Entropy

1305 KB

Open AccessArticle

Semantic Communication for Intelligent Transmission and Recognition of High-Resolution Satellite Images in Satellite-to-Ground Systems

by Jiaxin Liu, Qiwang Chen and Yijun Chen

Entropy 2026, 28(7), 803; https://doi.org/10.3390/e28070803 - 14 Jul 2026

Abstract

Very-high-resolution (VHR) multispectral satellite imagery contains rich semantic information, yet its real-time transmission is constrained by limited satellite-to-ground bandwidth and dynamic channel impairments. Conventional communication schemes prioritize pixel-level reconstruction, resulting in large transmission overhead and poor robustness under unfavorable channel conditions. To address [...] Read more.

Very-high-resolution (VHR) multispectral satellite imagery contains rich semantic information, yet its real-time transmission is constrained by limited satellite-to-ground bandwidth and dynamic channel impairments. Conventional communication schemes prioritize pixel-level reconstruction, resulting in large transmission overhead and poor robustness under unfavorable channel conditions. To address these challenges, an end-to-end task-oriented semantic communication framework for remote sensing downstream recognition tasks, termed Semantic Transmission Architecture for Remote Sensing (STARS), is proposed. To improve transmission efficiency for very-high-resolution remote sensing images with highly redundant background regions, a Semantic Feature Reweighting Module (SFRM) is introduced to dynamically evaluate token-level semantic importance and adaptively allocate transmission resources to task-critical features. Furthermore, vector quantization and a practical digital transmission chain are jointly integrated to achieve efficient semantic compression, while dynamic channel variations are incorporated during training to improve robustness under fading channel conditions. Experimental results on the DOTA dataset demonstrate that STARS consistently outperforms conventional schemes and existing semantic baselines under Rician fading channels, validating the effectiveness of semantic-aware feature allocation for bandwidth-efficient VHR imagery transmission. Full article

(This article belongs to the Special Issue AI for Wireless Communication Systems: From Semantic Communications to 6G)

► Show Figures

Figure 1

406 KB

Open AccessArticle

Quantum Cosmology in Krylov Space: Complexity and Entropy

by Meysam Motaharfar, Maxwell R. Siebersma and Parampreet Singh

Entropy 2026, 28(7), 802; https://doi.org/10.3390/e28070802 - 14 Jul 2026

Abstract

We study the quantum dynamics in Krylov space of a spatially flat, homogeneous, and isotropic universe sourced with a massless scalar field within Wheeler–DeWitt (WDW) quantum cosmology and loop quantum cosmology (LQC) frameworks. The availability of a physical Hilbert space and physical Hamiltonian [...] Read more.

We study the quantum dynamics in Krylov space of a spatially flat, homogeneous, and isotropic universe sourced with a massless scalar field within Wheeler–DeWitt (WDW) quantum cosmology and loop quantum cosmology (LQC) frameworks. The availability of a physical Hilbert space and physical Hamiltonian and the presence of an internal clock enable us to construct the Krylov basis analytically by applying the Lanczos algorithm. We then evaluate both the Krylov state and operator complexity for WDW quantum cosmology and LQC on this basis. In regimes where the wave function of the universe is sharply peaked, our results indicate that the Krylov complexity grows quadratically with the scalar field clock for the state and operator complexities in both the WDW quantum cosmology and LQC. We further show that the operator complexity is exactly twice the state complexity in these regimes. We discuss the interpretation of the global behavior of these systems by calculating the Krylov entropy for both quantum cosmological frameworks. We observe that in LQC, the Krylov complexity and entropy remain finite at the bounce, whereas in the WDW quantum cosmology, they diverge at the big bang/crunch singularity. Our work provides the first example of computing Krylov complexity for a system with a totally constrained Hamiltonian and no external time, a framework to calculate a purely quantum-mechanical entropy in quantum cosmology, and, to our knowledge, the first direct bridge between Krylov complexity and canonical quantum cosmology, as a first step toward understanding how polymerized quantum geometry modifies complexity and entropy. Full article

(This article belongs to the Section Multidisciplinary Applications)

► Show Figures

Figure 1

299 KB

Open AccessArticle

Topological Complexity of the Length-Constrained Systems of Finite Symbols

by Qingsong Wang, Cailing Yao, Jiaxing He and Bingzhe Hou

Entropy 2026, 28(7), 801; https://doi.org/10.3390/e28070801 - 14 Jul 2026

Abstract

In this paper, we consider a class of constrained systems named n-tuple upper bound

(m_{1}, \dots, m_{n})

-constrained systems (n-TUB systems briefly) for

2 \leq n < \infty

, which are subshifts of finite [...] Read more.

