General Kinetic Equation Research Articles

Wigner Function Method for Describing Electromagnetic Field in Plasma-Like Media with Spatial Dispersion and Resonant Dissipation

The paper gives a systematic presentation of the Wigner function method (Weyl formalism) for modeling the propagation and absorption of electromagnetic waves in anisotropic and gyrotropic dissipative media with spatial dispersion. A general kinetic equation for the Wigner function (tensor) is formulated, and its asymptotic expansion up to the second order for smoothly inhomogeneous and weakly dissipative media is constructed. As a result, a modification of the method of the kinetic equation for rays is proposed, based on the stochastic description of rays, making it possible to increase the accuracy of numerical modeling of wave problems with strong transverse inhomogeneity of the absorption coefficient without increasing the amount of calculations. The technique developed can be used to describe the propagation, absorption, and scattering of electron-cyclotron waves in high-temperature plasma of magnetic traps for controlled fusion in cases where standard modeling methods do not provide the necessary accuracy.

Can varying activation energy be determined reliably from thermogravimetric experiments?

Solutions of the general isoconversional kinetic equation were generated and compared assuming activation energies, E, which vary with the advance of the reaction, α. Series belonging to 4–5 heating rates were compared. TG curves simulated with highly varying activation energies could approximate well the curves simulated with first-order kinetics and constant E. This observation indicates that the information content of a series of TG curves at constant heating rates is not sufficient for the determination of activation energies that vary with the advance of the studied reactions. The problem proved to be smaller when differential curves were compared in the same way; the uncertainties decreased by factors 0.2–0.5. There is a standard procedure of ASTM International (ASTM E2958-19, 2019. https://doi.org/10.1520/E2958-21) that describes the estimation of E from experiments carried out at a specific modulated temperature program. The reliability of this procedure was also tested and found to be low, though not as low as that of the evaluation of TG curves at linear temperature programs with usual heating rates. The work continues and complements a recent study of the author (Várhegyi in J Therm Anal Calorim 148:12835–12843, 2023).

On discrete models of Boltzmann-type kinetic equations

The known nonlinear kinetic equations, in particular, the wave kinetic equation and the quantum Nordheim–Uehling–Uhlenbeck equations are considered as a natural generalization of the classical spatially homogeneous Boltzmann equation. To this goal we introduce the general Boltzmann-type kinetic equation that depends on a function of four real variables \(F(x,y;
 v,w)\). The function \(F\) is assumed to satisfy certain simple relations. The main properties of this kinetic equation are studied. It is shown that the above mentioned specific kinetic equations correspond to different polynomial forms of the function \(F\). Then the problem of discretization of the general Boltzmann-type kinetic equation is considered on the basis of ideas similar to those used for construction of discrete velocity models of the Boltzmann equation. The main attention is paid to discrete models of the wave kinetic equation. It is shown that such models have a monotone functional similarly to the Boltzmann \(H\)-function. The theorem of existence, uniqueness and convergence to equilibrium of solutions to the Cauchy problem with any positive initial conditions is formulated and discussed. The differences in long time behaviour between solutions of the wave kinetic equation and solutions of its discrete models are also briefly discussed.

Electrospherization of genistein@DNA core-shell nanospheres as a drug delivery system and theoretical study of the release mechanism

Encapsulation of poor water-soluble drug was considered as one of the widely used approaches to overcome such obstacle. The goal of this research was in situ encapsulate the hydrophobic genistein (GEN) during the electrospherization (Es) of deoxyribonucleic acid (DNA) nanospheres as a delivery system with appropriate drug release properties. The prepared nanospheres (free GEN sample; Es DNA and GEN loaded DNA; Es GEN@DNA) were characterized using uv–visible spectroscopy (UV–vis), transmission electron microscope (TEM), zeta potential, Fourier transform infrared spectroscopy (FTIR), x-ray diffraction analysis (XRD), and stability test. In addition, the drug encapsulation % was studied, the drug release efficiency was recorded and theoretically visualized to understand the mechanism and kinetics of GEN drug release. The results revealed that GEN was successfully encapsulated during the DNA electrospherization as a core (GEN)/shell (DNA) like structure with a wonderful stability against time. The UV–vis patterns and their deconvolutions introduced that the emerged peak around 372–500 nm of Es GEN@DNA nanospheres has an area under curve larger than that of the free GEN sample. Zeta potential value decreased from −17.4 to −14.3 mV. In addition, TEM images confirmed the formation of a core/shell structure with average size 109.2 nm. XRD pattern of Es GEN@DNA revealed that GEN lost its crystalline phase, which indicated the incorporation of the drug. On the other hand, %Encapsulation of GEN within DNA nanospheres was found to be 89.62%. Es GEN@DNA release profile explored that the well entrapped GEN within the DNA nanospheres could be a promising for sustained drug release. Besides, the dilemma of using a fractal or fractional kinetics model was overcame by introducing a general fractional kinetic equation that involves a time-dependent rate coefficient, which introduced that the solution of the fractional kinetic model can fit the release data profiles.

