the code length for the former is replaced by the description length for the latter. The analytical result is exemplified by calculating compression bounds of protein-encoding DNA sequences under different parsing models. Typically, compression is maximized when parsing aligns with amino acid codons, while pseudo-random sequences remain incompressible, as predicted by Kolmogorov complexity. Notably, the empirical bound becomes more accurate as the dictionary size increases. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
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18 pages, 737 KiB  
Article
Mutual Information and Quantum Coherence in Minimum Error Discrimination of N Pure Equidistant Quantum States
by Omar Jiménez
Entropy 2025, 27(8), 863; https://doi.org/10.3390/e27080863 - 14 Aug 2025
Abstract
We study the quantum state discrimination problem under the minimum error (ME) strategy for a set of N pure equidistant states. These states are characterized by the property that the inner product between any pair of states is given by a unique complex [...] Read more.
We study the quantum state discrimination problem under the minimum error (ME) strategy for a set of N pure equidistant states. These states are characterized by the property that the inner product between any pair of states is given by a unique complex number S. We provide the explicit form of the states and analyze their main structural properties. The optimal success probability for ME discrimination is evaluated as a function of the number of states, as well as the modulus and phase of the inner product S. Furthermore, we propose an experimental scheme for implementing the ME discrimination of equidistant states. We also investigate the quantum coherence consumed in the implementation of the minimum error discrimination of the equidistant states, which has an established operational interpretation as cryptographic randomness gain. As an application, we propose a quantum communication protocol in which Alice prepares and sends one of the equidistant states, while Bob applies the minimum error discrimination to extract the classical information encoded in the state. Finally, we discuss the optimal conditions under which the protocol achieves an optimal balance of classical correlations and quantum coherence, thereby ensuring effective information transfer and cryptographic security. Full article
(This article belongs to the Special Issue Insight into Entropy)
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24 pages, 3961 KiB  
Article
Hierarchical Multi-Scale Mamba with Tubular Structure-Aware Convolution for Retinal Vessel Segmentation
by Tao Wang, Dongyuan Tian, Haonan Zhao, Jiamin Liu, Weijie Wang, Chunpei Li and Guixia Liu
Entropy 2025, 27(8), 862; https://doi.org/10.3390/e27080862 - 14 Aug 2025
Abstract
Retinal vessel segmentation plays a crucial role in diagnosing various retinal and cardiovascular diseases and serves as a foundation for computer-aided diagnostic systems. Blood vessels in color retinal fundus images, captured using fundus cameras, are often affected by illumination variations and noise, making [...] Read more.
Retinal vessel segmentation plays a crucial role in diagnosing various retinal and cardiovascular diseases and serves as a foundation for computer-aided diagnostic systems. Blood vessels in color retinal fundus images, captured using fundus cameras, are often affected by illumination variations and noise, making it difficult to preserve vascular integrity and posing a significant challenge for vessel segmentation. In this paper, we propose HM-Mamba, a novel hierarchical multi-scale Mamba-based architecture that incorporates tubular structure-aware convolution to extract both local and global vascular features for retinal vessel segmentation. First, we introduce a tubular structure-aware convolution to reinforce vessel continuity and integrity. Building on this, we design a multi-scale fusion module that aggregates features across varying receptive fields, enhancing the model’s robustness in representing both primary trunks and fine branches. Second, we integrate multi-branch Fourier transform with the dynamic state modeling capability of Mamba to capture both long-range dependencies and multi-frequency information. This design enables robust feature representation and adaptive fusion, thereby enhancing the network’s ability to model complex spatial patterns. Furthermore, we propose a hierarchical multi-scale interactive Mamba block that integrates multi-level encoder features through gated Mamba-based global context modeling and residual connections, enabling effective multi-scale semantic fusion and reducing detail loss during downsampling. Extensive evaluations on five widely used benchmark datasets—DRIVE, CHASE_DB1, STARE, IOSTAR, and LES-AV—demonstrate the superior performance of HM-Mamba, yielding Dice coefficients of 0.8327, 0.8197, 0.8239, 0.8307, and 0.8426, respectively. Full article
(This article belongs to the Special Issue Methods in Artificial Intelligence and Information Processing, Third Edition)
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19 pages, 1692 KiB  
Article
Overview of Mathematical Relations Between Poincaré Plot Measures and Time and Frequency Domain Measures of Heart Rate Variability
by Arie M. van Roon, Mark M. Span, Joop D. Lefrandt and Harriëtte Riese
Entropy 2025, 27(8), 861; https://doi.org/10.3390/e27080861 - 14 Aug 2025
Abstract
The Poincaré plot was introduced as a tool to analyze heart rate variations caused by arrhythmias. Later, it was applied to time series with normal beats. The plot shows the relationship between the inter-beat interval (IBI) of one beat to the next. Several [...] Read more.
