During the last years some new variants of Permutation entropy have been introduced and applied to EEG analysis, among them a conditional variant and variants using some additional metric information or being based on entropies different from the Shannon entropy. In some situations it is not completely clear what kind of information the new measures and their algorithmic implementations provide. We discuss the new developments and illustrate them for EEG data.

This article presents multiple novel and compelling arguments to systematically disprove two widely accepted notions: one is that entropy is a measure of disorder, and the other is Schrödinger's notion of negative entropy (negentropy). These two notions have been cited hundreds of times annually in scientific literature for decades. To mathematically describe the second law of thermodynamics, the concept of entropy was created and defined in the 1860s as heat energy divided by thermodynamic temperature, and the entropy of an object increases when it absorbs heat. This is the uncontroversial and fundamental nature of entropy, which is pivotal to current thermodynamics and from which multiple facts are deduced. These facts consistently disprove the above two notions. Some biological facts, such as the one that life orderliness is encoded and provided by inherited genes rather than entropy decline, can also disprove Schrödinger's negentropy notion. Moreover, this article further clarifies the reasoning errors of the two notions. It also shows that the tendency of many natural or social systems to become more disordered over time does not result from the second law of thermodynamics.

Numerous scientific fields employ entropy, the Shannonian measure of disorder and inaccessible information. Current paper examines educational uses of entropy. The major objectives of this survey are to create a coherent perspective of the applicability of classical entropies and to provide a concise detailed explanation that blends technical pertinence with conceptual extraction from the most recent literature. What's more intriguing are the closing thoughts mixed with a few unsolved issues and the creation of a future work schedule in these areas.

Quantum physics is intrinsically probabilistic. The entropy of a quantum state quantifies the amount of randomness (or information loss) of such state. The degrees of freedom of a quantum state are position and spin, and upon their specification Born's rule in phase space defines randomness. We focus on the spin degree of freedom and elucidate the spin entropy. Then, we present some of its properties and show how entanglement increases spin entropy. Some speculative predictions on the decay of the $Z^0,W^{+,-}$ gauge bosons conclude the paper.

In addition to their importance in statistical thermodynamics, probabilistic entropy measurements are crucial for understanding and analyzing complex systems, with diverse applications in time series and one-dimensional profiles. However, extending these methods to two- and three-dimensional data still requires further development. In this study, we present a new method to classify spatiotemporal processes based on entropy measurements. To test and validate the method, we selected four classes of similar processes related to the evolution of random patterns: dynamic colored noises ((i)~white and (ii)~red); (iii)~weak turbulence from reaction-diffusion; (iv)~hydrodynamic fully developed turbulence, and (v)~plasma turbulence from MHD. Considering seven possible ways to measure entropy from a matrix, we present the method as a parameter space composed of the two best separating measures of the five selected classes. The results highlight better combined performance of Shannon Permutation Entropy ($S^p_H$) and a new approach based on Tsallis Spectral Permutation Entropy ($S^s_q$). Notably, our observations reveal the segregation of reaction terms in this $S^p_H \times S^s_q$ space, a result that identifies specific sectors for each class of dynamic process, and can be used to train machine learning models for automatic classification of complex spatiotemporal patterns.

In this work, we introduce a generalized measure of cumulative residual entropy and study its properties. We show that several existing measures of entropy such as cumulative residual entropy, weighted cumulative residual entropy and cumulative residual Tsallis entropy, are all special cases of the generalized cumulative residual entropy. We also propose a measure of generalized cumulative entropy, which includes cumulative entropy, weighted cumulative entropy and cumulative Tsallis entropy as special cases. We discuss generating function approach using which we derive different entropy measures. Finally, using the newly introduced entropy measures, we establish some relationships between entropy and extropy measures.

{Here, we propose a new tool to estimate the complexity of a time series: the entropy of difference (ED). The method is based solely on the sign of the difference between neighbouring values in a time series. This makes it possible to describe the signal as efficiently as prior proposed parameters such as permutation entropy (PE) or modified permutation entropy (mPE), but (1) reduces the size of the sample that is necessary to estimate the parameter value, and (2) enables the use of the Kullback-Leibler divergence to estimate the "distance" between the time series data and random signals.

Entropy generation, formulated by combining the first and second laws of thermodynamics with an appropriate thermodynamic potential, emerges as the difference between a phenomenological entropy function and a reversible entropy function. The phenomenological entropy function is evaluated over an irreversible path through thermodynamic state space via real time measurements of thermodynamic states. The reversible entropy function is calculated along an ideal reversible path through the same state space. Entropy generation models for various classes of systems—thermal, externally loaded, internally reactive, open and closed––are developed via selection of suitable thermodynamic potentials. Here we simplify thermodynamic principles to specify convenient and consistently accurate system governing equations and characterization models. The formulations introduce a new and universal Phenomenological Entropy Generation (PEG) theorem. The systems and methods presented––and demonstrated on grease degradation, battery charging and discharging, metal fatigue and pump flow––can be used for design, analysis, and support of diagnostic monitoring and optimization.

In this paper, under the symmetric entropy and the scale squared error loss functions, we consider the maximum likelihood (ML) estimation and Bayesian estimation of the Shannon entropy and Rényi entropy of the two-parameter inverse Weibull distribution. In the ML estimation, the dichotomy is used to solve likelihood equation. In addition, the approximation confidence interval is given by the Delta method. Because the form of estimation results is more complex in the Bayesian estimation, the Lindley approximation method is used to achieve the numerical calculation. Finally, Monte Carlo simulations and a real data set are used to illustrate the results derived. By comparing the mean square error between the estimated value and the real value, it can be found that the performance of ML estimation of Shannon entropy is better than that of Bayesian estimation, and there is no significant difference between the performance of ML estimation of Rényi entropy and that of Bayesian estimation.

Experimental recordings of the collective activity of interacting spiking neurons exhibit random behavior and memory effects, thus the stochastic process modeling the spiking activity is expected to show some degree of time irreversibility. We use the thermodynamic formalism to build a framework, in the context of spike train statistics, to quantify the degree of irreversibility of any parametric maximum entropy measure under arbitrary constraints, and provide an explicit formula for the information entropy production of the inferred Markov maximum entropy process. We provide examples to illustrate our results and discuss the importance of time irreversibility for modeling the spike train statistics.

Two heating surfaces with longitudinal fins are located within the closed channel inside the housing made of thermal insulation material. The air flow inside the channel was carried out by an axial fan. The heating of the fins is established by using two positive temperature coefficient heating elements (PTC). Air flows circularly from one finned surface to another, and after reaching the required temperature, it leaves the housing. In the heating system conceived in this way, a methodology based on transient thermal irreversibility of heating sources and circular air flow has been established. Volumetric air flow and circulation time within the channel were varied. The total entropy generated does not include hydraulic irreversibility, but transient thermal irreversibility of the air and heat sources. For conducting analytical modeling, the temperature of the PTC heater was considered constant at 423K and 473K. Volumetric air flow varied in the interval from 0.00001m3s-1 to 0.00006m3s-1. The experimental determination of the transient thermal entropy was performed at a much higher air flow rate of 0.005m3s-1 inside the closed channel. Due to the continuous increase in air temperature, the temperatures of both PTC heaters increased within the examined time interval of 300s. According to experimental testing, the total transient thermal entropy of the described heating system has a minimum after 175s from the start of the circular heating of the air. The minimum of transient entropy also implies the optimal time of channel opening and exit of heated air.

Double-spiral galaxies are common in the Universe. It is known that the logarithmic double spiral is a Maximum Entropy geometry and represents spiral galaxies well. It is also known that the virial mass of such a galaxy can be approximately determined from the entropy of its central supermassive black hole. But over time the black hole must accrete mass, and therefore the overall galactic entropy must increase. From the associated entropic Euler-Lagrange equation (forming the basis of the Principle of Least Exertion, and also enabling the application of Nöther’s theorem) we show that the galactic entropy production is a conserved quantity, and we derive an appropriate expression for the relativistic entropic Hamiltonian of an idealised spiral galaxy. We generalise Onsager’s celebrated expression for entropy production and demonstrate that galactic entropy production has two parts, one many orders of magnitude larger than the other, and where the smaller is comparable to the Hawking radiation of the central supermassive black hole. We conclude that galaxies cannot be isolated, since even idealised spiral galaxies have non-zero entropy production.

While entropy was introduced in the second half of the 19th century in the international vocabulary as a scientific term, in the 20th century it became common in colloquial use. Popular imagination has loaded “entropy” with almost every negative quality in the universe, in life and in society, with a dominant meaning of disorder and disorganization. Exploring the history of the term and many different approaches on it, we show that entropy has a universal stochastic definition which is not disorder. The accompanying principle of maximum entropy, which lies behind the Second Law, gives explanatory and inferential power to the concept and promotes entropy as the mother of creativity and evolution. As social sciences are often contaminated by subjectivity and ideological influences, we try to explore whether the maximum entropy, applied to the distribution of wealth quantified by annual income, can give an objective description. Using publicly available income data, we show that the income distribution is consistent with the principle of the maximum entropy. The increase of entropy is associated to increase of society’s wealth yet a standardized form of entropy can be used to quantify inequality. Historically, technology has played a major role in development and increase of the entropy of income. Such findings are contrary to the theories of ecological economics and other theories which use the term entropy in a Malthusian perspective.

How to build an optimal model where three terms “time, causality, and entropy” will be interrelated in a most correct way? This task has a semantic and physical aspect. We give in this article the solution that is based only on elementary perception. In physics, we experience time with our senses as the duration of material change running in space. This fact is the standpoint for our model where causality is only a principle, it is not a physical actuality and entropy runs only in space and not in time. Time is merely a duration of entropy increasing and is entering existence when measured by the observer.

The extensions of Boltzmann-Gibbs classical entropic form are presented. The focus is on the thermodynamical entropies which do not obey classical definition of additivity and extensity with main motivation coming from the gravitational systems admitting long-range interactions both in astrophysics (black holes) and in cosmology. Horizon entropies are the main concern. In the paper, the plethora of non-Boltzmann-Gibbs entropies are studied and the comparative analysis of their properties with respect to additivity and extensivity is presented.

