Entropy

24 pages, 9086 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Linking Optimization Success and Stability of Finite-Time Thermodynamics Heat Engines

by Julian Gonzalez-Ayala, David Pérez-Gallego, Alejandro Medina, José M. Mateos Roco, Antonio Calvo Hernández, Santiago Velasco and Fernando Angulo-Brown

Entropy 2025, 27(8), 822; https://doi.org/10.3390/e27080822 - 2 Aug 2025

Viewed by 1348

Abstract

In celebration of 50 years of the endoreversible Carnot-like heat engine, this work aims to link the thermodynamic success of the irreversible Carnot-like heat engine with the stability dynamics of the engine. This region of success is defined by two extreme configurations in [...] Read more.

In celebration of 50 years of the endoreversible Carnot-like heat engine, this work aims to link the thermodynamic success of the irreversible Carnot-like heat engine with the stability dynamics of the engine. This region of success is defined by two extreme configurations in the interaction between heat reservoirs and the working fluid. The first corresponds to a fully reversible limit, and the second one is the fully dissipative limit; in between both limits, the heat exchange between reservoirs and working fluid produces irreversibilities and entropy generation. The distance between these two extremal configurations is minimized, independently of the chosen metric, in the state where the efficiency is half the Carnot efficiency. This boundary encloses the region where irreversibilities dominate or the reversible behavior dominates (region of success). A general stability dynamics is proposed based on the endoreversible nature of the model and the operation parameter in charge of defining the operation regime. For this purpose, the maximum ecological and maximum Omega regimes are considered. The results show that for single perturbations, the dynamics rapidly directs the system towards the success region, and under random perturbations producing stochastic trajectories, the system remains always in this region. The results are contrasted with the case in which no restitution dynamics exist. It is shown that stability allows the system to depart from the original steady state to other states that enhance the system’s performance, which could favor the evolution and specialization of systems in nature and in artificial devices. Full article

(This article belongs to the Special Issue The First Half Century of Finite-Time Thermodynamics)

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25 pages, 8472 KB

Open AccessEditor’s ChoiceArticle

Harnessing the Power of Pre-Trained Models for Efficient Semantic Communication of Text and Images

by Emrecan Kutay and Aylin Yener

Entropy 2025, 27(8), 813; https://doi.org/10.3390/e27080813 - 29 Jul 2025

Cited by 4 | Viewed by 2471

Abstract

This paper investigates point-to-point multimodal digital semantic communications in a task-oriented setup, where messages are classified at the receiver. We employ a pre-trained transformer model to extract semantic information and propose three methods for generating semantic codewords. First, we propose semantic quantization that [...] Read more.

This paper investigates point-to-point multimodal digital semantic communications in a task-oriented setup, where messages are classified at the receiver. We employ a pre-trained transformer model to extract semantic information and propose three methods for generating semantic codewords. First, we propose semantic quantization that uses quantized embeddings of source realizations as a codebook. We investigate the fixed-length coding, considering the source semantic structure and end-to-end semantic distortion. We propose a neural network-based codeword assignment mechanism incorporating codeword transition probabilities to minimize the expected semantic distortion. Second, we present semantic compression that clusters embeddings, exploiting the inherent semantic redundancies to reduce the codebook size, i.e., further compression. Third, we introduce a semantic vector-quantized autoencoder (VQ-AE) that learns a codebook through training. In all cases, we follow this semantic source code with a standard channel code to transmit over the wireless channel. In addition to classification accuracy, we assess pre-communication overhead via a novel metric we term system time efficiency. Extensive experiments demonstrate that our proposed semantic source-coding approaches provide comparable accuracy and better system time efficiency compared to their learning-based counterparts. Full article

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

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23 pages, 758 KB

Open AccessEditor’s ChoiceArticle

Low-Complexity Automorphism Ensemble Decoding of Reed-Muller Codes Using Path Pruning

by Kairui Tian, Rongke Liu and Zheng Lu

Entropy 2025, 27(8), 808; https://doi.org/10.3390/e27080808 - 28 Jul 2025

Viewed by 1553

Abstract

The newly developed automorphism ensemble decoder (AED) leverages the rich automorphisms of Reed–Muller (RM) codes to achieve near maximum likelihood (ML) performance at short code lengths. However, the performance gain of AED comes at the cost of high complexity, as the ensemble size [...] Read more.

The newly developed automorphism ensemble decoder (AED) leverages the rich automorphisms of Reed–Muller (RM) codes to achieve near maximum likelihood (ML) performance at short code lengths. However, the performance gain of AED comes at the cost of high complexity, as the ensemble size required for near ML decoding grows exponentially with the code length. In this work, we address this complexity issue by focusing on the factor graph permutation group (FGPG), a subgroup of the full automorphism group of RM codes, to generate permutations for AED. We propose a uniform partitioning of FGPG based on the affine bijection permutation matrices of automorphisms, where each subgroup of FGPG exhibits permutation invariance (PI) in a Plotkin construction-based information set partitioning for RM codes. Furthermore, from the perspective of polar codes, we exploit the PI property to prove a subcode estimate convergence (SEC) phenomenon in the AED that utilizes successive cancellation (SC) or SC list (SCL) constituent decoders. Observing that strong SEC correlates with low noise levels, where the full decoding capacity of AED is often unnecessary, we perform path pruning to reduce the decoding complexity without compromising the performance. Our proposed SEC-aided path pruning allows only a subset of constituent decoders to continue decoding when the intensity of SEC exceeds a preset threshold during decoding. Numerical results demonstrate that, for the FGPG-based AED of various short RM codes, the proposed SEC-aided path pruning technique incurs negligible performance degradation, while achieving a complexity reduction of up to 67.6%. Full article

(This article belongs to the Special Issue Next-Generation Channel Coding: Theory and Applications)

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24 pages, 4106 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Visualizing Three-Qubit Entanglement

by Alfred Benedito and Germán Sierra

Entropy 2025, 27(8), 800; https://doi.org/10.3390/e27080800 - 27 Jul 2025

Viewed by 1282

Abstract

We present a graphical framework to represent entanglement in three-qubit states. The geometry associated with each entanglement class and type is analyzed, revealing distinct structural features. We explore the connection between this geometric perspective and the tangle, deriving bounds that depend on the [...] Read more.

We present a graphical framework to represent entanglement in three-qubit states. The geometry associated with each entanglement class and type is analyzed, revealing distinct structural features. We explore the connection between this geometric perspective and the tangle, deriving bounds that depend on the entanglement class. Based on these insights, we conjecture a purely geometric expression for both the tangle and Cayley’s hyperdeterminant for non-generic states. As an application, we analyze the energy eigenstates of physical Hamiltonians, identifying the sufficient conditions for genuine tripartite entanglement to be robust under symmetry-breaking perturbations and level repulsion effects. Full article

(This article belongs to the Special Issue Editorial Board Members' Collection Series on Quantum Entanglement)

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33 pages, 1024 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Graph-Theoretic Limits of Distributed Computation: Entropy, Eigenvalues, and Chromatic Numbers

by Mohammad Reza Deylam Salehi and Derya Malak

Entropy 2025, 27(7), 757; https://doi.org/10.3390/e27070757 - 15 Jul 2025

Cited by 1 | Viewed by 1834

Abstract

We address the problem of the distributed computation of arbitrary functions of two correlated sources,

X_{1}

and

X_{2}

, residing in two distributed source nodes, respectively. We exploit the structure of a computation task by coding source characteristic graphs (and multiple [...] Read more.

We address the problem of the distributed computation of arbitrary functions of two correlated sources,

X_{1}

and

X_{2}

, residing in two distributed source nodes, respectively. We exploit the structure of a computation task by coding source characteristic graphs (and multiple instances using the n-fold OR product of this graph with itself). For regular graphs and general graphs, we establish bounds on the optimal rate—characterized by the chromatic entropy for the n-fold graph products—that allows a receiver for asymptotically lossless computation of arbitrary functions over finite fields. For the special class of cycle graphs (i.e., 2-regular graphs), we establish an exact characterization of chromatic numbers and derive bounds on the required rates. Next, focusing on the more general class of d-regular graphs, we establish connections between d-regular graphs and expansion rates for n-fold graph products using graph spectra. Finally, for general graphs, we leverage the Gershgorin Circle Theorem (GCT) to provide a characterization of the spectra, which allows us to derive new bounds on the optimal rate. Our codes leverage the spectra of the computation and provide a graph expansion-based characterization to succinctly capture the computation structure, providing new insights into the problem of distributed computation of arbitrary functions. Full article

(This article belongs to the Special Issue Information Theory and Data Compression)

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15 pages, 5676 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Transverse Self-Propulsion Enhances the Aggregation of Active Dumbbells

by Pasquale Digregorio, Claudio Basilio Caporusso, Lucio Mauro Carenza, Giuseppe Gonnella, Daniela Moretti, Giuseppe Negro, Massimiliano Semeraro and Antonio Suma

Entropy 2025, 27(7), 692; https://doi.org/10.3390/e27070692 - 27 Jun 2025

Cited by 1 | Viewed by 1544

Abstract

We investigate a two-dimensional system of active Brownian dumbbells using molecular dynamics simulations. In this model, each dumbbell is driven by an active force oriented perpendicular to the axis connecting its two constituent beads. We characterize the resulting phase behavior and find that, [...] Read more.