In this paper, we consider a class of constrained systems named n-tuple upper bound

(m_{1}, \dots, m_{n})

-constrained systems (n-TUB systems briefly) for

2 \leq n < \infty

, which are subshifts of finite type. We determinate the topological entropies (Shannon capacities)

C (m_{1}, \dots, m_{n})

of all n-TUB systems and consequently order all n-TUB systems according to the size of the topological entropies. An algorithm is also presented to compute the transition matrix and topological entropy for n-TUB systems. Full article

(This article belongs to the Section Complexity)

110051 KB

Open AccessArticle

A Novel PQC-Based Image Encryption Scheme Using Seismic Wave Permutation

by Cemile İnce

Entropy 2026, 28(7), 800; https://doi.org/10.3390/e28070800 - 14 Jul 2026

Abstract

Image encryption schemes based on chaotic maps offer strong statistical properties but are vulnerable to quantum attacks, and their integration with post-quantum cryptography has not been sufficiently explored. This paper presents a post-quantum secure image encryption framework integrating ML-KEM (FIPS 203), standardized by [...] Read more.

Image encryption schemes based on chaotic maps offer strong statistical properties but are vulnerable to quantum attacks, and their integration with post-quantum cryptography has not been sufficiently explored. This paper presents a post-quantum secure image encryption framework integrating ML-KEM (FIPS 203), standardized by NIST in 2024, with a two-dimensional Sinh-Logistic chaotic map, HKDF-SHA256 nonce-based key derivation, feedback diffusion, and a novel Seismic Wave Permutation (SWP). The scheme derives channel-specific encryption keys from ML-KEM shared secrets using random, channel-specific nonces via HKDF-SHA256, ensuring plaintext independence and avoiding metadata-based leakage. The proposed SWP effectively breaks spatial correlations by displacing pixels according to a chaotic SWP model. RGB images are processed with independent ML-KEM encapsulation and HKDF-derived key material per channel, enabling multi-channel encryption without cross-channel leakage. Experiments on 512 × 512 test images have demonstrated Shannon entropy exceeding 7.999 bits per pixel across all channels, NPCR of at least 99.59%, UACI between 33.41% and 33.53%, and near-zero pixel correlations, further validated across 14 standard SIPI test images. An IND-CPA game simulation using four independent distinguishers, including a learned classifier trained via chosen-plaintext oracle access, over 5000 rounds per image, showed a maximum adversary advantage of 0.0186, consistent with random prediction. ML-KEM encapsulation contributes between 3.9% (ML-KEM-512) and 8.0% (ML-KEM-1024) of total encryption latency at 512 × 512 resolution, remaining a minority cost across all security levels while keeping the total encryption time within a narrow 227–258 ms range. The proposed architecture bridges standardized post-quantum cryptography with chaos-based image security for privacy-preserving image transmission. Full article

(This article belongs to the Section Multidisciplinary Applications)

► Show Figures

Figure 1

388 KB

Open AccessArticle

Exponent Spectrum of Lorenz Curves and Its Relation to a System’s Heterogeneity

by Soumyaditya Das and Soumyajyoti Biswas

Entropy 2026, 28(7), 799; https://doi.org/10.3390/e28070799 - 14 Jul 2026

Abstract

We analyze the effect of microscopic heterogeneity on the Lorenz curve of macroscopic observables. The Lorenz curve of a response function, being a cumulative and bounded quantity; it is often a more stable function than the corresponding probability density. We show here that [...] Read more.

We analyze the effect of microscopic heterogeneity on the Lorenz curve of macroscopic observables. The Lorenz curve of a response function, being a cumulative and bounded quantity; it is often a more stable function than the corresponding probability density. We show here that by doing an exponent spectrum analysis of the complementary Lorenz curve, it is possible to obtain a reflection of the underlying heterogeneity that causes the response function to depart from a power law behavior. We demonstrate this framework first by synthetic data and then by analyzing the avalanche statistics of a two dimensional, Random Field Ising Model (RFIM) at zero temperature. This method can lead to possible use in estimating the microscopic heterogeneity of a system from the analysis of an estimated Lorenz curve, particularly in socio-economic and physical contexts where the full probability distribution function is unavailable. Full article

(This article belongs to the Special Issue Ising Model—100 Years Old and Still Attractive)

► Show Figures

Figure 1

33 pages, 619 KB

Open AccessArticle

Polar Codes for Decomposed Multi-Input Multi-Output Gaussian Broadcast Channels

by Muhammed Yusuf Şener, Gerhard Kramer, Shlomo Shamai (Shitz), Ronald Böhnke and Wen Xu

Entropy 2026, 28(7), 798; https://doi.org/10.3390/e28070798 - 14 Jul 2026

Abstract

Dirty paper coding (DPC) is applied to multi-input multi-output (MIMO) broadcast channels with additive Gaussian noise and one message per receiver. The method decomposes each receiver MIMO channel into parallel scalar channels and applies modulo operators, amplitude-shift keying (ASK), and probabilistic shaping. The [...] Read more.