The Poincaré plot was introduced as a tool to analyze heart rate variations caused by arrhythmias. Later, it was applied to time series with normal beats. The plot shows the relationship between the inter-beat interval (IBI) of one beat to the next. Several parameters were developed to characterize this relationship. The short and long axis of the fitting ellipse, SD1 and SD2, respectively, their ratio, and their product are used. The difference between the IBI of a beat and m beats later are also studied, SD1(m) and SD2(m). We studied the mathematical relations between heart rate variability measures and the Poincaré measures in the time (standard deviation of IBI, SDNN, root mean square of successive differences, RMSSD) and frequency domain (power in low and high frequency band, and their ratio). We concluded that SD1 and SD2 do not provide new information compared to SDNN and RMSSD. Only the correlation coefficient r(m) provides new information for m > 1. Novel findings are that ln(SD2(m)/SD1(m)) = tanh−1(r(m)), which is an approximately normal distributed transformation of r(m), and that SD1(m) and SD2(m) can be calculated by multiplying the power spectrum by a weighing function that depends on m, revealing the relationship with spectral measures, but also the relationship between SD1(m) and SD2(m). Both lagged parameters are extremely difficult to interpret compared to low and high frequency power, which are more closely related to the functioning of the autonomic nervous system. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
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15 pages, 2607 KiB  
Article
Adaptive Feedback Compensation Algorithm for Quantum Random Number Generators
by Wei Deng, Kun Chen, Fei Hua, Jing Cheng, Banghong Guo and Huanwen Xie
Entropy 2025, 27(8), 860; https://doi.org/10.3390/e27080860 - 14 Aug 2025
Abstract
As a core component in quantum cryptography, Quantum Random Number Generators (QRNGs) face dual critical challenges: insufficient randomness enhancement and limited compatibility with post-processing algorithms. This study proposes an Adaptive Feedback Compensation Algorithm (AFCA) to address these limitations through dynamic parameter feedback and [...] Read more.
As a core component in quantum cryptography, Quantum Random Number Generators (QRNGs) face dual critical challenges: insufficient randomness enhancement and limited compatibility with post-processing algorithms. This study proposes an Adaptive Feedback Compensation Algorithm (AFCA) to address these limitations through dynamic parameter feedback and selective encryption strategies. The AFCA dynamically adjusts nonlinear transformation intensity based on real-time statistical deviations, retaining over 50% of original bits while correcting local imbalances. Experimental results demonstrate significant improvements across QRNG types: the Monobit Test p-value for continuous QRNGs increased from 0.1376 to 0.9743, and the 0/1 distribution deviation in discrete QRNGs decreased from 7.9% to 0.5%. Compared to traditional methods like von Neumann correction, AFCA reduces data discard rates by over 55% without compromising processing efficiency. These advancements provide a robust solution for high-security quantum communication systems requiring multi-layered encryption architectures. Full article
(This article belongs to the Section Quantum Information)
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17 pages, 386 KiB  
Article
A Horizon-as-Apparatus Model That Reproduces Black Hole Thermodynamics
by Daegene Song
Entropy 2025, 27(8), 859; https://doi.org/10.3390/e27080859 - 14 Aug 2025
Abstract
We present a measurement-driven model in which the black hole horizon functions as a classical apparatus, with Planck-scale patches acting as detectors for quantum field modes. This approach reproduces the Bekenstein–Hawking area law SBH=A4ℓp2 and provides [...] Read more.
We present a measurement-driven model in which the black hole horizon functions as a classical apparatus, with Planck-scale patches acting as detectors for quantum field modes. This approach reproduces the Bekenstein–Hawking area law SBH=A4ℓp2 and provides a concrete statistical interpretation of the 1/4 factor, while adhering to established principles rather than deriving the entropy anew from first principles. Each patch generates a thermal ensemble (∼0.25 nat per mode), and summing over area-scaling patches yields the total entropy. Quantum simulations incorporating a realistic Hawking spectrum produce ⟨Sk⟩=0.257 nat (3% above 0.25 nat), and we outline testable predictions for analogue systems. Our main contribution is the horizon-as-apparatus mechanism and its information-theoretic bookkeeping. Full article
(This article belongs to the Special Issue Coarse and Fine-Grained Aspects of Gravitational Entropy)
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23 pages, 418 KiB  
Article
Robust Stability and Robust Stabilization of Discrete-Time Markov Jump Linear Systems Under a Class of Stochastic Structured Nonlinear Uncertainties
by Vasile Dragan and Samir Aberkane
Entropy 2025, 27(8), 858; https://doi.org/10.3390/e27080858 - 13 Aug 2025
Abstract
Robust stability/stabilization for discrete-time time-varying Markovian jump linear systems subject to block-diagonal stochastic parameter perturbations is addressed in this paper. Using a scaling technique, we succeed in effectively addressing the multi-perturbations case. We obtain an estimation of the lower bound of the stability [...] Read more.
Robust stability/stabilization for discrete-time time-varying Markovian jump linear systems subject to block-diagonal stochastic parameter perturbations is addressed in this paper. Using a scaling technique, we succeed in effectively addressing the multi-perturbations case. We obtain an estimation of the lower bound of the stability radius in terms of the unique bounded and positive semidefinite solutions of adequately defined parameterized backward Lyapunov difference equations. In the time-invariant case, we show that such a lower bound is actually the exact value of the stability radius. Using the obtained result, we effectively address the state-feedback robust stabilization problem. Full article
(This article belongs to the Special Issue Information Theory in Control Systems, 2nd Edition)
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13 pages, 662 KiB  
Article
Phase-Space Approach for Topological Phase Transitions in Silicene
by Maciej Kalka, Piotr Pigoń and Bartłomiej J. Spisak
Entropy 2025, 27(8), 857; https://doi.org/10.3390/e27080857 - 12 Aug 2025
Viewed by 122
Abstract
Silicene is a two-dimensional silicon monolayer with a band gap caused by relatively strong spin–orbit coupling. This band gap can be steered using a vertical electric field. In turn, the change in this electric field value leads to a transition from a topological [...] Read more.