We investigate a two-dimensional system of active Brownian dumbbells using molecular dynamics simulations. In this model, each dumbbell is driven by an active force oriented perpendicular to the axis connecting its two constituent beads. We characterize the resulting phase behavior and find that, across all values of activity, the system undergoes phase separation between dilute and dense phases. The dense phase exhibits hexatic order, and for large enough activity, we observe a marked increase in local polarization, with dumbbells predominantly oriented towards the interior of the clusters. Compared to the case of axially self-propelled dumbbells, we find that the binodal region is enlarged towards lower densities at all activities. This shift arises because dumbbells with transverse propulsion can more easily form stable cluster cores, serving as nucleation seeds, and show a highly suppressed escaping rate from the cluster boundary. Finally, we observe that clusters exhibit spontaneous rotation, with the modulus of the angular velocity scaling as

ω \sim r_{g}^{- 2}

, where

r_{g}

is the cluster’s radius of gyration. This contrasts with axially propelled dumbbells, where the scaling follows

ω \sim r_{g}^{- 1}

. We develop a simplified analytical model to rationalize this scaling behavior. Full article

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

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14 pages, 4151 KB

Open AccessEditor’s ChoiceArticle

Fisher Information and the Dynamics of Multicellular Ageing

by Zachary F. Hale, Gonzalo A. Cánez and Thomas C. T. Michaels

Entropy 2025, 27(6), 638; https://doi.org/10.3390/e27060638 - 15 Jun 2025

Viewed by 2172

Abstract

Information theory has long been integrated into the study of biological ageing, for example, in examining the roles of genetic and epigenetic fidelity in cellular and organismal longevity. Here, we introduce a theoretical model that interprets ageing in multicellular systems through the lens [...] Read more.

Information theory has long been integrated into the study of biological ageing, for example, in examining the roles of genetic and epigenetic fidelity in cellular and organismal longevity. Here, we introduce a theoretical model that interprets ageing in multicellular systems through the lens of Fisher information. Previous theories have suggested that the ageing of multicellular organisms is an inevitable consequence of the inherent tension between individual cell reproduction and the homeostasis of the multicellular system. Utilising concepts from information theory and statistical mechanics, we show that Fisher information parametrises the dynamics of this tension through non-monotonic behaviour, which depends on an optimal balance of competition and cooperation between cells. Moreover, Fisher information suggests that the ability to infer true biological age from a sample evolves through complex dynamics over an organism’s lifespan. Full article

(This article belongs to the Collection Disorder and Biological Physics)

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24 pages, 5959 KB

Open AccessEditor’s ChoiceArticle

An Information Geometry-Based Track-Before-Detect Algorithm for Range-Azimuth Measurements in Radar Systems

by Jinguo Liu, Hao Wu, Zheng Yang, Xiaoqiang Hua and Yongqiang Cheng

Entropy 2025, 27(6), 637; https://doi.org/10.3390/e27060637 - 14 Jun 2025

Cited by 3 | Viewed by 2551

Abstract

The detection of weak moving targets in heterogeneous clutter backgrounds is a significant challenge in radar systems. In this paper, we propose a track-before-detect (TBD) method based on information geometry (IG) theory applied to range-azimuth measurements, which extends the IG detectors to multi-frame [...] Read more.

The detection of weak moving targets in heterogeneous clutter backgrounds is a significant challenge in radar systems. In this paper, we propose a track-before-detect (TBD) method based on information geometry (IG) theory applied to range-azimuth measurements, which extends the IG detectors to multi-frame detection through inter-frame information integration. The approach capitalizes on the distinctive benefits of the information geometry detection framework in scenarios with strong clutter, while enhancing the integration of information across multiple frames within the TBD approach. Specifically, target and clutter trajectories in multi-frame range-azimuth measurements are modeled on the Hermitian positive definite (HPD) and power spectrum (PS) manifolds. A scoring function based on information geometry, which uses Kullback–Leibler (KL) divergence as a geometric metric, is then devised to assess these motion trajectories. Moreover, this study devises a solution framework employing dynamic programming (DP) with constraints on state transitions, culminating in an integrated merit function. This algorithm identifies target trajectories by maximizing the integrated merit function. Experimental validation using real-recorded sea clutter datasets showcases the effectiveness of the proposed algorithm, yielding a minimum 3 dB enhancement in signal-to-clutter ratio (SCR) compared to traditional approaches. Full article

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

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14 pages, 1641 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Measurement-Induced Dynamical Quantum Thermalization

by Marvin Lenk, Sayak Biswas, Anna Posazhennikova and Johann Kroha

Entropy 2025, 27(6), 636; https://doi.org/10.3390/e27060636 - 14 Jun 2025

Viewed by 1285

Abstract

One of the fundamental problems of quantum statistical physics is how an ideally isolated quantum system can ever reach thermal equilibrium behavior despite the unitary time evolution of quantum-mechanical systems. Here, we study, via explicit time evolution for the generic model system of [...] Read more.

One of the fundamental problems of quantum statistical physics is how an ideally isolated quantum system can ever reach thermal equilibrium behavior despite the unitary time evolution of quantum-mechanical systems. Here, we study, via explicit time evolution for the generic model system of an interacting, trapped Bose gas with discrete single-particle levels, how the measurement of one or more observables subdivides the system into observed and non-observed Hilbert subspaces and the tracing over the non-measured quantum numbers defines an effective, thermodynamic bath, induces the entanglement of the observed Hilbert subspace with the bath, and leads to a bi-exponential approach of the entanglement entropy and of the measured observables to thermal equilibrium behavior as a function of time. We find this to be more generally fulfilled than in the scenario of the eigenstate thermalization hypothesis (ETH), namely for both local particle occupation numbers and non-local density correlation functions, and independent of the specific initial quantum state of the time evolution. Full article

(This article belongs to the Special Issue Non-Equilibrium Dynamics in Ultra-Cold Quantum Gases)

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13 pages, 820 KB

Open AccessFeature PaperEditor’s ChoiceArticle

An Efficient Algorithmic Way to Construct Boltzmann Machine Representations for Arbitrary Stabilizer Code

by Yuan-Hang Zhang, Zhian Jia, Yu-Chun Wu and Guang-Can Guo

Entropy 2025, 27(6), 627; https://doi.org/10.3390/e27060627 - 13 Jun 2025

Cited by 1 | Viewed by 1452

Abstract

Restricted Boltzmann machines (RBMs) have demonstrated considerable success as variational quantum states; however, their representational power remains incompletely understood. In this work, we present an analytical proof that RBMs can exactly and efficiently represent stabilizer code states—a class of highly entangled quantum states [...] Read more.

Restricted Boltzmann machines (RBMs) have demonstrated considerable success as variational quantum states; however, their representational power remains incompletely understood. In this work, we present an analytical proof that RBMs can exactly and efficiently represent stabilizer code states—a class of highly entangled quantum states that are central to quantum error correction. Given a set of stabilizer generators, we develop an efficient algorithm to determine both the RBM architecture and the exact values of its parameters. Our findings provide new insights into the expressive power of RBMs, highlighting their capability to encode highly entangled states, and may serve as a useful tool for the classical simulation of quantum error-correcting codes. Full article

(This article belongs to the Special Issue Quantum Information and Quantum Computation)

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29 pages, 3108 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Soft Classification in a Composite Source Model

by Yuefeng Cao, Jiakun Liu and Wenyi Zhang

Entropy 2025, 27(6), 620; https://doi.org/10.3390/e27060620 - 11 Jun 2025

Cited by 1 | Viewed by 1287

Abstract

A composite source model consists of an intrinsic state and an extrinsic observation. The fundamental performance limit of reproducing the intrinsic state is characterized by the indirect rate–distortion function. In a remote classification application, a source encoder encodes the extrinsic observation (e.g., image) [...] Read more.

A composite source model consists of an intrinsic state and an extrinsic observation. The fundamental performance limit of reproducing the intrinsic state is characterized by the indirect rate–distortion function. In a remote classification application, a source encoder encodes the extrinsic observation (e.g., image) into bits, and a source decoder plays the role of a classifier that reproduces the intrinsic state (e.g., label of image). In this work, we characterize the general structure of the optimal transition probability distribution, achieving the indirect rate–distortion function. This optimal solution can be interpreted as a “soft classifier”, which generalizes the conventionally adopted “classify-then-compress” scheme. We then apply the soft classification to aid the lossy compression of the extrinsic observation of a composite source. This leads to a coding scheme that exploits the soft classifier to guide reproduction, outperforming existing coding schemes without classification or with hard classification. Full article

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

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15 pages, 419 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Ordinal Random Processes

by Christoph Bandt

Entropy 2025, 27(6), 610; https://doi.org/10.3390/e27060610 - 7 Jun 2025

Cited by 2 | Viewed by 1275

Abstract

Ordinal patterns have proven to be a valuable tool in many fields. Here, we address the need for theoretical models. A paradigmatic example shows that a model for frequencies of ordinal patterns can be determined without any numerical values. We specify the important [...] Read more.

Ordinal patterns have proven to be a valuable tool in many fields. Here, we address the need for theoretical models. A paradigmatic example shows that a model for frequencies of ordinal patterns can be determined without any numerical values. We specify the important concept of stationary order and the fundamental problems to be solved in order to establish a genuine statistical methodology for ordinal time series. Full article

(This article belongs to the Special Issue Ordinal Patterns-Based Tools and Their Applications)

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8 pages, 379 KB

Open AccessEditor’s ChoiceArticle

Scaling Laws in Language Families

by Maelyson Rolim Fonseca dos Santos and Marcelo Andrade de Filgueiras Gomes

Entropy 2025, 27(6), 588; https://doi.org/10.3390/e27060588 - 31 May 2025

Cited by 1 | Viewed by 1498

Abstract

This article investigates scaling laws within language families using data from over six thousand languages and analyzes emergent patterns observed in Zipf-like classification graphs. Both macroscopic (based on the number of languages by family) and microscopic (based on the number of speakers by [...] Read more.

