Kinetic Differential Equations Research Articles

A statistical theory for self-condensing vinyl polymerization

The method of statistical mechanics is used to investigate several properties of self-condensing vinyl polymerization system. Under the framework of mean field theory, two types of canonical partition functions of the system are constructed from viewpoints of functional groups and polymers, and by which the explicit expressions of equilibrium free energy and the law of mass action are obtained. Based on the free energy, the same size distribution function of hyperbranched polymers is derived by two different methods, which is identical with the previous result given by solving the kinetic differential equations. This implies that the two partition functions are consistent with each other. Furthermore, in terms of the size distribution function, the kth radius of gyration as well as its scaling behavior near the critical point are studied, and the corresponding scaling law is given. As an application, the isothermal compressibility is derived on the basis of equation of state, which indicates that the spatial correlation between monomers increases with the increasing of conversion of the double bonds, and reaches the maximum at the critical point. In addition, it is shown that a usual treatment on the polydispersity index would lead it to infinite, which is not agreement with the true result of 1 at the end of the reaction system. To clarify this fact, we find that the correlation length plays an important role, and then by using asymptotic forms of the size distribution and the second moment, the reasonable result can be carried out.

Perturbation of cell-selfregulation through oncoproteins

A brief review is given of the mechanisms of human oncogeneactivation caused by carcinogens. The oncogene products perturb the selfregulation of the cell. To treat this perturbation a system of kinetic partial differential equations describing the key chemical reactions of a cell is written down. It is suggested that this formidable mathematical problem can be handled numerically by dividing the cell into different compartments and applying an iterative procedure. One expects in this way to Abstract from the solutions the most important regulation parameters and with the help of them account for homeostasis. Finally, on the basis of the understanding of the cell selfregulation, its perturbation through protein inactivation or/and overproduction as well as through other physical or chemical changes can be treated theoretically.

Patterns of Stochastic Behavior in Dynamically Unstable High-Dimensional Biochemical Networks

The question of dynamical stability and stochastic behavior of large biochemical networks is discussed. It is argued that stringent conditions of asymptotic stability have very little chance to materialize in a multidimensional system described by the differential equations of chemical kinetics. The reason is that the criteria of asymptotic stability (Routh-Hurwitz, Lyapunov criteria, Feinberg's Deficiency Zero theorem) would impose the limitations of very high algebraic order on the kinetic rates and stoichiometric coefficients, and there are no natural laws that would guarantee their unconditional validity. Highly nonlinear, dynamically unstable systems, however, are not necessarily doomed to collapse, as a simple Jacobian analysis would suggest. It is possible that their dynamics may assume the form of pseudo-random fluctuations quite similar to a shot noise, and, therefore, their behavior may be described in terms of Langevin and Fokker-Plank equations. We have shown by simulation that the resulting pseudo-stochastic processes obey the heavy-tailed Generalized Pareto Distribution with temporal sequence of pulses forming the set of constituent-specific Poisson processes. Being applied to intracellular dynamics, these properties are naturally associated with burstiness, a well documented phenomenon in the biology of gene expression.

Chapter 5 Discrete Stochastic Simulation Methods for Chemically Reacting Systems

Discrete stochastic chemical kinetics describe the time evolution of a chemically reacting system by taking into account the fact that, in reality, chemical species are present with integer populations and exhibit some degree of randomness in their dynamical behavior. In recent years, with the development of new techniques to study biochemistry dynamics in a single cell, there are increasing studies using this approach to chemical kinetics in cellular systems, where the small copy number of some reactant species in the cell may lead to deviations from the predictions of the deterministic differential equations of classical chemical kinetics. This chapter reviews the fundamental theory related to stochastic chemical kinetics and several simulation methods based on that theory. We focus on nonstiff biochemical systems and the two most important discrete stochastic simulation methods: Gillespie's stochastic simulation algorithm (SSA) and the tau-leaping method. Different implementation strategies of these two methods are discussed. Then we recommend a relatively simple and efficient strategy that combines the strengths of the two methods: the hybrid SSA/tau-leaping method. The implementation details of the hybrid strategy are given here and a related software package is introduced. Finally, the hybrid method is applied to simple biochemical systems as a demonstration of its application.

On the theory of recombination of ultracold plasma

Results are given of the solution of an algebroid set of kinetic differential equations of populations of states of excited atoms in recombining plasma. Homogeneous nonequilibrium two-temperature hydrogen plasma with initial electron temperatures Te0 = 1, 6, 13.33 K and initial density ne0 = (1–2.7) × 109 cm−3 is considered. Comparison is made with theoretical and experimental data. The question is considered of the validity of the formulas for the probabilities of quenching an excited atom by electron impact in ultracold plasma (Te0 < 100 K). The calculation is performed for weakly nonideal (γe0 < 1) and nonideal (γe0 ≥ 1) plasma. Dependences are obtained of the concentration of excited atoms and free electrons, the temperature of free electrons, and the nonideality parameter on the time and on the distribution of excited atoms with respect to the main quantum number.