The real systems are often complex, nonlinear, and noisy in various areas including mathematics, natural science, and social science. We present the symplectic entropy (SymEn) measure as well as an analysis method based on SymEn to estimate the nonlinearity of the complex system by analyzing the given time series. The SymEn estimation is a kind of entropy based on symplectic principal component analysis (SPCA) which represent organized but unpredictable behaviors of systems. The key to SPCA is to preserve the global submanifold geometrical properties of the systems through symplectic transform in the phase space, which is a kind of the measure-preserving transforms. The capability of preserving the global geometrical characteristics makes the SymEn a test statistic to detect the nonlinear characteristics in several typical chaotic time series and the stochastic characteristic in the Gaussian white noise. The results are in agreement with findings in the approximate entropy (ApEn), the sample entropy (SampEn) and the fuzzy entropy (FuzzyEn). Moreover, the SymEn method is also used to analyze the nonlinearities of the real signals (including the EEG signals for ASD and healthy subjects, and the sound and vibration signals for the mechanical systems). The results indicate that the SymEn estimation can be taken as a measure for describing the nonlinear characteristics in the data collected from the natural complex systems.

In the realm of molecular science, intricate relationships between molecular structures and their biomedical and pharmacological characteristics have been revealed through empirical experiments. This exploration hinges on the application of numerical descriptors, known as Topological Indices, which illuminate the inherent properties of diverse molecular structures. With particular significance in the medical and pharmaceutical domains, these indices facilitate the prediction of biological features for new chemical compounds and drugs by quantifying weighted entropies. In the context of this paper, we delve into the concept of graph entropy, weaving it intricately with the topological properties of the crystalline framework of the copper oxide molecule, denoted as Cu\textsubscript{2}O$[i,j,t]$. Our primary objective is to unravel the mathematical symphony underlying the structural intricacies of Cu\textsubscript{2}O$[i,j,t]$ and to imbue it with the conceptual essence of entropy. We accomplish this through the computation of entropy, leveraging various topological indices, including weight. In addition to our analytical journey, we present a graphical comparison that juxtaposes the computed indices and entropies, shedding light on the interplay between mathematical analysis and the structural elegance of Cu\textsubscript{2}O$[i,j,t]$. This endeavor contributes to a deeper understanding of the material's multifaceted applications, spanning domains from chemical sensors to solar cells, photocatalysis, and batteries, where Cu\textsubscript{2}O's crystallographic structure plays a pivotal role.

This study examined the association between pressure injuries and complexity of abdominal temperature measured in residents of a nursing facility. The temperature served as a proxy measure for skin thermoregulation. Refined multiscale sample entropy and bubble entropy were used to measure the complexity of the temperature time series measured over two days at 1-minute intervals. Robust summary measures were derived for the multiscale entropies and used in predictive models for pressure injuries that were built with adaptive lasso regression and neural networks. Both types of entropies were lower in the group of participants with pressure injuries (n=11) relative to the group of non-injured participants (n=15). This was generally true at the longer temporal scales, with the effect peaking at scale τ=23 minutes. Predictive models for pressure injury on the basis of refined multiscale sample entropy and bubble entropy yielded 92% accuracy, outperforming predictions based on any single measure of entropy. Combining entropy measures with a widely used risk assessment score led to the best prediction accuracy. Complexity of abdominal temperature series could therefore serve as an indicator of risk of pressure injury.

In this study, we evaluate the nonlinear dynamics of convection flow using the thermodynamic variational principle, focusing on scenarios where multiple external forces, such as a thermal gradient and rotational field, are applied to a shallow annular pool. We observed that with the increase in the thermal gradient, the flow changed from an axial flow to a rotational oscillatory flow with the wave amplitudes aligned. Further increasing the temperature difference led to a rotational oscillatory flow characterized by alternating wave generation and annihilation. Our analysis of the flow, considering heat fluxes orthogonal to the thermal gradient, allowed us to describe the flow state as a phase at equilibrium. The state transition of the flow was accompanied by a discontinuous jump in the heat flux, which occurred at the intersection of the entropy production curves.

In this paper, we study the relativistic correction to Bekenstein-Hawking entropy in the canonical ensemble and isothermal-isobaric ensemble and apply it to the cases of non-rotating BTZ and AdS-Schwarzschild black holes. This is realized by generalizing the equations obtained using Boltzmann-Gibbs(BG) statistics with its relativistic generalization, Kaniadakis statistics, or κ-statistics. The relativistic corrections are found to be logarithmic in nature and it is observed that their effect becomes appreciable only in the high-temperature limit suggesting that the entropy corrections must include these relativistically corrected terms while taking the aforementioned limit. The non-relativistic corrections are recovered in the κ → 0 limit.

The maximization of entropy S within a closed system is accepted as an inevitability (as the second law of thermodynamics) by statistical inference alone. The Maximum Entropy Production Principle (MEPP) states that not only S maximizes, but $\dot{S}$ as well: a system will dissipate as fast as possible. There is still no consensus on the general validity of this MEPP, even though it shows remarkable explanatory power (both qualitatively and quantitatively), and has been empirically demonstrated for many domains. In this theoretical paper I provide a generalization of entropy gradients, to show that the MEPP actually follows from the same statistical inference, as that of the 2nd law of thermodynamics. For this generalization I only use the concepts of super-statespaces and microstate-density. These concepts also allow for the abstraction of 'Self Organizing Criticality' to a bifurcating local difference in this density, and allow for a generalization of the fundamentally unresolved concepts of 'chaos' and 'order'.

Membrane fouling caused by many direct and indirect triggering factors has become an obstacle to the application of membrane bioreactor (MBR). The nonlinear relationship between those factors is subject to complex causality or affiliation, which is difficult to clarify for the diagnosis of membrane fouling. To solve this problem, this paper proposes a compressible diagnosis model (CDM) based on transfer entropy to facilitate the fault diagnosis of the root cause for membrane fouling. Firstly, a framework of CDM between membrane fouling and causal variables is built based on a feature extraction algorithm and mechanism analysis. The framework can identify fault transfer scenarios following the changes in operating conditions. Secondly, the fault transfer topology of CDM based on transfer entropy is constructed to describe the causal relationship between variables dynamically. Thirdly, an information compressible strategy is designed to simplify the fault transfer topology. This strategy can eliminate the repetitious affiliation relationship, which contributes to the root causal variables speedily and accurately. Finally, the effectiveness of the proposed CDM is verified by the measured data from an actual MBR.

The present work investigates primarily the morphology evolution of the body-centered-cubic (BCC)/B2 phases in AlxNiCoFeCr high-entropy alloys (HEAs) with increasing Al content, which has been neglected so far. There exist two types of microscopic morphologies of BCC and B2 phases in this HEA series: one is the weave-like morphology induced by the spinodal decomposition, and the other is the microstructure of a spherical disordered BCC precipitation on the ordered B2 matrix that appears in HEAs with a much higher Al content. The shape of coherent precipitates is found to be closely related to the lattice misfit between BCC and B2 phases, which is sensitive to Al. The mechanical properties, including the compressive yielding strength and microhardness of the AlxNiCoFeCr HEAs, are also discussed in light of the concept of the valence electron concentration (VEC).

People have always hoped to be able to fill an entire plane with ``single unit cells'' without periodicity. This wish was realized after the mathematician discovered a 13-sided ``single unit cell'' named `einstein'. These non-periodic tessellations generally exhibit anisotropic properties, making them superior in terms of mechanical performance compared to periodic structures. From the perspective of information entropy, the reason behind the improved mechanical properties of these structures is the higher entropy associated with non-periodic configurations. To quantify the disorder of non-periodic structures, we propose an entropy expression for the `einstein' metamaterial. To demonstrate the mechanical properties of these high-entropy structures, we fabricate specimens using 3D printing and conducte mechanical experiments. For comparative analysis, we also use ABAQUS to perform finite element analysis of the problem. The research results reveal that the mechanical properties of high-entropy structures created by the non-periodic stacking of cells are significantly improved compared to those of low-entropy structures created by periodic stacking. The conclusions drawn from the study of individual issues are generalizable and may be of assistance in future material and structural design.

An analysis of the processes of plastic deformation and acoustic relaxation in a high-entropy alloy Al0.5CoCrCuFeNi was carried out. The following have been established: dominant dislocation defects; types of barriers that prevent the movement of dislocations; mechanisms of thermally activated movement of various elements of dislocation lines through barriers at room and low temperatures. Based on modern dislocation theory, quantitative estimates have been obtained for the most important characteristics of dislocations and their interaction with barriers.

Different notions of entropy can be identified in different communities: (i) the thermodynamic sense, (ii) the information sense, (iii) the statistical sense, (iv) the disorder sense, and (v) the homogeneity sense. Especially the “disorder sense” and the “homogeneity sense” relate to and require the notion of space and time. One of the few prominent examples relating entropy to geometry and to space is the Bekenstein-Hawking entropy of a Black Hole. Although being developed for the description of a physics object—a black hole—having a mass, a momentum, a temperature, a charge etc. absolutely no information about these attributes of this object can eventually be found in the final formula. In contrast, the Bekenstein-Hawking entropy in its dimensionless form is a positive quantity only comprising geometric attributes like an area A—which is the area of the event horizon of the black hole-, a length L_P—which is the Planck length - and a factor ¼. A purely geometric approach towards this formula will be presented. The approach is based on a continuous 3D extension of the Heaviside function, with this extension drawing on the phase-field concept of diffuse interfaces. Entropy enters into the local, statistical description of contrast respectively gradient distributions in the transition region of the extended Heaviside function definition. The structure of the Bekenstein-Hawking formula eventually is derived for a geometric sphere based on mere geometric-statistic considerations.

This study introduces a method for quantitatively assessing the complexity and predictability of fish behavior in closed systems through the application of information entropy, offering a novel lens to understand how fish adapt to environmental changes. Utilizing simulations rooted in a random walk model for fish movement, we delve into entropy fluctuations under varying environmental conditions, including responses to feeding and external stimuli. Our findings underscore the utility of information entropy in capturing the intricacies of fish behavior, particularly highlighting the synchrony in collective actions and adaptations to environmental shifts. This research not only broadens our comprehension of fish behavior but also paves the way for its application in fields like aquaculture and resource management. Through our analysis, we discovered that smaller grid sizes in simulations capture detailed local fluctuations, while larger grids elucidate general trends, pinpointing a 2.5m grid as optimal for our study. Moreover, changes in swimming speeds and behavioral adaptations during feeding were quantitatively analyzed, with results illustrating significant behavior modifications. Additionally, employing a Gaussian Mixture Model helped to clarify the nuanced changes in fish behavior in response to altered light conditions, demonstrating the layered complexity of fish responses to environmental stimuli. This investigation confirms the efficacy of information entropy as a robust metric for evaluating fish shoal behavior, offering a fresh methodology for ecological and environmental studies, with promising implications for sustainable management practices.