This article investigates scaling laws within language families using data from over six thousand languages and analyzes emergent patterns observed in Zipf-like classification graphs. Both macroscopic (based on the number of languages by family) and microscopic (based on the number of speakers by language within a family) aspects of these classifications are examined. Particularly noteworthy is the discovery of a distinct division among the fourteen largest contemporary language families, excluding Afro-Asiatic and Nilo-Saharan languages. These families are found to be distributed across three language family quadruplets, each characterized by significantly different exponents in the Zipf graphs. This finding sheds light on the underlying structure and organization of major language families, revealing intriguing insights into the nature of linguistic diversity and distribution. Full article

(This article belongs to the Special Issue Complexity Characteristics of Natural Language)

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33 pages, 3843 KB

Open AccessEditor’s ChoiceArticle

Deep Hybrid Models: Infer and Plan in a Dynamic World

by Matteo Priorelli and Ivilin Peev Stoianov

Entropy 2025, 27(6), 570; https://doi.org/10.3390/e27060570 - 27 May 2025

Cited by 2 | Viewed by 2095

Abstract

To determine an optimal plan for complex tasks, one often deals with dynamic and hierarchical relationships between several entities. Traditionally, such problems are tackled with optimal control, which relies on the optimization of cost functions; instead, a recent biologically motivated proposal casts planning [...] Read more.

To determine an optimal plan for complex tasks, one often deals with dynamic and hierarchical relationships between several entities. Traditionally, such problems are tackled with optimal control, which relies on the optimization of cost functions; instead, a recent biologically motivated proposal casts planning and control as an inference process. Active inference assumes that action and perception are two complementary aspects of life whereby the role of the former is to fulfill the predictions inferred by the latter. Here, we present an active inference approach that exploits discrete and continuous processing, based on three features: the representation of potential body configurations in relation to the objects of interest; the use of hierarchical relationships that enable the agent to easily interpret and flexibly expand its body schema for tool use; the definition of potential trajectories related to the agent’s intentions, used to infer and plan with dynamic elements at different temporal scales. We evaluate this deep hybrid model on a habitual task: reaching a moving object after having picked a moving tool. We show that the model can tackle the presented task under different conditions. This study extends past work on planning as inference and advances an alternative direction to optimal control. Full article

(This article belongs to the Special Issue Active Inference in Cognitive Neuroscience)

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23 pages, 2439 KB

Open AccessEditor’s ChoiceArticle

The Origin of Shared Emergent Properties in Discrete Systems

by Les Hatton and Greg Warr

Entropy 2025, 27(6), 561; https://doi.org/10.3390/e27060561 - 26 May 2025

Cited by 2 | Viewed by 1855

Abstract

Here, we propose that the shared emergent properties reproducibly observed in discrete systems can be explained by a theory that embeds the Conservation of Hartley–Shannon Information (CoHSI) in a statistical mechanics framework. Specific predictions of global properties that represent the most likely equilibrium [...] Read more.

Here, we propose that the shared emergent properties reproducibly observed in discrete systems can be explained by a theory that embeds the Conservation of Hartley–Shannon Information (CoHSI) in a statistical mechanics framework. Specific predictions of global properties that represent the most likely equilibrium state should be apparent in all qualifying systems, regardless of provenance. We demonstrate that these predictions of emergent global properties hold true in systems as disparate as collections of software written in the programming language C and collections of proteins. The implication is that the emergence of such shared properties is not driven by any specific local mechanism as the systems are so different. This raises the interesting prospect that important properties of biological systems (exemplified here by the length and multiplicity distributions of proteins) have little, if anything, to do with natural selection. Similarly, the size distribution of components and the frequency of tokens observed in computer software in C emerge as the most likely states, and are thus properties that are divorced from human agency, regardless of functionality. Full article

(This article belongs to the Section Entropy and Biology)

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14 pages, 262 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Universal Encryption of Individual Sequences Under Maximal Information Leakage

by Neri Merhav

Entropy 2025, 27(6), 551; https://doi.org/10.3390/e27060551 - 24 May 2025

Viewed by 913

Abstract

We consider the Shannon cipher system in the framework of individual sequences and finite-state encrypters under the metric of maximal information leakage. A lower bound and an asymptotically matching upper bound on the leakage are derived, which lead to the conclusion that asymptotically [...] Read more.

We consider the Shannon cipher system in the framework of individual sequences and finite-state encrypters under the metric of maximal information leakage. A lower bound and an asymptotically matching upper bound on the leakage are derived, which lead to the conclusion that asymptotically minimum leakage can be attained by Lempel–Ziv compression followed by one-time pad encryption of the compressed bitstream. Full article

(This article belongs to the Special Issue Information Theory and Data Compression)

13 pages, 1060 KB

Open AccessEditor’s ChoiceArticle

A Two-State Random Walk Model of Sperm Search on Confined Domains

by Martin Bier, Maciej Majka and Cameron Schmidt

Entropy 2025, 27(5), 539; https://doi.org/10.3390/e27050539 - 19 May 2025

Cited by 1 | Viewed by 1552

Abstract

Mammalian fertilization depends on sperm successfully navigating a spatially and chemically complex microenvironment in the female reproductive tract. This process is often conceptualized as a competitive race, but is better understood as a collective random search. Sperm within an ejaculate exhibit a diverse [...] Read more.

Mammalian fertilization depends on sperm successfully navigating a spatially and chemically complex microenvironment in the female reproductive tract. This process is often conceptualized as a competitive race, but is better understood as a collective random search. Sperm within an ejaculate exhibit a diverse distribution of motility patterns, with some moving in relatively straight lines and others following tightly turning trajectories. Here, we present a two-state random walk model in which sperm switch from high-persistence-length to low-persistence-length motility modes. In reproductive biology, such a switch is often recognized as “hyperactivation”. We study a circularly symmetric setup with sperm emerging at the center and searching a finite-area disk. We explore the implications of switching on search efficiency. The first proposed model describes an adaptive search strategy in which sperm achieve improved spatial coverage without cell-to-cell or environment-to-cell communication. The second model that we study adds a small amount of environment-to-cell communication. The models resemble macroscopic search-and-rescue tactics, but without organization or networked communication. Our findings provide a quantitative framework linking sperm motility patterns to efficient search strategies, offering insights into sperm physiology and the stochastic search dynamics of self-propelled particles. 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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22 pages, 287 KB

Open AccessEditor’s ChoiceArticle

Two-Step Estimation Procedure for Parametric Copula-Based Regression Models for Semi-Competing Risks Data

by Qingmin Zhang, Bowen Duan, Małgorzata Wojtyś and Yinghui Wei

Entropy 2025, 27(5), 521; https://doi.org/10.3390/e27050521 - 13 May 2025

Cited by 1 | Viewed by 1846

Abstract

Non-terminal and terminal events in semi-competing risks data are typically associated and may be influenced by covariates. We employed regression modeling for semi-competing risks data under a copula-based framework to evaluate the effects of covariates on the two events and the association between [...] Read more.

Non-terminal and terminal events in semi-competing risks data are typically associated and may be influenced by covariates. We employed regression modeling for semi-competing risks data under a copula-based framework to evaluate the effects of covariates on the two events and the association between them. Due to the complexity of the copula structure, we propose a new method that integrates a novel two-step algorithm with the Bound Optimization by Quadratic Approximation (BOBYQA) method. This approach effectively mitigates the influence of initial values and demonstrates greater robustness. The simulations validate the performance of the proposed method. We further applied our proposed method to the Amsterdam Cohort Study (ACS) real data, where some improvements could be found. Full article

10 pages, 8363 KB

Open AccessEditor’s ChoiceArticle

Improved Reconstruction of Chaotic Signals from Ordinal Networks

by Antonio Politi and Leonardo Ricci

Entropy 2025, 27(5), 499; https://doi.org/10.3390/e27050499 - 6 May 2025

Viewed by 1165

Abstract

Permutation entropy is customarily implemented to quantify the intrinsic indeterminacy of complex time series, under the assumption that determinism manifests itself by lowering the (permutation) entropy of the resulting symbolic sequence. We expect this to be roughly true, but, in general, it is [...] Read more.

Permutation entropy is customarily implemented to quantify the intrinsic indeterminacy of complex time series, under the assumption that determinism manifests itself by lowering the (permutation) entropy of the resulting symbolic sequence. We expect this to be roughly true, but, in general, it is not clear to what extent a given ordinal pattern indeed provides a faithful reconstruction of the original signal. Here, we address this question by attempting the reconstruction of the original time series by invoking an ergodic Markov approximation of the symbolic dynamics, thereby inverting the encoding procedure. Using the Hénon map as a testbed, we show that a meaningful reconstruction can also be made in the presence of a small observational noise. Full article

(This article belongs to the Special Issue Ordinal Patterns-Based Tools and Their Applications)

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14 pages, 617 KB

Open AccessEditor’s ChoiceArticle

Iterative Forecasting of Financial Time Series: The Greek Stock Market from 2019 to 2024

by Evangelos Bakalis and Francesco Zerbetto

Entropy 2025, 27(5), 497; https://doi.org/10.3390/e27050497 - 4 May 2025

Cited by 3 | Viewed by 3153

Abstract

Predicting the evolution of financial data, if at all possible, would be very beneficial in revealing the ways in which different aspects of a global environment can impact local economies. We employ an iterative stochastic differential equation that accurately forecasts an economic time [...] Read more.