Dirty paper coding (DPC) is applied to multi-input multi-output (MIMO) broadcast channels with additive Gaussian noise and one message per receiver. The method decomposes each receiver MIMO channel into parallel scalar channels and applies modulo operators, amplitude-shift keying (ASK), and probabilistic shaping. The achievable rate tuples include all points inside the capacity region by choosing truncated Gaussian shaping, large ASK alphabets, and large modulo intervals. Simulations with short polar codes show significant rate and power gains from DPC compared to linear precoding, while maintaining similar encoding and decoding complexities. Full article

(This article belongs to the Special Issue Foundations and Frontiers of Information Theory—Dedicated to Professor H. Vincent Poor on the Occasion of His 75th Birthday)

► Show Figures

Figure 1

24 pages, 7202 KB

Open AccessArticle

Fault Feature Extraction of Rolling Bearings Based on Ordered Singular Spectrum Decomposition—Multipoint Optimal Minimum Entropy Deconvolution Adjusted

by Longlong Li, Wenhao Chen, Wenhui Li, Yan Zhang, Jiaxin Liu and Runlin Chen

Entropy 2026, 28(7), 797; https://doi.org/10.3390/e28070797 - 14 Jul 2026

Abstract

In the early fault stage of rolling bearings, the fault-induced impact signals in vibration data are often extremely weak and easily obscured by strong noise, making effective extraction and analysis challenging. To address this issue, this paper proposes a novel weak fault impact [...] Read more.

In the early fault stage of rolling bearings, the fault-induced impact signals in vibration data are often extremely weak and easily obscured by strong noise, making effective extraction and analysis challenging. To address this issue, this paper proposes a novel weak fault impact signal feature extraction method combining Ordered Singular Spectrum Decomposition (OSSD) and Multipoint Optimal Minimum Entropy Deconvolution Adjusted (MOMEDA). First, OSSD is employed to decompose the raw vibration signal, progressively extracting signal components across different frequency bands. The optimal signal components are adaptively selected based on mutual information criteria, effectively avoiding mode mixing issues. Subsequently, MOMEDA is applied to enhance the periodic impact features within the fault signal, improving its recognizability. To address the signal length reduction issue inherent in the MOMEDA process, a waveform extension strategy is introduced to compensate for the missing signal, ensuring signal integrity. Simulation and experimental results demonstrate that the proposed method exhibits robust noise resistance and can effectively extract early fault features of rolling bearings under strong noise conditions, validating its accuracy and effectiveness. Full article

► Show Figures

Figure 1

30 pages, 3920 KB

Open AccessArticle

Complexity and Target Preservation in Category Maps

by Christoph D. Dahl

Entropy 2026, 28(7), 796; https://doi.org/10.3390/e28070796 - 13 Jul 2026

Abstract

Categorisation is often treated as a form of compression: a high-dimensional stimulus space is reduced to a smaller set of behaviourally or cognitively useful classes. However, compression alone does not determine whether a category map is useful. The present manuscript develops an information-theoretic [...] Read more.

Categorisation is often treated as a form of compression: a high-dimensional stimulus space is reduced to a smaller set of behaviourally or cognitively useful classes. However, compression alone does not determine whether a category map is useful. The present manuscript develops an information-theoretic framework for evaluating categorisation in terms of both category complexity and target-relevant information preservation. Here, categorisation is treated as a many-to-one mapping from stimulus instances to category labels, and category entropy quantifies the distribution of the resulting labels. Across a set of synthetic demonstrations, alternative category maps over the same stimulus space are shown to preserve different target variables, including identity, action, nuisance, and hierarchical category structure. The framework is then extended to learned visual representations by analysing layer-derived category maps from a pretrained ResNet-50 network applied to CIFAR-10 images. Clean-only, strong- and mild-nuisance controls test whether layer-derived maps preserve object or nuisance information within nuisance conditions. The results show that category maps can have substantial entropy while preserving information about a variable that is not aligned with the specified target and that the value of a categorisation depends on the target variable to be preserved. The manuscript argues that categorisation should therefore be evaluated not only by compression or separability, but by the information retained about a specified cognitive, behavioural, or computational target. Full article