Predicting the outcome of a future game between two National Basketball Association (NBA) teams poses a challenging problem of interest to statistical scientists as well as the general public. In this article, we formalize the problem of predicting the game results as a classification problem and apply the principle of maximum entropy to construct NBA maximum entropy (NBAME) model that fits to discrete statistics for NBA games, and then predict the outcomes of NBA playoffs by the NBAME model. The best NBAME model is able to correctly predict the winning team 74.4 percent of the time as compared to some other machine learning algorithms which is correct 69.3 percent of the time.

Bonchev and Trinajstic defined two distance based entropy measures to measure the molecular branching of molecular graphs in 1977 [Information theory, distance matrix, and molecular branching, J. Chem. Phys., 38 (1977), 4517–4533]. In this paper we use these entropy measures which are based on distance matrices of graphs. The first one is based on distribution of distances in distance matrix and the second one is based on distribution of distances in upper triangular submatrix. We obtain the two entropy measures of paths, stars, complete graphs, cycles and complete bipartite graphs. Finally we obtain the minimal trees with respect to these entropy measures with fixed diameter.

In order to solve the problem of feature extraction of underwater acoustic signals in complex ocean environment, a new method for feature extraction from ship radiated noise is presented based on empirical mode decomposition theory and permutation entropy. It analyzes the separability for permutation entropies of the intrinsic mode functions of three types of ship radiated noise signals, and discusses the permutation entropy of the intrinsic mode function with the highest energy. In this study, ship radiated noise signals measured from three types of ships are decomposed into a set of intrinsic mode functions with empirical mode decomposition method. Then, the permutation entropies of all intrinsic mode functions are calculated with appropriate parameters. The permutation entropies are obviously different in the intrinsic mode functions with the highest energy, thus, the permutation entropy of the intrinsic mode function with the highest energy is regarded as a new characteristic parameter to extract the feature of ship radiated noise. After that, the characteristic parameters, namely, the energy difference between high and low frequency, permutation entropy, and multi-scale permutation entropy, are compared with the permutation entropy of the intrinsic mode function with the highest energy. It is discovered that the four characteristic parameters are at the same level for similar ships, however, there are differences in the parameters for different types of ships. The results demonstrate that the permutation entropy of the intrinsic mode function with the highest energy is better in separability as the characteristic parameter than the other three parameters by comparing their fluctuation ranges and the average values of the four characteristic parameters. Hence, the feature of ship radiated noise can be extracted efficiently with the method.

：Non-Euclidean data, such as social networks, and citation relationships between documents, has node information and structural information. Graph Convolutional Network(GCN) can automatically learn node features and association information between nodes. The core ideology of the graph convolutional network is to aggregate node information by using edge information, thereby generating a new node feature. In the process of updating node features, there are two core influencing factors. One is the number of neighboring nodes of the central node, the other is the contribution of the neighboring nodes to the central node. Due to the previous GCN methods not simultaneously considering the numbers and different contributions of neighboring nodes to the central node, we design the Adaptive Attention Mechanism(AAM). To further enhance the representational capability of the model, we utilize Multi-Head Graph Convolution(MHGC). Finally, we adopt cross-entropy(CE) loss function to describe the difference between the predicted results of node categories and the ground truth (GT). Combined with backpropagation, this ultimately achieves accurate node classification. Based on AAM, MHGC and CE, we contrive the novel Graph Adaptive Attention Network (GAAN). Experiments show that the classification accuracy has achieved outstanding performances on Cora, Citeseer and Pubmed datasets.

We address the unsolved question of how best to estimate the collision entropy, also called quadratic or second order Rényi entropy. Integer-order Rényi entropies are synthetic indices useful for the characterization of probability distributions. In recent decades, numerous studies have been conducted to arrive at their valid estimates starting from experimental data, so to derive suitable classification methods for the underlying processes, but optimal solutions have not been reached yet. Limited to the estimation of collision entropy, a one-line formula is presented here. The results of some specific Monte Carlo experiments give evidence of the validity of this estimator even for the very low densities of the data spread in high-dimensional sample spaces. The method strengths are unbiased consistency, generality and minimum computational cost.

We study the atom-field interaction at finite temperature and in the dispersive regime. We show that the master equation for this sytem may be solved with the use of superoperator techniques. We calculate the linear entropy in case the field is initially in a coherent state and the atom in a superposition of its ground and excited states.

Approximate entropy (ApEn) and sample entropy (SampEn) are statistical indices designed to quantify the regularity or predictability of time series data. Although ApEn has been a prominent choice for use in the analysis of non-linear data, it is currently unclear as to which method and parameter selection combination is optimal for its application in biomechanics. The goal of this research was to examine the differences between ApEn and SampEn related to center of pressure (COP) data during tandem standing balance tasks, while also changing the tolerance window, r. Six participants completed five, 30-second trials, feet-together and tandem standing with eyes-open and eyes-closed. COP data (fs = 60 Hz, downsampled from 1200 Hz) from ground reaction force platforms were collected. ApEn and SampEn were calculated using a constant vector length, i.e., m = 2, but differing values of r (tolerance window). For each of the participants, four separate one-way analysis of variance analyses (ANOVA) were conducted for ApEn and SampEn in the anterior-posterior (AP) and medial-lateral (ML) directions. Dunnett's intervals were applied to the one-way ANOVA analyses to determine which tandem conditions differed significantly from the baseline condition. ApEn and SampEn provided comparable results in the predictability of patterns for different stability conditions, with increasing instability, i.e., tandem eyes closed postures, being associated with greater unpredictability. The selection of r had a relatively consistent effect on mean ApEn and SampEn values across r = 0.15 * SD to r = 0.25 * SD, where both entropy methods tended to decrease as r increased. Mean SampEn values were generally lower than ApEn values. The results suggest that both ApEn and SampEn indices demonstrated relative consistency and were equally effective in quantifying the level of the center of pressure signal regularity during quiet tandem standing postural balance tests.

The choice associated with words is a fundamental property of natural languages. It lies at the heart of quantitative linguistics, computational linguistics, and language sciences more generally. Information-theory gives us tools at hand to measure precisely the average amount of choice associated with words—the word entropy. Here we use three parallel corpora—encompassing ca. 450 million words in 1916 texts and 1259 languages—to tackle some of the major conceptual and practical problems of word entropy estimation: dependence on text size, register, style and estimation method, as well as non-independence of words in co-text. We present three main results: 1) a text size of 50K tokens is sufficient for word entropies to stabilize throughout the text, 2) across languages of the world, word entropies display a unimodal distribution that is skewed to the right. This suggests that there is a trade-off between the learnability and expressivity of words across languages of the world. 3) There is a strong linear relationship between unigram entropies and entropy rates, suggesting that they are inherently linked. We discuss the implications of these results for studying the diversity and evolution of languages from an information-theoretic point of view.

Static standing postural stability has been measured by multiscale entropy (MSE), which is used to measure complexity. In this study, we used the average entropy (AE) to measure the static standing postural stability, as AE is a good measure of disorder. The center of pressure (COP) trajectories were collected from 11 subjects under four kinds of balance situation, from stable to unstable: bipedal with open eyes, bipedal with closed eyes, unipedal with open eyes, and unipedal with closed eyes. The AE, entropy of entropy (EoE), and MSE methods were used to analyze these COP data, and EoE was found to be a good measure of complexity. The AE of the 11 subjects sequentially increased by 100%as the balance situations progressed from stable to unstable, but the results of EoE and MSE did not follow this trend. Therefore, AE, rather than EoE or MSE, is a good measure of static standing postural stability. Furthermore, the comparison of EoE and AE plots exhibited an inverted U curve, which is another example of a complexity versus disorder inverted U curve.

Graph entropies have been used to interpret the complexity of the networks. There are various graph entropies in the literature. In this note a new graph entropy is defined which is based on the dominating sets of the graphs. Moreover the entropy of some classes of the graphs is obtained

The model of an oscillating universe proposed by A. Friedmann in 1922 dominated cosmology for a long time. R. Tolman put forward a thermodynamic argument against the cyclic cosmological model, believing that entropy should accumulate from cycle to cycle. J. Bekenstein and S. Hawking discovered a tremendous amount of entropy associated with black holes. This discovery led to the conclusion that the majority of the Universe's entropy is contained in black holes. N. Poplawski considers a Universe that oscillates within a black hole. This paper analyzes the relationship between the entropy of black holes and the oscillating Universe inside a large black hole. It is demonstrated that the entropy of a black hole can only be defined for an external observer. For an observer inside the black hole, its entropy is zero. This opens up the possibility of constructing a cyclic cosmology in which entropy and the size of the visible part of the Universe change strictly periodically. This does not violate the second law of thermodynamics because the continuous accumulation of entropy from cycle to cycle pertains to the surface of the big black hole, which is invisible to the internal observer.

This study examines the overall needs of the green construction scheme with ‘carbon neutrality’ as the centre in the Zhejiang provincial green development target area. By aggregating and organising the construction and development data of Zhejiang Province, the entropy weight TOPSIS model is formed according to the statistical modelling for quantitative examination of the data, and the scientific assessment scheme of ‘carbon neutrality’ in the regional construction industry of Zhejiang Province is developed. This study aids in completely exhibiting and dynamically understanding the advancement of the ‘carbon neutral’ capacity of the urban construction industry. The objective is to discover the weak link in the advancement of carbon neutrality in several regional construction industries, which is of great relevance for further examining and forecasting the strategic outlook of carbon neutrality and modifying the planning of carbon neutrality strategy in special regional construction industries.