Boltzmann-type kinetic equations and discrete models

The known non-linear kinetic equations (in particular, the wave kinetic equation and the quantum Nordheim-Uehling-Uhlenbeck equations) are considered as a natural generalization of the classical spatially homogeneous Boltzmann equation. To this goal we introduce the general Boltzmann-type kinetic equation that depends on a function of four real variables $F(x,y;v,w)$. The function $F$ is assumed to satisfy certain commutation relations. The general properties of this equation are studied. It is shown that the kinetic equations mentioned above correspond to different forms of the function (polynomial) $F$. Then the problem of discretization of the general Boltzmann-type kinetic equation is considered on the basis of ideas which are similar to those used for the construction of discrete models of the Boltzmann equation. The main attention is paid to discrete models of the wave kinetic equation. It is shown that such models have a monotone functional similar to the Boltzmann $H$-function. The existence and uniqueness theorem for global in time solution of the Cauchy problem for these models is proved. Moreover, it is proved that the solution converges to the equilibrium solution when time goes to infinity. The properties of the equilibrium solution and the connection with solutions of the wave kinetic equation are discussed. The problem of the approximation of the Boltzmann-type equation by its discrete models is also discussed. The paper contains a concise introduction to the Boltzmann equation and its main properties. In principle, it allows one to read the paper without any preliminary knowledge in kinetic theory. Bibliography: 61 titles.

Boltzmann-type kinetic equations and discrete models

The known non-linear kinetic equations (in particular, the wave kinetic equation and the quantum Nordheim-Uehling-Uhlenbeck equations) are considered as a natural generalization of the classical spatially homogeneous Boltzmann equation. To this end we introduce the general Boltzmann-type kinetic equation that depends on a function of four real variables $F(x,y;v,w)$. The function $F$ is assumed to satisfy certain commutation relations. The general properties of this equation are studied. It is shown that the kinetic equations mentioned above correspond to different forms of the function (polynomial) $F$. Then the problem of discretization of the general Boltzmann-type kinetic equation is considered on the basis of ideas which are similar to those used for the construction of discrete models of the Boltzmann equation. The main attention is paid to discrete models of the wave kinetic equation. It is shown that such models have a monotone functional similar to the Boltzmann $H$-function. The existence and uniqueness theorem for global in time solution of the Cauchy problem for these models is proved. Moreover, it is proved that the solution converges to the equilibrium solution when time goes to infinity. The properties of the equilibrium solution and the connection with solutions of the wave kinetic equation are discussed. The problem of the approximation of the Boltzmann-type equation by its discrete models is also discussed. The paper contains a concise introduction to the Boltzmann equation and its main properties. In principle, it allows one to read the paper without any preliminary knowledge in kinetic theory. Bibliography: 61 titles.

