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Editor’s Choice Articles

Editor’s Choice articles are based on recommendations by the scientific editors of MDPI journals from around the world. Editors select a small number of articles recently published in the journal that they believe will be particularly interesting to readers, or important in the respective research area. The aim is to provide a snapshot of some of the most exciting work published in the various research areas of the journal.

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12 pages, 690 KiB  
Article
Fundamental Relation for Gas of Interacting Particles in a Heat Flow
by Robert Hołyst, Karol Makuch, Konrad Giżyński, Anna Maciołek and Paweł J. Żuk
Entropy 2023, 25(9), 1295; https://doi.org/10.3390/e25091295 - 4 Sep 2023
Cited by 4 | Viewed by 1262
Abstract
There is a long-standing question of whether it is possible to extend the formalism of equilibrium thermodynamics to the case of nonequilibrium systems in steady-states. We have made such an extension for an ideal gas in a heat flow. Here, we investigated whether [...] Read more.
There is a long-standing question of whether it is possible to extend the formalism of equilibrium thermodynamics to the case of nonequilibrium systems in steady-states. We have made such an extension for an ideal gas in a heat flow. Here, we investigated whether such a description exists for the system with interactions: the van der Waals gas in a heat flow. We introduced a steady-state fundamental relation and the parameters of state, each associated with a single way of changing energy. The first law of nonequilibrium thermodynamics follows from these parameters. The internal energy U for the nonequilibrium states has the same form as in equilibrium thermodynamics. For the van der Waals gas, U(S*,V,N,a*,b*) is a function of only five parameters of state (irrespective of the number of parameters characterizing the boundary conditions): the effective entropy S*, volume V, number of particles N, and rescaled van der Waals parameters a*, b*. The state parameters, a*, b*, together with S*, determine the net heat exchange with the environment. The net heat differential does not have an integrating factor. As in equilibrium thermodynamics, the steady-state fundamental equation also leads to the thermodynamic Maxwell relations for measurable steady-state properties. Full article
(This article belongs to the Special Issue Entropy Production and Nonequilibrium Thermodynamics in Materials)
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11 pages, 1090 KiB  
Article
Ergodic Measure and Potential Control of Anomalous Diffusion
by Bao Wen, Ming-Gen Li, Jian Liu and Jing-Dong Bao
Entropy 2023, 25(7), 1012; https://doi.org/10.3390/e25071012 - 30 Jun 2023
Cited by 1 | Viewed by 851
Abstract
In statistical mechanics, the ergodic hypothesis (i.e., the long-time average is the same as the ensemble average) accompanying anomalous diffusion has become a continuous topic of research, being closely related to irreversibility and increasing entropy. While measurement time is finite for a given [...] Read more.
In statistical mechanics, the ergodic hypothesis (i.e., the long-time average is the same as the ensemble average) accompanying anomalous diffusion has become a continuous topic of research, being closely related to irreversibility and increasing entropy. While measurement time is finite for a given process, the time average of an observable quantity might be a random variable, whose distribution width narrows with time, and one wonders how long it takes for the convergence rate to become a constant. This is also the premise of ergodic establishment, because the ensemble average is always equal to the constant. We focus on the time-dependent fluctuation width for the time average of both the velocity and kinetic energy of a force-free particle described by the generalized Langevin equation, where the stationary velocity autocorrelation function is considered. Subsequently, the shortest time scale can be estimated for a system transferring from a stationary state to an effective ergodic state. Moreover, a logarithmic spatial potential is used to modulate the processes associated with free ballistic diffusion and the control of diffusion, as well as the minimal realization of the whole power-law regime. The results presented suggest that non-ergodicity mimics the sparseness of the medium and reveals the unique role of logarithmic potential in modulating diffusion behavior. Full article
(This article belongs to the Section Statistical Physics)
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23 pages, 4268 KiB  
Article
A Variational Synthesis of Evolutionary and Developmental Dynamics
by Karl Friston, Daniel A. Friedman, Axel Constant, V. Bleu Knight, Chris Fields, Thomas Parr and John O. Campbell
Entropy 2023, 25(7), 964; https://doi.org/10.3390/e25070964 - 21 Jun 2023
Cited by 9 | Viewed by 2950
Abstract
This paper introduces a variational formulation of natural selection, paying special attention to the nature of ‘things’ and the way that different ‘kinds’ of ‘things’ are individuated from—and influence—each other. We use the Bayesian mechanics of particular partitions to understand how slow phylogenetic [...] Read more.
This paper introduces a variational formulation of natural selection, paying special attention to the nature of ‘things’ and the way that different ‘kinds’ of ‘things’ are individuated from—and influence—each other. We use the Bayesian mechanics of particular partitions to understand how slow phylogenetic processes constrain—and are constrained by—fast, phenotypic processes. The main result is a formulation of adaptive fitness as a path integral of phenotypic fitness. Paths of least action, at the phenotypic and phylogenetic scales, can then be read as inference and learning processes, respectively. In this view, a phenotype actively infers the state of its econiche under a generative model, whose parameters are learned via natural (Bayesian model) selection. The ensuing variational synthesis features some unexpected aspects. Perhaps the most notable is that it is not possible to describe or model a population of conspecifics per se. Rather, it is necessary to consider populations of distinct natural kinds that influence each other. This paper is limited to a description of the mathematical apparatus and accompanying ideas. Subsequent work will use these methods for simulations and numerical analyses—and identify points of contact with related mathematical formulations of evolution. Full article
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23 pages, 2054 KiB  
Article
On Sequential Bayesian Inference for Continual Learning
by Samuel Kessler, Adam Cobb, Tim G. J. Rudner, Stefan Zohren and Stephen J. Roberts
Entropy 2023, 25(6), 884; https://doi.org/10.3390/e25060884 - 31 May 2023
Cited by 2 | Viewed by 2891
Abstract
Sequential Bayesian inference can be used for continual learning to prevent catastrophic forgetting of past tasks and provide an informative prior when learning new tasks. We revisit sequential Bayesian inference and assess whether using the previous task’s posterior as a prior for a [...] Read more.
Sequential Bayesian inference can be used for continual learning to prevent catastrophic forgetting of past tasks and provide an informative prior when learning new tasks. We revisit sequential Bayesian inference and assess whether using the previous task’s posterior as a prior for a new task can prevent catastrophic forgetting in Bayesian neural networks. Our first contribution is to perform sequential Bayesian inference using Hamiltonian Monte Carlo. We propagate the posterior as a prior for new tasks by approximating the posterior via fitting a density estimator on Hamiltonian Monte Carlo samples. We find that this approach fails to prevent catastrophic forgetting, demonstrating the difficulty in performing sequential Bayesian inference in neural networks. From there, we study simple analytical examples of sequential Bayesian inference and CL and highlight the issue of model misspecification, which can lead to sub-optimal continual learning performance despite exact inference. Furthermore, we discuss how task data imbalances can cause forgetting. From these limitations, we argue that we need probabilistic models of the continual learning generative process rather than relying on sequential Bayesian inference over Bayesian neural network weights. Our final contribution is to propose a simple baseline called Prototypical Bayesian Continual Learning, which is competitive with the best performing Bayesian continual learning methods on class incremental continual learning computer vision benchmarks. Full article
(This article belongs to the Special Issue Information Theory for Data Science)
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20 pages, 2936 KiB  
Article
Bandit Algorithm Driven by a Classical Random Walk and a Quantum Walk
by Tomoki Yamagami, Etsuo Segawa, Takatomo Mihana, André Röhm, Ryoichi Horisaki and Makoto Naruse
Entropy 2023, 25(6), 843; https://doi.org/10.3390/e25060843 - 25 May 2023
Cited by 2 | Viewed by 1853
Abstract
Quantum walks (QWs) have a property that classical random walks (RWs) do not possess—the coexistence of linear spreading and localization—and this property is utilized to implement various kinds of applications. This paper proposes RW- and QW-based algorithms for multi-armed-bandit (MAB) problems. We show [...] Read more.
Quantum walks (QWs) have a property that classical random walks (RWs) do not possess—the coexistence of linear spreading and localization—and this property is utilized to implement various kinds of applications. This paper proposes RW- and QW-based algorithms for multi-armed-bandit (MAB) problems. We show that, under some settings, the QW-based model realizes higher performance than the corresponding RW-based one by associating the two operations that make MAB problems difficult—exploration and exploitation—with these two behaviors of QWs. Full article
(This article belongs to the Special Issue Recent Advances in Quantum Information Processing)
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17 pages, 464 KiB  
Opinion
Senses along Which the Entropy Sq Is Unique
by Constantino Tsallis
Entropy 2023, 25(5), 743; https://doi.org/10.3390/e25050743 - 1 May 2023
Cited by 6 | Viewed by 1661
Abstract
The Boltzmann–Gibbs–von Neumann–Shannon additive entropy SBG=kipilnpi as well as its continuous and quantum counterparts, constitute the grounding concept on which the BG statistical mechanics is constructed. This magnificent theory has produced, [...] Read more.