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

► Show Figures

Figure 1

36 pages, 11385 KB

Open AccessArticle

A Color Image Encryption Using a 4D Variable-Order Fractional Hyperchaotic System and Chess-Gameplay-Inspired Dynamic Mechanism

by Xiaomeng Cui, Xiaoqiang Zhang and Jiaqi Ji

Entropy 2026, 28(7), 795; https://doi.org/10.3390/e28070795 - 13 Jul 2026

Abstract

With the widespread adoption of digital images in network transmission and storage, the demand for image privacy protection keeps rising. We propose a robust scheme combining a four-dimensional variable-order fractional hyperchaotic system (4D-VOFHS) and a chess-game play-inspired dynamic mechanism. Firstly, we construct 4D-VOFHS, [...] Read more.

With the widespread adoption of digital images in network transmission and storage, the demand for image privacy protection keeps rising. We propose a robust scheme combining a four-dimensional variable-order fractional hyperchaotic system (4D-VOFHS) and a chess-game play-inspired dynamic mechanism. Firstly, we construct 4D-VOFHS, to overcome inherent limitations of constant-order systems: unlike constant-order systems that are vulnerable to deep-learning-based parameter identification attacks, this system introduces time-varying orders and high-dimensional coupling to enrich nonlinear dynamics. Secondly, inspired by the dynamic strategic interactions within chess gameplay, we design a synchronous encryption framework with a tightly coupled permutation–diffusion mechanism. This design not only significantly enhances the nonlinear complexity, confusion and diffusion performance of the algorithm, but also enables parallel synchronous processing to improve computational throughput. Finally, we propose a block-based collaborative scrambling strategy with multi-chess-piece rules, wherein traversal rules and scrambling operations are not predefined; instead, they are dynamically updated according to the real-time state evolution of the 4D-VOFHS. Through comprehensive correlation analysis and differential attack tests, the presented encryption framework achieves outstanding performance metrics: an average NPCR of 99.6%, a UACI of 33.4%, and an average information entropy of 7.9993. Overall, these results verify the strong cryptographic robustness and practical applicability of the scheme, highlighting its great potential for deployment in real-world color image encryption systems. Full article

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

31 pages, 1777 KB

Open AccessArticle

Statistical Inference for the Entropy of the Transmuted Weibull Distribution Under Progressive Type-II Censored Samples

by Yanqiu Zeng, Xinyu Wu and Shixiao Xiao

Entropy 2026, 28(7), 794; https://doi.org/10.3390/e28070794 - 13 Jul 2026

Abstract

This paper investigates statistical inference for the Shannon entropy of the Transmuted Weibull Distribution under progressively Type-II censored samples. The Transmuted Weibull Distribution is obtained by applying the quadratic rank transmutation map to the cumulative distribution function of the two-parameter Weibull distribution, thereby [...] Read more.

This paper investigates statistical inference for the Shannon entropy of the Transmuted Weibull Distribution under progressively Type-II censored samples. The Transmuted Weibull Distribution is obtained by applying the quadratic rank transmutation map to the cumulative distribution function of the two-parameter Weibull distribution, thereby substantially enhancing its modeling flexibility while preserving the analytical tractability of the baseline distribution. Consequently, it provides greater flexibility for modeling lifetime data exhibiting pronounced skewness and complex hazard rate behaviors. First, a closed-form expression for the Shannon entropy of the Transmuted Weibull Distribution is derived. From a frequentist perspective, the maximum likelihood estimators of the model parameters are obtained numerically using the Newton–Raphson algorithm, and the corresponding maximum likelihood estimator of Shannon entropy is derived through the invariance property of maximum likelihood estimation. To quantify estimation uncertainty, asymptotic confidence intervals are constructed using the Delta method together with the observed Fisher information matrix, while Bootstrap confidence intervals are also developed to improve finite-sample inference. From a Bayesian perspective, posterior inference is conducted using a hybrid Gibbs sampling algorithm within the Markov chain Monte Carlo framework. Bayesian point estimators of Shannon entropy are obtained under the squared error loss function, the absolute error loss function, and the 0–1 loss function, corresponding to the posterior mean, posterior median, and posterior mode, respectively. In addition, highest posterior density credible intervals are constructed for the Shannon entropy. The proposed methods are evaluated through an extensive Monte Carlo simulation study under three representative progressively Type-II censoring schemes. Estimation performance is assessed in terms of bias, mean squared error, interval coverage probability, and average interval length. The simulation results demonstrate that the Bayesian estimators consistently outperform the maximum likelihood estimator, particularly for small sample sizes and heavy censoring, while the highest posterior density credible intervals achieve more accurate coverage probabilities and shorter interval lengths. Finally, the proposed inferential procedures are illustrated using a real dataset consisting of remission times from 128 bladder cancer patients, demonstrating their practical applicability and robustness. Full article

