Entropy Of Mixture Research Articles

Entropy-Based Volatility Analysis of Financial Log-Returns Using Gaussian Mixture Models

Volatility in financial markets refers to the variation in asset prices over time. High volatility indicates increased risk, making its evaluation essential for effective risk management. Various methods are used to assess volatility, with the standard deviation of log-returns being a common approach. However, this implicitly assumes that log-returns follow a Gaussian distribution, which is not always valid. In this paper, we explore the use of (differential) entropy to evaluate the volatility of financial log-returns. Estimation of entropy is obtained using a Gaussian mixture model to approximate the underlying density of log-returns. Following this modeling approach, popular risk measures such as Value at Risk and Expected Shortfall can also be computed. By integrating Gaussian mixture modeling and entropy into the analysis of log-returns, we aim to provide a more accurate and robust framework for assessing financial volatility and risk measures.

Average Entropy of Gaussian Mixtures.

We calculate the average differential entropy of a q-component Gaussian mixture in Rn. For simplicity, all components have covariance matrix σ21, while the means {Wi}i=1q are i.i.d. Gaussian vectors with zero mean and covariance s21. We obtain a series expansion in μ=s2/σ2 for the average differential entropy up to order O(μ2), and we provide a recipe to calculate higher-order terms. Our result provides an analytic approximation with a quantifiable order of magnitude for the error, which is not achieved in previous literature.

Machine learning-enabled prediction and optimization of hardness for Nb-Ti-V-Zr refractory high entropy alloy

High entropy alloy (HEA) design strategies have been limited to experimental trial-and-error approaches. However, due to the vast compositional space of HEAs, efficiently discovering new HEAs with exceptional performance using trial-and-error methodology is time-consuming and challenging. In this work, a machine learning (ML) framework was introduced for Nb-Ti-V-Zr refractory HEA (RHEA) design. This framework aims to streamline the design of Nb-Ti-V-Zr RHEA with a focus on improving hardness. This framework encompasses data collection, feature construction, four-step feature selection, and training of different ML models, followed by Fisher material pattern recognition and SHAP feature analysis. For the first time, three key features, including the atomic size difference (δ), the configurational entropy of mixture (Smix), and the average deviation of electronegativity (AdE) were identified as the most significant compositional factors impacting the hardness of Nb-Ti-V-Zr RHEA. Furthermore, the Support Vector Regression (SVR) model with a polynomial kernel was the best-trained model for hardness prediction compared to other ML models trained and tested in this work. The coefficient of determination (R) was 0.85 for the testing set and 0.91 for 5-fold cross-validation, while the root mean square error (RMSE) values were 0.9 and 0.74, respectively. Moreover, it was observed that high-hardness samples follow a particular combination of key features, providing important optimization insights. Finally, the analysis of key features revealed the negative effect of AdE, the positive effect of Smix, and the hybrid effect of δ on hardness.

On the entropy and information of Gaussian mixtures

AbstractWe establish several convexity properties for the entropy and Fisher information of mixtures of centred Gaussian distributions. Firstly, we prove that if are independent scalar Gaussian mixtures, then the entropy of is concave in , thus confirming a conjecture of Ball, Nayar and Tkocz (2016) for this class of random variables. In fact, we prove a generalisation of this assertion which also strengthens a result of Eskenazis, Nayar and Tkocz (2018). For the Fisher information, we extend a convexity result of Bobkov (2022) by showing that the Fisher information matrix is operator convex as a matrix‐valued function acting on densities of mixtures in . As an application, we establish rates for the convergence of the Fisher information matrix of the sum of weighted i.i.d. Gaussian mixtures in the operator norm along the central limit theorem under mild moment assumptions.

Entropy of Coherent States of Optical Fields

We consider statistical mixtures of two and three arbitrary coherent states that are the states of a data carrier in an optical communication channel. The eigenvalues and eigenvectors of the corresponding density operators are obtained in analytic form. For this purpose, an original parameterization of triangles on the complex phase plane is introduced. Analytic expressions are obtained for the von Neumann entropy of mixtures of two and three arbitrary coherent states, and its maximal value for a given average number of photons is calculated. It is found that the use of three coherent states for a given average number of photons ensures a larger capacity of an optical communication channel as compared to two states.