The Boltzmann–Gibbs–von Neumann–Shannon additive entropy SBG=kipilnpi as well as its continuous and quantum counterparts, constitute the grounding concept on which the BG statistical mechanics is constructed. This magnificent theory has produced, and will most probably keep producing in the future, successes in vast classes of classical and quantum systems. However, recent decades have seen a proliferation of natural, artificial and social complex systems which defy its bases and make it inapplicable. This paradigmatic theory has been generalized in 1988 into the nonextensive statistical mechanics—as currently referred to—grounded on the nonadditive entropy Sq=k1ipiqq1 as well as its corresponding continuous and quantum counterparts. In the literature, there exist nowadays over fifty mathematically well defined entropic functionals. Sq plays a special role among them. Indeed, it constitutes the pillar of a great variety of theoretical, experimental, observational and computational validations in the area of complexity—plectics, as Murray Gell-Mann used to call it. Then, a question emerges naturally, namely In what senses is entropy Sq unique? The present effort is dedicated to a—surely non exhaustive—mathematical answer to this basic question. Full article
(This article belongs to the Special Issue The Statistical Foundations of Entropy II)
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13 pages, 2488 KiB  
Article
On Two Non-Ergodic Reversible Cellular Automata, One Classical, the Other Quantum
by Tomaž Prosen
Entropy 2023, 25(5), 739; https://doi.org/10.3390/e25050739 - 30 Apr 2023
Cited by 1 | Viewed by 1174
Abstract
We propose and discuss two variants of kinetic particle models—cellular automata in 1 + 1 dimensions—that have some appeal due to their simplicity and intriguing properties, which could warrant further research and applications. The first model is a deterministic and reversible automaton describing [...] Read more.
We propose and discuss two variants of kinetic particle models—cellular automata in 1 + 1 dimensions—that have some appeal due to their simplicity and intriguing properties, which could warrant further research and applications. The first model is a deterministic and reversible automaton describing two species of quasiparticles: stable massless matter particles moving with velocity ±1 and unstable standing (zero velocity) field particles. We discuss two distinct continuity equations for three conserved charges of the model. While the first two charges and the corresponding currents have support of three lattice sites and represent a lattice analogue of the conserved energy–momentum tensor, we find an additional conserved charge and current with support of nine sites, implying non-ergodic behaviour and potentially signalling integrability of the model with a highly nested R-matrix structure. The second model represents a quantum (or stochastic) deformation of a recently introduced and studied charged hardpoint lattice gas, where particles of different binary charge (±1) and binary velocity (±1) can nontrivially mix upon elastic collisional scattering. We show that while the unitary evolution rule of this model does not satisfy the full Yang–Baxter equation, it still satisfies an intriguing related identity which gives birth to an infinite set of local conserved operators, the so-called glider operators. Full article
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11 pages, 647 KiB  
Article
Outlier-Robust Surrogate Modeling of Ion–Solid Interaction Simulations
by Roland Preuss and Udo von Toussaint
Entropy 2023, 25(4), 685; https://doi.org/10.3390/e25040685 - 19 Apr 2023
Viewed by 787
Abstract
Data for complex plasma–wall interactions require long-running and expensive computer simulations. Furthermore, the number of input parameters is large, which results in low coverage of the (physical) parameter space. Unpredictable occasions of outliers create a need to conduct the exploration of this multi-dimensional [...] Read more.
Data for complex plasma–wall interactions require long-running and expensive computer simulations. Furthermore, the number of input parameters is large, which results in low coverage of the (physical) parameter space. Unpredictable occasions of outliers create a need to conduct the exploration of this multi-dimensional space using robust analysis tools. We restate the Gaussian process (GP) method as a Bayesian adaptive exploration method for establishing surrogate surfaces in the variables of interest. On this basis, we expand the analysis by the Student-t process (TP) method in order to improve the robustness of the result with respect to outliers. The most obvious difference between both methods shows up in the marginal likelihood for the hyperparameters of the covariance function, where the TP method features a broader marginal probability distribution in the presence of outliers. Eventually, we provide first investigations, with a mixture likelihood of two Gaussians within a Gaussian process ansatz for describing either outlier or non-outlier behavior. The parameters of the two Gaussians are set such that the mixture likelihood resembles the shape of a Student-t likelihood. Full article
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15 pages, 500 KiB  
Article
Time Series of Counts under Censoring: A Bayesian Approach
by Isabel Silva, Maria Eduarda Silva, Isabel Pereira and Brendan McCabe
Entropy 2023, 25(4), 549; https://doi.org/10.3390/e25040549 - 23 Mar 2023
Viewed by 1272
Abstract
Censored data are frequently found in diverse fields including environmental monitoring, medicine, economics and social sciences. Censoring occurs when observations are available only for a restricted range, e.g., due to a detection limit. Ignoring censoring produces biased estimates and unreliable statistical inference. The [...] Read more.
Censored data are frequently found in diverse fields including environmental monitoring, medicine, economics and social sciences. Censoring occurs when observations are available only for a restricted range, e.g., due to a detection limit. Ignoring censoring produces biased estimates and unreliable statistical inference. The aim of this work is to contribute to the modelling of time series of counts under censoring using convolution closed infinitely divisible (CCID) models. The emphasis is on estimation and inference problems, using Bayesian approaches with Approximate Bayesian Computation (ABC) and Gibbs sampler with Data Augmentation (GDA) algorithms. Full article
(This article belongs to the Special Issue Discrete-Valued Time Series)
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15 pages, 2994 KiB  
Review
Quantum Chaos and Level Dynamics
by Jakub Zakrzewski
Entropy 2023, 25(3), 491; https://doi.org/10.3390/e25030491 - 13 Mar 2023
Cited by 5 | Viewed by 1504
Abstract
We review the application of level dynamics to spectra of quantally chaotic systems. We show that the statistical mechanics approach gives us predictions about level statistics intermediate between integrable and chaotic dynamics. Then we discuss in detail different statistical measures involving level dynamics, [...] Read more.
We review the application of level dynamics to spectra of quantally chaotic systems. We show that the statistical mechanics approach gives us predictions about level statistics intermediate between integrable and chaotic dynamics. Then we discuss in detail different statistical measures involving level dynamics, such as level avoided-crossing distributions, level slope distributions, or level curvature distributions. We show both the aspects of universality in these distributions and their limitations. We concentrate in some detail on measures imported from the quantum information approach such as the fidelity susceptibility, and more generally, geometric tensor matrix elements. The possible open problems are suggested. Full article
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27 pages, 462 KiB  
Article
Stochastic Expectation Maximization Algorithm for Linear Mixed-Effects Model with Interactions in the Presence of Incomplete Data
by Alandra Zakkour, Cyril Perret and Yousri Slaoui
Entropy 2023, 25(3), 473; https://doi.org/10.3390/e25030473 - 8 Mar 2023
Cited by 1 | Viewed by 1498
Abstract
The purpose of this paper is to propose a new algorithm based on stochastic expectation maximization (SEM) to deal with the problem of unobserved values when multiple interactions in a linear mixed-effects model (LMEM) are present. We test the effectiveness of the proposed [...] Read more.
The purpose of this paper is to propose a new algorithm based on stochastic expectation maximization (SEM) to deal with the problem of unobserved values when multiple interactions in a linear mixed-effects model (LMEM) are present. We test the effectiveness of the proposed algorithm with the stochastic approximation expectation maximization (SAEM) and Monte Carlo Markov chain (MCMC) algorithms. This comparison is implemented to highlight the importance of including the maximum effects that can affect the model. The applications are made on both simulated psychological and real data. The findings demonstrate that our proposed SEM algorithm is highly preferable to the other competitor algorithms. Full article
(This article belongs to the Special Issue Monte Carlo Simulation in Statistical Physics)
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24 pages, 6744 KiB  
Article
Rate Distortion Theory for Descriptive Statistics
by Peter Harremoës
Entropy 2023, 25(3), 456; https://doi.org/10.3390/e25030456 - 5 Mar 2023
Cited by 1 | Viewed by 1809
Abstract
Rate distortion theory was developed for optimizing lossy compression of data, but it also has applications in statistics. In this paper, we illustrate how rate distortion theory can be used to analyze various datasets. The analysis involves testing, identification of outliers, choice of [...] Read more.
Rate distortion theory was developed for optimizing lossy compression of data, but it also has applications in statistics. In this paper, we illustrate how rate distortion theory can be used to analyze various datasets. The analysis involves testing, identification of outliers, choice of compression rate, calculation of optimal reconstruction points, and assigning “descriptive confidence regions” to the reconstruction points. We study four models or datasets of increasing complexity: clustering, Gaussian models, linear regression, and a dataset describing orientations of early Islamic mosques. These examples illustrate how rate distortion analysis may serve as a common framework for handling different statistical problems. Full article
(This article belongs to the Collection Feature Papers in Information Theory)
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10 pages, 4198 KiB  
Article
Spectral Form Factor and Dynamical Localization
by Črt Lozej
Entropy 2023, 25(3), 451; https://doi.org/10.3390/e25030451 - 4 Mar 2023
Viewed by 1536
Abstract
Quantum dynamical localization occurs when quantum interference stops the diffusion of wave packets in momentum space. The expectation is that dynamical localization will occur when the typical transport time of the momentum diffusion is greater than the Heisenberg time. The transport time is [...] Read more.