Silicene is a two-dimensional silicon monolayer with a band gap caused by relatively strong spin–orbit coupling. This band gap can be steered using a vertical electric field. In turn, the change in this electric field value leads to a transition from a topological insulator to a bulk insulator regime. This study aims to develop a phase-space approach to detecting the topological phase transitions in silicene induced by the presence of parallel magnetic and electric fields with the aid of the concept of topological quantum number based on the Wigner–Rényi entropy. A reinterpreted definition of the Wigner distribution function is employed to determine this indicator. The topological phase transition in silicene as a function of the electric field in the presence of the magnetic field is confirmed through the use of the topological quantum number determined for the one-half, Shannon and collision entropies. Full article
(This article belongs to the Section Statistical Physics)
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19 pages, 1029 KiB  
Article
Scaling Invariance: A Gateway to Phase Transitions
by Edson Denis Leonel
Entropy 2025, 27(8), 856; https://doi.org/10.3390/e27080856 - 11 Aug 2025
Viewed by 126
Abstract
We explore the concept of scaling invariance in a type of dynamical systems that undergo a transition from regularity to chaos. The systems are described by a two-dimensional, nonlinear mapping that preserves the area in the phase space. The key variables are the [...] Read more.
We explore the concept of scaling invariance in a type of dynamical systems that undergo a transition from regularity to chaos. The systems are described by a two-dimensional, nonlinear mapping that preserves the area in the phase space. The key variables are the action and the angle, as usual from Hamiltonian systems. The transition is influenced by a control parameter giving the form of the order parameter. We observe a scaling invariance in the average squared action within the chaotic region, providing evidence that this change from regularity (integrability) to chaos (non-integrability) is akin to a second-order or continuous phase transition. As the order parameter approaches zero, its response against the variation in the control parameter (susceptibility) becomes increasingly pronounced (indeed diverging), resembling a phase transition. Full article
(This article belongs to the Special Issue From Order to Disorder: Superfluidity, Stochastic Processes, and the Dynamics of Life—Dedicated to Professor Peter McClintock on the Occasion of His 85th Birthday)
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24 pages, 1233 KiB  
Article
DRL-Based Scheduling for AoI Minimization in CR Networks with Perfect Sensing
by Juan Sun, Shubin Zhang and Xinjie Yu
Entropy 2025, 27(8), 855; https://doi.org/10.3390/e27080855 - 11 Aug 2025
Viewed by 90
Abstract
Age of Information (AoI) is a newly introduced metric that quantifies the freshness and timeliness of data, playing a crucial role in applications reliant on time-sensitive information. Minimizing AoI through optimal scheduling is challenging, especially in energy-constrained Internet of Things (IoT) networks. In [...] Read more.
Age of Information (AoI) is a newly introduced metric that quantifies the freshness and timeliness of data, playing a crucial role in applications reliant on time-sensitive information. Minimizing AoI through optimal scheduling is challenging, especially in energy-constrained Internet of Things (IoT) networks. In this work, we begin by analyzing a simplified cognitive radio network (CRN) where a single secondary user (SU) harvests RF energy from the primary user and transmits status update packets when the PU spectrum is available. Time is divided into equal time slots, and the SU performs either energy harvesting, spectrum sensing, or status update transmission in each slot. To optimize the AoI within the CRN, we formulate the sequential decision-making process as a partially observable Markov decision process (POMDP) and employ dynamic programming to determine optimal actions. Then, we extend our investigation to evaluate the long-term average weighted sum of AoIs for a multi-SU CRN. Unlike the single-SU scenario, decisions must be made regarding which SU performs sensing and which SU forwards the status update packs. Given the partially observable nature of the PU spectrum, we propose an enhanced Deep Q-Network (DQN) algorithm. Simulation results demonstrate that the proposed policies significantly outperform the myopic policy. Additionally, we analyze the effect of various parameter settings on system performance. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
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22 pages, 370 KiB  
Article
Tight Bounds Between the Jensen–Shannon Divergence and the Minmax Divergence
by Arseniy Akopyan, Herbert Edelsbrunner, Žiga Virk and Hubert Wagner
Entropy 2025, 27(8), 854; https://doi.org/10.3390/e27080854 - 11 Aug 2025
Viewed by 91
Abstract
Motivated by questions arising at the intersection of information theory and geometry, we compare two dissimilarity measures between finite categorical distributions. One is the well-known Jensen–Shannon divergence, which is easy to compute and whose square root is a proper metric. The other is [...] Read more.
Motivated by questions arising at the intersection of information theory and geometry, we compare two dissimilarity measures between finite categorical distributions. One is the well-known Jensen–Shannon divergence, which is easy to compute and whose square root is a proper metric. The other is what we call the minmax divergence, which is harder to compute. Just like the Jensen–Shannon divergence, it arises naturally from the Kullback–Leibler divergence. The main contribution of this paper is a proof showing that the minmax divergence can be tightly approximated by the Jensen–Shannon divergence. The bounds suggest that the square root of the minmax divergence is a metric, and we prove that this is indeed true in the one-dimensional case. The general case remains open. Finally, we consider analogous questions in the context of another Bregman divergence and the corresponding Burbea–Rao (Jensen–Bregman) divergence. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
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20 pages, 1876 KiB  
Article
Efficient AES Side-Channel Attacks Based on Residual Mamba Enhanced CNN
by Zhaobin Li, Chenchong Du and Xiaoyi Duan
Entropy 2025, 27(8), 853; https://doi.org/10.3390/e27080853 - 11 Aug 2025
Viewed by 178
Abstract
With the continuous advancement of side-channel attacks (SCA), deep learning-based methods have emerged as a prominent research focus due to their powerful feature extraction and nonlinear modeling capabilities. Traditional convolutional neural networks (CNNs) excel at capturing local temporal dependencies but struggle to model [...] Read more.