A mild-slope formulation based on Weyl rule of association with application to coastal wave modelling

Weyl rule of association, proposed by Hermann Weyl for quantum mechanics applications (Weyl, 1931), can be used to associate between the dispersion relation of water waves and a non-local pseudo-differential operator. The central result of this study is that this operator correctly approximates the Dirichlet-to-Neumann operator derived for linear waves over a slowly varying bathymetry. This opens the door to a formal use of Weyl’s operational calculus, and consequently, allowing straightforward derivations and generalizations of water waves’ models over mild slopes. Specifically, within the framework of linear wave theory, the formulation based on Weyl rule of association provides a generalized mild-slope model which does not impose a limit on the spectral width. Most significantly, the mild-slope formulation based on Weyl rule of association allows to derive a general linear kinetic equation for which the widely used energy balance equation (the central equation of forecasting models such as SWAN and WAVEWATCH) serves as a special case. This result not only provides a formal link between the deterministic description (i.e., Euler equations) and the stochastic description (i.e., the energy balance equation), but also establishes the theoretical foundations for the statistical description of bathymetry-induced wave interferences. Such a statistical description is especially important over coastal waters, where through the interaction with the bathymetry, waves are rapidly scattered and tend to form focal zones and associated interference patterns.

Flexible Kinetic Model Determination of Reactions in Materials under Isothermal Conditions

Kinetic analysis remains a powerful tool for studying a large variety of reactions, which lies at the core of material science and industry. It aims at obtaining the kinetic parameters and model that best describe a given process and using that information to make reliable predictions in a wide range of conditions. Nonetheless, kinetic analysis often relies on mathematical models derived assuming ideal conditions that are not necessarily met in real processes. The existence of nonideal conditions causes large modifications to the functional form of kinetic models. Therefore, in many cases, experimental data hardly obey any of these ideal models. In this work, we present a novel method for the analysis of integral data obtained under isothermal conditions without any type of assumption about the kinetic model. The method is valid both for processes that follow and for those that do not follow ideal kinetic models. It consists of using a general kinetic equation to find the functional form of the kinetic model via numerical integration and optimization. The procedure has been tested both with simulated data affected by nonuniform particle size and experimental data corresponding to the pyrolysis of ethylene-propylene-diene.

A Spatial Kinetic Model of Crowd Evacuation Dynamics with Infectious Disease Contagion.

This paper proposes a kinetic theory approach coupling together the modeling of crowd evacuation from a bounded domain with exit doors and infectious disease contagion. The spatial movement of individuals in the crowd is modeled by a proper description of the interactions with people in the crowd and the environment, including walls and exits. At the same time, interactions among healthy and infectious individuals may generate disease spreading if exposure time is long enough. Immunization of the population and individual awareness to contagion is considered as well. Interactions are modeled by tools of game theory, that let us propose the so-called tables of games that are introduced in the general kinetic equations. The proposed model is qualitatively studied and, through a series of case studies, we explore different scenarios related to crowding and gathering formation within indoor venues under the spread of a respiratory infectious disease, obtaining insights on specific policies to reduce contagion that may be implemented.

Studies on luminescence properties and photo-thermo transfer phenomena in fluorapatite

This study reports the structure, composition, photoluminescence (PL), thermoluminescence (TL), and optically stimulated luminescence (OSL) properties of the Brazilian natural fluorapatite. Also, the existence of photo-thermo transfer phenomena is investigated from TL/OSL emission mechanisms. X-ray diffraction (XRD) pattern reveals that the sample has a hexagonal system with a crystallite size of about 46 nm calculated using the Scherrer formula. The presence of several elements including the rare earths such as Ce, Nd, and Er are identified using total X-ray fluorescence (TXRF) spectroscopy results. Ce3+ ions are associated with the emission band at 371 nm when excited at 300 nm. Fluorapatite exhibits intense TL and OSL signals when irradiated with beta particles and shows linear dose-response from 1 up to 50 Gy. The TL glow curves are deconvoluted to six peaks centered at 99, 147, 188, 242, 329, and 431 °C using general order kinetic equation. The corresponding activation energies vary between 0.68 and 1.11 eV. An anomalous fading is noticed for the first TL peak, which decays completely in 6 h; the second TL glow peak decays about 59% within 12 h and other high temperature peaks are more stable. Continuous-wave OSL (CW-OSL) is constituted of three components, having decay times (τ) of 0.39, 5, and 64 s. The deconvoluted linearly modulated OSL (LM-OSL) has four peaks, with peak maxima (tm) at 2.85, 7.4, 27, and 130 s. Further, it is noticed the phenomenon of phototransferred TL (PTTL) at 30 °C, due to the optical transfer of electrons related to the traps of 329 and 431 °C peaks to the main peak at 242 °C. The temperature-induced PTTL using thermally assisted OSL (TA-OSL) at 170 °C with 400 s of blue stimulation depicted a transfer of trapped electrons from very deep traps (VDTs), located at temperatures higher than 500 °C, to the main peak. Here, the luminescence intensity increases with the beta dose.