Quantum dynamical localization occurs when quantum interference stops the diffusion of wave packets in momentum space. The expectation is that dynamical localization will occur when the typical transport time of the momentum diffusion is greater than the Heisenberg time. The transport time is typically computed from the corresponding classical dynamics. In this paper, we present an alternative approach based purely on the study of spectral fluctuations of the quantum system. The information about the transport times is encoded in the spectral form factor, which is the Fourier transform of the two-point spectral autocorrelation function. We compute large samples of the energy spectra (of the order of 106 levels) and spectral form factors of 22 stadium billiards with parameter values across the transition between the localized and extended eigenstate regimes. The transport time is obtained from the point when the spectral form factor transitions from the non-universal to the universal regime predicted by random matrix theory. We study the dependence of the transport time on the parameter value and show the level repulsion exponents, which are known to be a good measure of dynamical localization, depend linearly on the transport times obtained in this way. Full article
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12 pages, 484 KiB  
Article
Two Features of the GINAR(1) Process and Their Impact on the Run-Length Performance of Geometric Control Charts
by Manuel Cabral Morais
Entropy 2023, 25(3), 444; https://doi.org/10.3390/e25030444 - 2 Mar 2023
Viewed by 1048
Abstract
The geometric first-order integer-valued autoregressive process (GINAR(1)) can be particularly useful to model relevant discrete-valued time series, namely in statistical process control. We resort to stochastic ordering to prove that the GINAR(1) process is a discrete-time Markov chain governed by a totally positive [...] Read more.
The geometric first-order integer-valued autoregressive process (GINAR(1)) can be particularly useful to model relevant discrete-valued time series, namely in statistical process control. We resort to stochastic ordering to prove that the GINAR(1) process is a discrete-time Markov chain governed by a totally positive order 2 (TP2) transition matrix.Stochastic ordering is also used to compare transition matrices referring to pairs of GINAR(1) processes with different values of the marginal mean. We assess and illustrate the implications of these two stochastic ordering results, namely on the properties of the run length of geometric charts for monitoring GINAR(1) counts. Full article
(This article belongs to the Special Issue Discrete-Valued Time Series)
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15 pages, 418 KiB  
Article
Design and Analysis of Joint Group Shuffled Scheduling Decoding Algorithm for Double LDPC Codes System
by Qiwang Chen, Yanzhao Ren, Lin Zhou, Chen Chen and Sanya Liu
Entropy 2023, 25(2), 357; https://doi.org/10.3390/e25020357 - 15 Feb 2023
Cited by 3 | Viewed by 1422
Abstract
In this paper, a joint group shuffled scheduling decoding (JGSSD) algorithm for a joint source-channel coding (JSCC) scheme based on double low-density parity-check (D-LDPC) codes is presented. The proposed algorithm considers the D-LDPC coding structure as a whole and applies shuffled scheduling to [...] Read more.
In this paper, a joint group shuffled scheduling decoding (JGSSD) algorithm for a joint source-channel coding (JSCC) scheme based on double low-density parity-check (D-LDPC) codes is presented. The proposed algorithm considers the D-LDPC coding structure as a whole and applies shuffled scheduling to each group; the grouping relies on the types or the length of the variable nodes (VNs). By comparison, the conventional shuffled scheduling decoding algorithm can be regarded as a special case of this proposed algorithm. A novel joint extrinsic information transfer (JEXIT) algorithm for the D-LDPC codes system with the JGSSD algorithm is proposed, by which the source and channel decoding are calculated with different grouping strategies to analyze the effects of the grouping strategy. Simulation results and comparisons verify the superiority of the JGSSD algorithm, which can adaptively trade off the decoding performance, complexity and latency. Full article
(This article belongs to the Special Issue Coding and Entropy)
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23 pages, 2911 KiB  
Article
The Typical Set and Entropy in Stochastic Systems with Arbitrary Phase Space Growth
by Rudolf Hanel and Bernat Corominas-Murtra
Entropy 2023, 25(2), 350; https://doi.org/10.3390/e25020350 - 14 Feb 2023
Cited by 1 | Viewed by 1568
Abstract
The existence of the typical set is key for data compression strategies and for the emergence of robust statistical observables in macroscopic physical systems. Standard approaches derive its existence from a restricted set of dynamical constraints. However, given its central role underlying the [...] Read more.
The existence of the typical set is key for data compression strategies and for the emergence of robust statistical observables in macroscopic physical systems. Standard approaches derive its existence from a restricted set of dynamical constraints. However, given its central role underlying the emergence of stable, almost deterministic statistical patterns, a question arises whether typical sets exist in much more general scenarios. We demonstrate here that the typical set can be defined and characterized from general forms of entropy for a much wider class of stochastic processes than was previously thought. This includes processes showing arbitrary path dependence, long range correlations or dynamic sampling spaces, suggesting that typicality is a generic property of stochastic processes, regardless of their complexity. We argue that the potential emergence of robust properties in complex stochastic systems provided by the existence of typical sets has special relevance to biological systems. Full article
(This article belongs to the Special Issue The Statistical Foundations of Entropy II)
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12 pages, 574 KiB  
Article
Toward Prediction of Financial Crashes with a D-Wave Quantum Annealer
by Yongcheng Ding, Javier Gonzalez-Conde, Lucas Lamata, José D. Martín-Guerrero, Enrique Lizaso, Samuel Mugel, Xi Chen, Román Orús, Enrique Solano and Mikel Sanz
Entropy 2023, 25(2), 323; https://doi.org/10.3390/e25020323 - 10 Feb 2023
Cited by 22 | Viewed by 3277
Abstract
The prediction of financial crashes in a complex financial network is known to be an NP-hard problem, which means that no known algorithm can efficiently find optimal solutions. We experimentally explore a novel approach to this problem by using a D-Wave quantum annealer, [...] Read more.
The prediction of financial crashes in a complex financial network is known to be an NP-hard problem, which means that no known algorithm can efficiently find optimal solutions. We experimentally explore a novel approach to this problem by using a D-Wave quantum annealer, benchmarking its performance for attaining a financial equilibrium. To be specific, the equilibrium condition of a nonlinear financial model is embedded into a higher-order unconstrained binary optimization (HUBO) problem, which is then transformed into a spin-1/2 Hamiltonian with at most, two-qubit interactions. The problem is thus equivalent to finding the ground state of an interacting spin Hamiltonian, which can be approximated with a quantum annealer. The size of the simulation is mainly constrained by the necessity of a large number of physical qubits representing a logical qubit with the correct connectivity. Our experiment paves the way for the codification of this quantitative macroeconomics problem in quantum annealers. Full article
(This article belongs to the Special Issue Quantum Control and Quantum Computing)
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17 pages, 3668 KiB  
Article
Sample, Fuzzy and Distribution Entropies of Heart Rate Variability: What Do They Tell Us on Cardiovascular Complexity?
by Paolo Castiglioni, Giampiero Merati, Gianfranco Parati and Andrea Faini
Entropy 2023, 25(2), 281; https://doi.org/10.3390/e25020281 - 2 Feb 2023
Cited by 6 | Viewed by 2664
Abstract
Distribution Entropy (DistEn) has been introduced as an alternative to Sample Entropy (SampEn) to assess the heart rate variability (HRV) on much shorter series without the arbitrary definition of distance thresholds. However, DistEn, considered a measure of cardiovascular complexity, differs substantially from SampEn [...] Read more.
Distribution Entropy (DistEn) has been introduced as an alternative to Sample Entropy (SampEn) to assess the heart rate variability (HRV) on much shorter series without the arbitrary definition of distance thresholds. However, DistEn, considered a measure of cardiovascular complexity, differs substantially from SampEn or Fuzzy Entropy (FuzzyEn), both measures of HRV randomness. This work aims to compare DistEn, SampEn, and FuzzyEn analyzing postural changes (expected to modify the HRV randomness through a sympatho/vagal shift without affecting the cardiovascular complexity) and low-level spinal cord injuries (SCI, whose impaired integrative regulation may alter the system complexity without affecting the HRV spectrum). We recorded RR intervals in able-bodied (AB) and SCI participants in supine and sitting postures, evaluating DistEn, SampEn, and FuzzyEn over 512 beats. The significance of “case” (AB vs. SCI) and “posture” (supine vs. sitting) was assessed by longitudinal analysis. Multiscale DistEn (mDE), SampEn (mSE), and FuzzyEn (mFE) compared postures and cases at each scale between 2 and 20 beats. Unlike SampEn and FuzzyEn, DistEn is affected by the spinal lesion but not by the postural sympatho/vagal shift. The multiscale approach shows differences between AB and SCI sitting participants at the largest mFE scales and between postures in AB participants at the shortest mSE scales. Thus, our results support the hypothesis that DistEn measures cardiovascular complexity while SampEn/FuzzyEn measure HRV randomness, highlighting that together these methods integrate the information each of them provides. Full article
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13 pages, 1425 KiB  
Article
Quantum Bounds on the Generalized Lyapunov Exponents
by Silvia Pappalardi and Jorge Kurchan
Entropy 2023, 25(2), 246; https://doi.org/10.3390/e25020246 - 30 Jan 2023
Cited by 10 | Viewed by 1841
Abstract
We discuss the generalized quantum Lyapunov exponents Lq, defined from the growth rate of the powers of the square commutator. They may be related to an appropriately defined thermodynamic limit of the spectrum of the commutator, which plays the role of [...] Read more.
