Entropy — Open Access Journal

This paper adopts the information and fractional calculus tools for studying the dynamics of a national soccer league. A soccer league season is treated as a complex system (CS) with a state observable at discrete time instants, that is, at the time of rounds. The CS state, consisting of the goals scored by the teams, is processed by means of different tools, namely entropy, mutual information and Jensen–Shannon divergence. The CS behavior is visualized in 3-D maps generated by multidimensional scaling. The points on the maps represent rounds and their relative positioning allows for a direct interpretation of the results. Full article

Polymer nanocomposite materials, consisting of a polymer matrix embedded with nanoscale fillers or additives that reinforce the inherent properties of the matrix polymer, play a key role in many industrial applications. Understanding of the relation between thermodynamic interactions and macroscopic morphologies of the composites allow for the optimization of design and mechanical processing. This review article summarizes the recent advancement in various aspects of entropic effects in polymer nanocomposites, and highlights molecular methods used to perform numerical simulations, morphologies and phase behaviors of polymer matrices and fillers, and characteristic parameters that significantly correlate with entropic interactions in polymer nanocomposites. Experimental findings and insight obtained from theories and simulations are combined to understand how the entropic effects are turned into effective interparticle interactions that can be harnessed for tailoring nanostructures of polymer nanocomposites. Full article

The device-independent approach to physics is one where conclusions about physical systems (and hence of Nature) are drawn directly and solely from the observed correlations between measurement outcomes. This operational approach to physics arose as a byproduct of Bell’s seminal work to distinguish, via a Bell test, quantum correlations from the set of correlations allowed by local-hidden-variable theories. In practice, since one can only perform a finite number of experimental trials, deciding whether an empirical observation is compatible with some class of physical theories will have to be carried out via the task of hypothesis testing. In this paper, we show that the prediction-based-ratio method—initially developed for performing a hypothesis test of local-hidden-variable theories—can equally well be applied to test many other classes of physical theories, such as those constrained only by the nonsignaling principle, and those that are constrained to produce any of the outer approximation to the quantum set of correlations due to Navascués-Pironio-Acín. We numerically simulate Bell tests using hypothetical nonlocal sources of correlations to illustrate the applicability of the method in both the independent and identically distributed (i.i.d.) scenario and the non-i.i.d. scenario. As a further application, we demonstrate how this method allows us to unveil an apparent violation of the nonsignaling conditions in certain experimental data collected in a Bell test. This, in turn, highlights the importance of the randomization of measurement settings, as well as a consistency check of the nonsignaling conditions in a Bell test. Full article

Recurrent neural networks (RNNs) are nonlinear dynamical models commonly used in the machine learning and dynamical systems literature to represent complex dynamical or sequential relationships between variables. Recently, as deep learning models have become more common, RNNs have been used to forecast increasingly complicated systems. Dynamical spatio-temporal processes represent a class of complex systems that can potentially benefit from these types of models. Although the RNN literature is expansive and highly developed, uncertainty quantification is often ignored. Even when considered, the uncertainty is generally quantified without the use of a rigorous framework, such as a fully Bayesian setting. Here we attempt to quantify uncertainty in a more formal framework while maintaining the forecast accuracy that makes these models appealing, by presenting a Bayesian RNN model for nonlinear spatio-temporal forecasting. Additionally, we make simple modifications to the basic RNN to help accommodate the unique nature of nonlinear spatio-temporal data. The proposed model is applied to a Lorenz simulation and two real-world nonlinear spatio-temporal forecasting applications. Full article

Fixed sample entropy (fSampEn) has been successfully applied to myographic signals for inspiratory muscle activity estimation, attenuating interference from cardiac activity. However, several values have been suggested for fSampEn parameters depending on the application, and there is no consensus standard for optimum values. This study aimed to perform a thorough evaluation of the performance of the most relevant fSampEn parameters in myographic respiratory signals, and to propose, for the first time, a set of optimal general fSampEn parameters for a proper estimation of inspiratory muscle activity. Different combinations of fSampEn parameters were used to calculate fSampEn in both non-invasive and the gold standard invasive myographic respiratory signals. All signals were recorded in a heterogeneous population of healthy subjects and chronic obstructive pulmonary disease patients during loaded breathing, thus allowing the performance of fSampEn to be evaluated for a variety of inspiratory muscle activation levels. The performance of fSampEn was assessed by means of the cross-covariance of fSampEn time-series and both mouth and transdiaphragmatic pressures generated by inspiratory muscles. A set of optimal general fSampEn parameters was proposed, allowing fSampEn of different subjects to be compared and contributing to improving the assessment of inspiratory muscle activity in health and disease. Full article