27 pages, 8112 KB

Open AccessArticle

HH-MAPPO: A Hierarchical Reinforcement Learning Framework for Dynamic-Scale Target–Attacker–Defender Games

by Junhui Huang, Yan Guo, Xiliang Chen, Jianyu Wei, Jiawei Yi, Xinliang Chen and Lifeng Chen

Entropy 2026, 28(7), 793; https://doi.org/10.3390/e28070793 - 13 Jul 2026

Abstract

The Target–Attacker–Defender (TAD) pursuit–evasion game is a core challenge in multi-agent cooperative control, yet real-world settings involving dynamic team scaling and strict energy constraints remain largely unaddressed. When scalable shared-parameter policies are adopted to cope with the varying number of agents, severe policy [...] Read more.

The Target–Attacker–Defender (TAD) pursuit–evasion game is a core challenge in multi-agent cooperative control, yet real-world settings involving dynamic team scaling and strict energy constraints remain largely unaddressed. When scalable shared-parameter policies are adopted to cope with the varying number of agents, severe policy homogeneity emerges, preventing effective division of labor. This paper proposes a Hierarchical Heterogeneous Multi-Agent Proximal Policy Optimization (HH-MAPPO) framework to resolve these challenges. Both levels employ actor–critic networks with Role-Aware Embedding (RAE). In this mechanism, each agent is assigned a unique, learnable role embedding derived from its identity. These embeddings serve as conditioning inputs to the shared policy network, enabling it to generate differentiated behaviors and effectively mitigating policy homogeneity. The upper-level policy determines the number of defenders to deploy and assigns interception targets, while the lower-level policy handles continuous control of each defender and the ground moving target (GMT). This hierarchy resolves dynamic observation spaces via a target-matching mechanism, where each defender’s observation includes only its own state and its assigned attacker’s state, keeping observation dimension constant. Experiments in a 3D TAD simulation with continuous attacker arrivals and energy-constrained defenders show the following: (1) HH-MAPPO achieves superior interception performance compared to baseline methods in both symmetric and asymmetric scenarios; (2) ablation studies confirm RAE increases policy diversity, raising Sequence-Based Action Dissimilarity (SBAD) by 15.5%; and (3) Pareto analysis demonstrates a superior performance–energy trade-off, maintaining about 70% interception rate even under an extreme energy cap (E = 30). Full article

(This article belongs to the Section Multidisciplinary Applications)

► Show Figures

Figure 1

21 pages, 894 KB

Open AccessReview

The Evolution of Physical Laws and the Entropic Measure of Time

by Leonid M. Martyushev

Entropy 2026, 28(7), 792; https://doi.org/10.3390/e28070792 - 13 Jul 2026

Abstract

The traditional paradigm of natural science treats the laws of nature as eternal and immutable. This review examines a powerful alternative tradition that views these laws as historically evolving and constructed entities, tracing this shift from ancient roots to evolutionary epistemology, radical constructivism [...] Read more.

The traditional paradigm of natural science treats the laws of nature as eternal and immutable. This review examines a powerful alternative tradition that views these laws as historically evolving and constructed entities, tracing this shift from ancient roots to evolutionary epistemology, radical constructivism and physics. Specifically, it provides a chronological analysis of how ideas about the variability of laws developed from ancient Greek philosophy through Enlightenment thinkers to contemporary physicists like Ilya Prigogine and Lee Smolin. We address the resulting methodological crisis—where different branches of science optimize their own laws and isolate from one another—by proposing a strict hierarchical framework. Under this method, invariant basic concepts are strictly separated from flexible models. Crucially, the Entropic Measure of Time (EMT) is presented as the central operational tool. By defining time through entropy production, EMT enables the deductive derivation of physical laws from specific models, restoring a unified, cohesive structure to modern science and offering a robust strategy to counteract the fragmentation of scientific disciplines. Full article

(This article belongs to the Special Issue Symmetry and Its Applications in Complex Systems)

► Show Figures

Figure 1

2 pages, 137 KB

Open AccessCorrection

Correction: Neukart et al. Extending the QMM Framework to the Strong and Weak Interactions. Entropy 2025, 27, 153

by Florian Neukart, Eike Marx and Valerii Vinokur

Entropy 2026, 28(7), 791; https://doi.org/10.3390/e28070791 - 13 Jul 2026

Abstract

In the original publication [...] Full article

39 pages, 3060 KB

Open AccessArticle

Concentration, Information, and Distributional Stability in High-Dimensional Portfolios: A Talagrand Stability Index Approach

by Irina Georgescu and Jani Kinnunen

Entropy 2026, 28(7), 790; https://doi.org/10.3390/e28070790 - 12 Jul 2026

Abstract

This paper investigates the stability of high-dimensional financial portfolios using concentration inequalities, information-theoretic measures, optimal transport metrics, and financial network analysis. Asset returns are generated under both multivariate Gaussian and multivariate Student-t distributions. Equal Weight and Regularized Minimum Variance portfolios are evaluated across [...] Read more.