We discuss the generalized quantum Lyapunov exponents Lq, defined from the growth rate of the powers of the square commutator. They may be related to an appropriately defined thermodynamic limit of the spectrum of the commutator, which plays the role of a large deviation function, obtained from the exponents Lq via a Legendre transform. We show that such exponents obey a generalized bound to chaos due to the fluctuation–dissipation theorem, as already discussed in the literature. The bounds for larger q are actually stronger, placing a limit on the large deviations of chaotic properties. Our findings at infinite temperature are exemplified by a numerical study of the kicked top, a paradigmatic model of quantum chaos. Full article
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16 pages, 1935 KiB  
Article
Measurement-Based Quantum Thermal Machines with Feedback Control
by Bibek Bhandari, Robert Czupryniak, Paolo Andrea Erdman and Andrew N. Jordan
Entropy 2023, 25(2), 204; https://doi.org/10.3390/e25020204 - 20 Jan 2023
Cited by 6 | Viewed by 2402
Abstract
We investigated coupled-qubit-based thermal machines powered by quantum measurements and feedback. We considered two different versions of the machine: (1) a quantum Maxwell’s demon, where the coupled-qubit system is connected to a detachable single shared bath, and (2) a measurement-assisted refrigerator, where the [...] Read more.
We investigated coupled-qubit-based thermal machines powered by quantum measurements and feedback. We considered two different versions of the machine: (1) a quantum Maxwell’s demon, where the coupled-qubit system is connected to a detachable single shared bath, and (2) a measurement-assisted refrigerator, where the coupled-qubit system is in contact with a hot and cold bath. In the quantum Maxwell’s demon case, we discuss both discrete and continuous measurements. We found that the power output from a single qubit-based device can be improved by coupling it to the second qubit. We further found that the simultaneous measurement of both qubits can produce higher net heat extraction compared to two setups operated in parallel where only single-qubit measurements are performed. In the refrigerator case, we used continuous measurement and unitary operations to power the coupled-qubit-based refrigerator. We found that the cooling power of a refrigerator operated with swap operations can be enhanced by performing suitable measurements. Full article
(This article belongs to the Special Issue Thermodynamics in Quantum and Mesoscopic Systems)
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22 pages, 2693 KiB  
Article
Scarring in Rough Rectangular Billiards
by Felix M. Izrailev, German A. Luna-Acosta and J. A. Mendez-Bermudez
Entropy 2023, 25(2), 189; https://doi.org/10.3390/e25020189 - 18 Jan 2023
Viewed by 1343
Abstract
We study the mechanism of scarring of eigenstates in rectangular billiards with slightly corrugated surfaces and show that it is very different from that known in Sinai and Bunimovich billiards. We demonstrate that there are two sets of scar states. One set is [...] Read more.
We study the mechanism of scarring of eigenstates in rectangular billiards with slightly corrugated surfaces and show that it is very different from that known in Sinai and Bunimovich billiards. We demonstrate that there are two sets of scar states. One set is related to the bouncing ball trajectories in the configuration space of the corresponding classical billiard. A second set of scar-like states emerges in the momentum space, which originated from the plane-wave states of the unperturbed flat billiard. In the case of billiards with one rough surface, the numerical data demonstrate the repulsion of eigenstates from this surface. When two horizontal rough surfaces are considered, the repulsion effect is either enhanced or canceled depending on whether the rough profiles are symmetric or antisymmetric. The effect of repulsion is quite strong and influences the structure of all eigenstates, indicating that the symmetric properties of the rough profiles are important for the problem of scattering of electromagnetic (or electron) waves through quasi-one-dimensional waveguides. Our approach is based on the reduction of the model of one particle in the billiard with corrugated surfaces to a model of two artificial particles in the billiard with flat surfaces, however, with an effective interaction between these particles. As a result, the analysis is conducted in terms of a two-particle basis, and the roughness of the billiard boundaries is absorbed by a quite complicated potential. Full article
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14 pages, 809 KiB  
Article
Geometrical Bounds on Irreversibility in Squeezed Thermal Bath
by Chen-Juan Zou, Yue Li, Jia-Kun Xu, Jia-Bin You, Ching Eng Png and Wan-Li Yang
Entropy 2023, 25(1), 128; https://doi.org/10.3390/e25010128 - 9 Jan 2023
Cited by 2 | Viewed by 1853
Abstract
Irreversible entropy production (IEP) plays an important role in quantum thermodynamic processes. Here, we investigate the geometrical bounds of IEP in nonequilibrium thermodynamics by exemplifying a system coupled to a squeezed thermal bath subject to dissipation and dephasing, respectively. We find that the [...] Read more.
Irreversible entropy production (IEP) plays an important role in quantum thermodynamic processes. Here, we investigate the geometrical bounds of IEP in nonequilibrium thermodynamics by exemplifying a system coupled to a squeezed thermal bath subject to dissipation and dephasing, respectively. We find that the geometrical bounds of the IEP always shift in a contrary way under dissipation and dephasing, where the lower and upper bounds turning to be tighter occur in the situation of dephasing and dissipation, respectively. However, either under dissipation or under dephasing, we may reduce both the critical time of the IEP itself and the critical time of the bounds for reaching an equilibrium by harvesting the benefits of squeezing effects in which the values of the IEP, quantifying the degree of thermodynamic irreversibility, also become smaller. Therefore, due to the nonequilibrium nature of the squeezed thermal bath, the system–bath interaction energy has a prominent impact on the IEP, leading to tightness of its bounds. Our results are not contradictory with the second law of thermodynamics by involving squeezing of the bath as an available resource, which can improve the performance of quantum thermodynamic devices. Full article
(This article belongs to the Special Issue Quantum Thermodynamics: Fundamentals and Applications)
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15 pages, 832 KiB  
Article
Implications of Spectral Interlacing for Quantum Graphs
by Junjie Lu, Tobias Hofmann, Ulrich Kuhl and Hans-Jürgen Stöckmann
Entropy 2023, 25(1), 109; https://doi.org/10.3390/e25010109 - 4 Jan 2023
Cited by 1 | Viewed by 1488
Abstract
Quantum graphs are ideally suited to studying the spectral statistics of chaotic systems. Depending on the boundary conditions at the vertices, there are Neumann and Dirichlet graphs. The latter ones correspond to totally disassembled graphs with a spectrum being the superposition of the [...] Read more.
Quantum graphs are ideally suited to studying the spectral statistics of chaotic systems. Depending on the boundary conditions at the vertices, there are Neumann and Dirichlet graphs. The latter ones correspond to totally disassembled graphs with a spectrum being the superposition of the spectra of the individual bonds. According to the interlacing theorem, Neumann and Dirichlet eigenvalues on average alternate as a function of the wave number, with the consequence that the Neumann spectral statistics deviate from random matrix predictions. There is, e.g., a strict upper bound for the spacing of neighboring Neumann eigenvalues given by the number of bonds (in units of the mean level spacing). Here, we present analytic expressions for level spacing distribution and number variance for ensemble averaged spectra of Dirichlet graphs in dependence of the bond number, and compare them with numerical results. For a number of small Neumann graphs, numerical results for the same quantities are shown, and their deviations from random matrix predictions are discussed. Full article
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17 pages, 354 KiB  
Article
Invariant-Parameterized Exact Evolution Operator for SU(2) Systems with Time-Dependent Hamiltonian
by Hiromichi Nakazato, Alessandro Sergi, Agostino Migliore and Antonino Messina
Entropy 2023, 25(1), 96; https://doi.org/10.3390/e25010096 - 3 Jan 2023
Cited by 5 | Viewed by 1482
Abstract
We report the step-by-step construction of the exact, closed and explicit expression for the evolution operator U(t) of a localized and isolated qubit in an arbitrary time-dependent field, which for concreteness we assume to be a magnetic field. Our approach [...] Read more.
We report the step-by-step construction of the exact, closed and explicit expression for the evolution operator U(t) of a localized and isolated qubit in an arbitrary time-dependent field, which for concreteness we assume to be a magnetic field. Our approach is based on the existence of two independent dynamical invariants that enter the expression of SU(2) by means of two strictly related time-dependent, real or complex, parameters. The usefulness of our approach is demonstrated by exactly solving the quantum dynamics of a qubit subject to a controllable time-dependent field that can be realized in the laboratory. We further discuss possible applications to any SU(2) model, as well as the applicability of our method to realistic physical scenarios with different symmetry properties. Full article
(This article belongs to the Special Issue Quantum Nonstationary Systems)
10 pages, 660 KiB  
Article
Entanglement Dynamics and Classical Complexity
by Jiaozi Wang, Barbara Dietz, Dario Rosa and Giuliano Benenti
Entropy 2023, 25(1), 97; https://doi.org/10.3390/e25010097 - 3 Jan 2023
Viewed by 1242
Abstract
We study the dynamical generation of entanglement for a two-body interacting system, starting from a separable coherent state. We show analytically that in the quasiclassical regime the entanglement growth rate can be simply computed by means of the underlying classical dynamics. Furthermore, this [...] Read more.
We study the dynamical generation of entanglement for a two-body interacting system, starting from a separable coherent state. We show analytically that in the quasiclassical regime the entanglement growth rate can be simply computed by means of the underlying classical dynamics. Furthermore, this rate is given by the Kolmogorov–Sinai entropy, which characterizes the dynamical complexity of classical motion. Our results, illustrated by numerical simulations on a model of coupled rotators, establish in the quasiclassical regime a link between the generation of entanglement, a purely quantum phenomenon, and classical complexity. Full article
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18 pages, 1344 KiB  
Article
COVID-19 Effects on the Relationship between Cryptocurrencies: Can It Be Contagion? Insights from Econophysics Approaches
by Dora Almeida, Andreia Dionísio, Isabel Vieira and Paulo Ferreira
Entropy 2023, 25(1), 98; https://doi.org/10.3390/e25010098 - 3 Jan 2023
Cited by 5 | Viewed by 1741
Abstract
Cryptocurrencies are relatively new and innovative financial assets. They are a topic of interest to investors and academics due to their distinctive features. Whether financial or not, extraordinary events are one of the biggest challenges facing financial markets. The onset of the COVID-19 [...] Read more.