Given the intensity and frequency of environmental change, the linked and cross-scale nature of social-ecological systems, and the proliferation of big data, methods that can help synthesize complex system behavior over a geographical area are of great value. Fisher information evaluates order in data and has been established as a robust and effective tool for capturing changes in system dynamics, including the detection of regimes and regime shifts. The methods developed to compute Fisher information can accommodate multivariate data of various types and requires no a priori decisions about system drivers, making it a unique and powerful tool. However, the approach has primarily been used to evaluate temporal patterns. In its sole application to spatial data, Fisher information successfully detected regimes in terrestrial and aquatic systems over transects. Although the selection of adjacently positioned sampling stations provided a natural means of ordering the data, such an approach limits the types of questions that can be answered in a spatial context. Here, we expand the approach to develop a method for more fully capturing spatial dynamics. The results reflect changes in the index that correspond with geographical patterns and demonstrate the utility of the method in uncovering hidden spatial trends in complex systems. Full article

In this paper first, we review the physical root bases of chemical reaction networks as a Markov process in multidimensional vector space. Then we study the chemical reactions from a microscopic point of view, to obtain the expression for the propensities for the different reactions that can happen in the network. These chemical propensities, at a given time, depend on the system state at that time, and do not depend on the state at an earlier time indicating that we are dealing with Markov processes. Then the Chemical Master Equation (CME) is deduced for an arbitrary chemical network from a probability balance and it is expressed in terms of the reaction propensities. This CME governs the dynamics of the chemical system. Due to the difficulty to solve this equation two methods are studied, the first one is the probability generating function method or z-transform, which permits to obtain the evolution of the factorial moment of the system with time in an easiest way or after some manipulation the evolution of the polynomial moments. The second method studied is the expansion of the CME in terms of an order parameter (system volume). In this case we study first the expansion of the CME using the propensities obtained previously and splitting the molecular concentration into a deterministic part and a random part. An expression in terms of multinomial coefficients is obtained for the evolution of the probability of the random part. Then we study how to reconstruct the probability distribution from the moments using the maximum entropy principle. Finally, the previous methods are applied to simple chemical networks and the consistency of these methods is studied. Full article

Cracks and defects, which could result in lower reflectivity and larger full width at half maximum (FWHM), are the major obstacles for obtaining highly ordered structures of colloidal crystals (CCs). The high-quality CCs with high reflectivity (more than 90%) and 9.2 nm narrow FWHM have been successfully fabricated using a fixed proportion of a soft matter system composed of silica particles (SPs), polyethylene glycol diacrylate (PEGDA), and ethanol. The influences of refractivity difference, volume fractions, and particle dimension on FWHM were illuminated. Firstly, we clarified the influences of the planar interface and the bending interface on the self-assembly. The CCs had been successfully fabricated on the planar interface and presented unfavorable results on the bending interface. Secondly, a hard sphere system consisting of SPs, PEGDA, and ethanol was established, and the entropy-driven phase transition mechanism of a polydisperse system was expounded. The FWHM and reflectivity of CCs showed an increasing trend with increasing temperature. Consequently, high-quality CCs were obtained by adjusting temperatures (ordered structure formed at 90 °C and solidified at 0 °C) based on the surface phase rule of the system. We acquired a profound understanding of the principle and process of self-assembly, which is significant for preparation and application of CCs such as optical filters. Full article

We review and improve previous work on non-equilibrium classical and quantum statistical systems, subject to potentials, without ab initio dissipation. We treat classical closed three-dimensional many-particle interacting systems without any “heat bath” ($hb$), evolving through the Liouville equation for the non-equilibrium classical distribution $W_c$, with initial states describing thermal equilibrium at large distances but non-equilibrium at finite distances. We use Boltzmann’s Gaussian classical equilibrium distribution $W_{c,eq}$, as weight function to generate orthogonal polynomials ($H_n$’s) in momenta. The moments of $W_c$, implied by the $H_n$’s, fulfill a non-equilibrium hierarchy. Under long-term approximations, the lowest moment dominates the evolution towards thermal equilibrium. A non-increasing Liapunov function characterizes the long-term evolution towards equilibrium. Non-equilibrium chemical reactions involving two and three particles in a $hb$ are studied classically and quantum-mechanically (by using Wigner functions W). Difficulties related to the non-positivity of W are bypassed. Equilibrium Wigner functions $W_{eq}$ generate orthogonal polynomials, which yield non-equilibrium moments of W and hierarchies. In regimes typical of chemical reactions (short thermal wavelength and long times), non-equilibrium hierarchies yield approximate Smoluchowski-like equations displaying dissipation and quantum effects. The study of three-particle chemical reactions is new. Full article