This paper investigates the stability of high-dimensional financial portfolios using concentration inequalities, information-theoretic measures, optimal transport metrics, and financial network analysis. Asset returns are generated under both multivariate Gaussian and multivariate Student-t distributions. Equal Weight and Regularized Minimum Variance portfolios are evaluated across alternative portfolio dimensions. The results show that increasing portfolio dimension reduces portfolio risk, tail probabilities, and risk estimation errors, indicating stronger concentration and higher stability in high-dimensional settings. Entropy and mutual information measures reveal improved diversification and weaker dependence structures as portfolio size increases. To assess distributional robustness, a novel Talagrand Stability Index (TSI), combining Wasserstein distance and Kullback–Leibler divergence, is introduced. The results show that TSI decreases with portfolio dimension. Heavy-tailed Student-t returns generate weaker concentration effects, stronger dependence structures, and lower distributional stability than Gaussian returns. Mutual information-based financial networks reveal sparse and moderately interconnected dependence structures. To illustrate the practical applicability of the proposed framework, an empirical application based on daily returns of ten large U.S. equities during 2020–2025 is conducted, showing that the Regularized Minimum Variance portfolio achieves a marginally lower TSI than the Equal Weight portfolio. Robustness checks reported further indicate that this advantage is modest and outcome-dependent rather than decisive. Full article

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

20 pages, 565 KB

Open AccessArticle

An Endogenous Quantum–Classical Crossover Temperature in the van der Waals Fluid: Quantumness as an Emergent Behavior

by Flavia Pennini and Angelo Plastino

Entropy 2026, 28(7), 789; https://doi.org/10.3390/e28070789 - 12 Jul 2026

Abstract

The onset of quantum behavior in gases is traditionally established through a criterion that is external to classical statistical mechanics. One introduces the thermal de Broglie wavelength and compares it with the mean intermolecular separation, concluding that quantum effects become relevant when [...] Read more.

The onset of quantum behavior in gases is traditionally established through a criterion that is external to classical statistical mechanics. One introduces the thermal de Broglie wavelength and compares it with the mean intermolecular separation, concluding that quantum effects become relevant when

n λ_{T}^{3} \sim 1

. This condition originates in quantum statistical mechanics and is absent from the classical ideal-gas or van der Waals partition functions. In this work, we show that a grand-canonical treatment of the van der Waals fluid naturally generates an interaction-corrected crossover temperature

T_{3} (a, b, m, n)

determined by the particle mass, density, and van der Waals interaction parameters. While the thermal de Broglie wavelength provides the standard quantum crossover scale, the interaction-induced correction leading to

T_{3}

is obtained without invoking the explicit form of the Bose–Einstein or Fermi–Dirac distributions. Instead,

T_{3}

follows from a self-consistent condition within the grand-canonical van der Waals description. We demonstrate that, below this temperature, the statistical assumptions underlying the classical theory become self-inconsistent, indicating the breakdown of the classical description and the onset of the quantum-degenerate regime. The resulting temperature scale therefore provides an interaction-corrected boundary of validity of the classical van der Waals description. These findings provide a new perspective on how intermolecular interactions modify the crossover to the quantum-degenerate regime and clarify the limits of applicability of the classical van der Waals theory. Full article

► Show Figures

Figure 1

20 pages, 1275 KB

Open AccessArticle

Quantum Cournot Triopoly Game with Heterogeneous Expectations: Dynamics and Chaos Control with Isoelastic Demand

by Longfei Wei, Shouli Wang and Jing Wang

Entropy 2026, 28(7), 788; https://doi.org/10.3390/e28070788 - 12 Jul 2026

Abstract

This paper investigates how quantum entanglement and heterogeneous expectations jointly affect the stability, complexity, and controllability of a Cournot triopoly with isoelastic demand. Based on the Li–Du–Massar quantization scheme, we construct a discrete-time quantum Cournot triopoly in which three firms adopt different updating [...] Read more.