Cryptocurrencies are relatively new and innovative financial assets. They are a topic of interest to investors and academics due to their distinctive features. Whether financial or not, extraordinary events are one of the biggest challenges facing financial markets. The onset of the COVID-19 pandemic crisis, considered by some authors a “black swan”, is one of these events. In this study, we assess integration and contagion in the cryptocurrency market in the COVID-19 pandemic context, using two entropy-based measures: mutual information and transfer entropy. Both methodologies reveal that cryptocurrencies exhibit mixed levels of integration before and after the onset of the pandemic. Cryptocurrencies displaying higher integration before the event experienced a decline in such link after the world became aware of the first cases of pneumonia in Wuhan city. In what concerns contagion, mutual information provided evidence of its presence solely for the Huobi Token, and the transfer entropy analysis pointed out Tether and Huobi Token as its main source. As both analyses indicate no contagion from the pandemic turmoil to these financial assets, cryptocurrencies may be good investment options in case of real global shocks, such as the one provoked by the COVID-19 outbreak. Full article
(This article belongs to the Special Issue Cryptocurrency Behavior under Econophysics Approaches)
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31 pages, 1844 KiB  
Article
Extreme Eigenvalues and the Emerging Outlier in Rank-One Non-Hermitian Deformations of the Gaussian Unitary Ensemble
by Yan V. Fyodorov, Boris A. Khoruzhenko and Mihail Poplavskyi
Entropy 2023, 25(1), 74; https://doi.org/10.3390/e25010074 - 30 Dec 2022
Cited by 4 | Viewed by 1489
Abstract
Complex eigenvalues of random matrices J=GUE+iγdiag(1,0,,0) provide the simplest model for studying resonances in wave scattering from a quantum chaotic system via a single open channel. It is [...] Read more.
Complex eigenvalues of random matrices J=GUE+iγdiag(1,0,,0) provide the simplest model for studying resonances in wave scattering from a quantum chaotic system via a single open channel. It is known that in the limit of large matrix dimensions N1 the eigenvalue density of J undergoes an abrupt restructuring at γ=1, the critical threshold beyond which a single eigenvalue outlier (“broad resonance”) appears. We provide a detailed description of this restructuring transition, including the scaling with N of the width of the critical region about the outlier threshold γ=1 and the associated scaling for the real parts (“resonance positions”) and imaginary parts (“resonance widths”) of the eigenvalues which are farthest away from the real axis. In the critical regime we determine the density of such extreme eigenvalues, and show how the outlier gradually separates itself from the rest of the extreme eigenvalues. Finally, we describe the fluctuations in the height of the eigenvalue outlier for large but finite N in terms of the associated large deviation function. Full article
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52 pages, 9913 KiB  
Review
Quantum Chaos in the Dynamics of Molecules
by Kazuo Takatsuka
Entropy 2023, 25(1), 63; https://doi.org/10.3390/e25010063 - 29 Dec 2022
Cited by 3 | Viewed by 3088
Abstract
Quantum chaos is reviewed from the viewpoint of “what is molecule?”, particularly placing emphasis on their dynamics. Molecules are composed of heavy nuclei and light electrons, and thereby the very basic molecular theory due to Born and Oppenheimer gives a view that quantum [...] Read more.
Quantum chaos is reviewed from the viewpoint of “what is molecule?”, particularly placing emphasis on their dynamics. Molecules are composed of heavy nuclei and light electrons, and thereby the very basic molecular theory due to Born and Oppenheimer gives a view that quantum electronic states provide potential functions working on nuclei, which in turn are often treated classically or semiclassically. Therefore, the classic study of chaos in molecular science began with those nuclear dynamics particularly about the vibrational energy randomization within a molecule. Statistical laws in probabilities and rates of chemical reactions even for small molecules of several atoms are among the chemical phenomena requiring the notion of chaos. Particularly the dynamics behind unimolecular decomposition are referred to as Intra-molecular Vibrational energy Redistribution (IVR). Semiclassical mechanics is also one of the main research fields of quantum chaos. We herein demonstrate chaos that appears only in semiclassical and full quantum dynamics. A fundamental phenomenon possibly giving birth to quantum chaos is “bifurcation and merging” of quantum wavepackets, rather than “stretching and folding” of the baker’s transformation and the horseshoe map as a geometrical foundation of classical chaos. Such wavepacket bifurcation and merging are indeed experimentally measurable as we showed before in the series of studies on real-time probing of nonadiabatic chemical reactions. After tracking these aspects of molecular chaos, we will explore quantum chaos found in nonadiabatic electron wavepacket dynamics, which emerges in the realm far beyond the Born-Oppenheimer paradigm. In this class of chaos, we propose a notion of Intra-molecular Nonadiabatic Electronic Energy Redistribution (INEER), which is a consequence of the chaotic fluxes of electrons and energy within a molecule. Full article
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25 pages, 544 KiB  
Review
Diffusion Coefficient of a Brownian Particle in Equilibrium and Nonequilibrium: Einstein Model and Beyond
by Jakub Spiechowicz, Ivan G. Marchenko, Peter Hänggi and Jerzy Łuczka
Entropy 2023, 25(1), 42; https://doi.org/10.3390/e25010042 - 26 Dec 2022
Cited by 12 | Viewed by 4764
Abstract
The diffusion of small particles is omnipresent in many processes occurring in nature. As such, it is widely studied and exerted in almost all branches of sciences. It constitutes such a broad and often rather complex subject of exploration that we opt here [...] Read more.
The diffusion of small particles is omnipresent in many processes occurring in nature. As such, it is widely studied and exerted in almost all branches of sciences. It constitutes such a broad and often rather complex subject of exploration that we opt here to narrow our survey to the case of the diffusion coefficient for a Brownian particle that can be modeled in the framework of Langevin dynamics. Our main focus centers on the temperature dependence of the diffusion coefficient for several fundamental models of diverse physical systems. Starting out with diffusion in equilibrium for which the Einstein theory holds, we consider a number of physical situations outside of free Brownian motion and end by surveying nonequilibrium diffusion for a time-periodically driven Brownian particle dwelling randomly in a periodic potential. For this latter situation the diffusion coefficient exhibits an intriguingly non-monotonic dependence on temperature. Full article
(This article belongs to the Collection Foundations of Statistical Mechanics)
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12 pages, 296 KiB  
Article
Entanglement-Assisted Quantum Codes from Cyclic Codes
by Francisco Revson F. Pereira and Stefano Mancini
Entropy 2023, 25(1), 37; https://doi.org/10.3390/e25010037 - 24 Dec 2022
Cited by 10 | Viewed by 1546
Abstract
Entanglement-assisted quantum-error-correcting (EAQEC) codes are quantum codes which use entanglement as a resource. These codes can provide better error correction than the (entanglement unassisted) codes derived from the traditional stabilizer formalism. In this paper, we provide a general method to construct EAQEC codes [...] Read more.
Entanglement-assisted quantum-error-correcting (EAQEC) codes are quantum codes which use entanglement as a resource. These codes can provide better error correction than the (entanglement unassisted) codes derived from the traditional stabilizer formalism. In this paper, we provide a general method to construct EAQEC codes from cyclic codes. Afterwards, the method is applied to Reed–Solomon codes, BCH codes, and general cyclic codes. We use the Euclidean and Hermitian construction of EAQEC codes. Three families have been created: two families of EAQEC codes are maximal distance separable (MDS), and one is almost MDS or almost near MDS. The comparison of the codes in this paper is mostly based on the quantum Singleton bound. Full article
(This article belongs to the Special Issue Advances in Quantum Computing)
21 pages, 5934 KiB  
Article
Chaos and Thermalization in the Spin-Boson Dicke Model
by David Villaseñor, Saúl Pilatowsky-Cameo, Miguel A. Bastarrachea-Magnani, Sergio Lerma-Hernández, Lea F. Santos and Jorge G. Hirsch
Entropy 2023, 25(1), 8; https://doi.org/10.3390/e25010008 - 21 Dec 2022
Cited by 13 | Viewed by 1919
Abstract
We present a detailed analysis of the connection between chaos and the onset of thermalization in the spin-boson Dicke model. This system has a well-defined classical limit with two degrees of freedom, and it presents both regular and chaotic regions. Our studies of [...] Read more.
We present a detailed analysis of the connection between chaos and the onset of thermalization in the spin-boson Dicke model. This system has a well-defined classical limit with two degrees of freedom, and it presents both regular and chaotic regions. Our studies of the eigenstate expectation values and the distributions of the off-diagonal elements of the number of photons and the number of excited atoms validate the diagonal and off-diagonal eigenstate thermalization hypothesis (ETH) in the chaotic region, thus ensuring thermalization. The validity of the ETH reflects the chaotic structure of the eigenstates, which we corroborate using the von Neumann entanglement entropy and the Shannon entropy. Our results for the Shannon entropy also make evident the advantages of the so-called “efficient basis” over the widespread employed Fock basis when investigating the unbounded spectrum of the Dicke model. The efficient basis gives us access to a larger number of converged states than what can be reached with the Fock basis. Full article
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25 pages, 2133 KiB  
Article
Canonical Density Matrices from Eigenstates of Mixed Systems
by Mahdi Kourehpaz, Stefan Donsa, Fabian Lackner, Joachim Burgdörfer and Iva Březinová
Entropy 2022, 24(12), 1740; https://doi.org/10.3390/e24121740 - 29 Nov 2022
Cited by 3 | Viewed by 5932
Abstract
One key issue of the foundation of statistical mechanics is the emergence of equilibrium ensembles in isolated and closed quantum systems. Recently, it was predicted that in the thermodynamic (N) limit of large quantum many-body systems, canonical density matrices [...] Read more.