Kinetics of confined water freezing – An application to the frost-resistance of porous materials

In the manuscript one presents a regular study aiming to determine the relation describing the freezing kinetics as a function of temperature employing the following cooling rates: 2.5 °C/min, 5 °C/min, 7.5 °C/min and 10 °C/min. The increase in ice mass is determined using differential scanning calorimeter by integrating the net heat flow for two building materials: red clay brick and cement paste (w/c = 0.5). The isoconversional method, which states that the transition rate (the rate of ice mass) at constant extent of conversion (ice mass) depends only on temperature, is applied to determine a transition model. We select the most appropriate model and determine the kinetic triplet using the method proposed by Málek and recommended by International Committee for Thermal Analysis and Calorimetry (ICTAC). The activation energy is estimated using the differential Friedman's method. Both analysed materials evinced different microstructures, but the kinetic parameters determined for both of them are relatively similar. Hence, one might conclude that the kinetic of inpore water freezing is not very sensitive to size of voids. The results of experimental analysis enabled one to formulate the general kinetic equations for both analysed materials. The comparison of theoretical and experimental results evinces good agreement. Therefore, the formulas derived in the manuscript might be used for further analysis of the porous building materials durability, which experience significant temperature fluctuations. Furthermore, they can be used to design the most efficient testing procedure for the determination of frost resistance of concrete. The proposed procedure might be used for other problems related with the phase transition of water in porous materials, i.e., food freezing geotechnics, etc.Concluding, the results calculated using the proposed models for both materials are in good agreement with the experimental data. Therefore, the formulas derived in the manuscript might be used for further analysis of the porous building materials durability, which experience significant temperature fluctuations. Furthermore, they can be used to design the most efficient testing procedure for the determination of frost resistance of concrete. The proposed procedure might be used for other problems related with the phase transition of water in porous materials, i.e. food freezing geotechnics, etc.

Investigation of the efficacy of the UV/Chlorine process for the removal of trimethoprim: Effects of operational parameters and artificial neural networks modelling

The UV/Cl2 process (also known as chlorine photolysis, which is the combination of chlorine and simultaneous irradiation of UV light) is conventionally applied at acidic mediums for drinking water treatment and further treatment of wastewater effluents for secondary reuse. This is because the quantum yield of HO• from HOCl (ϕHO•, 254 = 1.4) is greater than the one from OCl- (ϕHO•, 254 = 0.278) by approximately 5 times. Moreover, chlorine photolysis in acidic mediums also tends to have lower radical quenching rates than that of their alkaline counterparts by up to 1000 times. The aim of this research is to investigate the applicability of the UV/Cl2 process by assessing its efficacy on the removal of trimethoprim (TMP) at not only acidic to neutral conditions (pH 6-7), but also alkaline mediums (pH 8-9). At alkaline pH, free chlorine exists as OCl- and since OCl- has a higher molar absorption coefficient as compared to HOCl at higher wavelengths, there would be higher reactive chlorine species (RCS) formation and contribution. TMP removal followed pseudo-first order kinetics and depicted that a maximum fluence based constant (kf′ = 0.275 cm2/mJ) was obtained using 42.25 μM (3 mg/L) of chlorine at pH 9, with an irradiation of 275 nm. At alkaline conditions, chlorine photolysis performance followed the trend of UV (275)/Cl2 > UV (265)/Cl2 > UV (310)/Cl2 > UV (254)/Cl2. RCS like Cl•, Cl2−• and ClO• contributed to the degradation of TMP. When the pH was increased from 6 to 8, contribution from hydroxyl radicals (HO• ) was decreased whilst that of RCS was increased. Application of UV (310)/Cl2 had the highest HO• generation, contributing to TMP removals up to 13% to 48% as compared to 5% to 27% in UV (254, 265, 275)/Cl2 systems at pH 6-9. Artificial neural networks modelling was found to be able to verify and predict the contribution of HO• and RCS conventionally calculated via the general kinetic equations in the UV/Cl2 system at 254, 265, 275 and 310 nm.