With the continuous advancement of side-channel attacks (SCA), deep learning-based methods have emerged as a prominent research focus due to their powerful feature extraction and nonlinear modeling capabilities. Traditional convolutional neural networks (CNNs) excel at capturing local temporal dependencies but struggle to model long-range sequential information effectively, limiting attack efficiency and generalization. In this paper, we propose a hybrid deep neural network architecture that integrates Residual Mamba blocks with multi-layer perceptrons (MLP) to enhance the modeling of side-channel information from AES implementations. The Residual Mamba module leverages state-space modeling to capture long-range dependencies, improving the model’s global temporal perception, while the MLP module further fuses high-dimensional features. Experiments conducted on the publicly available ASCAD dataset targeting the second byte of AES demonstrate that our model achieves guessing entropy (GE) rank 1 with fewer than 100 attack traces, significantly outperforming traditional CNNs and recent Transformer-based models. The proposed approach exhibits fast convergence and high attack efficiency, offering an effective new paradigm for deep learning in side-channel analysis with important theoretical and practical implications. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
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19 pages, 8180 KiB  
Article
Weighted Color Image Encryption Algorithm Based on RNA Extended Dynamic Coding and Quantum Chaotic System
by Xiangyu Zhang, Heping Wen, Wei Feng, Shenghao Kang, Zhiyu Xie, Xuexi Zhang and Yiting Lin
Entropy 2025, 27(8), 852; https://doi.org/10.3390/e27080852 - 11 Aug 2025
Viewed by 154
Abstract
The rapid development of Internet technology, while providing convenient services for users, has also aroused deep concern among the public about the issue of privacy leakage during image data transmission. To address this situation, this article proposes a color image encryption algorithm based [...] Read more.
The rapid development of Internet technology, while providing convenient services for users, has also aroused deep concern among the public about the issue of privacy leakage during image data transmission. To address this situation, this article proposes a color image encryption algorithm based on RNA extended dynamic coding and quantum chaos (CIEA-RQ). This algorithm significantly improves the ability of the system to withstand cryptographic attacks by introducing RNA extended dynamic encoding with 384 encoding rules. The employed quantum chaotic map improves the randomness of chaotic sequences and increases the key space. First, the algorithm decomposes the plaintext image into bit planes and obtains two parts, high 4-bit and low 4-bit planes, based on different weights of information. Then, the high 4-bit planes are partitioned into blocks and scrambled, and the scrambled planes are confused using RNA extended coding rules. Meanwhile, the low 4-bit planes employ a lightweight XOR operation to improve encryption efficiency. Finally, the algorithm performs cross-iterative diffusion on the processed high 4-bit and low 4-bit planes and then synthesizes a color ciphertext image. Experimental simulations and security assessments demonstrate the superior numerical statistical outcomes of the CIEA-RQ. According to the criteria of cryptanalysis, it can effectively resist known-plaintext attacks and chosen-plaintext attacks. Therefore, the CIEA-RQ presented in this article serves as an efficient digital image privacy safeguard technique, promising extensive applications in image secure transmission for the upcoming generation of networks. Full article
(This article belongs to the Section Multidisciplinary Applications)
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11 pages, 1243 KiB  
Article
Fast and Robust Optical Cooling via Shortcut to Adiabaticity
by Zhiyu Wang and Jie Lu
Entropy 2025, 27(8), 851; https://doi.org/10.3390/e27080851 - 11 Aug 2025
Viewed by 108
Abstract
Optical cooling is a key technique for preparing ultracold atoms in quantum technologies and precision experiments. We employ shortcut-to-adiabaticity (STA) techniques to accelerate and stabilize laser-based atomic cooling protocols. This approach improves the performance of conventional adiabatic momentum transfer schemes by addressing key [...] Read more.
Optical cooling is a key technique for preparing ultracold atoms in quantum technologies and precision experiments. We employ shortcut-to-adiabaticity (STA) techniques to accelerate and stabilize laser-based atomic cooling protocols. This approach improves the performance of conventional adiabatic momentum transfer schemes by addressing key limitations such as Doppler shifts, laser intensity fluctuations, and spontaneous emission. We first examine two- and three-level atomic systems subjected to counter-propagating laser pulses that induce momentum reduction through photon recoil. STA methods are then employed to construct pulse sequences that are robust against detuning errors and amplitude noise, outperforming standard π-pulse schemes in resilience. Meanwhile, we analyze the dissipative dynamics during the momentum transfer and demonstrate the superiority of the STA protocol in enhancing momentum transfer efficiency via accelerated control. The results demonstrate that STA can significantly improve both the efficiency and robustness of cooling. These findings have implications for applications in atomic physics, quantum information processing, and precision metrology. Full article
(This article belongs to the Special Issue Shortcut to Adiabaticity in Classical and Quantum Systems)
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20 pages, 1350 KiB  
Article
Beyond the Second Law: Darwinian Evolution as a Tendency for Entropy Production to Increase
by Charles H. Lineweaver
Entropy 2025, 27(8), 850; https://doi.org/10.3390/e27080850 - 11 Aug 2025
Viewed by 235
Abstract
There is much confusion about the apparent opposition between Darwinian evolution and the second law of thermodynamics. Both entropy and entropy production play more fundamental roles in the origin of life and Darwinian evolution than is generally recognized. I argue that Darwinian evolution [...] Read more.