One key issue of the foundation of statistical mechanics is the emergence of equilibrium ensembles in isolated and closed quantum systems. Recently, it was predicted that in the thermodynamic (N) limit of large quantum many-body systems, canonical density matrices emerge for small subsystems from almost all pure states. This notion of canonical typicality is assumed to originate from the entanglement between subsystem and environment and the resulting intrinsic quantum complexity of the many-body state. For individual eigenstates, it has been shown that local observables show thermal properties provided the eigenstate thermalization hypothesis holds, which requires the system to be quantum-chaotic. In the present paper, we study the emergence of thermal states in the regime of a quantum analog of a mixed phase space. Specifically, we study the emergence of the canonical density matrix of an impurity upon reduction from isolated energy eigenstates of a large but finite quantum system the impurity is embedded in. Our system can be tuned by means of a single parameter from quantum integrability to quantum chaos and corresponds in between to a system with mixed quantum phase space. We show that the probability for finding a canonical density matrix when reducing the ensemble of energy eigenstates of the finite many-body system can be quantitatively controlled and tuned by the degree of quantum chaos present. For the transition from quantum integrability to quantum chaos, we find a continuous and universal (i.e., size-independent) relation between the fraction of canonical eigenstates and the degree of chaoticity as measured by the Brody parameter or the Shannon entropy. Full article
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13 pages, 356 KiB  
Article
Entropy Optimization, Generalized Logarithms, and Duality Relations
by Angel R. Plastino, Constantino Tsallis, Roseli S. Wedemann and Hans J. Haubold
Entropy 2022, 24(12), 1723; https://doi.org/10.3390/e24121723 - 25 Nov 2022
Cited by 5 | Viewed by 1617
Abstract
Several generalizations or extensions of the Boltzmann–Gibbs thermostatistics, based on non-standard entropies, have been the focus of considerable research activity in recent years. Among these, the power-law, non-additive entropies [...] Read more.
Several generalizations or extensions of the Boltzmann–Gibbs thermostatistics, based on non-standard entropies, have been the focus of considerable research activity in recent years. Among these, the power-law, non-additive entropies Sqk1ipiqq1(qR;S1=SBGkipilnpi) have harvested the largest number of successful applications. The specific structural features of the Sq thermostatistics, therefore, are worthy of close scrutiny. In the present work, we analyze one of these features, according to which the q-logarithm function lnqxx1q11q(ln1x=lnx) associated with the Sq entropy is linked, via a duality relation, to the q-exponential function characterizing the maximum-entropy probability distributions. We enquire into which entropic functionals lead to this or similar structures, and investigate the corresponding duality relations. Full article
(This article belongs to the Special Issue Non-additive Entropy Formulas: Motivation and Derivations)
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23 pages, 5468 KiB  
Article
Functional Connectome of the Human Brain with Total Correlation
by Qiang Li, Greg Ver Steeg, Shujian Yu and Jesus Malo
Entropy 2022, 24(12), 1725; https://doi.org/10.3390/e24121725 - 25 Nov 2022
Cited by 5 | Viewed by 3340
Abstract
Recent studies proposed the use of Total Correlation to describe functional connectivity among brain regions as a multivariate alternative to conventional pairwise measures such as correlation or mutual information. In this work, we build on this idea to infer a large-scale (whole-brain) connectivity [...] Read more.
Recent studies proposed the use of Total Correlation to describe functional connectivity among brain regions as a multivariate alternative to conventional pairwise measures such as correlation or mutual information. In this work, we build on this idea to infer a large-scale (whole-brain) connectivity network based on Total Correlation and show the possibility of using this kind of network as biomarkers of brain alterations. In particular, this work uses Correlation Explanation (CorEx) to estimate Total Correlation. First, we prove that CorEx estimates of Total Correlation and clustering results are trustable compared to ground truth values. Second, the inferred large-scale connectivity network extracted from the more extensive open fMRI datasets is consistent with existing neuroscience studies, but, interestingly, can estimate additional relations beyond pairwise regions. And finally, we show how the connectivity graphs based on Total Correlation can also be an effective tool to aid in the discovery of brain diseases. Full article
(This article belongs to the Special Issue Applications of Information Theory in Neuroscience)
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11 pages, 3187 KiB  
Article
Quantum Coherence in Loopless Superconductive Networks
by Massimiliano Lucci, Valerio Campanari, Davide Cassi, Vittorio Merlo, Francesco Romeo, Gaetano Salina and Matteo Cirillo
Entropy 2022, 24(11), 1690; https://doi.org/10.3390/e24111690 - 18 Nov 2022
Cited by 3 | Viewed by 1279
Abstract
Measurements indicating that planar networks of superconductive islands connected by Josephson junctions display long-range quantum coherence are reported. The networks consist of superconducting islands connected by Josephson junctions and have a tree-like topological structure containing no loops. Enhancements of superconductive gaps over specific [...] Read more.
Measurements indicating that planar networks of superconductive islands connected by Josephson junctions display long-range quantum coherence are reported. The networks consist of superconducting islands connected by Josephson junctions and have a tree-like topological structure containing no loops. Enhancements of superconductive gaps over specific branches of the networks and sharp increases in pair currents are the main signatures of the coherent states. In order to unambiguously attribute the observed effects to branches being embedded in the networks, comparisons with geometrically equivalent—but isolated—counterparts are reported. Tuning the Josephson coupling energy by an external magnetic field generates increases in the Josephson currents, along the above-mentioned specific branches, which follow a functional dependence typical of phase transitions. Results are presented for double comb and star geometry networks, and in both cases, the observed effects provide positive quantitative evidence of the predictions of existing theoretical models. Full article
(This article belongs to the Special Issue Quantum Information and Quantum Optics)
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17 pages, 3674 KiB  
Article
Universal Single-Mode Lasing in Fully Chaotic Billiard Lasers
by Mengyu You, Daisuke Sakakibara, Kota Makino, Yonosuke Morishita, Kazutoshi Matsumura, Yuta Kawashima, Manao Yoshikawa, Mahiro Tonosaki, Kazutaka Kanno, Atsushi Uchida, Satoshi Sunada, Susumu Shinohara and Takahisa Harayama
Entropy 2022, 24(11), 1648; https://doi.org/10.3390/e24111648 - 14 Nov 2022
Cited by 4 | Viewed by 1845
Abstract
By numerical simulations and experiments of fully chaotic billiard lasers, we show that single-mode lasing states are stable, whereas multi-mode lasing states are unstable when the size of the billiard is much larger than the wavelength and the external pumping power is sufficiently [...] Read more.
By numerical simulations and experiments of fully chaotic billiard lasers, we show that single-mode lasing states are stable, whereas multi-mode lasing states are unstable when the size of the billiard is much larger than the wavelength and the external pumping power is sufficiently large. On the other hand, for integrable billiard lasers, it is shown that multi-mode lasing states are stable, whereas single-mode lasing states are unstable. These phenomena arise from the combination of two different nonlinear effects of mode-interaction due to the active lasing medium and deformation of the billiard shape. Investigations of billiard lasers with various shapes revealed that single-mode lasing is a universal phenomenon for fully chaotic billiard lasers. Full article
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18 pages, 733 KiB  
Article
Dependency Structures in Cryptocurrency Market from High to Low Frequency
by Antonio Briola and Tomaso Aste
Entropy 2022, 24(11), 1548; https://doi.org/10.3390/e24111548 - 28 Oct 2022
Cited by 4 | Viewed by 2042
Abstract
We investigate logarithmic price returns cross-correlations at different time horizons for a set of 25 liquid cryptocurrencies traded on the FTX digital currency exchange. We study how the structure of the Minimum Spanning Tree (MST) and the Triangulated Maximally Filtered Graph (TMFG) evolve [...] Read more.
We investigate logarithmic price returns cross-correlations at different time horizons for a set of 25 liquid cryptocurrencies traded on the FTX digital currency exchange. We study how the structure of the Minimum Spanning Tree (MST) and the Triangulated Maximally Filtered Graph (TMFG) evolve from high (15 s) to low (1 day) frequency time resolutions. For each horizon, we test the stability, statistical significance and economic meaningfulness of the networks. Results give a deep insight into the evolutionary process of the time dependent hierarchical organization of the system under analysis. A decrease in correlation between pairs of cryptocurrencies is observed for finer time sampling resolutions. A growing structure emerges for coarser ones, highlighting multiple changes in the hierarchical reference role played by mainstream cryptocurrencies. This effect is studied both in its pairwise realizations and intra-sector ones. Full article
(This article belongs to the Special Issue Recent Trends and Developments in Econophysics)
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18 pages, 381 KiB  
Article
An Overview on Irreversible Port-Hamiltonian Systems
by Hector Ramirez and Yann Le Gorrec
Entropy 2022, 24(10), 1478; https://doi.org/10.3390/e24101478 - 17 Oct 2022
Cited by 4 | Viewed by 1834
Abstract
A comprehensive overview of the irreversible port-Hamiltonian system’s formulation for finite and infinite dimensional systems defined on 1D spatial domains is provided in a unified manner. The irreversible port-Hamiltonian system formulation shows the extension of classical port-Hamiltonian system formulations to cope with irreversible [...] Read more.