On Regularized Systems of Equations for Gas Mixture Dynamics with New Regularizing Velocities and Diffusion Fluxes

We deal with multidimensional regularized systems of equations for the one-velocity and one-temperature inert gas mixture dynamics consisting of the balance equations for the mass of components and the momentum and total energy of the mixture, with diffusion fluxes between the components as well as the viscosity and heat conductivity terms. The regularizations are kinetically motivated and aimed at constructing conditionally stable symmetric in space discretizations without limiters. We consider a new combined form of regularizing velocities containing the total pressure of the mixture. To confirm the physical correctness of the regularized systems, we derive the balance equation for the mixture entropy with the non-negative entropy production, under generalized assumptions on the diffusion fluxes. To confirm nice regularizing properties, we derive the systems of equations linearized at constant solutions and provide the existence, uniqueness and L2-dissipativity of weak solutions to an initial-boundary problem for them. For the original systems, we also discuss the related Petrovskii parabolicity property and its important corollaries. In addition, in the one-dimensional case, we also present the special three-point and symmetric finite-difference discretization in space of the regularized systems and prove that it inherits the entropy correctness property. We also give results of numerical experiments confirming that the discretization is able to simulate well various dynamic problems of contact between two different gases.

Predicting vapor-liquid equilibria of CO<sub>2</sub>+HFC binary mixtures by the PR EOS combined with a group contribution model

CO<sub>2</sub>+HFC binary mixtures have good performance and environmental friendliness and are considered good alternative working fluids in cooling and power cycle systems. The vapor-liquid phase equilibrium properties are key to the calculation of the enthalpy and entropy of mixtures, which is critical for the analysis of cooling and power cycle systems. To accurately predict the vapor-liquid equilibrium of CO<sub>2</sub> and HFC (R23, R32, R41, R125, R134a, R143a, R152a, R161, and R227ea) binary mixtures, a group contribution model based on the excess free energy (<i>G</i><sup>E</sup>) mixing rules (PR+MHV1+UNIFAC and PR+LCVM+UNIFAC) is established in this paper. The interaction parameters between groups such as -CO<sub>2</sub>, -Alkane, -CHF, and -CHF<sub>3</sub> are obtained by the vapor-liquid phase equilibrium experiment of CO<sub>2</sub> and HFC refrigerants, and these group parameters are critical for predicting their vapor-liquid phase equilibrium properties (the pressures and vapor phase molar fractions). The AARD<i>p</i> value calculated by the PR+LCVM+UNIFAC model is 5.53%, the value of AAD<i>y</i><sub>1</sub> is 0.0132, and the AARD<i>p</i> and AAD<i>y</i><sub>1</sub> values of the PR+MHV1+UNIFAC model are 7.40% and 0.0229, respectively. However, for the CO<sub>2</sub>+R32 system, the PR+MHV+UNIFAC prediction model can reproduce the experimental data with lower deviations, and the values of AARD<i>p</i> and AAD<i>y</i><sub>1</sub> are 1.53% and 0.0045, respectively. In summary, for CO<sub>2</sub>+HFC binary mixtures, the PR+LCVM+UNIFAC group contribution model can reproduce the experimental data with lower deviations, but for individual CO<sub>2</sub> binary mixtures (such as CO<sub>2</sub>+R32), the PR+MHV1+UNIFAC model also has unique advantages. According to the prediction results of the group contribution model, the PR+LCVM+UNIFAC model has significantly improved the calculation progress compared with the PR+MHV1+UNIFAC model used in the previous system.

Study of relaxation processes in a two-phase flow model

This work concerns the analysis of the relaxation processes toward thermodynamical equilibrium arising in a compressible immiscible two-phase flow. Classically the relaxation processes are taken into account through dynamical systems which are coupled to the dynamics of the flow. The present paper compares two types of source terms which are commonly used: a BGK-like system and a mixture entropy gradient type. For both systems, main properties are investigated (agreement with second principle of thermodynamics, existence of solutions, maximum principle,...) and numerical experiments illustrate their asymptotic behaviour.