There is much confusion about the apparent opposition between Darwinian evolution and the second law of thermodynamics. Both entropy and entropy production play more fundamental roles in the origin of life and Darwinian evolution than is generally recognized. I argue that Darwinian evolution can be understood as a tendency for entropy production to increase. Since the second law is about the increase in entropy, this hypothesis goes beyond the second law because it is about the increase in entropy production. This hypothesis can explain some aspects of biology that Darwinism struggles with, such as the origin of life, the origin of Darwinism, ecological successions, and an apparent general trend towards biological complexity. Gould proposed a wall of minimal complexity to explain this apparent increase in biological complexity. I argue that the apparent increase in biological complexity can be understood as a tendency for biological entropy production to increase through a broader range of free energy transduction mechanisms. In the context of a simple universe-in-a-cup-of-coffee model, entropy production is proposed as a more quantifiable replacement for the notion of complexity. Finally, I sketch the cosmic history of entropy production, which suggests that increases and decreases of free energy availability constrain the tendency for entropy production to increase. Full article
(This article belongs to the Special Issue Thermodynamics of Dissipative Structures and Related Emergent Phenomena)
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24 pages, 1117 KiB  
Article
Adsorption of Ternary Mixtures in the Presence of Multisite Occupancy: Theory and Monte Carlo Simulations
by Pablo Jesús Longone and Antonio José Ramirez-Pastor
Entropy 2025, 27(8), 849; https://doi.org/10.3390/e27080849 - 10 Aug 2025
Viewed by 124
Abstract
Adsorption of multicomponent mixtures on solid substrates is essential to numerous technological processes and provides key insights into surface phenomena. Despite advancements in theoretical modeling, many approaches still assume that each adsorbate occupies a single site, thereby neglecting important effects arising from molecules [...] Read more.
Adsorption of multicomponent mixtures on solid substrates is essential to numerous technological processes and provides key insights into surface phenomena. Despite advancements in theoretical modeling, many approaches still assume that each adsorbate occupies a single site, thereby neglecting important effects arising from molecules that span multiple adsorption sites. In this work, we broaden the theoretical description of such systems by considering the adsorption of j distinct polyatomic species on triangular lattices. Our approach is based on (i) exact thermodynamic results for polyatomic gases on one-dimensional lattices, extended here to account for substrates with higher coordination numbers, and (ii) the “0D cavity” functional theory originally developed by Lafuente and Cuesta, which reduces to the well-known Guggenheim–DiMarzio model in the limit of rigid rods. As a case study, we explore the behavior of a three-component system consisting of dimers, linear trimers, and triangular trimers adsorbing onto a triangular lattice. This model captures the interplay between structural simplicity, multisite occupancy, configurational diversity, and competition for space, key factors in many practical scenarios involving size-asymmetric molecules. We characterize the system using total and partial isotherms, energy of adsorption, and configurational entropy of the adsorbed phase. To ensure the reliability of our theoretical predictions, we perform Monte Carlo simulations, which show excellent agreement with the analytical approaches. Our findings demonstrate that even complex adsorption systems can be efficiently described using this generalized framework, offering new insights into multicomponent surface adsorption. Full article
(This article belongs to the Section Statistical Physics)
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14 pages, 405 KiB  
Article
Quantum Coherence and Purity in Dissipative Hydrogen Atoms: Insights from the Lindblad Master Equation
by Kamal Berrada and Smail Bougouffa
Entropy 2025, 27(8), 848; https://doi.org/10.3390/e27080848 - 10 Aug 2025
Viewed by 233
Abstract
In this work, we investigate the quantum coherence and purity in hydrogen atoms under dissipative dynamics, with a focus on the hyperfine structure states arising from the electron–proton spin interaction. Using the Lindblad master equation, we model the time evolution of the density [...] Read more.
In this work, we investigate the quantum coherence and purity in hydrogen atoms under dissipative dynamics, with a focus on the hyperfine structure states arising from the electron–proton spin interaction. Using the Lindblad master equation, we model the time evolution of the density matrix of the system, incorporating both the unitary dynamics driven by the hyperfine Hamiltonian and the dissipative effects due to environmental interactions. Quantum coherence is quantified using the L1 norm and relative entropy measures, while purity is assessed via von Neumann entropy, for initial states, including a maximally entangled Bell state and a separable state. Our results reveal distinct dynamics: for the Bell states, both coherence and purity decay exponentially with a rate proportional to the dissipation parameter, whereas for a kind of separable state, coherence exhibits oscillatory behavior modulated via the hyperfine coupling constant, superimposed on an exponential decay, and accompanied by a steady increase in entropy. Higher dissipation rates accelerate the loss of coherence and the growth of von Neumann entropy, underscoring the environment’s role in suppressing quantum superposition and driving the system towards mixed states. These findings enhance our understanding of coherence and purity preservation in atomic systems and offer insights for quantum information applications where robustness against dissipation is critical. Full article
(This article belongs to the Special Issue Entropy in Classical and Quantum Information Theory with Applications)
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14 pages, 2356 KiB  
Article
The Synergistic Effects of Structural Evolution and Attack Strategies on Network Matching Robustness
by Xu Na, Junying Cui, Chang Su, Shimin Cai and Linyuan Lü
Entropy 2025, 27(8), 847; https://doi.org/10.3390/e27080847 - 9 Aug 2025
Viewed by 192
Abstract
Research on network robustness has long focused on changes in the structure connectivity of networks under attacks, effectively depicting structural integrity while ignoring the exploration of functional integrity. When the core path of the network is attacked, even if it remains connected, the [...] Read more.