Study background and aims: Laparoscopic cholecystectomy is one of the most frequently performed interventions in departments of general surgery. One of the most important aims in achieving perioperative stability of these patients is diminishing the impact of general anesthesia on the hemodynamic stability and the optimization of anesthetic drug doses based on the individual clinical profile of each patient. The objective of this study is the evaluation of the impact monitoring the depth of anesthesia through Entropy (state entropy – SE and response entropy -RE) has on the hemodynamic stability and on the doses of volatile anesthetic. Material and Methods: This is a prospective, observational, randomized, monocentric study carried out between January 2019 and December 2019 in the Clinic of Anesthesia and Intensive Care from the “Pius Brînzeu” Emergency County Hospital in Timișoara, Romania. The patients included in the study were divided in two study groups; patients in Group A (target group) received multimodal monitoring that included monitoring of standard parameters and of Entropy (SE and RE), while patients in Group B (control group) only received standard monitoring. Anesthetic dose in group A were optimized to achieve a target entropy of 40-60. Results: 68 patients met the inclusion criteria and were allocated to one of the two study groups, Group A (N=43) and Group B (N=25). There were no statistically significant differences identified between the two groups for both demographical and clinical data (p>0.05). Statistically significant differences have been identified for the number of hypotensive episodes (p = 0.011, 95% CI 0.1851 to 0.7042) and for the number of episodes of bradycardia (p < 0.0001, 95% CI 0.3296 to 0.7923). Moreover, there was a significant difference in the Sevoflurane consumption between the two study groups (p = 0.0498, 95% CI -0.3942 to 0.9047). Conclusions: The implementation of the multimodal monitoring protocol that includes the standard parameters and the measurement of Entropy for determining the depth of anesthesia (SE and RE) lead to a considerable improvement in perioperative hemodynamic stability. Optimizing the doses of anesthetic drugs based on the individual clinical profile of each patient leads to a considerable decrease in drug consumption as well as to a lower incidence of hemodynamic side-effects.

Models with age as a variable but actual dynamics invariant under age structure allow use of MaxEnt to predict uniform age structure. Interpretations include a major role for accepting randomness as a causal principle in ecology (in the face of ignorance, in a further interpretative stance).

Strong yet ductile alloys have long been desired for industrial applications to enhance structural reliability. This work produced two (CoCrNi)93.5Al3Ti3C0.5 medium-entropy alloys with exceptional strength-ductility combinations, via short/medium (3 min/30 min)-time annealing after hot-rolling. Three types of intergranular precipitates including MC, M23C6 carbides, and L12 phase were detected in both the samples. Noticeably, the high density of intragranular L12 precipitates were only founded in the medium-time annealed sample but not in another one. Upon inspection of deformed substructure, it was revealed that the plane slip is the dominant deformation mechanism of both the alloys. This is related to the lower stacking fault energy, higher lattice friction induced by C solute, and slip plane softening caused by intragranular dense L12 precipitates. Additionally, we noted that the stacking fault and twinning act as the mediated mechanisms in deformation of the short-time annealed alloy, while only the former in the medium-time annealed one. It is attributed to the higher energy stacking faults and the increased critical twinning stress caused by intragranular dense L12 precipitates that inhibited the twinning tendency. Our present findings provide not only a guidance for optimizing the mechanical properties of high/medium-entropy alloys, but also a fundamental understanding of deformation mechanisms.

It has been recognized for some time that even for perfect conductors, the interaction Casimir entropy, due to quantum/thermal fluctuations, can be negative. This result was not considered problematic because it was thought that the self-entropies of the bodies would cancel this negative interaction entropy, yielding a total entropy that was positive. In fact, this cancellation seems not to occur. The positive self entropy of a perfectly conducting sphere does indeed just cancel the negative interaction entropy of a system consisting of a perfectly conducting sphere and plate, but a model with weaker coupling in general possesses a regime where negative self-entropy appears. The physical meaning of this surprising result remains obscure. In this paper we re-examine these issues, using improved physical and mathematical techniques, partly based on the Abel-Plana formula, and present numerical results for arbitrary temperatures and couplings, which exhibit the same remarkable features.

The study of the change in entropy during the formation of stars in collapsing molecular clouds is an important area of research in astrophysics. As the cloud decreases in size, the presence of gases leads to an increase in entropy. However, the extent of this change in entropy is influenced by the amount of space available for the gases. In this scientific investigation, I conduct a detailed analysis of entropy on a small scale within the contracting cloud, following the principles of the second law of thermodynamics. Initially, my focus is on a turbulent cloud composed of N particles with a mass of M, which collapses under the force of gravity. It becomes clear that in order for the collapse to occur, gravitational energy must overcome other pressure forces that oppose it. This process results in the rapid movement of particles within the cloud, ultimately leading to a change in entropy

Attempts to find and quantify the supposed low entropy of organisms and its preservation are revised. Absolute entropy of the mixed components of non-living biomass (around -1.6 x 103 J K-1 L-1) is the reference to which other entropy decreases would be ascribed to life. Compartmentation of metabolites and departure from the equilibrium of metabolic reactions account for 1 and 40-50 J K-1 L-1, respectively, decreases of entropy and, though small, are distinctive features of living tissues. DNA and proteins do not supply significant decreases of thermodynamic entropy, but their low informational entropy is relevant for life and its evolution. No other living feature contributes significantly to the low entropy associated to life. The photosynthetic conversion of radiant energy to biomass energy accounts for the most of entropy (2.8 x 105 J K-1 carbon kg-1) produced by living beings. The comparative very low entropy produced in other processes (around 4.8 x102 J K-1 L-1 day-1 in human body) must be rapidly exported outside as heat to preserve the low entropy decreases due to compartmentation and non-equilibrium metabolism. Enzymes and genes are described whose control minimize the rate of production of entropy and could explain selective pressures in biological evolution and the rapid proliferation of cancer cells.

Entropy is usually understood as the quantitative measure of “chaos” or “disorder”. However, the notions of “chaos” and “disorder” are definitely obscure. This leads to numerous misinterpretations of entropy. We propose to see the disorder as an absence of symmetry and to identify “ordering” with symmetrizing of a physical system; in other words, introducing the elements of symmetry into initially disordered physical system. We demonstrate with the binary system of elementary magnets that introducing elements of symmetry necessarily diminishes its entropy. This is true for 1D and 2D systems of elementary magnets. Imposing symmetry does not influence the Landauer Principle valid for the addressed systems. Imposing the symmetry restrictions onto the system built of particles contained within the chamber divided by the permeable partition also diminishes its entropy.

In this paper, we propose a method for assessing the effectiveness of entropy features using several chaotic mappings. We anlyze fuzzy entropy (FuzzyEn) and neural network entropy (NNetEn) on four discrete mappings: the logistic map, the sine map, the Planck map, and the two-memristor-based map, with a base length time series of 300 elements. FuzzyEn is shown to have improved global efficiency (GEFMCC) in the classification task compared to NNetEn. At the same time, there are local areas of the time series dynamics in which the classification efficiency NNetEn is higher than FuzzyEn. The results of using horizontal visibility graphs (HVG) instead of the raw time series are also shown. GEFMCC decreases after HVG time series transformation, but there are local areas of the time series dynamics in which the classification efficiency increases after including the HVG. The scientific community can use the results to explore the efficiency of entropy-based classification of time series.

The Carnot cycle and the attendant notions of reversibility and entropy are examined. It is shown how the modern view of these concepts still correspond to the ideas Clausius laid down in the nineteenth century. As such, they reflect the outmoded idea current at the time that heat is motion. It is shown how this view of heat led Clausius to develop the entropy of a body based on the work that could be done in a reversible process rather than the work that was actually done. In consequence, Clausius built into entropy a conflict with energy conservation, which is concerned with actual changes in energy. In this paper, a macroscopic formulation of internal mechanisms of damping based on rate equations for the distribution of energy within a gas. It is shown that work processes involving a step-change in external pressure, however small, are intrinsically irreversible. However, under idealised conditions of zero damping the gas inside a piston expands and traces out a trajectory through the space of equilibrium states. Therefore, the entropy change due to heat flow from the reservoir matches the entropy change of the equilibrium states. This trajectory can traced out in reverse as the piston reverses direction, but if the external conditions are adjusted appro-priately, the gas can be made to trace out a Carnot cycle in P-V space. The cycle is dynamic as opposed to quasi-static as the piston has kinetic energy equal in difference to the work done in-ternally and externally.

The role of energy and entropy in decomposition of turbulent velocity flow-fields is to be shown in the paper. The decomposition methods based on energy concept are taken into account, namely the Proper Orthogonal Decomposition (POD) and its extension Bi-Orthogonal Decomposition (BOD). Entropy motivated view on the decomposed modes could offer new possibilities in the modes physical interpretation and in Reduced Order Modelling (ROM) strategy efficiency evaluation.

We investigate some information properties of consecutive k-out-of-n:G systems in light of fractional generalized cumulative residual entropy. We firstly derive a formula to compute fractional generalized cumulative residual entropy related to the system's lifetime and explore its preservation properties in terms of established stochastic orders. Additionally, we obtain useful bounds. To aid practical applications, we propose two non parametric estimators for the fractional generalized cumulative residual entropy in these systems. The efficiency and performance of these estimators are illustrated using simulated and real datasets.

Tsallis entropy has been widely used in image thresholding because of its non-extensive properties. The non-extensive parameter q contained in this entropy plays an important role in various adaptive algorithms and has been successfully applied in bi-level image thresholding. In this paper, the relationships between parameter q and pixels’ long-range correlations have been further studied within multi-threshold image segmentation. It is found that the pixels’ correlations are remarkable and stable for images generated by a known physical principle, such as infrared images and medical CT images. And the corresponding non-extensive parameter q can be evaluated by using self-adaptive Tsallis entropy algorithm. The results of this algorithm are compared with those of Shannon entropy algorithm and the original Tsallis entropy algorithm, in terms of quantitative image quality evaluation metrics PSNR (Peak Signal-to-Noise Ratio) and SSIM (Structural Similarity). Furthermore, we observed that for image series with the same background, the q values determined by the adaptive algorithm are consistently kept in a narrow range. Therefore, similar or identical scenes during imaging would produce similar strength of long-range correlations, which provides potential applications for unsupervised image processing.

Recently, there has been growing interest in alternative measures of uncertainty, including cumulative residual entropy. In this paper, we consider a mixed system consisting of n components, assuming that all components are operational at time t. We use the system signature to calculate the cumulative residual entropy of the mixed system’s remaining lifetime. This measure is a useful criterion for assessing the predictability of a system’s lifetime. We study several results related to the cumulative residual entropy of mixed systems, including expressions, limits, and order properties. These results shed light on the behavior of the measure and provide insights into the predictability of mixed systems. In addition, we propose a criterion for selecting a preferred system based on the relative residual cumulative entropy. This criterion is closely related to the parallel system and provides a practical way to choose the best system configuration. Overall, the present study of cumulative residual entropy and the proposed selection criterion provide valuable insights into the predictability of mixed systems and can be applied in various fields.