Entropy Correct Spatial Discretizations for 1D Regularized Systems of Equations for Gas Mixture Dynamics

One-dimensional regularized systems of equations for the general (multi-velocity and multi-temperature) and one-velocity and one-temperature compressible multicomponent gas mixture dynamics are considered in the absence of chemical reactions. Two types of the regularization are taken. For the latter system, diffusion fluxes between the components of the mixture are taken into account. For both the systems, the important mixture entropy balance equations with non-negative entropy productions are valid. By generalizing a discretization constructed previously in the case of a single-component gas, we suggest new nonstandard symmetric three-point spatial discretizations for both the systems which are not only conservative in mass, momentum, and total energy but also satisfy semi-discrete counterparts of the mentioned entropy balance equations with non-negative entropy productions. Importantly, the basic discretization in the one-velocity and one-temperature case is not constructed directly but by aggregation of the discretization in the case of general mixture, and that is a new approach. In this case, the results of numerical experiments are also presented for contact problems between two different gases for initial pressure jumps up to 2500.

Microstructure and Mechanical Properties of Rexnbmotaw Thin Films Prepared By RF Magnetron Sputtering

High Entropy Alloys (HEAs) and Refractory High-Entropy Alloys (RHEAs) are novel potential materials for high-temperature applications [1]. NbMoTaW, a RHEAs, such as the one in Fig. 1, possess superior mechanical properties, combining high strength, outstanding thermal stability, and resistance to softening at high temperatures [2]. The influence of adding Vanadium [3], Tantalum [4], and Rhenium [5] to NbMoTaW alloys on the bulk mechanical behavior was investigated by the previous studies. In this research, the effect of adding Rhenium with different (Re0, Re0.5, Re1) on the microstructure and the mechanical properties of NbMoTaW thin films will be nvestigated. The larger the enthalpy of mixing in negative values, the higher the binding force between the elements, which has been enhanced with Rhenium addition. As for the entropy of mixing, it increased from 11.53 to 13.38 for Re0 and Re1, respectively. The higher the mixture entropy suggests a more stable solid solution. This is confirmed by calculating the thermodynamic parameters proposed by Yang et. Al [6] where they suggested that a mixture with δ < 6.6 and Ω > 1.1 correspond to solid solution formation. The anticipated crystal structure for the three samples is BCC structure according to the valence electron concentration (VEC) theory, where the VEC values were less than 6.87 as suggested by Gou et. Al [7]. The RexNbMoTaW HEAs thin films are fabricated using RF magnetron sputtering. The film thickness, deposition power, and temperature impact on the films’ characteristics are studied. Field emission (FE-SEM), X-ray diffraction (XRD), and nanoindentation are used to investigate the RexNbMoTaW thin films’ crystal structure, surface morphology, and mechanical properties. The mechanical properties of the thin films will be compared with their counterpart bulk materials.References Kim, H., et al., International Journal of Refractory Metals and Hard Materials, 2019. 80: p. 286-291.Feng, X., et al., Materials Letters, 2018. 210: p. 84-87.Senkov, O.N., et al., Intermetallics, 2011. 19(5): p. 698-706.Han, Z.D., et al., Materials Science and Engineering: A, 2018. 712: p. 380-385.Zhang, J., et al., Journal of Alloys and Compounds, 2020. 827.Yang, X. and Y. Zhang, Materials Chemistry and Physics, 2012. 132(2-3): p. 233-238.Guo, S., et al., Journal of Applied Physics, 2011. 109(10): p. 103505. Figure 1

Application of Flory-Huggins model in experimental and theoretical study of stability of amphotericin B on nanocarrier based on functionalized carbon nanotube