Research on network robustness has long focused on changes in the structure connectivity of networks under attacks, effectively depicting structural integrity while ignoring the exploration of functional integrity. When the core path of the network is attacked, even if it remains connected, the rapid increase in energy consumption may still trigger systematic risks. Existing studies mainly use random networks and scale-free networks as comparative models, which has become a classic research paradigm. However, real-world networks often exhibit mixed topological features. To address the above issues, this paper introduces the concept of energy from physics into bipartite networks and establishes an evaluation framework for assessing the synergistic effects of structural evolution and attack strategies on network matching robustness. We first introduce a structural parameter u to construct a structural evolution model, where the network’s minimal matching energy distribution evolves from topological heterogeneity to random features. When u approaches 0, edges with the minimal matching energy concentrate on a few candidates, manifesting scale-free network features. When u approaches 1, the uniform distribution of the minimum-matching-energy edges corresponds to random network features. We then design three types of edge attack strategies—minimum-energy (min-E), random-energy (ran-E), and maximum-energy (max-E) attacks—simulating the impacts of critical path destruction, uniform perturbation, and redundancy removal, respectively. In addition, we construct two evaluation indicators, the average matching energy and the matching retention rate. The results show that structural evolution significantly affects network matching robustness in a nonlinear manner. Different attack strategies also exert different influence on matching robustness. Furthermore, the findings reveal the synergistic effects of the two factors on network matching robustness. The synergistic effects of redundancy capacity and network structure on matching robustness are also explored. The research deepens the understanding of network matching robustness and provides a theoretical basis for resource allocation systems to combat network attacks. Full article
(This article belongs to the Special Issue Complexity, Entropy and the Physics of Information II)
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23 pages, 8311 KiB  
Article
Active Inference with Dynamic Planning and Information Gain in Continuous Space by Inferring Low-Dimensional Latent States
by Takazumi Matsumoto, Kentaro Fujii, Shingo Murata and Jun Tani
Entropy 2025, 27(8), 846; https://doi.org/10.3390/e27080846 - 9 Aug 2025
Viewed by 154
Abstract
Active inference offers a unified framework in which agents can exhibit both goal-directed and epistemic behaviors. However, implementing policy search in high-dimensional continuous action spaces presents challenges in terms of scalability and stability. Our previously proposed model, T-GLean, addressed this issue by enabling [...] Read more.
Active inference offers a unified framework in which agents can exhibit both goal-directed and epistemic behaviors. However, implementing policy search in high-dimensional continuous action spaces presents challenges in terms of scalability and stability. Our previously proposed model, T-GLean, addressed this issue by enabling efficient goal-directed planning through low-dimensional latent space search, further reduced by conditioning on prior habituated behavior. However, the lack of an epistemic term in minimizing expected free energy limited the agent’s ability to engage in information-seeking behavior that can be critical for attaining preferred outcomes. In this study, we present EFE-GLean, an extended version of T-GLean that overcomes this limitation by integrating epistemic value into the planning process. EFE-GLean generates goal-directed policies by inferring low-dimensional future posterior trajectories while maximizing expected information gain. Simulation experiments using an extended T-maze task—implemented in both discrete and continuous domains—demonstrate that the agent can successfully achieve its goals by exploiting hidden environmental information. Furthermore, we show that the agent is capable of adapting to abrupt environmental changes by dynamically revising plans through simultaneous minimization of past variational free energy and future expected free energy. Finally, analytical evaluations detail the underlying mechanisms and computational properties of the model. Full article
(This article belongs to the Special Issue Active Inference in Cognitive Neuroscience)
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26 pages, 7587 KiB  
Article
PAC–Bayes Guarantees for Data-Adaptive Pairwise Learning
by Sijia Zhou, Yunwen Lei and Ata Kabán
Entropy 2025, 27(8), 845; https://doi.org/10.3390/e27080845 - 8 Aug 2025
Viewed by 226
Abstract
We study the generalization properties of stochastic optimization methods under adaptive data sampling schemes, focusing on the setting of pairwise learning, which is central to tasks like ranking, metric learning, and AUC maximization. Unlike pointwise learning, pairwise methods must address statistical dependencies between [...] Read more.
We study the generalization properties of stochastic optimization methods under adaptive data sampling schemes, focusing on the setting of pairwise learning, which is central to tasks like ranking, metric learning, and AUC maximization. Unlike pointwise learning, pairwise methods must address statistical dependencies between input pairs—a challenge that existing analyses do not adequately handle when sampling is adaptive. In this work, we extend a general framework that integrates two algorithm-dependent approaches—algorithmic stability and PAC–Bayes analysis for this purpose. Specifically, we examine (1) Pairwise Stochastic Gradient Descent (Pairwise SGD), widely used across machine learning applications, and (2) Pairwise Stochastic Gradient Descent Ascent (Pairwise SGDA), common in adversarial training. Our analysis avoids artificial randomization and leverages the inherent stochasticity of gradient updates instead. Our results yield generalization guarantees of order n−1/2 under non-uniform adaptive sampling strategies, covering both smooth and non-smooth convex settings. We believe these findings address a significant gap in the theory of pairwise learning with adaptive sampling. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
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18 pages, 1207 KiB  
Article
Rate-Distortion Analysis of Distributed Indirect Source Coding
by Jiancheng Tang and Qianqian Yang
Entropy 2025, 27(8), 844; https://doi.org/10.3390/e27080844 - 8 Aug 2025
Viewed by 101
Abstract
Motivated by task-oriented semantic communication and distributed learning systems, this paper studies a distributed indirect source coding problem where M correlated sources are independently encoded for a central decoder. The decoder has access to correlated side information in addition to the messages received [...] Read more.
Motivated by task-oriented semantic communication and distributed learning systems, this paper studies a distributed indirect source coding problem where M correlated sources are independently encoded for a central decoder. The decoder has access to correlated side information in addition to the messages received from the encoders and aims to recover a latent random variable under a given distortion constraint rather than recovering the sources themselves. We characterize the exact rate-distortion function for the case where the sources are conditionally independent given the side information. Furthermore, we develop a distributed Blahut–Arimoto (BA) algorithm to numerically compute the rate-distortion function. Numerical examples are provided to demonstrate the effectiveness of the proposed approach in calculating the rate-distortion region. Full article
(This article belongs to the Special Issue Semantic Information Theory)
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15 pages, 2066 KiB  
Article
Multifractal Nonlinearity in Behavior During a Computer Task with Increasing Difficulty: What Does It Teach Us?
by Alix Bouni, Laurent M. Arsac, Olivier Chevalerias and Veronique Deschodt-Arsac
Entropy 2025, 27(8), 843; https://doi.org/10.3390/e27080843 - 8 Aug 2025
Viewed by 155
Abstract
The complex systems approach to cognitive–motor processing values multifractal nonlinearity as a key formalism in understanding internal interactions across multiple scales that preserve adequate task-directed behaviors. By using a computer task with increasing difficulty, we focused on the potential link between the difficulty [...] Read more.