A convenient practical model for accurately estimating the total entropy (ΣSi) of atmospheric gases based on physical action is proposed. This realistic approach is fully consistent with statistical mechanics, but reinterprets its partition functions as measures of translational, rotational and vibrational action or quantum states, to estimate the entropy. With all kinds of molecular action expressed as logarithmic functions, the total heat required for warming a chemical system from 0o K (ΣSiT) to a given temperature and pressure can be computed, yielding results identical with published experimental third law values of entropy. All thermodynamic properties of gases including entropy, enthalpy, Gibbs energy and Helmholtz energy are directly estimated using simple algorithms based on simple molecular and physical properties, without resource to tables of standard values; both free energies are measures of quantum field states and of minimal statistical degeneracy, decreasing with temperature and declining density. We propose that this more realistic approach has heuristic value for thermodynamic computation of atmospheric profiles, based on steady state heat flows equilibrating with gravity. Potentially, this application of an action principle can provide better understanding of emergent properties of many natural or evolving complex systems, including modelling of predictions for global warming.

Recent works in mechanical fatigue consider that a threshold of entropy exists, the fracture fatigue entropy. The determination of this quantity is usually done considering empirical models for the mechanical power estimation. In this paper, we experimentally observe the existence of a threshold of entropy and exergy in low cycle fatigue for a flat Al-2024 specimen avoiding the use of a model, solely measuring the heat generated during a fatigue test. Results are then compared considering various hypotheses (1D heat dissipation with convection and radiation considered as heat sources, and, heat transfer from a fin with convection and radiation as boundary conditions) to an empirical mechanical model known in the literature and deviations between them are discussed.

We consider that the relationship between entropy in statistical mechanics, which is the Boltzmann principle, and the complex velocity potential in complex fluid dynamics. We define the generalized complex entropy which expanded entropy from real space to complex space. We show that the complex entropy can be expressed by the composition of sources, sinks and laminar flows of the complex velocity potential. Therefore, the complex entropy is considered a special case of the complex fluid dynamics,that is, the complex velocity potential. In other words, we show that the complex entropy is expressed by the complex velocity potential. Moreover, we show that a complex number is expressed by the complex entropy. Thus, we show that the complex velocity potential is expressed by the complex entropy. Namely, we will expand entropy to complex space and see that the complex entropy is expressed by the complex velocity potential. Furthermore, we examine the possibility that it is an expansion of Boltzmann's principle and Planck distribution.

The purpose of this study is to provide an accurate formula for calculating entropy for short DNA sequences and to demonstrate how to use it to examine leukemia patient surviving. We used IDIBAPS leukemia patient’s data base with 117 anonymized records. The generalized form of the Robust Entropy Estimator (EnRE) for short DNA sequences was proposed and key EnRE futures was showed. The Survival Analysis has been done using statistical package IBM SPSS. Entropy EnRE were calculated for leukemia patients for two samples: A. 2 groups divided by median EnRE and B. 2 groups of patients were formed according to their belonging to 1st and 4th quartiles of EnRE. The result of survival analysis are statistically significant: A. p &lt; 0.05; B. p &lt; 0.005. The death hazard for a patient with EnRE below median is 1.556 times that of a patient with EnRE over median and that the death hazard for a patient of 1st quartile (lowest EnRE) is 2.143 times that of a patient of 4th quartile (highest EnRE). The transition from median to quartile patients’ groups with more EnRE differentiation confirmed the unique significance of the entropy of DNA sequences for leukemia patients surviving.

Synergy, emerges from synchronized reciprocal positive feedback loops between a network of diverse actors. For this process to proceed, compatible information from different sources synchronically coordinates the actions of the actors resulting in a nonlinear increase in the useful work or potential energy the system can manage. In contrast noise is produced when incompatible information is mixed. This synergy produced from the coordination of different agents achieves non-linear gains in free energy and in information (negentropy) that are greater than the sum of the parts. The final product of new synergies is an increase in individual autonomy of an organism that achieves increased emancipation from the environment with increases in productivity, efficiency, capacity for flexibility, self-regulation and self-control of behavior through a synchronized division of ever more specialized labor. Examples that provide quantitative data for this phenomenon are presented. Results show that increases in free energy density require decreases in entropy density. This is proposed as a law of thermodynamics.

The existing segmentation-based scene text detection methods mostly need complicated post-processing, and the post-processing operation is separated from the training process, which greatly reduces the detection performance. The previous method, DBNet successfully simplified post-processing and integrated the post-processing into a segmentation network. However, the training process of the model took a long time for 1200 epochs and the sensitivity to texts of various scales was lacking, leading to some text instances being missed. Considering the above two problems, we design the text detection Network with Binarization of Hyperbolic Tangent(HTBNet). First of all, we propose Binarization of Hyperbolic Tangent (HTB), optimized along with which, the segmentation network can expedite the initial convergent speed by reducing the amount of epochs from 1200 to 600. Because features of different channels in the same scale feature map focus on the information of different regions in the image, to better represent the important features of all objects in the image, we devise the Multi-Scale Channel Attention(MSCA). Meanwhile considering that multi-scale objects in the image cannot be simultaneously detected, we propose a novel module named Fused Module with Channel and Spatial(FMCS), which can fuse the multi-scale feature maps from channel and spatial dimension. Finally we adopt cross entropy as the loss function, which measures the difference between predicted values and ground truths. The experimental results show that HTBNet compared with lightweight models has achieved competitive performance and speed on Total-Text(F-measure:86.0%, FPS:30) and MSRA-TD500 (F-measure:87.5%, FPS:30).

High entropy alloys have excellent wear resistance, so they have great application prospects in the fields of wear resistance and surface protection. In this study, the wear resistance of FeNiCrCoCu high entropy alloy coating was systematically analyzed by molecular dynamics method. FeNiCrCoCu high entropy alloy was used as a coating material to adhere to the surface of Cu matrix. The friction and nano indentation simulation of this coating material were carried out by controlling the ambient temperature. The influence of temperature on its friction properties was analyzed from five aspects: lattice structure, dislocation evolution, friction coefficient, hardness and elastic modulus. The results show that with the increase of temperature, the disorder of the lattice structure increases, which leads to the increase of the tangential force and the friction coefficient in the friction process. At 300K and 600K, the ordered lattice structure of the high entropy alloy coating material is basically the same, and thus their hardness is basically the same. However, the dislocation density at 600K is significantly reduced compared with that at 300K, resulting in the increase of the elastic modulus of the material from 173GPa to 219GPa. At the temperatures of 900K and 1200k, lattice disorder take place rapidly, and dislocation density also decreases significantly, resulting in a significant decrease in the hardness and elastic modulus of the material. When the temperature reaches 900K, the wear resistance of the FeNiCrCoCu high entropy alloy coating decreases sharply. This work is of great value to the analysis of wear resistance of high entropy alloy at high temperature.

High temperature Ni-superalloys are limited by reliance on critical elements and complex processing steps. High entropy alloys (HEAs) have demonstrated stable single/dual phase microstructures with strong ordering tendencies. A dual phase (FCC+BCC) Al0.5Co0.5CrFeNi1.5Ti0.25 alloy (developed using CALPHAD) was tested for room and high temperature compression behavior. The microstructure mainly comprises of coarse ordered FCC(L12) surrounding decomposed lamellae of ordered FCC(L12) and BCC(L21) phases, which coarsened upon annealing at high temperature (1100 °C) into a bi-continuous structure of disordered FCC+BCC(B2). Additionally, cold rollability of the cast alloy increased from &lt;5% to ~40% after annealing, with a corresponding softening observed in room temperature compression mainly due to disordering of FCC and BCC phases. Peak flow stresses of 793.45MPa, 535.12MPa, and 324.40MPa were achieved at 800oC, 900oC, and 1000oC respectively for the high temperature annealed alloy.

The Schelling model of segregation allows for a general description of residential movements in an environment modeled by a lattice. The key factor is that occupants change positions until they are surrounded by a designated minimum number of similarly labeled residents. An analogy to the Ising model has been made in previous research, primarily due the assumption of state changes being dependent upon the adjacent cell positions. This allows for concepts produced in statistical mechanics to be applied to the Schelling model. Here is presented a methodology to estimate the entropy of the model for different states of the simulation. A Monte Carlo estimate is obtained for the set of macrostates defined as the different aggregate homogeneity satisfaction values across all residents, which allows for the entropy value to be produced for each state. This produces a trace of the estimated entropy value for the states of the lattice configurations to be displayed with each iteration. The results show that the initial random placements of residents have larger entropy values than the final states of the simulation when the overall homogeneity of the residential locality is increased.

The Rényi entropy functional was optimised for the novel findings of this investigation under various conditions. Their research's findings suggest that the complete spectrum of Lévy stable distributions is covered by generalised t-distributions. Additionally, an exposition was undertaken to prove that the Lévy distribution generalizes Tsallisian distribution, rather than the reverse. The current study is a strong generalization of an existing research work in the literature.

In this paper, the elemental construct of an I-shaped fin cooling a heat generating volume is studied through the entropy production perspective. Using the constructal design method framework with entropy production minimization as an objective leads to different optimal shape ratio, different from the ones obtained using the excess of temperature. Moreover, the optimal shape ratio evolution is not necessary the same for local entropy production minimization (excess of entropy production) and global entropy production (entropy produced in the whole area). Finally, an analysis on a possible dimensionless number is introduced for the study of constructal plates.

Market uncertainty has a significant impact on market performance. Previous studies put much efforts in investigations on market uncertainty related to information asymmetry and risk. However, they have neglected the uncertainty inherent in market transactions, which is also an important aspect of market performance, besides quantity of transactions and market efficiency. In this paper, we put forward the concept of transaction entropy to measure the market uncertainty and see how it changes with price. The definition of transaction entropy is the ratio of total information entropy of all traders to quantity of transactions, reflecting the level of uncertainty in making transactions. And the transaction entropy is the lowest at equilibrium, it will be decreasing in a shortage market, and increasing in a surplus market. Additionally, we make a comparison of total entropy of the centralized market with that of the decentralized market, revealing that the price filtering mechanism could effectively reduce market uncertainty. Overall, the introduction of transaction entropy enriches our understanding of market uncertainty and facilitates a more comprehensive assessment of market performance.