This paper investigates how quantum entanglement and heterogeneous expectations jointly affect the stability, complexity, and controllability of a Cournot triopoly with isoelastic demand. Based on the Li–Du–Massar quantization scheme, we construct a discrete-time quantum Cournot triopoly in which three firms adopt different updating mechanisms: boundedly rational adjustment, naïve and adaptive expectations. The quantum boundary equilibrium and the unique interior quantum Nash equilibrium are derived explicitly. By linearizing the resulting three-dimensional nonlinear map and applying the Jury criterion, we obtain analytical local stability conditions for the interior equilibrium. The results show that increasing the entanglement level reduces the admissible range of the adjustment speed, thereby shrinking the stability domain and making the market dynamics more prone to bifurcation and chaos. Numerical simulations further reveal a typical transition from stable convergence to flip bifurcation, period-doubling cascades, chaotic attractors, and sensitive dependence on initial conditions. Finally, a control parameter is introduced to rescale the effective adjustment speed of the boundedly rational firm. This mechanism preserves the equilibrium set while restoring convergence to a stable fixed point once the control intensity exceeds a critical threshold. The findings highlight the joint role of entanglement, expectation heterogeneity, and nonlinear demand in shaping complex quantum oligopoly dynamics. Full article

(This article belongs to the Section Quantum Information)

► Show Figures

Figure 1

46 pages, 9452 KB

Open AccessArticle

Hopf Bifurcation in an Incommensurate Caputo Fractional-Order Computer Virus Epidemic Model with Multiple Time Delays

by Ailing Zhong and Chengqiang Wang

Entropy 2026, 28(7), 787; https://doi.org/10.3390/e28070787 - 12 Jul 2026

Abstract

Complex nonlinear dynamical systems, often associated with high-entropy time series, have been widely employed to describe and predict intricate dynamic phenomena in real-world systems. Motivated by the need to better understand such complex dynamics in network-based epidemic processes, this paper investigates bifurcation dynamics [...] Read more.

Complex nonlinear dynamical systems, often associated with high-entropy time series, have been widely employed to describe and predict intricate dynamic phenomena in real-world systems. Motivated by the need to better understand such complex dynamics in network-based epidemic processes, this paper investigates bifurcation dynamics in a fractional-order extension of the classical Susceptible–Latent–Breaking–Out model for computer virus propagation. The proposed framework incorporates two distinct transmission-related time delays and employs Caputo fractional derivatives of incommensurate orders, with the delays associated with infection rate and latent period selected as the primary bifurcation parameters. Due to the combined influence of multiple delays and incommensurate fractional exponents, the resulting system exhibits a complexity that goes beyond most existing models in the literature. By linearizing the model around its endemic equilibrium and analyzing the associated characteristic roots, we characterize how the system’s qualitative behavior depends on the magnitudes of the time delays, and establish explicit sufficient conditions for bifurcation to occur. In particular, the endemic equilibrium remains asymptotically stable as long as each delay stays below a certain critical value; once any delay exceeds its threshold, the system undergoes a Hopf bifurcation, leading to sustained periodic oscillations in virus prevalence. Numerical simulations are provided to support the analytical results, and they show strong agreement between predicted and observed system responses. These findings enhance theoretical insight into bifurcation mechanisms in fractional-order delay models of epidemic dynamics on networks, and may offer useful guidance for designing containment strategies in large-scale interconnected systems. Full article

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

► Show Figures

Figure 1

25 pages, 9417 KB

Open AccessArticle

Robust Sparse Underwater Acoustic Channel Estimation Using a Bidirectional Proportionate Recursive Maximum Correntropy Criterion Algorithm

by Xiao-Chen Chen, Guan-Quan Dai, Yang Shi and Fei-Yun Wu

Entropy 2026, 28(7), 786; https://doi.org/10.3390/e28070786 - 12 Jul 2026

Abstract

Aiming at the problem that sparse channel estimation in underwater acoustic communication is susceptible to complex multipath propagation, non-Gaussian impulsive noise, and channel time variations, this paper proposes a bidirectional proportionate recursive maximum correntropy criterion algorithm, referred to as Bi-PRMCC. By introducing a [...] Read more.