The complex systems approach to cognitive–motor processing values multifractal nonlinearity as a key formalism in understanding internal interactions across multiple scales that preserve adequate task-directed behaviors. By using a computer task with increasing difficulty, we focused on the potential link between the difficulty threshold during a task, assessed by the individual’s score ceiling, and the corresponding level of multifractal nonlinearity in movement behavior, assessed based on a time series of cursor displacements. Entropy-based multifractality (MF) and multifractal nonlinearity obtained using a t-test comparison between the original and linearized surrogate series (tMF) of the time series characterized individual adaptive capacity. A time-varying increase in the score helped in assessing performance when facing increasing difficulty. Twenty-one participants performed a herding task (7 min), which involves keeping three moving sheep near the center of a screen by controlling the mouse pointer as a repelling shepherd dog. The more the score increased, the more the increased herd movement amplitude amplified task difficulty. The time course of the score, score dynamics (score-dyn), markedly diverged across participants, exhibiting a ceiling effect in some during the last third of the task (phase 3). This observation led us to arbitrarily distinguish three phases of the same duration and focus on phase 3, where marked differences in score-dyn emerged. Hierarchical clustering of principal components, starting with principal component analysis, identified three clusters among the participants: cluster 1 was defined by an underrepresentation of score-dyn, MF, and tMF; cluster 2 was defined by an overrepresentation of MF; and, as a critical outcome, cluster 3 was defined by an overrepresentation of score-dyn and tMF. Accordingly, participants belonging to cluster 3 had the highest score-dyn and tMF. Our interpretative hypothesis is that internal interactions that adequately perform the task are reflected in a high degree of multifractal nonlinearity. These findings extend the notion that multifractal nonlinearity is a useful conceptual framework for shedding light on adaptive behavior during complex tasks. Full article
(This article belongs to the Section Complexity)
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15 pages, 847 KiB  
Article
Compressor Power and Efficiency Optimization: A Finite-Time Thermodynamics Approach
by François Lanzetta
Entropy 2025, 27(8), 842; https://doi.org/10.3390/e27080842 - 8 Aug 2025
Viewed by 113
Abstract
This paper presents a theoretical optimization of an endoreversible compressor under steady-state conditions. A parametric study using finite-time thermodynamic principles highlights the effect of external irreversibilities on compressor performance. A compressor efficiency metric is established based on heat pump theory’s analogous performance coefficient [...] Read more.
This paper presents a theoretical optimization of an endoreversible compressor under steady-state conditions. A parametric study using finite-time thermodynamic principles highlights the effect of external irreversibilities on compressor performance. A compressor efficiency metric is established based on heat pump theory’s analogous performance coefficient concept. The external irreversibilities are characterized as functions of the conductance coefficients between the compressor and the low- and high-pressure reservoirs. In particular, the influence of suction and discharge tube diameters and gas pressures is investigated to determine the optimum compressor operating performance for a given gas mass flow rate. The results highlight the importance of selecting optimal suction and discharge tube diameters to improve compressor power efficiency and minimize energy consumption during gas compression. Full article
(This article belongs to the Special Issue The First Half Century of Finite-Time Thermodynamics)
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13 pages, 639 KiB  
Review
Heider Balance—A Continuous Dynamics
by Krzysztof Kułakowski
Entropy 2025, 27(8), 841; https://doi.org/10.3390/e27080841 - 8 Aug 2025
Viewed by 160
Abstract
This paper is a short review on applications of non-linear dynamics in the concept of Heider balance, known also as structural balance. In all the papers listed here, the basic tools are ordinary differential equations. All papers pay attention to real social phenomena, [...] Read more.
This paper is a short review on applications of non-linear dynamics in the concept of Heider balance, known also as structural balance. In all the papers listed here, the basic tools are ordinary differential equations. All papers pay attention to real social phenomena, which play the role of illustrations of the mathematical formalisms. Full article
(This article belongs to the Special Issue Computational and Statistical Physics Approaches for Complex Systems and Social Phenomena, 3rd Edition)
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17 pages, 3234 KiB  
Article
Including the Magnitude Variability of a Signal in the Ordinal Pattern Analysis
by Melvyn Tyloo, Joaquín González and Nicolás Rubido
Entropy 2025, 27(8), 840; https://doi.org/10.3390/e27080840 - 7 Aug 2025
Viewed by 271
Abstract
One of the most popular and innovative methods to analyse signals is by using Ordinal Patterns (OPs). The OP encoding is based on transforming a (univariate) signal into a symbolic sequence of OPs, where each OP represents the number of permutations needed to [...] Read more.