Predicting the evolution of financial data, if at all possible, would be very beneficial in revealing the ways in which different aspects of a global environment can impact local economies. We employ an iterative stochastic differential equation that accurately forecasts an economic time series’s next value by analysing its past. The input financial data are assumed to be consistent with an

α

-stable Lévy motion. The computation of the scaling exponent and the value of

α

, which characterises the type of the

α

-stable Lévy motion, are crucial for the iterative scheme. These two indices can be determined at each iteration from the form of the structure function, for the computation of which we use the method of generalised moments. Their values are used for the creation of the corresponding

α

-stable Lévy noise, which acts as a seed for the stochastic component. Furthermore, the drift and diffusion terms are calculated at each iteration. The proposed model is general, allowing the kind of stochastic process to vary from one iterative step to another, and its applicability is not restricted to financial data. As a case study, we consider Greece’s stock market general index over a period of five years, from September 2019 to September 2024, after the completion of bailout programmes. Greece’s economy changed from a restricted to a free market over the chosen era, and its stock market trading increments are likely to be describable by an

α

-stable L’evy motion. We find that

α = 2

and the scaling exponent H varies over time for every iterative step we perform. The forecasting points follow the same trend, are in good agreement with the actual data, and for most of the forecasts, the percentage error is less than 2%. Full article

(This article belongs to the Special Issue Entropy-Based Applications in Sociophysics II)

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12 pages, 526 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Time Dilation of Quantum Clocks in a Relativistic Gravitational Potential

by Tommaso Favalli and Augusto Smerzi

Entropy 2025, 27(5), 489; https://doi.org/10.3390/e27050489 - 1 May 2025

Cited by 4 | Viewed by 4635

Abstract

We study the dynamical evolution of two quantum clocks interacting with a relativistic gravitational potential. We find a time dilation effect for the clocks in agreement with the gravitational time dilation as obtained from the Schwarzschild solution in General Relativity. We perform our [...] Read more.

We study the dynamical evolution of two quantum clocks interacting with a relativistic gravitational potential. We find a time dilation effect for the clocks in agreement with the gravitational time dilation as obtained from the Schwarzschild solution in General Relativity. We perform our investigation via the Page and Wootters quantum-time formalism, exploring the dynamics of clocks assuming them in both a product state and a more general (entangled) state. The gravitational redshift, as emerging from our framework, is also proposed and discussed. Full article

(This article belongs to the Special Issue Time, Change, Observables, and Quantum Gravity)

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17 pages, 722 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Thermal Fisher Information for a Rotating BTZ Black Hole

by Everett A. Patterson and Robert B. Mann

Entropy 2025, 27(5), 478; https://doi.org/10.3390/e27050478 - 28 Apr 2025

Viewed by 1903

Abstract

Relativistic quantum metrology provides a framework within which we can quantify the quality of measurement and estimation procedures while accounting for both quantum and relativistic effects. The chief measure for describing such procedures is the Fisher information, which quantifies how sensitive a given [...] Read more.

Relativistic quantum metrology provides a framework within which we can quantify the quality of measurement and estimation procedures while accounting for both quantum and relativistic effects. The chief measure for describing such procedures is the Fisher information, which quantifies how sensitive a given estimation is to a variance in some underlying parameter. Recently, the Fisher information has been used to quantify the spacetime information accessible to two-level quantum particle detectors. We have previously shown that such a system is capable of discerning black hole mass for static black holes in 2 + 1 dimensions. Here, we extend these results to the astrophysically interesting case of rotating black holes and show that the Fisher information is also sensitive to the rotation of a black hole. Full article

(This article belongs to the Special Issue Applications of Fisher Information in Sciences II)

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37 pages, 996 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Kolmogorov Capacity with Overlap

by Anshuka Rangi and Massimo Franceschetti

Entropy 2025, 27(5), 472; https://doi.org/10.3390/e27050472 - 27 Apr 2025

Cited by 1 | Viewed by 1413

Abstract

The notion of

δ

-mutual information between non-stochastic uncertain variables is introduced as a generalization of Nair’s non-stochastic information functional. Several properties of this new quantity are illustrated and used in a communication setting to show that the largest

δ

-mutual information between [...] Read more.

The notion of

δ

-mutual information between non-stochastic uncertain variables is introduced as a generalization of Nair’s non-stochastic information functional. Several properties of this new quantity are illustrated and used in a communication setting to show that the largest

δ

-mutual information between received and transmitted codewords over

ϵ

-noise channels equals the

(ϵ, δ)

-capacity. This notion of capacity generalizes the Kolmogorov

ϵ

-capacity to packing sets of overlap at most

δ

and is a variation of a previous definition proposed by one of the authors. Results are then extended to more general noise models, including non-stochastic, memoryless, and stationary channels. The presented theory admits the possibility of decoding errors, as in classical information theory, while retaining the worst-case, non-stochastic character of Kolmogorov’s approach. Full article

(This article belongs to the Collection Feature Papers in Information Theory)

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50 pages, 2918 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Classical Data in Quantum Machine Learning Algorithms: Amplitude Encoding and the Relation Between Entropy and Linguistic Ambiguity

by Jurek Eisinger, Ward Gauderis, Lin de Huybrecht and Geraint A. Wiggins

Entropy 2025, 27(4), 433; https://doi.org/10.3390/e27040433 - 16 Apr 2025

Cited by 7 | Viewed by 4934

Abstract

The Categorical Compositional Distributional (DisCoCat) model has been proven to be very successful in modelling sentence meaning as the interaction of word meanings. Words are modelled as quantum states, interacting guided by grammar. This model of language has been extended to density matrices [...] Read more.

The Categorical Compositional Distributional (DisCoCat) model has been proven to be very successful in modelling sentence meaning as the interaction of word meanings. Words are modelled as quantum states, interacting guided by grammar. This model of language has been extended to density matrices to account for ambiguity in language. Density matrices describe probability distributions over quantum states, and in this work we relate the mixedness of density matrices to ambiguity in the sentences they represent. The von Neumann entropy and the fidelity are used as measures of this mixedness. Via the process of amplitude encoding, we introduce classical data into quantum machine learning algorithms. First, the findings suggest that in quantum natural language processing, amplitude-encoding data onto a quantum computer can be a useful tool to improve the performance of the quantum machine learning models used. Second, the effect that these encoded data have on the above-introduced relation between entropy and ambiguity is investigated. We conclude that amplitude-encoding classical data in quantum machine learning algorithms makes the relation between the entropy of a density matrix and ambiguity in the sentence modelled by this density matrix much more intuitively interpretable. Full article

(This article belongs to the Collection Feature Papers in Information Theory)

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17 pages, 9977 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Statistical Properties of Correlated Semiclassical Bands in Tight-Binding Small-World Networks

by Natalya Almazova, Giorgos P. Tsironis and Efthimios Kaxiras

Entropy 2025, 27(4), 420; https://doi.org/10.3390/e27040420 - 12 Apr 2025

Viewed by 900

Abstract

Linear tight-binding models with long-range interactions and small-world geometry have a broad energy spectrum in the nearest neighbor coupling limit, while the spectrum becomes narrow in the fully connected limit due to the emergence of flat bands. A transition to a Wigner-like density [...] Read more.

Linear tight-binding models with long-range interactions and small-world geometry have a broad energy spectrum in the nearest neighbor coupling limit, while the spectrum becomes narrow in the fully connected limit due to the emergence of flat bands. A transition to a Wigner-like density of states appears at a low fraction of long-range bonds. Adding nonlinearity to the model introduces correlations among the stationary states, while multiple new states are generated as a result of the nonlinearity. In this work, we study the effect of band correlations on the local density of states for small-world networks as a function of the number of long-range bonds. We find that close to the nearest neighbor limit, the onset of correlations shifts the nonlinear density of states towards the band edge of the spectrum. Close to the opposite limit of the fully connected model, the band collapses in the band center, accompanied by a large increase in the new states induced by the nonlinearity. While in both limits the effect of correlations is to flatten the band, close to the mean field fully connected limit, the states are correlated and generally have distinct localized features. These effects may have implications for the dynamics of electrons in two-dimensional moiré structures and the onset of superconductivity in these systems. Full article

(This article belongs to the Special Issue New Challenges in Contemporary Statistical Physics)

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24 pages, 1419 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Measurement-Induced Symmetry Restoration and Quantum Mpemba Effect

by Giuseppe Di Giulio, Xhek Turkeshi and Sara Murciano

Entropy 2025, 27(4), 407; https://doi.org/10.3390/e27040407 - 10 Apr 2025

Cited by 20 | Viewed by 3817

Abstract

Monitoring a quantum system can profoundly alter its dynamical properties, leading to non-trivial emergent phenomena. In this work, we demonstrate that dynamical measurements strongly influence the evolution of symmetry in many-body quantum systems. Specifically, we demonstrate that monitored systems governed by non-Hermitian dynamics [...] Read more.

Monitoring a quantum system can profoundly alter its dynamical properties, leading to non-trivial emergent phenomena. In this work, we demonstrate that dynamical measurements strongly influence the evolution of symmetry in many-body quantum systems. Specifically, we demonstrate that monitored systems governed by non-Hermitian dynamics exhibit a quantum Mpemba effect, where systems with stronger initial asymmetry relax faster to a symmetric state. Crucially, this phenomenon is purely measurement-induced: in the absence of measurements, we find states where the corresponding unitary evolution does not display any Mpemba effect. Furthermore, we uncover a novel measurement-induced symmetry restoration mechanism: below a critical measurement rate, the symmetry remains broken, but beyond a threshold, it is fully restored in the thermodynamic limit—along with the emergence of the quantum Mpemba effect. Full article

(This article belongs to the Special Issue Entanglement Entropy in Quantum Field Theory)

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21 pages, 375 KB

Open AccessEditor’s ChoiceArticle

Weak Identification Robust Tests for Subvectors Using Implied Probabilities

by Marine Carrasco and Saraswata Chaudhuri

Entropy 2025, 27(4), 396; https://doi.org/10.3390/e27040396 - 8 Apr 2025

Viewed by 1443

Abstract

This paper develops tests for hypotheses concerning subvectors of parameters in models defined by moment conditions. It is well known that conventional tests such as Wald, Likelihood-ratio and Score tests tend to over-reject when the identification is weak. To prevent uncontrolled size distortion [...] Read more.