• Water activities for binary and ternary systems were measured. • The interaction between species were obtained based on the Flory-Huggins parameter. • The Gibbs free energy and entropy of ternary mixtures were calculated. • The HSPs ( δ p , δ h , and δ d ) for each component were calculated by GCM. The objective of this study is to investigate the interaction between components in a novel and stable drug delivery system for delivering amphotericin B (Amb) as a model drug. A newly designed drug delivery system will be set up based on a single-wall carbon nanotube (SWCNT), which will be functionalized with a biocompatible polymer called 1,2-Distearoyl-sn-Glycerol-3-Phosphoethanolamine Polyethylene Glycol Carboxylic Acid (DSPE-PEG-COOH), and then amine groups are created on its surface by adding ethylenediamine (EDA). It is used as a novel nanocarrier vehicle for delivering Amb for treatment of potentially life threatening fungal infections. It is imperative that the conjugation of Amb to the functionalized single wall carbon nanotube (FCNT) remains stable to reach the required release profile during the drug administration process. The stabilities of conjugations are investigated through analyses of interactions and miscibility of drug delivery system components. The Flory-Huggins parameter ( χ FH ) is an important criterion that will estimate interactions between the drug delivery system (DDS) components. Each binary system has two components selected from a pool of three entities (FCNT, Amb and water), and the χ FH of each system is calculated by group contribution method (GCM) through Hansen solubility parameter (HSP) and the measurement of water activity. Calculating the χ FH for FCNT and Amb pair shows that the interactions and miscibility between them is favorable, and will provide the required stability of nanocarrier-drug conjugation in FCNT-Amb. The average absolute deviation percentages for (drug-water), (FCNT-water) and (FCNT-water-Amb) systems are 0.542, 0.549 and 0.449, respectively.

Vehicle Counting Method Based On Gaussian Mixture Models And Blob Analysis

Combination of a Gaussian Mixture Models and Entropy to detect vehicles in motion from any sort of video sequences. The proposed method is composed of four major steps: background subtraction, improved GMM, Blob Analysis, Voting Scheme processing. An implementation of proposed technique has been performed using MATLAB R2017a. Vehicle detection process provides relevant information about traffic patterns, crash occurrences and traffic peak times in roadways. Detection of number of vehicles and traffic monitoring will be helpful by implementing motion and object detection using background subtraction, Monitoring traffic from video footages will provide a greater reality in terms of vehicle count. Gaussian Mixture Model is most advanced model in motion detection due to the reliability that it has shown in the background extraction and foreground segmentation process. Here we are going to find motion of object in video sequences under both static camera arrangement and dynamic background sequences by using background subtraction method and blob analysis approach and analyzing them

Investigation on two-phase fluid mixture flow, heat transfer and entropy generation of a non-Newtonian water-CMC/CuO nanofluid inside a twisted tube with variable twist pitch: Numerical and evolutionary machine learning simulation

A 3-D numerical investigation is performed to evaluate the effect of variable twist pitch on the hydrothermal behavior and entropy generation features of non-Newtonian water-CMC/CuO nanofluid (NF) flow inside a twisted tube with a square cross-section. Three twisted tubes with a length of 500 mm, each of which has 3 twists, are considered. The first tube (Case I) has a constant twist pitch of 100 mm, while the twist pitch of Case II (150.0, 127.5, 100.8, 74.2, and 47.5 mm) and Case III (190, 144.6, 97.9, 51.3, and 6.2 mm) are variable. The simulations are performed using the two-phase mixture method considering different nanoparticle concentrations (ϕs) of 0–3% and Reynolds numbers (Res) of 600–1500. Based on the results, the highest and lowest overall hydrothermal performance was obtained for Case II and Case I, respectively. Moreover, the lowest ratios of thermal and frictional entropies of NF flow in the twisted tube to those of the plain tube were obtained for Case II. As another novelty of the current work, an evolutionary machine learning approach, namely, gene expression programming (GEP), was adopted to simulate the first law and second law performances of the NF in Case III.

The Differential Entropy of Mixtures: New Bounds and Applications

Mixture distributions are extensively used as a modeling tool in diverse areas from machine learning to communications engineering to physics, and obtaining bounds on the entropy of mixture distributions is of fundamental importance in many of these applications. This article provides sharp bounds on the entropy concavity deficit, which is the difference between the differential entropy of the mixture and the weighted sum of differential entropies of constituent components. Toward establishing lower and upper bounds on the concavity deficit, results that are of importance in their own right are obtained. In order to obtain nontrivial upper bounds, properties of the skew-divergence are developed and notions of “skew” <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f$ </tex-math></inline-formula> -divergences are introduced; a reverse Pinsker inequality and a bound on Jensen-Shannon divergence are obtained along the way. Complementary lower bounds are derived with special attention paid to the case that corresponds to independent summation of a continuous and a discrete random variable. Several applications of the bounds are delineated, including to mutual information of additive noise channels, thermodynamics of computation, and functional inequalities.