In this paper, we establish a clear distinction between two processes yielding anomalous diffusion and $1/f$ noise. The first process is called Stationary Fractional Brownian Motion (SFBM) and is characterized by the use of stationary correlation functions. The second process rests on the action of crucial events generating ergodicity breakdown and aging effects. We refer to the latter as Aging Fractional Brownian Motion (AFBM). To settle the confusion between these different forms of Fractional Brownian Motion (FBM) we use an entropic approach properly updated to incorporate the recent advances of biology and psychology sciences on cognition. We show that although the joint action of crucial and non-crucial events may have the effect of making the crucial events virtually invisible, the entropic approach allows us to detect their action. The results of this paper lead us to the conclusion that the communication between the heart and the brain is accomplished by AFBM processes. Full article

The entropy production in stochastic dynamical systems is linked to the structure of their causal representation in terms of Bayesian networks. Such a connection was formalized for bipartite (or multipartite) systems with an integral fluctuation theorem in [Phys. Rev. Lett. 111, 180603 (2013)]. Here we introduce the information thermodynamics for time series, that are non-bipartite in general, and we show that the link between irreversibility and information can only result from an incomplete causal representation. In particular, we consider a backward transfer entropy lower bound to the conditional time series irreversibility that is induced by the absence of feedback in signal-response models. We study such a relation in a linear signal-response model providing analytical solutions, and in a nonlinear biological model of receptor-ligand systems where the time series irreversibility measures the signaling efficiency. Full article

To improve the recognition accuracy of ship-radiated noise, a feature extraction method based on regenerated phase-shifted sinusoid-assisted empirical mode decomposition (RPSEMD), mutual information (MI), and differential symbolic entropy (DSE) is proposed in this paper. RPSEMD is an improved empirical mode decomposition (EMD) that alleviates the mode mixing problem of EMD. DSE is a new tool to quantify the complexity of nonlinear time series. It not only has high computational efficiency, but also can measure the nonlinear complexity of short time series. Firstly, the ship-radiated noise is decomposed into a series of intrinsic mode functions (IMFs) by RPSEMD, and the DSE of each IMF is calculated. Then, the MI between each IMF and the original signal is calculated; the sum of MIs is taken as the denominator; and each normalized MI (norMI) is obtained. Finally, each norMI is used as the weight coefficient to weight the corresponding DSE, and the weighted DSE (WDSE) is obtained. The WDSEs are sent into the support vector machine (SVM) classifier to classify and recognize three types of ship-radiated noise. The experimental results demonstrate that the recognition rate of the proposed method reaches 98.3333%. Consequently, the proposed WDSE method can effectively achieve the classification of ships. Full article

In this review, we outline some principal theoretical knowledge of the properties of frustrated spin systems and magnetic thin films. The two points we would like to emphasize: (i) the physics in low dimensions where exact solutions can be obtained; (ii) the physics at phase boundaries where interesting phenomena can occur due to competing interactions of the two phases around the boundary. This competition causes a frustration. We will concentrate our attention on magnetic thin films and phenomena occurring near the boundary of two phases of different symmetries. Two-dimensional (2D) systems are in fact the limiting case of thin films with a monolayer. Naturally, we will treat this case at the beginning. We begin by defining the frustration and giving examples of frustrated 2D Ising systems that we can exactly solve by transforming them into vertex models. We will show that these simple systems already contain most of the striking features of frustrated systems such as the high degeneracy of the ground state (GS), many phases in the GS phase diagram in the space of interaction parameters, the reentrance occurring near the boundaries of these phases, the disorder lines in the paramagnetic phase, and the partial disorder coexisting with the order at equilibrium. Thin films are then presented with different aspects: surface elementary excitations (surface spin waves), surface phase transition, and criticality. Several examples are shown and discussed. New results on skyrmions in thin films and superlattices are also displayed. By the examples presented in this review we show that the frustration when combined with the surface effect in low dimensions gives rise to striking phenomena observed in particular near the phase boundaries. Full article