This paper develops tests for hypotheses concerning subvectors of parameters in models defined by moment conditions. It is well known that conventional tests such as Wald, Likelihood-ratio and Score tests tend to over-reject when the identification is weak. To prevent uncontrolled size distortion and introduce refined finite-sample performance, we extend the projection-based test to a modified version of the score test using implied probabilities obtained by information theoretic criteria. Our test is performed in two steps, where the first step reduces the space of parameter candidates, while the second one involves the modified score test mentioned earlier. We derive the asymptotic properties of this procedure for the entire class of Generalized Empirical Likelihood implied probabilities. Simulations show that the test has very good finite-sample size and power. Finally, we apply our approach to the veteran earnings and find a negative impact of the veteran status. Full article

(This article belongs to the Special Issue Maximum Entropy Principle and Applications)

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27 pages, 15276 KB

Open AccessEditor’s ChoiceArticle

The Dynamics of Shannon Entropy in Analyzing Climate Variability for Modeling Temperature and Precipitation Uncertainty in Poland

by Bernard Twaróg

Entropy 2025, 27(4), 398; https://doi.org/10.3390/e27040398 - 8 Apr 2025

Cited by 7 | Viewed by 3796

Abstract

The aim of this study is to quantitatively analyze the long-term climate variability in Poland during the period 1901–2010, using Shannon entropy as a measure of uncertainty and complexity within the atmospheric system. The analysis is based on the premise that variations in [...] Read more.

The aim of this study is to quantitatively analyze the long-term climate variability in Poland during the period 1901–2010, using Shannon entropy as a measure of uncertainty and complexity within the atmospheric system. The analysis is based on the premise that variations in temperature and precipitation reflect the dynamic nature of the climate, understood as a nonlinear system sensitive to fluctuations. This study focuses on monthly distributions of temperature and precipitation, modeled using the bivariate Clayton copula function. A normal marginal distribution was adopted for temperature and a gamma distribution for precipitation, both validated using the Anderson–Darling test. To improve estimation accuracy, a bootstrap resampling technique and numerical integration were applied to calculate Shannon entropy at each of the 396 grid points, with a spatial resolution of 0.25° × 0.25°. The results indicate a significant increase in Shannon entropy during the summer months, particularly in July (+0.203 bits) and January (+0.221 bits), compared to the baseline period (1901–1971), suggesting a growing unpredictability of the climate. The most pronounced trend changes were identified in the years 1985–1996 (as indicated by the Pettitt test), while seasonal trends were confirmed using the Mann–Kendall test. A spatial analysis of entropy at the levels of administrative regions and catchments revealed notable regional disparities—entropy peaked in January in the West Pomeranian Voivodeship (4.919 bits) and reached its minimum in April in Greater Poland (3.753 bits). Additionally, this study examined the relationship between Shannon entropy and global climatic indicators, including the Land–Ocean Temperature Index (NASA GISTEMP) and the ENSO index (NINO3.4). Statistically significant positive correlations were observed between entropy and global temperature anomalies during both winter (

ρ

= 0.826) and summer (

ρ

= 0.650), indicating potential linkages between local climate variability and global warming trends. To explore the direction of this relationship, a Granger causality test was conducted, which did not reveal statistically significant causality between NINO3.4 and Shannon entropy (

p

> 0.05 for all lags tested), suggesting that the observed relationships are likely co-varying rather than causal in the Granger sense. Further phase–space analysis (with a delay of

τ

= 3 months) allowed for the identification of attractors characteristic of chaotic systems. The entropy trajectories revealed transitions from equilibrium states (average entropy: 4.124–4.138 bits) to highly unstable states (up to 4.768 bits), confirming an increase in the complexity of the climate system. Shannon entropy thus proves to be a valuable tool for monitoring local climatic instability and may contribute to improved risk modeling of droughts and floods in the context of climate change in Poland. Full article

(This article belongs to the Special Issue 25 Years of Sample Entropy)

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18 pages, 314 KB

Open AccessEditor’s ChoiceArticle

The POVM Theorem in Bohmian Mechanics

by Christian Beck and Dustin Lazarovici

Entropy 2025, 27(4), 391; https://doi.org/10.3390/e27040391 - 7 Apr 2025

Cited by 1 | Viewed by 3467

Abstract

The POVM theorem is a central result in Bohmian mechanics, grounding the measurement formalism of standard quantum mechanics in a statistical analysis based on the quantum equilibrium hypothesis (the Born rule for Bohmian particle positions). It states that the outcome statistics of an [...] Read more.

The POVM theorem is a central result in Bohmian mechanics, grounding the measurement formalism of standard quantum mechanics in a statistical analysis based on the quantum equilibrium hypothesis (the Born rule for Bohmian particle positions). It states that the outcome statistics of an experiment are described by a positive operator-valued measure (POVM) acting on the Hilbert space of the measured system. In light of recent debates about the scope and status of this result, we provide a systematic presentation of the POVM theorem and its underlying assumptions with a focus on their conceptual foundations and physical justifications. We conclude with a brief discussion of the scope of the POVM theorem—especially the sense in which it does (and does not) place limits on what is “measurable” in Bohmian mechanics. Full article

(This article belongs to the Special Issue Quantum Foundations: 100 Years of Born’s Rule)

14 pages, 265 KB

Open AccessEditor’s ChoiceArticle

Successive Refinement for Lossy Compression of Individual Sequences

by Neri Merhav

Entropy 2025, 27(4), 370; https://doi.org/10.3390/e27040370 - 31 Mar 2025

Cited by 2 | Viewed by 1059

Abstract

We consider the problem of successive-refinement coding for lossy compression of individual sequences, namely, compression in two stages, where in the first stage, a coarse description at a relatively low rate is sent from the encoder to the decoder, and in the second [...] Read more.

We consider the problem of successive-refinement coding for lossy compression of individual sequences, namely, compression in two stages, where in the first stage, a coarse description at a relatively low rate is sent from the encoder to the decoder, and in the second stage, an additional coding rate is allocated in order to refine the description and thereby improve the reproduction. Our main result is in establishing outer bounds (converse theorems) for the rate region where we limit the encoders to be finite-state machines in the spirit of Ziv and Lempel’s 1978 model. The matching achievability scheme is conceptually straightforward. We also consider the more general multiple description coding problem on a similar footing and propose achievability schemes that are analogous to the well-known El Gamal–Cover and the Zhang–Berger achievability schemes of memoryless sources and additive distortion measures. Full article

(This article belongs to the Collection Feature Papers in Information Theory)

25 pages, 1002 KB

Open AccessEditor’s ChoiceArticle

InfoMat: Leveraging Information Theory to Visualize and Understand Sequential Data

by Dor Tsur and Haim Permuter

Entropy 2025, 27(4), 357; https://doi.org/10.3390/e27040357 - 28 Mar 2025

Cited by 1 | Viewed by 1575

Abstract

Despite the widespread use of information measures in analyzing probabilistic systems, effective visualization tools for understanding complex dependencies in sequential data are scarce. In this work, we introduce the information matrix (InfoMat), a novel and intuitive matrix representation of information transfer in sequential [...] Read more.

Despite the widespread use of information measures in analyzing probabilistic systems, effective visualization tools for understanding complex dependencies in sequential data are scarce. In this work, we introduce the information matrix (InfoMat), a novel and intuitive matrix representation of information transfer in sequential systems. InfoMat provides a structured visual perspective on mutual information decompositions, enabling the discovery of new relationships between sequential information measures and enhancing interpretability in time series data analytics. We demonstrate how InfoMat captures key sequential information measures, such as directed information and transfer entropy. To facilitate its application in real-world datasets, we propose both an efficient Gaussian mutual information estimator and a neural InfoMat estimator based on masked autoregressive flows to model more complex dependencies. These estimators make InfoMat a valuable tool for uncovering hidden patterns in data analytics applications, encompassing neuroscience, finance, communication systems, and machine learning. We further illustrate the utility of InfoMat in visualizing information flow in real-world sequential physiological data analysis and in visualizing information flow in communication channels under various coding schemes. By mapping visual patterns in InfoMat to various modes of dependence structures, we provide a data-driven framework for analyzing causal relationships and temporal interactions. InfoMat thus serves as both a theoretical and empirical tool for data-driven decision making, bridging the gap between information theory and applied data analytics. Full article

(This article belongs to the Special Issue Information-Theoretic Methods in Data Analytics)

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21 pages, 737 KB

Open AccessEditor’s ChoiceArticle

A Model for the Formation of Beliefs and Social Norms Based on the Satisfaction Problem (SAT)

by Bastien Chopard, Franck Raynaud and Julien Stalhandske

Entropy 2025, 27(4), 358; https://doi.org/10.3390/e27040358 - 28 Mar 2025

Cited by 3 | Viewed by 1174

Abstract

We propose a numerical representation of beliefs in social systems based on the so-called SAT problem in computer science. The main idea is that a belief system is a set of true/false values associated with claims or propositions. Each individual assigns these values [...] Read more.