Modeling the Transport of Fuel Mixture Perturbations and Entropy Waves in the Linearized Framework

Abstract In this paper, a new method is introduced to model the transport of entropy waves and equivalence ratio fluctuations in turbulent flows. The model is based on the Navier–Stokes equations and includes a transport equation for a passive scalar, which may stand for entropy or equivalence ratio fluctuations. The equations are linearized around the mean turbulent fields. These serve as the input to the model in addition to a turbulent eddy viscosity, which accounts for turbulent diffusion of the perturbations. Based on these inputs, the framework is able to predict the linear response of the flow velocity and passive scalar to harmonic perturbations that are imposed at the boundaries of the computational domain. These, in this study, are fluctuations in the passive scalar and/or velocities at the inlet of a channel flow. The code is first validated against analytic results, showing very good agreement. Then, the method is applied to predict the convection, mean flow dispersion, and turbulent mixing of passive scalar fluctuations in a turbulent channel flow, which has been studied in previous work with direct numerical simulations (DNS). Results show that our code reproduces the dynamics of coherent passive scalar transport in the DNS with very high accuracy and low numerical costs. Furthermore, we demonstrate that turbulent mixing has a significant effect on the transport of the passive scalar fluctuations. Finally, we apply the method to explain experimental observations of transport of equivalence ratio fluctuations in the mixing duct of a model burner.

Solubility, dissolution thermodynamics and preferential solvation of 4-nitroaniline in (ethanol + water) mixtures

ABSTRACT In this research, the equilibrium mole fraction solubility of 4-nitroaniline (4-NA) in some aqueous-ethanolic mixtures was determined at seven temperatures from (293.15 to 323.15) K. The respective apparent thermodynamic functions (Gibbs energy, enthalpy, and entropy) of the dissolution processes were computed using the van’t Hoff and Gibbs equations. The enthalpy-entropy relationship for 4-NA was non-linear in the plot of enthalpy vs. Gibbs energy of solution with negative slopes from neat water to the mixture w 1 = 0.10 and from w 1 = 0.20 to neat ethanol but positive slope from w 1 = 0.10 to w 1 = 0.20. Therefore, the driving mechanism for 4-NA transfer processes is the entropy in water-rich mixtures. Additionally, by means of the inverse Kirkwood-Buff integrals is observed that 4-NA is preferentially solvated by water molecules in water-rich mixtures but preferentially solvated by ethanol molecules in the mixtures of 0.23 ≤ x 1 ≤ 1.00.

Dissolution thermodynamics and preferential solvation of 2,4-dinitrotoluene in (ethanol + water) mixtures

In this research, the equilibrium mole fraction solubility of 2,4-dinitrotoluene (2,4-DNT) in some aqueous-ethanolic mixtures was determined at seven temperatures from (293.15 to 323.15) K. The respective apparent thermodynamic functions (Gibbs energy, enthalpy, and entropy of the dissolution processes) were computed using the van't Hoff and Gibbs equations. The enthalpy-entropy relationship for 2,4-DNT was non-linear in the plot of enthalpy vs. Gibbs energy of solution with negative slope in the composition region 0.00 ≤ w1 ≤ 0.40 but positive slope in the region 0.40 ≤ w1 ≤ 1.00. Therefore, the driving mechanism for 2,4-DNT transfer processes is the entropy in water-rich mixtures and the enthalpy in ethanol-rich mixtures. In addition, by means of the inverse Kirkwood-Buff integrals is observed that 2,4-DNT is preferentially solvated by water molecules in water-rich mixtures but preferentially solvated by ethanol molecules in the mixtures of 0.23 ≤ x1 ≤ 1.00.