In this paper, we concentrate on the study of the properties of residual Tsallis entropy for order statistics. Order statistics have an important role in reliability structural engineering for example for modelling lifetimes of series and parallel systems. The residual Tsallis entropy of ith order statistic from a continuous distribution function and its deviation from the residual Tsallis entropy of ith order statistics from a uniform distribution is investigated. In a mathematical framework, a method to express the residual Tsallis entropy of the ith order statistic from a continuous distribution in terms of the residual Tsallis entropy of the ith order statistic from a uniform distribution is provided. This approach may provide insight into the behavior and properties of the residual Tsallis entropy for order statistics. Further, we study the monotonicity properties of the residual Tsallis entropy of order statistics under dierent conditions. By studying these properties, deeper understanding of the relationship between the position of order statistics and the resulting residual Tsallis entropy is gained.

For a system that turns out to be inactive in time t, the past entropy is considered as an uncertainty measure for the past lifetime distribution. In this study, we consider a coherent system that includes n components and has the property that all the components of the system have failed at time t. To assess the predictability of the coherent system’s lifetime, we use the system’s signature to determine the Rényi entropy of its past lifetime. We study several analytical results, including expressions, bounds, and order properties for this measure.

Although most biological processes are characterized by a strong temporal asymmetry, several popular mathematical models neglect this issue. Maximum entropy methods provide a principled way of addressing time irreversibility, which leverages powerful results and ideas from the3literature of non-equilibrium statistical mechanics. This article provides a comprehensive overview of these issues, with a focus in the case of spike train statistics. We provide a detailed account of the5mathematical foundations and work out examples to illustrate the key concepts and results from non-equilibrium statistical mechanics

The historical idea of entropy as a property of a body has been reviewed and shown to arise from Clausius’s view of heat as motion. This view of heat, being intermediate between the now defunct idea of heat as substance and the modern view that heat represents an exchange of energy, implied that a body contains a definite quantity of heat. Heat, and therefore entropy, was thus considered a property of a body. These ideas led Clausius to develop his famous inequality and the idea that entropy always increases in irreversible, non-cyclic processes. In this paper, the notion of the entropy as a property of a body is examined in detail. A physical meaning is attached to the thermodynamic entropy by showing that a change in the function Q/T can be understood as representing a change in the number of ways that energy can be distributed among the degrees of freedom active in the system at any given temperature. The idea is illustrated with reference to solids and simple liquids. It is shown that the total thermodynamic entropy is, in general, less than the Boltzmann entropy, except for the case of a monatomic classical ideal gas in which the number of degrees of freedom is independent of temperature. Finally, entropy as a state function is discussed. It is argued that this is entirely mathematical in nature and that the entropy of a state represented in p-V space is not equivalent to a physical property of a physical system in the same thermodynamic state.

The fundamental objective is to study the application of multivariate sets of data in Gaussian distribution. This paper examines broad measurements of structure for both Gaussian and non-Gaussian distributions, which shows that they can be described in terms of the infor-mation-theoretic between the given covariance matrix and correlated random variables (in terms of relative entropy). In order to develop the multivariate Gaussian distribution with entropy and mutual information, several significant methodologies are presented through the discussion supported by illustrations, both technically and statistically. The content obtained allows readers to better perceive concepts, comprehend techniques, and properly execute software programs for future study on the topic's science and implementations. It also helps readers grasp the themes' fundamental concepts. Involving the relative entropy and mutual information as well as the potential correlated covariance analysis based on differential equations, a wide range of information is addressed, including basic to application concerns.

Following a recent proof of Shannon's entropy power inequality (EPI), a comprehensive framework for deriving various EPIs for the Rényi entropy is presented that uses transport arguments from normal densities and a change of variable by rotation. Simple arguments are given to recover the previously known Rényi EPIs and derive new ones, by unifying a multiplicative form with constant c and a modification with exponent α of previous works. In particular, for log-concave densities, we obtain a simple transportation proof of a sharp varentropy bound.

Shannon entropy quantifying bi-colored Ramsey complete graphs is introduced. Complete graphs in which vertices are connected with two types of links, labeled as α-links and β-links are considered. Shannon-entropy is introduced according to the classical Shannon formula considering the fractions of monochromatic convex α-colored polygons with n α-sides or edges, and the fraction of monochromatic β-colored convex polygons with m β-sides in the given complete graph. Introduced Shannon entropy is insensitive to the exact shape of the graph, but it is sensitive to the distribution of monochromatic polygons in a given graph. The introduced Shannon Entropies Sα and Sβ are interpreted as follows: Sα is interpreted as an average uncertainty to find the green α-polygon in the given graph, Sβ is, in turn, an average uncertainty to find the red β-polygon in the same graph. The re-shaping of the Ramsey theorem in terms of the Shannon Entropy is suggested. Various measures quantifying the Shannon Entropy of the entire complete bi-colored graphs are suggested. Physical interpretations of the suggested Shannon Entropies are discussed.

In this work, satellite data from the Clouds and Earth’s Radiant Energy System (CERES) and Moderate Resolution Imaging Spectroradiometer (MODIS) instruments is analyzed to determine how the global absorbed sunlight and global entropy generation rate have changed during the time period from 2002 to 2023. The data is used to test hypotheses derived from the Maximum Power Principle (MPP) and Maximum Entropy Production Principle (MEP) about the evolution of earth’s surface and atmosphere. The results indicate that the amount of absorbed sunlight has increased over the last 20 years but rising surface temperatures have caused the entropy generation rate to remain approximately constant despite that decrease in albedo. The results are simpler to explain using the MPP but do not completely rule out the MEP. Given the acceptance of the MPP or MEP, some extensions and nuances are discussed.

The notion of the informational measure of symmetry is introduced according to: HsymG=-i=1kPGilnPGi, where PGi is the probability of appearance of the symmetry operation Gi within the given 2D pattern. HsymG is interpreted as an averaged uncertainty in the presence of symmetry elements from the group G in the given pattern. The informational measure of symmetry of the “ideal” pattern built of identical equilateral triangles is established as HsymD3=1.792. The informational measure of symmetry of the random, completely disordered pattern is zero, Hsym=0. Informational measure of symmetry is calculated for the patterns generated by the P3 Penrose tessellation. Informational measure of symmetry does not correlate neither with the Voronoi entropy of the studied patterns nor with the continuous measure of symmetry of the patterns.

Maximum entropy production principle (MEPP) has been formulated in the mid-twentieth century, and today it has acquired the status of an important principle of science, which is extremely effective in considering various non-equilibrium problems. In this study, for the first time, definition of life is based on an easily measurable physical quantity that is entropy production. Life and evolution are discussed from the point of view of MEPP and the Universe, but not a human

This paper is devoted to change-point detection using only the ordinal structure of a time series. A statistic based on the conditional entropy of ordinal patterns characterizing the local up and down in a time series is introduced and investigated. The statistic requires only minimal a priori information on given data and shows good performance in numerical experiments.

The residual multiparticle entropy (RMPE) of a fluid is defined as the difference, Δs, between the excess entropy per particle (relative to an ideal gas with the same temperature and density), s_ex, and the pair-correlation contribution, s₂. Thus, the RMPE represents the net contribution to s_ex due to spatial correlations involving three, four, or more particles. A heuristic “ordering” criterion identifies the vanishing of the RMPE as an underlying signature of an impending structural or thermodynamic transition of the system from a less ordered to a more spatially organized condition (freezing is a typical example). Regardless of this, the knowledge of the RMPE is important to assess the impact of non-pair multiparticle correlations on the entropy of the fluid. Recently, an accurate and simple proposal for the thermodynamic and structural properties of a hard-sphere fluid in fractional dimension 1 < d < 3 has been proposed [Santos, A.; López de Haro, M. Phys. Rev. E 2016, 93, 062126]. The aim of this work is to use this approach to evaluate the RMPE as a function of both d and the packing fraction ϕ. It is observed that, for any given dimensionality d, the RMPE takes negative values for small densities, reaches a negative minimum Δs_min at a packing fraction ϕ_min, and then rapidly increases, becoming positive beyond a certain packing fraction ϕ₀. Interestingly, while both ϕ_min and ϕ₀ monotonically decrease as dimensionality increases, the value of Δs_min exhibits a nonmonotonic behavior, reaching an absolute minimum at a fractional dimensionality d ≃ 2.38. A plot of the scaled RMPE Δs/|Δs_min| shows a quasiuniversal behavior in the region −0.14 ≲ ϕ − ϕ₀ ≲ 0.02.

We examined a series of real-world, pictorial photographs with varying characteristics, along with their modification by noise addition and unsharp masking. As response metrics we used three different versions of the aesthetic measure originally proposed by Birkhoff. The first aesthetic measure, which has been used in other studies, and which we used in our previous work as well, showed a preference for the least complex of the images. It provided no justification for noise addition, but did reveal enhancement on unsharp masking. Optimum level of unsharp masking varied with the image, but was predictable from the individual image’s GIF compressibility. We expect this result to be useful for guiding the processing of pictorial photographic imagery. The second aesthetic measure, that of informational aesthetics based on entropy alone failed to provide useful discrimination among the images or the conditions of their modification. A third measure, derived from the concepts of entropy maximization, as well as the hypothesized preference of observers for “simpler”, i.e., more compressible, images, yielded qualitatively the same results as the more traditional version of the measure. Differences among the photographs and the conditions of their modification were more clearly defined with this metric, however.