A Focused Transport-based Kinetic Fractional Diffusion-advection Equation for Energetic Particle Trapping and Reconnection-related Acceleration by Small-scale Magnetic Flux Ropes in the Solar Wind

Abstract Analysis of energetic particle inner heliospheric spacecraft data increasingly suggests the existence of anomalous diffusion phenomena that should be addressed to achieve a better understanding of energetic particle transport and acceleration in the expanding solar wind medium. Related to this is fast-growing observational evidence supporting the long-standing prediction from magnetohydrodynamic (MHD) theory and simulations of the presence of an inner heliospheric, dominant quasi-two-dimensional MHD turbulence component that contains coherent contracting and merging (reconnecting) small-scale magnetic flux rope (SMFR) structures. This suggests that energetic particle trapping in SMFRs should play a role in anomalous diffusion in the solar wind that warrants further investigation. However, progress in studying such anomalous energetic particle transport phenomena in the solar wind is hampered by the lack of a fundamental derivation of a general fractional kinetic transport equation linking macroscopic energetic particle fractional transport to the microscopic physics of energetic particle interaction with SMFR structures. Here, we outline details of how one can derive a closed ensemble-averaged focused transport equation in the form of a general kinetic fractional diffusion-advection equation from first principles following the nonlinear Eulerian correlation function closure approach of Sanchez et al. With this equation one can model the anomalous diffusion of energetic particles in ordinary, momentum, and pitch-angle space in response to particle trapping in numerous SMFRs advected with the solar wind flow.

Boltzmann-type equations for multi-agent systems with label switching

<p style='text-indent:20px;'>In this paper, we propose a Boltzmann-type kinetic description of mass-varying interacting multi-agent systems. Our agents are characterised by a microscopic state, which changes due to their mutual interactions, and by a label, which identifies a group to which they belong. Besides interacting within and across the groups, the agents may change label according to a state-dependent Markov-type jump process. We derive general kinetic equations for the joint interaction/label switch processes in each group. For prototypical birth/death dynamics, we characterise the transient and equilibrium kinetic distributions of the groups via a Fokker-Planck asymptotic analysis. Then we introduce and analyse a simple model for the contagion of infectious diseases, which takes advantage of the joint interaction/label switch processes to describe quarantine measures.</p>

Diffusive Plus Convective Mass Transport, Accompanied by Biochemical Reaction, Across Capillary Membrane

This study theoretically analyzes the mass transport through capillary, asymmetric, biocatalytic membrane reactor, where the diffusive plus convective mass transport is accompanied by biochemical reaction with Michaelis-Menten kinetics. An approach mathematical model was developed that provides the mass transfer properties in closed, explicit mathematical forms. The inlet and outlet mass transfer rates can then put into the differential mass transport expressions of the lumen and the shell fluid phases as boundary values. The approach solution was obtained by dividing the membrane layer into very thin sub-layers with constant transport and reaction kinetic parameters and the obtained second-order differential equation with constant parameters, given for every sublayer, could be solved analytically. Two operating modes are analyzed in this paper, namely, with and without a sweeping phase on the permeating side. These models deviate by the boundary conditions, only, defined them for the outlet membrane surface. The main purpose of this study is to show how the cylindrical space affects the transport process, concentration distribution, mass transfer rates and conversion in presence of a biochemical reaction. It is shown that the capillary transport can significantly be affected by the lumen radius, by the biocatalytic reactor thickness and the convective flow. Decreasing values of the lumen radius reduce the effect of the biochemical/chemical reaction; the increasing reactor thickness also decreases the physical mass transfer rate and, with it, increases the effect of reaction rate. The model can also be applied to reactions with more general kinetic equations with variable parameters.