Aiming at the problem that sparse channel estimation in underwater acoustic communication is susceptible to complex multipath propagation, non-Gaussian impulsive noise, and channel time variations, this paper proposes a bidirectional proportionate recursive maximum correntropy criterion algorithm, referred to as Bi-PRMCC. By introducing a bidirectional filtering structure into the proportionate recursive maximum correntropy criterion (PRMCC) framework, the proposed algorithm jointly exploits the information from forward and backward data sequences, thereby improving the estimation accuracy and block-based channel variation tracking capability for sparse underwater acoustic channels. Meanwhile, the maximum correntropy criterion enhances the robustness of the algorithm against non-Gaussian impulsive noise and outlier error samples, while the proportionate update mechanism improves its identification capability for dominant taps in sparse channels. To verify the effectiveness of the proposed algorithm, short-range sparse underwater acoustic channels and long-range complex multipath underwater acoustic channels are constructed based on the Bellhop ray-tracing model. Simulation experiments are then conducted under three typical non-Gaussian noise environments, namely Cauchy noise,

α

-stable distribution noise, and Middleton noise. The experimental results show that, compared with recursive least squares (RLS), bidirectional recursive least squares (Bi-RLS), proportionate recursive least squares (PRLS), recursive maximum correntropy criterion (RMCC), and PRMCC, Bi-PRMCC achieves a lower steady-state normalized mean square deviation (NMSD) under different non-Gaussian noise conditions, indicating stronger robustness against impulsive noise. Under different signal-to-noise ratio conditions, the proposed algorithm still maintains superior steady-state estimation performance. In addition, in the channel abrupt-change tracking experiment, Bi-PRMCC can rapidly reconverge after channel variations occur, demonstrating favorable reconvergence capability under abrupt channel variations. The ablation study further verifies the stable performance gain brought by the bidirectional structure to PRMCC. Overall, the proposed Bi-PRMCC algorithm exhibits high estimation accuracy, robustness, and reconvergence capability under complex non-Gaussian noise and abrupt channel variation conditions. Full article

(This article belongs to the Section Signal and Data Analysis)

► Show Figures

Figure 1

22 pages, 1747 KB

Open AccessArticle

Structural Characteristics and Controllability Analysis of China’s Provincial-Industrial Embodied Carbon Emission Transfer Network

by Yixin Bao, Wenxia Chen, Chenhao Qian, Titi Zhang and Zidan Zhou

Entropy 2026, 28(7), 785; https://doi.org/10.3390/e28070785 - 11 Jul 2026

Abstract

In the context of global climate change and China’s “Dual Carbon” target, the misallocation of carbon emission reduction responsibilities and low regulatory efficiency urgently require analysis and resolution. Based on China’s 2020 MRIO and carbon emission inventory data, this study integrates multi-regional input–output [...] Read more.

In the context of global climate change and China’s “Dual Carbon” target, the misallocation of carbon emission reduction responsibilities and low regulatory efficiency urgently require analysis and resolution. Based on China’s 2020 MRIO and carbon emission inventory data, this study integrates multi-regional input–output models and complex network theory to construct an embodied carbon emission (ECE) transfer network at the provincial-industrial level and analyze its structural characteristics. Drawing on complex network control theory, this paper proposes a heuristic node-ranking strategy to identify driver nodes for full controllability of the ECE transfer network and compare its regulatory effect with other topological indicators. The findings reveal: (1) At the provincial level, embodied carbon emissions show a distinct transfer pattern from central provinces to southeast coastal or economically developed regions. Jiangxi, Anhui, Shandong, etc., are net outflow provinces, while Jiangsu, Beijing, Guangdong, etc., are net inflow provinces. (2) At the industrial level, secondary industry is the main net inflow industry, and primary industry is the main net outflow industry. The secondary industries in Guangdong, Henan, etc., have high betweenness centrality, acting as “hub” nodes for carbon transmission. Community detection shows that the largest community in China is centered on the secondary and tertiary industries of Jiangsu, Henan, Guangdong, etc., and the network overall exhibits small-world characteristics. (3) Compared with other control strategies, the designed algorithm achieves the best control effect: it realizes full network controllability with the minimum number of control nodes (26), and the shortest reachable paths from the control node set to non-control nodes, meaning policy signals imposed on control nodes transmit at the fastest speed. (4) Among the control node set, 22 key control nodes are mostly secondary and tertiary industries, located at the center of the transfer network and ranking high in net outflow or inflow, belonging to the core nodes of the ECE transfer network. This study provides a scientific basis and methodological support for clarifying the attribution of carbon transfer responsibilities and formulating differentiated collaborative regulatory policies. This paper establishes a qualitative matching mechanism between network control inputs and carbon tax, emission quotas and industrial regulation to connect controllability theory and practical carbon governance. Full article

(This article belongs to the Section Complexity)

► Show Figures

Figure 1

Journal Description

Entropy

Latest Articles

Journal Menu

Journal Browser

Highly Accessed Articles

Latest Books

E-Mail Alert

News

Topics

Conferences

Special Issues

Topical Collections

Further Information

Guidelines

MDPI Initiatives

Follow MDPI