The fractal dimension of is identified in this paper. More importantly, the undertaken research has shown how affects the Koch snowflake . The study has demonstrated the existence of negative values of (Koch snowflake), which connects to a considerable number of extremely intriguing mathematical and physical research areas. The information-theoretic impact on the Sierpiniski Gasket dimension , which corresponds to , was treated in a more fundamental way. Notably, the credibility of this current study adds to the existing kThe fractal dimension of is identified in this paper. More importantly, the undertaken research has shown how affects the Koch snowflake . The study has demonstrated the existence of negative values of (Koch snowflake), which connects to a considerable number of extremely intriguing mathematical and physical research areas. The information-theoretic impact on the Sierpiniski Gasket dimension , which corresponds to , was treated in a more fundamental way. Notably, the credibility of this current study adds to the existing knowledge for the information- theoretic fractal dimensions. On another applicative note, this paper offers some potential applications of fractal dimension to smart cities. The investigation of the threshold theorems for and in terms of the triad is part of the next study phase. Additionally, investigating the newly obtained 's corresponding arbitrary Sierpinski Gasket for the first time.applicative note, this paper offers some potential applications of fractal dimension to smart cities. The investigation of the threshold theorems for and in terms of the triad is part of the next study phase. Additionally, investigating the newly obtained 's corresponding arbitrary Sierpinski Gasket for the first time.

In this paper, we introduce generalized entropy, the acceleration of its entropy and its the partial entropy. We assume that generalized entropy can be expressed as a second-order polynomial by applying the idea of logistics function to its entropy. In other words, we assume that the acceleration of generalized entropy is a constant. Besides, we show that the negative inverse of the partial entropy can express Newton's classical gravity, which is an inverse square law. By applying these concepts, we attempt to explain that 1) Gravity is constant within small distances with some conditions. It is possible that gravity have 6-states within small distance. Furthermore, within small distance, we show the possibility that the gravitational potential and the Coulomb potential can be treated in the same way, that 2) The rotation speed of a galaxy does not depend on its radius if the radius is within the size level of the universe. (The galaxy rotation curve problem), and that 3) The gravitational acceleration toward the center changes at long distances compared to the classical theory of gravity. Furthermore, we show that the possibility of the existence of some constants which controls gravity and the speed of galaxies, and that gravity may relate on entropy.

In quantum many-body systems, a Hamiltonian is called an ``extensive entropy generator'' if starting from a random product state the entanglement entropy obeys a volume law at long times with overwhelming probability. We prove that (i) any Hamiltonian whose spectrum has non-degenerate gaps is an extensive entropy generator; (ii) in the space of (geometrically) local Hamiltonians, the non-degenerate gap condition is satisfied almost everywhere. Specializing to many-body localized systems, these results imply the observation stated in the title of Bardarson et al. [PRL 109, 017202 (2012)].

We study the existence or absence of non-Shannon inequalities for variables that are related by functional dependencies. Although the power-set on four variables is the smallest Boolean lattice with non-Shannon inequalities there exist lattices with many more variables without non-Shannon inequalities. We search for conditions that excludes the existence of non-Shannon inequalities. It is demonstrated that planar modular lattices cannot have non-Shannon inequalities. The existence of non-Shannon inequalities is related to the question of whether a lattice is isomorphic to a lattice of subgroups of a group.

Diagnostics and decomposition of atmospheric disturbances in a planar flow are considered and applied to numerical modeling results with the direct possibility to use in atmosphere monitoring especially in such strong events which follow magnetic storms. The study examines a situation in which the stationary equilibrium temperature of a gas may depend on a vertical coordinate, that seriously complicates the problem solution. The relations connecting perturbations for acoustic and entropy modes are analytically established and led to the solvable diagnostic equations. These perturbation structures, found as the equation solutions specify acoustic and entropy modes in an arbitrary stratified gas under the condition of stability. These time-independent diagnostic relations link gas perturbation variables of the acoustic and the entropy modes. Hence, they provide the ability to decompose the total vector of perturbations into acoustic and non-acoustic (entropy) parts uniquely at any instant within the all accessible heights range. As a prospective model, we consider the diagnostics at the height interval [120;180] km, where the equilibrium temperature of a gas depends linearly on the vertical coordinate. For such a heights range it is possible to proceed with analytical expressions for pressure and entropy perturbations of gas variables. Individual profiles of acoustic and entropy parts for some data, obtained by numerical experiment, are illustrated by the plots for the pure numerical data against ones obtained by the model. The total energy of a flow is determined for both approaches and its height profiles are compared.

The thermocontextual interpretation (TCI) establishes a system’s exergy, entropic energy, and thermal entropy as thermocontextual properties of state, defined with respect to its positive temperature surroundings. This work extends previous applications of the TCI to irreversible and statistical transitions. The TCI defines statistical entropy as a transactional process of derandomization and transition to a negative-entropy state. Statistical measurements of a confined quantum particle’s position are detailed in terms of reversible processes of instantiation and actualization. The TCI then formalizes the MaxEnt as a fundamental physical principle. We apply MaxEnt and statistical entropy measurements to the double-slit experiment. Particles passing through parallel slits record a wave interference pattern, but a “which-slit” detector eliminates wave interference. Richard Feynman called the double-slit experiment the only mystery, at the heart of quantum mechanics. The TCI offers a simple explanation. The which-slit detector breaks the system’s symmetry, enabling particles to pass through one slit or the other, and MaxEnt selects the asymmetrical transition, which has no wave interference and a higher statistical entropy.

This paper studies the dividend optimization problem in the entropy regularization framework by following the same continuous-time reinforcement learning setting as in Wang et al. (2020). The exploratory HJB is established and the optimal exploratory dividend policy is a truncated exponential distribution. We show that, for suitable choices of the maximal dividend paying rate and the temperature parameter, the value function of the exploratory dividend optimization problem could be significantly different from the value function in the classical dividend optimization problem. In particular, the value function of the exploratory dividend optimization problem could be classified into three cases based on its monotonicity. Numerical examples are also presented to show the impact of temperature parameter on the solution.

The prevailing interpretations of physics are based on deeply entrenched assumptions, rooted in classical mechanics. Logical implications include: the denial of entropy as a fundamental physical property, and the inability to explain irreversible change, random quantum measurements, or nonlocality without untestable and implausible metaphysical implications. We propose a conceptual model that is based on empirically justifiable assumptions and consistent with observations. The WYSIWYG Conceptual Model (WCM) assumes no hidden properties: “What You can See Is What You Get.” The WCM contextually defines a system’s state with respect to its actual ambient background, and it extends existing models of physical reality by defining entropy and exergy as objective contextual properties of state. The WCM establishes the irreversible dissipation of exergy and the Second law of thermodynamics as a fundamental law of physics, it recognizes physical randomness, and it provides a physical explanation for nonlocality, consistent with Special Relativity, without hidden variables, superdeterminism, or “spooky action.”

Combinatoric measures of entropy capture the complexity of a graph, but rely upon the calculation of its independent sets, or collections of non-adjacent vertices. This decomposition of the vertex set is a known NP-Complete problem and for most real world graphs is an inaccessible calculation. Recent work by Dehmer et al. and Tee et al. identified a number of alternative vertex level measures of entropy that do not suffer from this pathological computational complexity. It can be demonstrated that they are still effective at quantifying graph complexity. It is intriguing to consider whether there is a fundamental link between local and global entropy measures. In this paper, we investigate the existence of correlation between vertex level and global measures of entropy, for a narrow subset of random graphs. We use the greedy algorithm approximation for calculating the chromatic information and therefore Körner entropy. We are able to demonstrate close correlation for this subset of graphs and outline how this may arise theoretically.

The use of entropy concepts in the field of stochastic electrodynamics is briefly reviewed here. Entropy calculations to date that have been fully carried out are discussed in two main situations: first, where electric dipole oscillators interact with zero-point, or zero-point plus Planckian, or Rayleigh-Jeans radiation, and second where only these radiation fields exist within a cavity. The emphasis here will be on the first more difficult situation where both charged particles and radiation fields are present and interacting. Unlike the usual exposition on entropy in clasical statistical mechanics involving probabilistic notions of phase space occupation, the calculations to date for both particles and fields or for fields alone, follow the caloric entropy method where the notions of heat flow, adiabatic surfaces, and isothermal conditions, are utilized. Probability notions certainly still enter into the calculations, as the fields and charged particles interact stochastically together, following Maxwellian electrodynamics. Examples of phase-space calculations for harmonic oscillators and classical hydrogen atoms are carried out, emphasizing how much farther caloric entropy calculations have been successfully utilized.

With the development of computer science and software engineering, software becomes more and more complex. Traditional software reliability assurance techniques including software testing and evaluation can't ensure software reliable execution after being deployed. Software failure prediction techniques based on failure indicators can predict software failures according to abnormal indicator values. The latter can be collected using runtime monitoring techniques. An essential part of this method is finding proper indicators which have strong correlation with software failures. We propose a novel type of indicators in this work named software runtime entropy, which takes both software module execution time and call times into consideration. Three common open source software, grep, flex and gzip are used as study cases for finding the relationships between the indicators and software failures. Firstly, a series of fault injection experiments are conducted on those three software respectively. The decision tree algorithm is used to train those data to build the correlation models between software runtime entropy and software failures. Several common measures in machine learning domains such as accuracy, recall rates, and F-measure are used to evaluate the models. The decision tree models can be used as failure mechanisms to assist the failure prediction work. One can examine the value of runtime entropy and make a warning report when it ranges from the normal interval to abnormal one.

Autofocus is an essential part of the SAR imaging process. Multi-subaperture autofocus algorithm is a commonly used autofocus algorithm for processing SAR stripmap mode data. The multi-subaperture autofocus algorithm has two main steps, the first is to estimate the phase error gradient within the subaperture, the second is to splice the phase error gradient, that is, to remove the shift amount between the estimated adjacent subapertures’ error gradients. Previous gradient-splicing algorithms assume that the estimation of subaperture error is accurate, but when the estimation of subaperture phase error gradients is not accurate enough, these algorithm performance will be degraded. A new phase error gradient splicing algorithm is proposed in this paper. It roughly estimates the shift amount first, and then finely estimates the shift amount based on the minimum-entropy criterion, which can improve the robustness of splicing especially when the estimation of the phase error gradients of the subaperture is not accurate enough. To speed up the algorithm, a variable-step-size search method is used. Simulation and experimental results show that the algorithm has enough accuracy and still has good performance when other splicing algorithms doesn’t perform well.

We consider the maximum entropy Markov chain inference approach to characterize the collective statistics of neuronal spike trains, focusing on the statistical properties of the inferred model. We review large deviations techniques useful in this context to describe properties of accuracy and convergence in terms of sampling size. We use these results to study the statistical fluctuation of correlations, distinguishability and irreversibility of maximum entropy Markov chains. We illustrate these applications using simple examples where the large deviation rate function is explicitly obtained for maximum entropy models of relevance in this field.