One of the most popular and innovative methods to analyse signals is by using Ordinal Patterns (OPs). The OP encoding is based on transforming a (univariate) signal into a symbolic sequence of OPs, where each OP represents the number of permutations needed to order a small subset of the signal’s magnitudes. This implies that OPs are conceptually clear, methodologically simple to implement, and robust to noise, and that they can be applied to short signals. Moreover, they simplify the statistical analyses that can be carried out on a signal, such as entropy and complexity quantifications. However, because of the relative ordering, information about the magnitude of the signal at each timestamp is lost—this being one of the major drawbacks of this method. Here, we propose a way to use the signal magnitudes discarded in the OP encoding as a complementary variable to its permutation entropy. To illustrate our approach, we analyse synthetic trajectories from logistic and Hénon maps—with and without added noise—and real-world signals, including intracranial electroencephalographic recordings from rats in different sleep-wake states and frequency fluctuations in power grids. Our results show that, when complementing the permutation entropy with the variability in the signal magnitudes, the characterisation of these signals is improved and the results remain explainable. This implies that our approach can be useful for feature engineering and improving AI classifiers, as typical machine learning algorithms need complementary signal features as inputs to improve classification accuracy. Full article
(This article belongs to the Special Issue Ordinal Patterns-Based Tools and Their Applications)
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12 pages, 545 KiB  
Article
Signal Detection Based on Separable CNN for OTFS Communication Systems
by Ying Wang, Zixu Zhang, Hang Li, Tao Zhou and Zhiqun Cheng
Entropy 2025, 27(8), 839; https://doi.org/10.3390/e27080839 - 7 Aug 2025
Viewed by 213
Abstract
This paper proposes a low-complexity signal detection method for orthogonal time frequency space (OTFS) communication systems, based on a separable convolutional neural network (SeCNN), termed SeCNN-OTFS. A novel SeparableBlock architecture is introduced, which integrates residual connections and a channel attention mechanism to enhance [...] Read more.
This paper proposes a low-complexity signal detection method for orthogonal time frequency space (OTFS) communication systems, based on a separable convolutional neural network (SeCNN), termed SeCNN-OTFS. A novel SeparableBlock architecture is introduced, which integrates residual connections and a channel attention mechanism to enhance feature discrimination and training stability under high Doppler conditions. By decomposing standard convolutions into depthwise and pointwise operations, the model achieves a substantial reduction in computational complexity. To validate its effectiveness, simulations are conducted under a standard OTFS configuration with 64-QAM modulation, comparing the proposed SeCNN-OTFS with conventional CNN-based models and classical linear estimators, such as least squares (LS) and minimum mean square error (MMSE). The results show that SeCNN-OTFS consistently outperforms LS and MMSE, and when the signal-to-noise ratio (SNR) exceeds 12.5 dB, its bit error rate (BER) performance becomes nearly identical to that of 2D-CNN. Notably, SeCNN-OTFS requires only 19% of the parameters compared to 2D-CNN, making it highly suitable for resource-constrained environments such as satellite and IoT communication systems. For scenarios where higher accuracy is required and computational resources are sufficient, the CNN-OTFS model—with conventional convolutional layers replacing the separable convolutional layers—can be adopted as a more precise alternative. Full article
(This article belongs to the Special Issue Delay-Doppler Domain Communications for Future Wireless Networks 2nd Edition)
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23 pages, 3561 KiB  
Article
Chaos-Based Color Image Encryption with JPEG Compression: Balancing Security and Compression Efficiency
by Wei Zhang, Xue Zheng, Meng Xing, Jingjing Yang, Hai Yu and Zhiliang Zhu
Entropy 2025, 27(8), 838; https://doi.org/10.3390/e27080838 - 6 Aug 2025
Viewed by 237
Abstract
In recent years, most proposed digital image encryption algorithms have primarily focused on encrypting raw pixel data, often neglecting the integration with image compression techniques. Image compression algorithms, such as JPEG, are widely utilized in internet applications, highlighting the need for encryption methods [...] Read more.
In recent years, most proposed digital image encryption algorithms have primarily focused on encrypting raw pixel data, often neglecting the integration with image compression techniques. Image compression algorithms, such as JPEG, are widely utilized in internet applications, highlighting the need for encryption methods that are compatible with compression processes. This study introduces an innovative color image encryption algorithm integrated with JPEG compression, designed to enhance the security of images susceptible to attacks or tampering during prolonged transmission. The research addresses critical challenges in achieving an optimal balance between encryption security and compression efficiency. The proposed encryption algorithm is structured around three key compression phases: Discrete Cosine Transform (DCT), quantization, and entropy coding. At each stage, the algorithm incorporates advanced techniques such as block segmentation, block replacement, DC coefficient confusion, non-zero AC coefficient transformation, and RSV (Run/Size and Value) pair recombination. Extensive simulations and security analyses demonstrate that the proposed algorithm exhibits strong robustness against noise interference and data loss, effectively meeting stringent security performance requirements. Full article
(This article belongs to the Section Multidisciplinary Applications)
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23 pages, 723 KiB  
Article
Multivariate Modeling of Some Datasets in Continuous Space and Discrete Time
by Rigele Te and Juan Du
Entropy 2025, 27(8), 837; https://doi.org/10.3390/e27080837 - 6 Aug 2025
Viewed by 180
Abstract
Multivariate space–time datasets are often collected at discrete, regularly monitored time intervals and are typically treated as components of time series in environmental science and other applied fields. To effectively characterize such data in geostatistical frameworks, valid and practical covariance models are essential. [...] Read more.
Multivariate space–time datasets are often collected at discrete, regularly monitored time intervals and are typically treated as components of time series in environmental science and other applied fields. To effectively characterize such data in geostatistical frameworks, valid and practical covariance models are essential. In this work, we propose several classes of multivariate spatio-temporal covariance matrix functions to model underlying stochastic processes whose discrete temporal margins correspond to well-known autoregressive and moving average (ARMA) models. We derive sufficient and/or necessary conditions under which these functions yield valid covariance matrices. By leveraging established methodologies from time series analysis and spatial statistics, the proposed models are straightforward to identify and fit in practice. Finally, we demonstrate the utility of these multivariate covariance functions through an application to Kansas weather data, using co-kriging for prediction and comparing the results to those obtained from traditional spatio-temporal models. Full article
(This article belongs to the Special Issue Statistical Methods for Modeling High-Dimensional and Complex Data: Second Edition)
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