We propose a numerical representation of beliefs in social systems based on the so-called SAT problem in computer science. The main idea is that a belief system is a set of true/false values associated with claims or propositions. Each individual assigns these values according to its cognitive system in order to minimize logical contradictions, thus trying to solve a satisfaction problem. Social interactions between agents that disagree on a proposition can be introduced in order to see how, in the long term, social norms and competing belief systems build up in a population. Among other metrics, entropy is used to characterize the diversity of belief systems. Full article

(This article belongs to the Special Issue Entropy-Based Applications in Sociophysics II)

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17 pages, 1129 KB

Open AccessEditor’s ChoiceArticle

High-Dimensional and Multi-Intensity One-Photon-Interference Quantum Secure Direct Communication

by Yu-Ting Lei, Xiang-Jie Li, Xing-Bo Pan, Yun-Rong Zhang and Gui-Lu Long

Entropy 2025, 27(4), 332; https://doi.org/10.3390/e27040332 - 22 Mar 2025

Cited by 2 | Viewed by 1945

Abstract

As a novel paradigm in quantum communication, quantum secure direct communication (QSDC) enables secure, reliable, and deterministic information transmission, leveraging the principles of quantum mechanics. One-photon-interference QSDC is particularly attractive because it mitigates the vulnerabilities in measurement devices while extending transmission distances. In [...] Read more.

As a novel paradigm in quantum communication, quantum secure direct communication (QSDC) enables secure, reliable, and deterministic information transmission, leveraging the principles of quantum mechanics. One-photon-interference QSDC is particularly attractive because it mitigates the vulnerabilities in measurement devices while extending transmission distances. In this paper, we propose a high-dimensional one-photon-interference QSDC protocol that exploits the advantages of high-dimensional encoding in the phase of weak coherent pluses to further enhance transmission distances and improve secrecy channel capacity. The security of this protocol is analyzed using quantum wiretap channel theory, and its resistance to common quantum threats is discussed. Numerical simulations demonstrate that our protocol outperforms its predecessor in terms of its secrecy capacity and extends the maximum communication distance achievable up to 494 km, which is over 13% longer than the two-dimensional case, effectively doubling the transmission length of traditional protocols. These improvements highlight the protocol’s potential for use in quantum communication applications in this era of frequent data breaches and information leaks. Full article

(This article belongs to the Special Issue Quantum Information: Working Towards Applications)

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13 pages, 2121 KB

Open AccessEditor’s ChoiceArticle

Structural Transitions and Melting of Two-Dimensional Ion Crystals in RF Traps

by Boris V. Pashinsky, Alexander Kato and Boris B. Blinov

Entropy 2025, 27(4), 325; https://doi.org/10.3390/e27040325 - 21 Mar 2025

Cited by 2 | Viewed by 2763

Abstract

We investigate the structural properties and melting behaviors of two-dimensional ion crystals in an RF trap, focusing on the effects of ion temperature and trap potential symmetry. We identify distinct crystal structures that form under varying trapping conditions and temperatures through experimental observations [...] Read more.

We investigate the structural properties and melting behaviors of two-dimensional ion crystals in an RF trap, focusing on the effects of ion temperature and trap potential symmetry. We identify distinct crystal structures that form under varying trapping conditions and temperatures through experimental observations and theoretical analyses. As the temperature increases or the trap potential becomes more symmetric, we observe a transition from a lattice arrangement to elongated ring-like formations aligned along the trap axes. Our experimental and theoretical efforts enhance our understanding of phase transitions in low-dimensional, confined systems, offering insights into the controlled formation of quantum crystals for applications in quantum simulations and many-body physics. Full article

(This article belongs to the Special Issue Quantum Computing with Trapped Ions)

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12 pages, 689 KB

Open AccessEditor’s ChoiceArticle

Modeling Diffusion of Elongated Particles Through a Narrowing Channel

by Anna Strzelewicz, Michał Cieśla, Bartłomiej Dybiec and Monika Krasowska

Entropy 2025, 27(3), 293; https://doi.org/10.3390/e27030293 - 12 Mar 2025

Cited by 2 | Viewed by 1734

Abstract

Simulations of the Brownian dynamics of diffusing particles in complex environments provide important information about the characteristics of the medium and the properties of biological processes. Notable examples include the diffusion of ions and macromolecular solutes through channels of varying cross-section, such as [...] Read more.

Simulations of the Brownian dynamics of diffusing particles in complex environments provide important information about the characteristics of the medium and the properties of biological processes. Notable examples include the diffusion of ions and macromolecular solutes through channels of varying cross-section, such as pores in biological membranes, living tissues, zeolites, carbon nanotubes, and synthetic porous materials. In these systems, the observed diffusion can exhibit anomalous behavior characterized by a nonlinear increase in the mean squared displacement. In this article, we present a toy model of the diffusion of rod-shaped particles through a narrowing, conical pore with a trapezoidal longitudinal cross-section. Particles of different sizes undergo a random walk due to interactions with the environment (modeled as noise). We study how the diffusion properties change with particle size as a function of pore width. The numerical analysis of diffusion-driven transport through narrowing conical channels reveals its effective subdiffusive, i.e., anomalous, character. Full article

(This article belongs to the Collection Disorder and Biological Physics)

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17 pages, 991 KB

Open AccessFeature PaperEditor’s ChoiceArticle

The Statistical Thermodynamics of Generative Diffusion Models: Phase Transitions, Symmetry Breaking, and Critical Instability

by Luca Ambrogioni

Entropy 2025, 27(3), 291; https://doi.org/10.3390/e27030291 - 11 Mar 2025

Cited by 26 | Viewed by 5434

Abstract

Generative diffusion models have achieved spectacular performance in many areas of machine learning and generative modeling. While the fundamental ideas behind these models come from non-equilibrium physics, variational inference, and stochastic calculus, in this paper we show that many aspects of these models [...] Read more.

Generative diffusion models have achieved spectacular performance in many areas of machine learning and generative modeling. While the fundamental ideas behind these models come from non-equilibrium physics, variational inference, and stochastic calculus, in this paper we show that many aspects of these models can be understood using the tools of equilibrium statistical mechanics. Using this reformulation, we show that generative diffusion models undergo second-order phase transitions corresponding to symmetry breaking phenomena. We show that these phase transitions are always in a mean-field universality class, as they are the result of a self-consistency condition in the generative dynamics. We argue that the critical instability arising from these phase transitions lies at the heart of their generative capabilities, which are characterized by a set of mean-field critical exponents. Finally, we show that the dynamic equation of the generative process can be interpreted as a stochastic adiabatic transformation that minimizes the free energy while keeping the system in thermal equilibrium. Full article

(This article belongs to the Special Issue The Statistical Physics of Generative Diffusion Models)

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18 pages, 456 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Optimal Control of an Electromechanical Energy Harvester

by Dario Lucente, Alessandro Manacorda, Andrea Plati, Alessandro Sarracino and Marco Baldovin

Entropy 2025, 27(3), 268; https://doi.org/10.3390/e27030268 - 5 Mar 2025

Cited by 7 | Viewed by 2113

Abstract

Many techniques originally developed in the context of deterministic control theory have recently been applied to the quest for optimal protocols in stochastic processes. Given a system subject to environmental fluctuations, one may ask what is the best way to change its controllable [...] Read more.

Many techniques originally developed in the context of deterministic control theory have recently been applied to the quest for optimal protocols in stochastic processes. Given a system subject to environmental fluctuations, one may ask what is the best way to change its controllable parameters in time in order to maximize, on average, a certain reward function, while steering the system between two pre-assigned states. In this work, we study the problem of optimal control for a wide class of stochastic systems, inspired by a model of an energy harvester. The stochastic noise in this system is due to the mechanical vibrations, while the reward function is the average power extracted from them. We consider the case in which the electrical resistance of the harvester can be changed in time, and we exploit the tools of control theory to work out optimal solutions in a perturbative regime, close to the stationary state. Our results show that it is possible to design protocols that perform better than any possible solution with constant resistance. Full article

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

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10 pages, 586 KB

Open AccessEditor’s ChoiceArticle

The Quantum Relative Entropy of the Schwarzschild Black Hole and the Area Law

by Ginestra Bianconi

Entropy 2025, 27(3), 266; https://doi.org/10.3390/e27030266 - 4 Mar 2025

Cited by 6 | Viewed by 4868

Abstract

The area law obeyed by the thermodynamic entropy of black holes is one of the fundamental results relating gravity to statistical mechanics. In this work, we provide a derivation of the area law for the quantum relative entropy of the Schwarzschild black hole [...] Read more.

The area law obeyed by the thermodynamic entropy of black holes is one of the fundamental results relating gravity to statistical mechanics. In this work, we provide a derivation of the area law for the quantum relative entropy of the Schwarzschild black hole for an arbitrary Schwarzschild radius. The quantum relative entropy between the metric of the manifold and the metric induced by the geometry and the matter field has been proposed in G. Bianconi as the action for entropic quantum gravity leading to modified Einstein equations. The quantum relative entropy generalizes Araki’s entropy and treats the metrics between zero-forms, one-forms, and two-forms as quantum operators. Although the Schwarzschild metric is not an exact solution of the modified Einstein equations of the entropic quantum gravity, it is an approximate solution valid in the low-coupling, small-curvature limit. Here, we show that the quantum relative entropy associated to the Schwarzschild metric obeys the area law for a large Schwarzschild radius. We provide a full statistical mechanics interpretation of the results. Full article

(This article belongs to the Special Issue Entropy in Quantum Systems and Quantum Field Theory (QFT): Gravity, 3rd Edition)

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47 pages, 4735 KB

Open AccessEditor’s ChoiceTutorial

Principles Entailed by Complexity, Crucial Events, and Multifractal Dimensionality

by Bruce J. West and Senthil Mudaliar

Entropy 2025, 27(3), 241; https://doi.org/10.3390/e27030241 - 26 Feb 2025

Cited by 5 | Viewed by 3592

Abstract

Complexity is one of those descriptive terms adopted in science that we think we understand until it comes time to form a coherent definition upon which everyone can agree. Suddenly, we are awash in conditions that qualify this or that situation, much like [...] Read more.