The calculation of the standard enthalpies of vaporization, sublimation and solvation of organic molecules is presented using a common computer algorithm on the basis of a group-additivity method. The same algorithm is also shown to enable the calculation of their entropy of fusion as well as the total phase-change entropy of liquid crystals. The present method is based on the complete break-down of the molecules into their constituting atoms and their immediate neighbourhood; the respective calculations of the contribution of the atomic groups by means of the Gauss-Seidel fitting method is based on experimental data collected from literature. The feasibility of the calculations for each of the mentioned descriptors was verified by means of a 10-fold cross-validation procedure proving the good to high quality of the predicted values for the three mentioned enthalpies and for the entropy of fusion, whereas the predictive quality for the total phase-change entropy of liquid crystals was poor. The goodness of fit (Q²) and the standard deviation (σ) of the cross-validation calculations for the five descriptors was as follows: 0.9641 and 4.56 kJ/mol (N=3386 test molecules) for the enthalpy of vaporization, 0.8657 and 11.39 kJ/mol (N=1791) for the enthalpy of sublimation, 0.9546 and 4.34 kJ/mol (N=373) for the enthalpy of solvation, 0.8727 and 17.93 J/mol/K (N=2637) for the entropy of fusion and 0.5804 and 32.79 J/mol/K (N=2643) for the total phase-change entropy of liquid crystals. The large discrepancy between the results of the two closely related entropies is discussed in detail. Molecules, for which both the standard enthalpies of vaporization and sublimation were calculable, enabled the estimation of their standard enthalpy of fusion by simple subtraction of the former from the latter enthalpy. For 990 of them the experimental enthalpy-of-fusion values are also known, allowing their comparison with predictions, yielding a correlation coefficient R² of 0.6066.

Complex systems can be described as the network of interactions between their components. In directed networks, information, energy and matter flow in dominant directions. Undirected networks, on the other hand, cannot easily capture these asymmetries. The local flow of information depends on the node spin, which is a function of the number of incoming and outgoing links attached to it. And yet, how the heterogeneous association between in-degree and out-degree constraints global dynamics remains poorly understood. Here, we develop a new theoretical model to study the relationship between node directionality and global flow dynamics. Our computational analysis of random and scale-free directed networks indicates that structural diversity constrains the dimensionality of reachable dynamical space. These results suggest how local directionality might be an universal driver of global dynamics in many systems, from social networks, to technological systems to the connectome.

Abstract: The high entropy alloy (HEA) of equiatomic composition CrNiFeCoMn and with FCC 11 crystal structure was prepared by either induction melting and additive manufacturing with a se- 12 lective laser melting (SLM) process, starting from mechanically alloyed powders. Unlike induction 13 melting, the as-produced SLM alloy showed fine nitride and Cr-rich s phase precipitates. Mechan- 14 ical spectroscopy measurements were performed on specimens cold worked and/or recrystallized. 15 Young’s modulus values of (140±10) GPa and (90±10) GPa were measured at 300 K for induction 16 melted and SLM samples, respectively. The values increased to (160±10) GPa and (170±10) GPa for 17 recrystallized samples. Damping measurements as a function of temperature showed two peaks 18 which were attributed to dislocations bending and grain boundaries sliding. The peaks were super- 19 posed on a temperature increasing background.

Several optimization models of irreversible reverse cycle machines have been developed based on different optimization criteria in the literature, most of them using linear heat transfer laws at the source and sink. This raises the issue how close to actual operation conditions they are, since the heat transfer law on the phase-change processes is dependent on ΔT3. This paper addresses this issue by proposing a general model for study and optimization of thermal machines with two heat reservoirs applied to a Carnot-like refrigerator, with non-linear heat transfer laws and internal and external irreversibility. The optimization was performed using First and Second Law of Thermodynamics and Lagrange multipliers method. Thus, several constraints were imposed to the system, also different objective functions were considered, allowing finding the optimum operating conditions, as well as the limited variation ranges of the system parameters. Results show that the nature of the heat transfer laws affects the optimum values of system parameters for obtaining maximum performances and also their magnitude. Sensitivity studies with respect to system several parameters are presented. The results contribute to the understanding of the system limits in operation under different constraints and allow choosing the most convenient variables in given circumstances.

The diffusion coefficient--a measure of dissipation, and the entropy--a measure of fluctuation are found to be intimately correlated in many physical systems. Unlike the fluctuation dissipation theorem in linear response theory, the correlation is often strongly non-linear. To understand this complex dependence, we consider the classical Brownian diffusion in this work. Under certain rational assumption, i.e. in the bi-component fluid mixture, the mass of the Brownian particle $M$ is far greater than that of the bath molecule $m$, we can adopt the weakly couple limit. Only considering the first-order approximation of the mass ratio $m/M$, we obtain a linear motion equation in the reference frame of the observer as a Brownian particle. Based on this equivalent equation, we get the Hamiltonian at equilibrium. Finally, using canonical ensemble method, we define a new entropy that is similar to the Kolmogorov-Sinai entropy. Further, we present an analytic expression of the relationship between the diffusion coefficient $D$ and the entropy $S$ in the thermal equilibrium, that is to say, $D =\frac{\hbar}{eM} \exp{[S/(k_Bd)]}$, where $d$ is the dimension of the space, $k_B$ the Boltzmann constant, $\hbar $ the reduced Planck constant and $e$ the Euler number. This kind of scaling relation has been well-known and well-tested since the similar one for single component is firstly derived by Rosenfeld with the expansion of volume ratio.

Terms related to gradients of scalar fields are introduced as scalar products into the formulation of entropy. A Lagrange density is then formulated by adding constraints based on known conservation laws. Applying the Lagrange formalism to the resulting Lagrange density leads to the Poisson equation of gravitation and also includes terms being related to curvature of space. The formalism further leads to terms possibly explaining nonlinear extensions known from modified Newtonian dynamics approaches. The article concludes with a short discussion of the presented methodology and provides an outlook on other phenomena, which might be tackled using this new approach.

In the fast-evolving landscape of web services, success relies on optimizing efficiency, reliability, and user satisfaction. Decision-making in real-world applications demands a blend of quantitative (numerical values of quality of Service factors) and qualitative data of experts’ ratings of the quality of service factors. This paper proposes an optimization technique that integrates Hesitant and Intuitionistic Fuzzy Linguistic Terms with TOPSIS, leveraging Entropy Objective weights. There is limited research on Hesitant Intuitionistic Fuzzy Linguistic terms and the need to model fuzziness, haste and intuition in data precisely from experts rating. Thus, this article introduces an Entropy-based model tailored for web service optimization. Using the WS-DREAM dataset, quantitative data was extracted and objective weights were determined through the Entropy algorithm and qualitative evaluations from five experts, employing Hesitant Intuitionistic Fuzzy Linguistic Terms, were integrated. The process involved computing criteria weights and experts' assessment of Quality of Service factors for four services (S1, S2, S3, and S4) to form the Fuzzy Decision Matrix using modified Online output software. Applying the proposed technique to a web service optimization problem validated its effectiveness. The results indicated optimal selection of S1 as the best web service, supported by Coefficient Index values (0.785). This study introduces a novel approach, Entrop – Hesitationistic - Fuzz-TOPSIS, for web service optimization, integrating quantitative and qualitative data through an Entropy-based model. The effectiveness of the proposed technique, validated through application on the WS-DREAM dataset, underscores its potential in addressing real-world decision-making challenges. The optimal selection of S1 as the best web service suggests practical utility. The study encourages further exploration of this technique's applicability to diverse Multiple Criteria Decision Making (MCDM) problems, opening avenues for future research and application.

We present a new method for assessing the strength of fuzzy rules with respect to a dataset, based on the measures of the greatest energy and smallest entropy of a fuzzy relation. Considering a fuzzy automaton (relation) in which A is the input fuzzy set and B the output fuzzy set, the fuzzy relation R1 with greatest energy provides information about the greatest strength of the input-output and the fuzzy relation R2 with the smallest entropy provides information about uncertainty of the relationship input-output. We consider a new index of the fuzziness of the input-output based over R1 and R2. In our method this index is calculated for each pair of input and output fuzzy sets in a fuzzy rule. A threshold value is set for choosing the most relevant fuzzy rules with respect to the data.

The emergence of various electronic devices and equipment such as electric vehicles and drones requires higher energy density energy storage devices. Lithium-sulfur batteries (LSBs) are considered as the most promising new generation energy storage system owing to high theoretical specific capacity and energy density. However, the severe shuttle behaviors of soluble lithium polysulfides (LiPSs) and the slow redox kinetics lead to low sulfur utilization and poor cycling stability, which seriously hinder the commercial application of LSBs. Therefore, various catalytic materials have been employed to solve these troublesome problems. High entropy materials (HEMs), as advanced materials, can provide unique surface and electronic structures that expose plentiful catalytic active sites, which opens new ideas for the regulation of LiPSs redox kinetics. Notwithstanding many instructive reviews in the land of LSBs, while this review aims to offer a complete and shrewd summary of the current progresses in HEMs-based LSBs, including in-depth interpretation of design principles and mechanistic electrocatalysis functions as well as pragmatic perspectives.

The utilization of selective laser melting (SLM) for high entropy alloys (HEAs) holds significant promise in commercial applications, and substantial experimental research efforts have been directed toward this domain. To take advantage of the reported experimental data of SLM manufactured (SLM-ed) HEAs and reduce unnecessary experimentation, this study incorporates machine learning (ML) techniques for the phase and tensile properties prediction of SLM-ed HEAs, thus presenting a novel avenue for accelerating the discovery of new SLM-ed HEAs. Through the adjustment of material descriptors and machine learning models, a model has been developed with an impressive accuracy of 93.8% in distinguishing between face-centered cubic (FCC), body-centered cubic (BCC), and dual-phase (FCC+BCC). Additionally, optimized models have been devised for the prediction of tensile properties, namely ultimate tensile strength (UTS), yield strength (YS), and elongation (δ), achieving noteworthy MAPEs (mean absolute percentage errors) of 10.33%, 8.55%, and 28.48%, respectively. Furthermore, several HEAs were fabricated using SLM, and the experimental outcomes exhibited a favorable alignment with the predicted results. These efforts carry significant implications in advancing the utilization of SLM for HEAs.