Research on the coagulation law of dielectric nanoparticles in Stokes regime via Taylor expansion moment method

For the collision and coagulation of dielectric nanoparticles randomly oriented and moving in an electric field, the collision frequency function of the particles in Stokes regime is derived. The numerical solution obtained by the Taylor expansion moment method in the Stokes regime coagulation problem is very consistent with the numerical solution obtained by the previous study of the TEMOM model. In this paper, the first-order ordinary differential equations are constructed to obtain closed-form expressions. The fourth-order Runge-Kutta algorithm is used to derive the particle concentration of the dielectric nanoparticles under different electric field strengths and different polarities. The law of variation of the polydispersity of the system during coagulation and rupture. These expressions are used in simple overall balance expressions to illustrate the importance of their range of physical parameter values encountered in real systems. The results show that this method can be used to solve the accuracy of the general kinetic equation in the process of particles coagulation under electric field, and the calculation cost is low.

Modeling Asymmetric Interactions in Economy

We consider a general nonlinear kinetic type equation that can describe the time evolution of a variable related to an economical state of an individual agent of the system. We assume asymmetric interactions between the agents. We show that in a corresponding limit, it is asymptotically equivalent to a nonlinear inviscid Burgers type equation.

Determination of Thermoluminescence Kinetic Parameters of La2O3 Doped with Dy3+ and Eu3.

Thermoluminescence (TL) properties of La2O3: Dy3+, Li+, and La2O3: Eu3+, Li+, exposed to 5.12 Gy of beta radiation, and recorded at different heating rates 0.5, 1, 2, 3, 4, and 5 °C s−1 (from Molefe et al., paper 2019), were analyzed and the trap parameters were determined in this study. These parameters include the order of kinetics b, the activation energy E (eV), the frequency factor S (s−1), or the pre-exponential factor S″ (s−1), and the initial concentration of trapped electrons no (cm−3). A new non-linear curve fitting technique, based on the general order kinetic equation and the outcomes of Hoogenstraaten’s Method, was established and applied on the TL glow peaks of La2O3: Dy3+, Li+. The fitting technique was evaluated by calculating the R-square and figure of merit (FOM) values. The results revealed that the FOM values are <1%, and the R-square values are >0.997, which demonstrates an excellent convergence between experimental and fitted curves. A modified technique based on the three-points analysis method was exploited to deconvolute complex TL glow curves of La2O3: Eu3+, Li+, and in turn, to determine the trap parameters the method disclosed that each TL glow curve consists of four peaks. The trap parameters of the individual peaks were numerically determined. The fading, as a function of storage temperature and time, from the TL signals of the investigated materials was predicted and discussed based on the calculated trap parameters. The results support the value of the materials for employment in radiation dosimeter applications with a low fading fraction.

ERROR ESTIMATE OF A BIFIDELITY METHOD FOR KINETIC EQUATIONS WITH RANDOM PARAMETERS AND MULTIPLE SCALES

In this paper, we conduct uniform error estimates of the bifidelity method for multiscale kinetic equations. We take the Boltzmann and the linear transport equations as important examples, and discuss various choices of low-fidelity models for more general kinetic equations. The main analytic tool is the hypocoercivity analysis for kinetic equations, considering solutions in a perturbative setting close to the global equilibrium. This allows us to obtain the error estimates in both kinetic and hydrodynamic regimes.

The effect of the reaction equilibrium on the kinetics of gas-solid reactions — A non-parametric modeling study

The viability of thermochemical energy storage for a given application is often determined by the reaction kinetics under process conditions. For high exergetic efficiency the process needs to operate in close proximity to the reaction equilibrium. Thus, accurate kinetic models that include the effect of the reaction equilibrium are required.In the present work, different parametrization methods for the equilibrium term in the General Kinetic Equation are evaluated by modeling the kinetics of two reaction systems relevant for thermochemical energy storage (CaC2O4 and CuO) from experimental data. A non-parametric modeling method based on tensor decompositions is used that allows for a purely data driven assessment of different parametrization methods.Our analysis shows that including a suitable equilibrium term is crucial. Omitting the equilibrium term when modeling formation reactions can lead to seemingly negative activation energies. Our tests also show that for formation reactions, the reaction rate decreases much faster towards the equilibrium than theory predicts. We present an empirical modeling approach that can predict the reaction rate of gas-solid reactions, regardless of the shortcomings of theory. In this way, non-parametric modeling offers a powerful tool for applied research and may contribute to the advancement of the thermochemical energy storage technology.