The calculation models of vapor-liquid equilibrium, enthalpy and entropy for R152a/R1234ze(E)

To mitigate the impact of human activities on the environment, R22 with ODP not equal to 0 and high GWP must be weeded out. R1234ze has excellent environmental performance but insufficient refrigeration capacity per unit. R152a has excellent refrigeration performance but flammability. Their mixture will make up for each other’s defects and can be used as a new alternative refrigerant. Thermodynamic properties of the mixture are studied in this paper. The vapor–liquid equilibrium model, based on Peng–Robinson equation of state and VDWS mixing law, is established for R152a/R1234ze with temperature from 258.15 to 288.15 K. When the results are compared to the values of experiments, the maximal and average errors for pressures are 0.617% and 0.135%, respectively. Based on vapor–liquid equilibrium model combining residual property, enthalpy and entropy models are also established to calculate enthalpy and entropy of mixture (R152a and R1234ze with the mass ratio 60:40), which enable provide theoretical basis for the application of R152a/R1234ze.

Solubility of meloxicam in (Carbitol® + water) mixtures: Determination, correlation, dissolution thermodynamics and preferential solvation

The equilibrium solubility of meloxicam (3) in {Carbitol® (1) + water (2)} mixtures at several temperatures from 293.15 K to 313.15 K has been determined and correlated by means of some well-known thermodynamic correlation models. Solubility increases 1144 times, from x3 = 1.137 × 10−6 in neat water to x3 = 1.300 × 10−3 in neat Carbitol® at 298.15 K. By using the van't Hoff and Gibbs equations the respective apparent thermodynamic quantities of the dissolution and mixing processes, namely Gibbs energy, enthalpy, and entropy, were calculated. Apparent dissolution enthalpies vary from a minimum of 7.7 kJ mol−1 in neat water to a maximum of 32.3 kJ mol−1 in the mixture x1 = 0.10. Moreover, apparent dissolution entropies vary from a minimum of −88.0 J mol−1 K−1 in neat water to a maximum of 9.9 J mol−1 K−1 in the mixture x1 = 0.10. Non-linear enthalpy–entropy relationship was observed for meloxicam in the plot of enthalpy vs. Gibbs energy exhibiting negative slope in the composition region 0.00 < x1 < 0.10 and variant but mainly positive slopes in the other mixtures. Hence, the driving mechanism for meloxicam transfer process from more polar to less polar solvent systems is the entropy in water-rich mixtures and the enthalpy in the other solvent compositions. Furthermore, the preferential solvation of meloxicam by Carbitol® and water was analysed by using the inverse Kirkwood-Buff integrals. Meloxicam is preferentially solvated by water molecules in water-rich mixtures exhibiting a maximum in the mixture x1 = 0.10 with δx1,3 = −3.19 × 10−2 but preferentially solvated by Carbitol® molecules in mixtures 0.12 < x1 < 1.00 exhibiting a maximum in the mixture x1 = 0.20 with δx1,3 = 6.87 × 10−2.

Dissolution thermodynamics and preferential solvation of gliclazide in (Transcutol® + water) mixtures

ABSTRACT The equilibrium solubility of gliclazide (3) in {Transcutol (1) + water (2)} mixtures at several temperatures was reported in the literature. However, no deep attempt was made to correlate gliclazide solubility with the composition of the aqueous-cosolvent mixtures. Moreover, no apparent thermodynamic calculations for the dissolution processes were performed. Thus, the reported solubility data were modelled by means of some well-known correlation models. By using the van’t Hoff and Gibbs equations, the respective apparent thermodynamic quantities of dissolution and mixing processes, namely Gibbs energy, enthalpy, and entropy, were calculated. Non-linear enthalpy–entropy relationship was observed for gliclazide in the plot of enthalpy vs Gibbs energy exhibiting negative slope in the composition region 0.00 < w 1 < 0.10 and variant but mainly positive slopes in the other mixtures. Hence, the driving mechanism for gliclazide transfer process from more polar to less polar solvent systems is the entropy in water-rich mixtures and the enthalpy in the other solvent compositions. Furthermore, the preferential solvation of gliclazide by both solvents was analysed by using the inverse Kirkwood-Buff integrals observing that gliclazide is preferentially solvated by water molecules in water-rich mixtures but preferentially solvated by Transcutol molecules in mixtures 0.12 < x 1 < 1.00.