A comprehensive overview of the irreversible port-Hamiltonian system’s formulation for finite and infinite dimensional systems defined on 1D spatial domains is provided in a unified manner. The irreversible port-Hamiltonian system formulation shows the extension of classical port-Hamiltonian system formulations to cope with irreversible thermodynamic systems for finite and infinite dimensional systems. This is achieved by including, in an explicit manner, the coupling between irreversible mechanical and thermal phenomena with the thermal domain as an energy-preserving and entropy-increasing operator. Similarly to Hamiltonian systems, this operator is skew-symmetric, guaranteeing energy conservation. To distinguish from Hamiltonian systems, the operator depends on co-state variables and is, hence, a nonlinear-function in the gradient of the total energy. This is what allows encoding the second law as a structural property of irreversible port-Hamiltonian systems. The formalism encompasses coupled thermo-mechanical systems and purely reversible or conservative systems as a particular case. This appears clearly when splitting the state space such that the entropy coordinate is separated from other state variables. Several examples have been used to illustrate the formalism, both for finite and infinite dimensional systems, and a discussion on ongoing and future studies is provided. Full article
(This article belongs to the Special Issue Geometric Structure of Thermodynamics: Theory and Applications)
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29 pages, 670 KiB  
Article
“Yet Once More”: The Double-Slit Experiment and Quantum Discontinuity
by Arkady Plotnitsky
Entropy 2022, 24(10), 1455; https://doi.org/10.3390/e24101455 - 12 Oct 2022
Cited by 5 | Viewed by 1749
Abstract
This article reconsiders the double-slit experiment from the nonrealist or, in terms of this article, “reality-without-realism” (RWR) perspective, grounded in the combination of three forms of quantum discontinuity: (1) “Heisenberg discontinuity”, defined by the impossibility of a representation or even conception of how [...] Read more.
This article reconsiders the double-slit experiment from the nonrealist or, in terms of this article, “reality-without-realism” (RWR) perspective, grounded in the combination of three forms of quantum discontinuity: (1) “Heisenberg discontinuity”, defined by the impossibility of a representation or even conception of how quantum phenomena come about, even though quantum theory (such as quantum mechanics or quantum field theory) predicts the data in question strictly in accord with what is observed in quantum experiments); (2) “Bohr discontinuity”, defined, under the assumption of Heisenberg discontinuity, by the view that quantum phenomena and the data observed therein are described by classical and not quantum theory, even though classical physics cannot predict them; and (3) “Dirac discontinuity” (not considered by Dirac himself, but suggested by his equation), according to which the concept of a quantum object, such as a photon or electron, is an idealization only applicable at the time of observation and not to something that exists independently in nature. Dirac discontinuity is of particular importance for the article’s foundational argument and its analysis of the double-slit experiment. Full article
(This article belongs to the Special Issue Quantum Information and Probability: From Foundations to Engineering)
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35 pages, 988 KiB  
Article
Revisiting Chernoff Information with Likelihood Ratio Exponential Families
by Frank Nielsen
Entropy 2022, 24(10), 1400; https://doi.org/10.3390/e24101400 - 1 Oct 2022
Cited by 6 | Viewed by 4151
Abstract
The Chernoff information between two probability measures is a statistical divergence measuring their deviation defined as their maximally skewed Bhattacharyya distance. Although the Chernoff information was originally introduced for bounding the Bayes error in statistical hypothesis testing, the divergence found many other applications [...] Read more.
The Chernoff information between two probability measures is a statistical divergence measuring their deviation defined as their maximally skewed Bhattacharyya distance. Although the Chernoff information was originally introduced for bounding the Bayes error in statistical hypothesis testing, the divergence found many other applications due to its empirical robustness property found in applications ranging from information fusion to quantum information. From the viewpoint of information theory, the Chernoff information can also be interpreted as a minmax symmetrization of the Kullback–Leibler divergence. In this paper, we first revisit the Chernoff information between two densities of a measurable Lebesgue space by considering the exponential families induced by their geometric mixtures: The so-called likelihood ratio exponential families. Second, we show how to (i) solve exactly the Chernoff information between any two univariate Gaussian distributions or get a closed-form formula using symbolic computing, (ii) report a closed-form formula of the Chernoff information of centered Gaussians with scaled covariance matrices and (iii) use a fast numerical scheme to approximate the Chernoff information between any two multivariate Gaussian distributions. Full article
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17 pages, 399 KiB  
Article
Information Theory for Biological Sequence Classification: A Novel Feature Extraction Technique Based on Tsallis Entropy
by Robson P. Bonidia, Anderson P. Avila Santos, Breno L. S. de Almeida, Peter F. Stadler, Ulisses Nunes da Rocha, Danilo S. Sanches and André C. P. L. F. de Carvalho
Entropy 2022, 24(10), 1398; https://doi.org/10.3390/e24101398 - 1 Oct 2022
Cited by 2 | Viewed by 2045
Abstract
In recent years, there has been an exponential growth in sequencing projects due to accelerated technological advances, leading to a significant increase in the amount of data and resulting in new challenges for biological sequence analysis. Consequently, the use of techniques capable of [...] Read more.
In recent years, there has been an exponential growth in sequencing projects due to accelerated technological advances, leading to a significant increase in the amount of data and resulting in new challenges for biological sequence analysis. Consequently, the use of techniques capable of analyzing large amounts of data has been explored, such as machine learning (ML) algorithms. ML algorithms are being used to analyze and classify biological sequences, despite the intrinsic difficulty in extracting and finding representative biological sequence methods suitable for them. Thereby, extracting numerical features to represent sequences makes it statistically feasible to use universal concepts from Information Theory, such as Tsallis and Shannon entropy. In this study, we propose a novel Tsallis entropy-based feature extractor to provide useful information to classify biological sequences. To assess its relevance, we prepared five case studies: (1) an analysis of the entropic index q; (2) performance testing of the best entropic indices on new datasets; (3) a comparison made with Shannon entropy and (4) generalized entropies; (5) an investigation of the Tsallis entropy in the context of dimensionality reduction. As a result, our proposal proved to be effective, being superior to Shannon entropy and robust in terms of generalization, and also potentially representative for collecting information in fewer dimensions compared with methods such as Singular Value Decomposition and Uniform Manifold Approximation and Projection. Full article
(This article belongs to the Special Issue Information Theory in Computational Biology)
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22 pages, 560 KiB  
Article
On Quantum Entropy
by Davi Geiger and Zvi M. Kedem
Entropy 2022, 24(10), 1341; https://doi.org/10.3390/e24101341 - 23 Sep 2022
Cited by 4 | Viewed by 4604
Abstract
Quantum physics, despite its intrinsically probabilistic nature, lacks a definition of entropy fully accounting for the randomness of a quantum state. For example, von Neumann entropy quantifies only the incomplete specification of a quantum state and does not quantify the probabilistic distribution of [...] Read more.
Quantum physics, despite its intrinsically probabilistic nature, lacks a definition of entropy fully accounting for the randomness of a quantum state. For example, von Neumann entropy quantifies only the incomplete specification of a quantum state and does not quantify the probabilistic distribution of its observables; it trivially vanishes for pure quantum states. We propose a quantum entropy that quantifies the randomness of a pure quantum state via a conjugate pair of observables/operators forming the quantum phase space. The entropy is dimensionless, it is a relativistic scalar, it is invariant under canonical transformations and under CPT transformations, and its minimum has been established by the entropic uncertainty principle. We expand the entropy to also include mixed states. We show that the entropy is monotonically increasing during a time evolution of coherent states under a Dirac Hamiltonian. However, in a mathematical scenario, when two fermions come closer to each other, each evolving as a coherent state, the total system’s entropy oscillates due to the increasing spatial entanglement. We hypothesize an entropy law governing physical systems whereby the entropy of a closed system never decreases, implying a time arrow for particle physics. We then explore the possibility that as the oscillations of the entropy must by the law be barred in quantum physics, potential entropy oscillations trigger annihilation and creation of particles. Full article
(This article belongs to the Special Issue Nature of Entropy and Its Direct Metrology)
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30 pages, 633 KiB  
Article
Assessing, Testing and Estimating the Amount of Fine-Tuning by Means of Active Information
by Daniel Andrés Díaz-Pachón and Ola Hössjer
Entropy 2022, 24(10), 1323; https://doi.org/10.3390/e24101323 - 21 Sep 2022
Cited by 8 | Viewed by 1629
Abstract
A general framework is introduced to estimate how much external information has been infused into a search algorithm, the so-called active information. This is rephrased as a test of fine-tuning, where tuning corresponds to the amount of pre-specified knowledge that the algorithm makes [...] Read more.