The methanol synthesis via CO₂ hydrogenation (MSCH) reaction is a useful CO₂ utilization strategy, and this synthesis path has also been widely applied commercially for many years. In this work the performance of a MSCH reactor with the minimum entropy generation rate (EGR) as the objective function is optimized by using finite time thermodynamic and optimal control theory. The exterior wall temperature (EWR) is taken as the control variable, and the fixed methanol yield and conservation equations are taken as the constraints in the optimization problem. Compared with the reference reactor with a constant EWR, the total EGR of the optimal reactor decreases by 20.5%, and the EGR caused by the heat transfer decreases by 68.8%. In the optimal reactor, the total EGRs mainly distribute in the first 30% reactor length, and the EGRs caused by the chemical reaction accounts for more than 84% of the total EGRs. The selectivity of CH₃OH can be enhanced by increasing the inlet molar flow rate of CO, and the CO₂ conversion rate can be enhanced by removing H₂O from the reaction system. The results obtained herein are in favor of optimal designs of practical tubular MSCH reactors. Full article

An increase of seismic activity is often observed before large earthquakes. Events responsible for this increase are usually named foreshock and their occurrence probably represents the most reliable precursory pattern. Many foreshocks statistical features can be interpreted in terms of the standard mainshock-to-aftershock triggering process and are recovered in the Epidemic Type Aftershock Sequence ETAS model. Here we present a statistical study of instrumental seismic catalogs from four different geographic regions. We focus on some common features of foreshocks in the four catalogs which cannot be reproduced by the ETAS model. In particular we find in instrumental catalogs a significantly larger number of foreshocks than the one predicted by the ETAS model. We show that this foreshock excess cannot be attributed to catalog incompleteness. We therefore propose a generalized formulation of the ETAS model, the ETAFS model, which explicitly includes foreshock occurrence. Statistical features of aftershocks and foreshocks in the ETAFS model are in very good agreement with instrumental results. Full article

We study thermodynamics in $f(R)$ gravity with the disformal transformation. The transformation applied to the matter Lagrangian has the form of ${\gamma }_{\mu \nu }=A(\varphi ,X)g_{\mu \nu }+B(\varphi ,X){\partial }_{\mu }\varphi {\partial }_{\nu }\varphi$ with the assumption of the Minkowski matter metric ${\gamma }_{\mu \nu }={\eta }_{\mu \nu }$, where $\varphi$ is the disformal scalar and X is the corresponding kinetic term of $\varphi$. We verify the generalized first and second laws of thermodynamics in this disformal type of $f(R)$ gravity in the Friedmann-Lemaître-Robertson-Walker (FLRW) universe. In addition, we show that the Hubble parameter contains the disformally induced terms, which define the effectively varying equations of state for matter. Full article

We study Bell scenarios with binary outcomes supplemented by one bit of classical communication. We developed a method to find facet inequalities for such scenarios even when direct facet enumeration is not possible, or at least difficult. Using this method, we partially solved the scenario where Alice and Bob choose between three inputs, finding a total of 668 inequivalent facet inequalities (with respect to relabelings of inputs and outputs). We also show that some of these inequalities are constructed from facet inequalities found in scenarios without communication, that is, the well-known Bell inequalities. Full article

Multi-scale permutation entropy (MPE) is a statistic indicator to detect nonlinear dynamic changes in time series, which has merits of high calculation efficiency, good robust ability, and independence from prior knowledge, etc. However, the performance of MPE is dependent on the parameter selection of embedding dimension and time delay. To complete the automatic parameter selection of MPE, a novel parameter optimization strategy of MPE is proposed, namely optimized multi-scale permutation entropy (OMPE). In the OMPE method, an improved Cao method is proposed to adaptively select the embedding dimension. Meanwhile, the time delay is determined based on mutual information. To verify the effectiveness of OMPE method, a simulated signal and two experimental signals are used for validation. Results demonstrate that the proposed OMPE method has a better feature extraction ability comparing with existing MPE methods. Full article

High entropy or compositionally complex alloys provide opportunities for optimization towards new high-temperature materials. Improvements in the equiatomic alloy Al₁₇Co₁₇Cr₁₇Cu₁₇Fe₁₇Ni₁₇ (at.%) led to the base alloy for this work with the chemical composition Al₁₀Co₂₅Cr₈Fe₁₅Ni₃₆Ti₆ (at.%). Characterization of the beneficial particle-strengthened microstructure by scanning electron microscopy (SEM) and observation of good mechanical properties at elevated temperatures arose the need of accomplishing further optimization steps. For this purpose, the refractory metals hafnium and molybdenum were added in small amounts (0.5 and 1.0 at.% respectively) because of their well-known positive effects on mechanical properties of Ni-based superalloys. By correlation of microstructural examinations using SEM with tensile tests in the temperature range of room temperature up to 900 °C, conclusions could be drawn for further optimization steps. Full article