Complexity is one of those descriptive terms adopted in science that we think we understand until it comes time to form a coherent definition upon which everyone can agree. Suddenly, we are awash in conditions that qualify this or that situation, much like we were in the middle of the last century when it came time to determine the solutions to differential equations that were not linear. Consequently, this tutorial is not an essay on the mathematics of complexity nor is it a rigorous review of the recent growth spurt of complexity science, but is rather an exploration of how physiologic time series (PTS) in the life sciences that have eluded traditional mathematical modeling become less mysterious when certain historical assumptions are discarded and so-called ordinary statistical events in PTS are replaced with crucial events (CEs) using mutifractal dimensionality as the working measure of complexity. The empirical datasets considered include respiration, electrocardiograms (ECGs), and electroencephalograms (EEGs), and as different as these time series appear from one another when recorded, they are in fact shown to be in synchrony when properly processed using the technique of modified diffusion entropy analysis (MDEA). This processing reveals a new synchronization mechanism among the time series which simultaneously measures their complexity by means of the multifractal dimension of each time series and are shown to track one another across time. These results reveal a set of priciples that capture the manner in which information is exchanged among physiologic organ networks. Full article

(This article belongs to the Section Complexity)

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17 pages, 660 KB

Open AccessEditor’s ChoiceArticle

User-Centric Cell-Free Massive Multiple-Input-Multiple-Output System with Noisy Channel Gain Estimation and Line of Sight: A Beckmann Distribution Approach

by Danilo B. T. Almeida, Marcelo S. Alencar, Wamberto J. L. Queiroz, Rafael M. Duarte and Francisco Madeiro

Entropy 2025, 27(3), 223; https://doi.org/10.3390/e27030223 - 21 Feb 2025

Cited by 2 | Viewed by 2139

Abstract

This paper analyzes for the first time how the Beckmann distribution can be used to characterize the random variable that represents the envelope of the effective channel gain experienced by the k-th user equipment (UE) of a user-centric (UC) cell-free (CF) system [...] Read more.

This paper analyzes for the first time how the Beckmann distribution can be used to characterize the random variable that represents the envelope of the effective channel gain experienced by the k-th user equipment (UE) of a user-centric (UC) cell-free (CF) system in a scenario with noisy channel state information (CSI) estimation and line of sight (LoS). Additionally, it is shown how the Beckmann probability density function (PDF) can be used to derive the PDF and the cumulative density function (CDF) of the instantaneous signal-to-interference-plus-noise ratio (SINR) of the UC CF k-th UE, followed by applications in the ergodic capacity (EC) and outage probability (OP) expression derivations. It is shown that, regardless of the type of distribution considered for the channel gain between each access point (AP) and UE links, the effective gain presents a Beckmann distribution. Full article

(This article belongs to the Special Issue Wireless Communications: Signal Processing Perspectives)

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59 pages, 715 KB

Open AccessFeature PaperEditor’s ChoiceArticle

New Formulas of Feedback Capacity for AGN Channels with Memory: A Time-Domain Sufficient Statistic Approach

by Charalambos D. Charalambous, Christos Kourtellaris and Stelios Louka

Entropy 2025, 27(2), 207; https://doi.org/10.3390/e27020207 - 15 Feb 2025

Cited by 1 | Viewed by 2195

Abstract

Recently, several papers identified technical issues related to equivalent time-domain and frequency-domain “characterization of the n–block or transmission” feedback capacity formula and its asymptotic limit, the feedback capacity, of additive Gaussian noise (AGN) channels, first introduce by Cover and Pombra in 1989 [...] Read more.

Recently, several papers identified technical issues related to equivalent time-domain and frequency-domain “characterization of the n–block or transmission” feedback capacity formula and its asymptotic limit, the feedback capacity, of additive Gaussian noise (AGN) channels, first introduce by Cover and Pombra in 1989 (IEEE Transactions on Information Theory). The main objective of this paper is to derive new results on the Cover and Pombra characterization of the n–block feedback capacity formula, and to clarify the main points of confusion regarding the time-domain results that appeared in the literature. The first part of this paper derives new equivalent time-domain sequential characterizations of feedback capacity of AGN channels driven by non-stationary and non-ergodic Gaussian noise. It is shown that the optimal channel input processes of the new equivalent sequential characterizations are expressed as functionals of a sufficient statistic and a Gaussian orthogonal innovations process. Further, the Cover and Pombra n–block capacity formula is expressed as a functional of two generalized matrix difference Riccati equations (DREs) of the filtering theory of Gaussian systems, contrary to results that appeared in the literature and involve only one DRE. It is clarified that prior literature deals with a simpler problem that presupposes the state of the noise is known to the encoder and the decoder. In the second part of this paper, the existence of the asymptotic limit of the n–block feedback capacity formula is shown to be equivalent to the convergence properties of solutions of the two generalized DREs. Further, necessary and or sufficient conditions are identified for the existence of asymptotic limits, for stable and unstable Gaussian noise, when the optimal input distributions are asymptotically time-invariant but not necessarily stationary. This paper contains an in-depth analysis, with various examples, and identifies the technical conditions on the feedback code and state space noise realization, so that the time-domain capacity formulas that appeared in the literature, for AGN channels with stationary noises, are indeed correct. Full article

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

22 pages, 750 KB

Open AccessEditor’s ChoiceArticle

Levy Noise Affects Ornstein–Uhlenbeck Memory

by Iddo Eliazar

Entropy 2025, 27(2), 157; https://doi.org/10.3390/e27020157 - 2 Feb 2025

Cited by 3 | Viewed by 2222

Abstract

This paper investigates the memory of the Ornstein–Uhlenbeck process (OUP) via three ratios of the OUP increments: signal-to-noise, noise-to-noise, and tail-to-tail. Intuition suggests the following points: (1) changing the noise that drives the OUP from Gauss to Levy will not affect the memory, [...] Read more.

This paper investigates the memory of the Ornstein–Uhlenbeck process (OUP) via three ratios of the OUP increments: signal-to-noise, noise-to-noise, and tail-to-tail. Intuition suggests the following points: (1) changing the noise that drives the OUP from Gauss to Levy will not affect the memory, as both noises share the common ‘independent increments’ property; (2) changing the auto-correlation of the OUP from exponential to slowly decaying will affect the memory, as the change yields a process with long-range correlations; and (3) with regard to Levy driving noise, the greater the noise fluctuations, the noisier the prediction of the OUP increments. This paper shows that intuition is plain wrong. Indeed, a detailed analysis establishes that for each of the three above-mentioned points, the very converse holds. Hence, Levy noise has a significant and counter-intuitive effect on Ornstein–Uhlenbeck memory. Full article

(This article belongs to the Collection Foundations of Statistical Mechanics)

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15 pages, 4130 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Protocells Either Synchronize or Starve

by Marco Villani and Roberto Serra

Entropy 2025, 27(2), 154; https://doi.org/10.3390/e27020154 - 2 Feb 2025

Cited by 2 | Viewed by 1681

Abstract

Two different processes take place in self-reproducing protocells, i.e., (i) cell reproduction by fission and (ii) duplication of the genetic material. One major problem is indeed that of assuring that the two processes take place at the same pace, i.e., that they synchronize, [...] Read more.

Two different processes take place in self-reproducing protocells, i.e., (i) cell reproduction by fission and (ii) duplication of the genetic material. One major problem is indeed that of assuring that the two processes take place at the same pace, i.e., that they synchronize, which is a necessary condition for sustainable growth. In previous theoretical works, using dynamical models, we had shown that such synchronization can spontaneously emerge, generation after generation, under a broad set of hypotheses about the architecture of the protocell, the nature of the self-replicating molecules, and the types of kinetic equations. However, an important class of cases (quadratic or higher-order self-replication) did not synchronize in the models we had used, but could actually lead to divergence of the concentration of replicators. We show here that this behavior is due to a simplification of the previous models, i.e., the “buffering” hypothesis, which assumes instantaneous equilibrium of the internal and external concentrations of those compounds which can cross the cell membrane. That divergence disappears if we make use of more realistic dynamical models, with finite transmembrane diffusion rates of the precursors of replicators. Full article

(This article belongs to the Section Entropy and Biology)

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30 pages, 1295 KB

Open AccessEditor’s ChoiceArticle

Why Uncertainty Is Essential for Consciousness: Local Prospect Theory vs. Predictive Processing

by Francis Heylighen and Shima Beigi

Entropy 2025, 27(2), 140; https://doi.org/10.3390/e27020140 - 28 Jan 2025

Cited by 1 | Viewed by 6838

Abstract

We present and develop local prospect theory (LPT), a novel framework for understanding consciousness, and, in particular, subjective experience and free will. While predictive processing (PP) theories model the brain as trying to optimize the accuracy of predictions, LPT sees uncertainty as an [...] Read more.