A general framework is introduced to estimate how much external information has been infused into a search algorithm, the so-called active information. This is rephrased as a test of fine-tuning, where tuning corresponds to the amount of pre-specified knowledge that the algorithm makes use of in order to reach a certain target. A function f quantifies specificity for each possible outcome x of a search, so that the target of the algorithm is a set of highly specified states, whereas fine-tuning occurs if it is much more likely for the algorithm to reach the target as intended than by chance. The distribution of a random outcome X of the algorithm involves a parameter θ that quantifies how much background information has been infused. A simple choice of this parameter is to use θf in order to exponentially tilt the distribution of the outcome of the search algorithm under the null distribution of no tuning, so that an exponential family of distributions is obtained. Such algorithms are obtained by iterating a Metropolis–Hastings type of Markov chain, which makes it possible to compute their active information under the equilibrium and non-equilibrium of the Markov chain, with or without stopping when the targeted set of fine-tuned states has been reached. Other choices of tuning parameters θ are discussed as well. Nonparametric and parametric estimators of active information and tests of fine-tuning are developed when repeated and independent outcomes of the algorithm are available. The theory is illustrated with examples from cosmology, student learning, reinforcement learning, a Moran type model of population genetics, and evolutionary programming. Full article
(This article belongs to the Special Issue Recent Advances in Statistical Theory and Applications)
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10 pages, 2358 KiB  
Article
Statistical Mechanics of Political Polarization
by Miron Kaufman, Sanda Kaufman and Hung T. Diep
Entropy 2022, 24(9), 1262; https://doi.org/10.3390/e24091262 - 8 Sep 2022
Cited by 8 | Viewed by 2708
Abstract
Rapidly increasing political polarization threatens democracies around the world. Scholars from several disciplines are assessing and modeling polarization antecedents, processes, and consequences. Social systems are complex and networked. Their constant shifting hinders attempts to trace causes of observed trends, predict their consequences, or [...] Read more.
Rapidly increasing political polarization threatens democracies around the world. Scholars from several disciplines are assessing and modeling polarization antecedents, processes, and consequences. Social systems are complex and networked. Their constant shifting hinders attempts to trace causes of observed trends, predict their consequences, or mitigate them. We propose an equivalent-neighbor model of polarization dynamics. Using statistical physics techniques, we generate anticipatory scenarios and examine whether leadership and/or external events alleviate or exacerbate polarization. We consider three highly polarized USA groups: Democrats, Republicans, and Independents. We assume that in each group, each individual has a political stance s ranging between left and right. We quantify the noise in this system as a “social temperature” T. Using energy E, we describe individuals’ interactions in time within their own group and with individuals of the other groups. It depends on the stance s as well as on three intra-group and six inter-group coupling parameters. We compute the probability distributions of stances at any time using the Boltzmann probability weight exp(−E/T). We generate average group-stance scenarios in time and explore whether concerted interventions or unexpected shocks can alter them. The results inform on the perils of continuing the current polarization trends, as well as on possibilities of changing course. Full article
(This article belongs to the Special Issue Statistical Physics of Opinion Formation and Social Phenomena)
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17 pages, 4479 KiB  
Article
Elastic Entropic Forces in Polymer Deformation
by Vladimir I. Kartsovnik and Dimitri Volchenkov
Entropy 2022, 24(9), 1260; https://doi.org/10.3390/e24091260 - 7 Sep 2022
Cited by 6 | Viewed by 2030
Abstract
The entropic nature of elasticity of long molecular chains and reticulated materials is discussed concerning the analysis of flows of polymer melts and elastomer deformation in the framework of Frenkel–Eyring molecular kinetic theory. Deformation curves are calculated in line with the simple viscoelasticity [...] Read more.
The entropic nature of elasticity of long molecular chains and reticulated materials is discussed concerning the analysis of flows of polymer melts and elastomer deformation in the framework of Frenkel–Eyring molecular kinetic theory. Deformation curves are calculated in line with the simple viscoelasticity models where the activation energy of viscous flow depends on the magnitude of elastic entropic forces of the stretched macromolecules. The interconnections between deformation processes and the structure of elastomer networks, as well as their mutual influence on each other, are considered. Full article
(This article belongs to the Special Issue Entropic Forces in Complex Systems II)
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11 pages, 1415 KiB  
Article
Decoding ‘Maximum Entropy’ Deconvolution
by Long V. Le, Tae Jung Kim, Young Dong Kim and David E. Aspnes
Entropy 2022, 24(9), 1238; https://doi.org/10.3390/e24091238 - 2 Sep 2022
Cited by 2 | Viewed by 3441
Abstract
For over five decades, the mathematical procedure termed “maximum entropy” (M-E) has been used to deconvolve structure in spectra, optical and otherwise, although quantitative measures of performance remain unknown. Here, we examine this procedure analytically for the lowest two orders for a Lorentzian [...] Read more.
For over five decades, the mathematical procedure termed “maximum entropy” (M-E) has been used to deconvolve structure in spectra, optical and otherwise, although quantitative measures of performance remain unknown. Here, we examine this procedure analytically for the lowest two orders for a Lorentzian feature, obtaining expressions for the amount of sharpening and identifying how spurious structures appear. Illustrative examples are provided. These results enhance the utility of this widely used deconvolution approach to spectral analysis. Full article
(This article belongs to the Special Issue Concepts of Entropy and Their Applications III)
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15 pages, 1850 KiB  
Article
A Bayesian Analysis of Plant DNA Length Distribution via κ-Statistics
by Maxsuel M. F. de Lima, Dory H. A. L. Anselmo, Raimundo Silva, Glauber H. S. Nunes, Umberto L. Fulco, Manoel S. Vasconcelos and Vamberto D. Mello
Entropy 2022, 24(9), 1225; https://doi.org/10.3390/e24091225 - 1 Sep 2022
Cited by 4 | Viewed by 1476
Abstract
We report an analysis of the distribution of lengths of plant DNA (exons). Three species of Cucurbitaceae were investigated. In our study, we used two distinct κ distribution functions, namely, κ-Maxwellian and double-κ, to fit the length distributions. To determine [...] Read more.
We report an analysis of the distribution of lengths of plant DNA (exons). Three species of Cucurbitaceae were investigated. In our study, we used two distinct κ distribution functions, namely, κ-Maxwellian and double-κ, to fit the length distributions. To determine which distribution has the best fitting, we made a Bayesian analysis of the models. Furthermore, we filtered the data, removing outliers, through a box plot analysis. Our findings show that the sum of κ-exponentials is the most appropriate to adjust the distribution curves and that the values of the κ parameter do not undergo considerable changes after filtering. Furthermore, for the analyzed species, there is a tendency for the κ parameter to lay within the interval (0.27;0.43). Full article
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22 pages, 446 KiB  
Article
Efficiency of the Moscow Stock Exchange before 2022
by Andrey Shternshis, Piero Mazzarisi and Stefano Marmi
Entropy 2022, 24(9), 1184; https://doi.org/10.3390/e24091184 - 25 Aug 2022
Cited by 5 | Viewed by 2111
Abstract
This paper investigates the degree of efficiency for the Moscow Stock Exchange. A market is called efficient if prices of its assets fully reflect all available information. We show that the degree of market efficiency is significantly low for most of the months [...] Read more.
This paper investigates the degree of efficiency for the Moscow Stock Exchange. A market is called efficient if prices of its assets fully reflect all available information. We show that the degree of market efficiency is significantly low for most of the months from 2012 to 2021. We calculate the degree of market efficiency by (i) filtering out regularities in financial data and (ii) computing the Shannon entropy of the filtered return time series. We developed a simple method for estimating volatility and price staleness in empirical data in order to filter out such regularity patterns from return time series. The resulting financial time series of stock returns are then clustered into different groups according to some entropy measures. In particular, we use the Kullback–Leibler distance and a novel entropy metric capturing the co-movements between pairs of stocks. By using Monte Carlo simulations, we are then able to identify the time periods of market inefficiency for a group of 18 stocks. The inefficiency of the Moscow Stock Exchange that we have detected is a signal of the possibility of devising profitable strategies, net of transaction costs. The deviation from the efficient behavior for a stock strongly depends on the industrial sector that it belongs to. Full article
(This article belongs to the Special Issue Applications of Statistical Physics in Finance and Economics)
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19 pages, 5449 KiB  
Article
Monte Carlo Simulation of Stochastic Differential Equation to Study Information Geometry
by Abhiram Anand Thiruthummal and Eun-jin Kim
Entropy 2022, 24(8), 1113; https://doi.org/10.3390/e24081113 - 12 Aug 2022
Cited by 7 | Viewed by 1906
Abstract
Information Geometry is a useful tool to study and compare the solutions of a Stochastic Differential Equations (SDEs) for non-equilibrium systems. As an alternative method to solving the Fokker–Planck equation, we propose a new method to calculate time-dependent probability density functions (PDFs) and [...] Read more.
Information Geometry is a useful tool to study and compare the solutions of a Stochastic Differential Equations (SDEs) for non-equilibrium systems. As an alternative method to solving the Fokker–Planck equation, we propose a new method to calculate time-dependent probability density functions (PDFs) and to study Information Geometry using Monte Carlo (MC) simulation of SDEs. Specifically, we develop a new MC SDE method to overcome the challenges in calculating a time-dependent PDF and information geometric diagnostics and to speed up simulations by utilizing GPU computing. Using MC SDE simulations, we reproduce Information Geometric scaling relations found from the Fokker–Planck method for the case of a stochastic process with linear and cubic damping terms. We showcase the advantage of MC SDE simulation over FPE solvers by calculating unequal time joint PDFs. For the linear process with a linear damping force, joint PDF is found to be a Gaussian. In contrast, for the cubic process with a cubic damping force, joint PDF exhibits a bimodal structure, even in a stationary state. This suggests a finite memory time induced by a nonlinear force. Furthermore, several power-law scalings in the characteristics of bimodal PDFs are identified and investigated. Full article
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