We present and develop local prospect theory (LPT), a novel framework for understanding consciousness, and, in particular, subjective experience and free will. While predictive processing (PP) theories model the brain as trying to optimize the accuracy of predictions, LPT sees uncertainty as an essential feature of conscious decision-making. This is achieved by creating a “local prospect”—a range of potential developments colored by subjective experience from which an agent can freely choose how to react. Drawing on global workspace theory, LPT conceptualizes consciousness as a self-maintaining process of circulating neural activation, creating a temporary working memory where thoughts and feelings coming from different brain modules enter into an asynchronous, non-linear interaction. This contrasts with unconscious processes, which operate automatically and deterministically. LPT proposes entropy-based measures, including the determination of actions by conditions and the breadth of prospect, to quantify the range of potential developments considered. This framework allows us to understand Buddhist practices and concepts, such as mindfulness, liberation from attachments, and meditation, which broaden consciousness and de-automatize reactions by reducing the influence of conditioning. The proposed prospect measure may be operationalized by indicators such as the variety of action, breadth of perception, and unpredictability of behavior, thus allowing for the empirical testing of the theory. Full article

(This article belongs to the Special Issue Complexity and Evolution, 2nd Edition)

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18 pages, 3516 KB

Open AccessEditor’s ChoiceArticle

Temperature Gradients as a Data Storage Principle

by Jeroen Schoenmaker, Pâmella Gonçalves Martins and Julio Carlos Teixeira

Entropy 2025, 27(2), 129; https://doi.org/10.3390/e27020129 - 26 Jan 2025

Cited by 2 | Viewed by 2822

Abstract

In this work, we analyze the thermodynamic principles underlying modern data storage systems, including Random Access Memory (RAM), hard disk drive (HDD), flash memory, magnetic RAM (MRAM), ferroelectric RAM (FeRAM), and phase-change RAM (PCRAM), as well as other less well-known data storage mechanisms. [...] Read more.

In this work, we analyze the thermodynamic principles underlying modern data storage systems, including Random Access Memory (RAM), hard disk drive (HDD), flash memory, magnetic RAM (MRAM), ferroelectric RAM (FeRAM), and phase-change RAM (PCRAM), as well as other less well-known data storage mechanisms. The analysis is conducted in the context of data storage and processing in relation to Landauer’s principle, with special emphasis on hysteresis. Analogous to how heat engines are characterized by thermodynamic cycles, data storage systems are examined in terms of the hysteresis loop of their fundamental data unit. We explore the role of heat in data storage systems. Afterward, we introduce the concept of temperature gradient memory (TeGraM) along with a detailed layout of a realizable device. Experimental results demonstrating this technology are also presented. Full article

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

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20 pages, 2127 KB

Open AccessFeature PaperEditor’s ChoiceReview

Entropy and Complexity Tools Across Scales in Neuroscience: A Review

by Rodrigo Cofré and Alain Destexhe

Entropy 2025, 27(2), 115; https://doi.org/10.3390/e27020115 - 24 Jan 2025

Cited by 14 | Viewed by 9996

Abstract

Understanding the brain’s intricate dynamics across multiple scales—from cellular interactions to large-scale brain behavior—remains one of the most significant challenges in modern neuroscience. Two key concepts, entropy and complexity, have been increasingly employed by neuroscientists as powerful tools for characterizing the interplay between [...] Read more.

Understanding the brain’s intricate dynamics across multiple scales—from cellular interactions to large-scale brain behavior—remains one of the most significant challenges in modern neuroscience. Two key concepts, entropy and complexity, have been increasingly employed by neuroscientists as powerful tools for characterizing the interplay between structure and function in the brain across scales. The flexibility of these two concepts enables researchers to explore quantitatively how the brain processes information, adapts to changing environments, and maintains a delicate balance between order and disorder. This review illustrates the main tools and ideas to study neural phenomena using these concepts. This review does not delve into the specific methods or analyses of each study. Instead, it aims to offer a broad overview of how these tools are applied within the neuroscientific community and how they are transforming our understanding of the brain. We focus on their applications across scales, discuss the strengths and limitations of different metrics, and examine their practical applications and theoretical significance. Full article

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

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21 pages, 22357 KB

Open AccessEditor’s ChoiceArticle

Dynamical Systems on Generalised Klein Bottles

by Peter Grindrod and Ka Man Yim

Entropy 2025, 27(2), 119; https://doi.org/10.3390/e27020119 - 24 Jan 2025

Cited by 1 | Viewed by 2239

Abstract

We propose a high-dimensional generalisation of the standard Klein bottle extending beyond those considered previously. We address the problem of generating continuous scalar fields (distributions) and dynamical systems (flows) on such state spaces, which can provide a rich source of examples for future [...] Read more.

We propose a high-dimensional generalisation of the standard Klein bottle extending beyond those considered previously. We address the problem of generating continuous scalar fields (distributions) and dynamical systems (flows) on such state spaces, which can provide a rich source of examples for future investigations. We consider a class of high-dimensional dynamical systems that model distributed information processing within the human cortex, which may be capable of exhibiting some Klein bottle symmetries. We deploy topological data analytic methods in order to analyse their resulting dynamical behaviour and suggesting future challenges. Full article

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

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13 pages, 13005 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Entropy Production in a System of Janus Particles

by Andrés Arango-Restrepo, Juan David Torrenegra-Rico and J. Miguel Rubi

Entropy 2025, 27(2), 112; https://doi.org/10.3390/e27020112 - 23 Jan 2025

Cited by 5 | Viewed by 2915

Abstract

Entropy production is a key descriptor of out-of-equilibrium behavior in active matter systems, providing insights into both single-particle dynamics and emergent collective phenomena. It helps determine transport coefficients and phoretic velocities and serves as a crucial tool for understanding collective phenomena such as [...] Read more.

Entropy production is a key descriptor of out-of-equilibrium behavior in active matter systems, providing insights into both single-particle dynamics and emergent collective phenomena. It helps determine transport coefficients and phoretic velocities and serves as a crucial tool for understanding collective phenomena such as structural transitions, regime shifts, clustering, and self-organization. This study investigates the role of entropy production for individual active (catalytic Janus) particles and in systems of active particles interacting with one another and their environment. We employ a multiscale framework to bridge microscopic particle dynamics and macroscopic behavior, offering a thermodynamic perspective on active matter. These findings enhance our understanding of the fundamental principles governing active particle systems and create new opportunities for addressing unresolved questions in non-equilibrium thermodynamics. Full article

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

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25 pages, 374 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Joint Identification and Sensing for Discrete Memoryless Channels

by Wafa Labidi, Yaning Zhao, Christian Deppe and Holger Boche

Entropy 2025, 27(1), 12; https://doi.org/10.3390/e27010012 - 27 Dec 2024

Cited by 2 | Viewed by 2591

Abstract

In the identification (ID) scheme proposed by Ahlswede and Dueck, the receiver’s goal is simply to verify whether a specific message of interest was sent. Unlike Shannon’s transmission codes, which aim for message decoding, ID codes for a discrete memoryless channel (DMC) are [...] Read more.

In the identification (ID) scheme proposed by Ahlswede and Dueck, the receiver’s goal is simply to verify whether a specific message of interest was sent. Unlike Shannon’s transmission codes, which aim for message decoding, ID codes for a discrete memoryless channel (DMC) are far more efficient; their size grows doubly exponentially with the blocklength when randomized encoding is used. This indicates that when the receiver’s objective does not require decoding, the ID paradigm is significantly more efficient than traditional Shannon transmission in terms of both energy consumption and hardware complexity. Further benefits of ID schemes can be realized by leveraging additional resources such as feedback. In this work, we address the problem of joint ID and channel state estimation over a DMC with independent and identically distributed (i.i.d.) state sequences. State estimation functions as the sensing mechanism of the model. Specifically, the sender transmits an ID message over the DMC while simultaneously estimating the channel state through strictly causal observations of the channel output. Importantly, the random channel state is unknown to both the sender and the receiver. For this system model, we present a complete characterization of the ID capacity–distortion function. Full article

(This article belongs to the Special Issue Integrated Sensing and Communications)

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42 pages, 984 KB

Open AccessFeature PaperEditor’s ChoiceReview

Applications of Entropy in Data Analysis and Machine Learning: A Review

by Salomé A. Sepúlveda-Fontaine and José M. Amigó

Entropy 2024, 26(12), 1126; https://doi.org/10.3390/e26121126 - 23 Dec 2024

Cited by 40 | Viewed by 14200

Abstract

Since its origin in the thermodynamics of the 19th century, the concept of entropy has also permeated other fields of physics and mathematics, such as Classical and Quantum Statistical Mechanics, Information Theory, Probability Theory, Ergodic Theory and the Theory of Dynamical Systems. Specifically, [...] Read more.

Since its origin in the thermodynamics of the 19th century, the concept of entropy has also permeated other fields of physics and mathematics, such as Classical and Quantum Statistical Mechanics, Information Theory, Probability Theory, Ergodic Theory and the Theory of Dynamical Systems. Specifically, we are referring to the classical entropies: the Boltzmann–Gibbs, von Neumann, Shannon, Kolmogorov–Sinai and topological entropies. In addition to their common name, which is historically justified (as we briefly describe in this review), another commonality of the classical entropies is the important role that they have played and are still playing in the theory and applications of their respective fields and beyond. Therefore, it is not surprising that, in the course of time, many other instances of the overarching concept of entropy have been proposed, most of them tailored to specific purposes. Following the current usage, we will refer to all of them, whether classical or new, simply as entropies. In particular, the subject of this review is their applications in data analysis and machine learning. The reason for these particular applications is that entropies are very well suited to characterize probability mass distributions, typically generated by finite-state processes or symbolized signals. Therefore, we will focus on entropies defined as positive functionals on probability mass distributions and provide an axiomatic characterization that goes back to Shannon and Khinchin. Given the plethora of entropies in the literature, we have selected a representative group, including the classical ones. The applications summarized in this review nicely illustrate the power and versatility of entropy in data analysis and machine learning. Full article

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

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