The correctional kinetic equation for the peak temperature in the differential thermal analysis

The peak temperature (T p) and different temperature (ΔT) are the basic information in the differential thermal analysis (DTA). Considering the kinetic relation and the heat equilibrium in DTA, a correctional differential kinetic equation (containing T p and ΔT parameter) is proposed. In the dehydration reaction of CaC2O4·H2O, the activation energy calculated from the new equation showed some smaller than that from Kissinger equation, but some bigger than that from Piloyan equation.

Simulations of the predose technique for retrospective dosimetry and authenticity testing

The predose technique of thermoluminescence (TL) for quartz has been used extensively for retrospective dosimetry and archaeological authenticity testing. In this paper, we use a previously published comprehensive kinetic model for quartz, to simulate the complete sequence of experimental steps taken during the additive dose and the multiple activation versions of the predose technique. The simulation results show how both versions of the predose technique can reproduce the paleodose received by the sample with an accuracy of 1–5% in the low dose region of 0–2 Gy. For doses greater than ∼ 2 Gy the non-linear dose dependence of the sensitivity of the “ 110 ∘ C ” TL peak causes significant inaccuracies in the technique. The solution of the kinetic differential equations elucidates several electron and hole processes taking place during the experimental predose procedure; these processes include the thermal transfer of holes from the Zimmerman hole reservoirs to the luminescence center, the radiation quenching of the TL sensitivity and the radiation-enhanced sensitivity of quartz samples. Specific numerical examples are given for samples exhibiting the thermal activation characteristics of “low- S 0 ” and “high- S 0 ” values. Quantitative results are presented for the effect of the test dose and of the calibration beta dose β on the accuracy of both versions of the predose technique. Results are also presented for the sensitivity of the predose technique to the natural variations of the hole concentrations in the luminescence center. Finally, the results of the predose technique simulations are compared with those from simulating the popular single aliquot SAR/OSL technique based on optically stimulated luminescence signals.

Modeling of Paste Extrusion in Semi-Solid State

A constitutive rheological equation is proposed for the paste extrusion of polytetrafluoroethylene (PTFE) that takes into account the continuous change of the microstructure during flow, through fibril formation. The mechanism of fibrillation is captured through a microscopic model for a structural parameter, ξ. This model essentially represents a balance of fibrillated and unfibrillated domains in the PTFE paste through a first-order kinetic differential equation. The rate of fibril formation is assumed to be a function of the strain rate and a flow type parameter, which describes the relative strength of straining and rotation in mixed type flows. The proposed constitutive equation consists of shear-thinning and shear-thickening terms, the relative contribution of the two being a function of ξ. Finite element simulations using the proposed constitutive relation predict correctly the variations of the extrusion pressure with the apparent shear rate and die geometrical parameters.

Studies of thermal aging of insulating fluids: Part 1. Mechanisms and kinetics of aging

Using an accurate description of the experiments on the thermal aging of insulating fluids and the results of a gas analysis, an accurate description of the mechanisms of physicochemical processes is developed in the form of kinetic schemes and differential equations that fully reflect results of experiments in the presence and absence of oxygen.

Control oriented 1D electrochemical model of lithium ion battery

Lithium ion (Li-ion) batteries provide high energy and power density energy storage for diverse applications ranging from cell phones to hybrid electric vehicles (HEVs). For efficient and reliable systems integration, low order dynamic battery models are needed. This paper introduces a general method to generate numerically a fully observable/controllable state variable model from electrochemical kinetic, species and charge partial differential equations that govern the discharge/charge behavior of a Li-ion battery. Validated against a 313th order nonlinear CFD model of a 6 Ah HEV cell, a 12th order state variable model predicts terminal voltage to within 1% for pulse and constant current profiles at rates up to 50 C. The state equation is constructed in modal form with constant negative real eigenvalues distributed in frequency space from 0 to 10 Hz. Open circuit potential, electrode surface concentration/reaction distribution coupling and electrolyte concentration/ionic conductivity nonlinearities are explicitly approximated in the model output equation on a local, electrode-averaged and distributed basis, respectively. The balanced realization controllability/observability gramian indicates that the fast electrode surface concentration dynamics are more observable/controllable than the electrode bulk concentration dynamics (i.e. state of charge).

Stochastic Simulation of Chemical Kinetics

Stochastic chemical kinetics describes the time evolution of a well-stirred chemically reacting system in a way that takes into account the fact that molecules come in whole numbers and exhibit some degree of randomness in their dynamical behavior. Researchers are increasingly using this approach to chemical kinetics in the analysis of cellular systems in biology, where the small molecular populations of only a few reactant species can lead to deviations from the predictions of the deterministic differential equations of classical chemical kinetics. After reviewing the supporting theory of stochastic chemical kinetics, I discuss some recent advances in methods for using that theory to make numerical simulations. These include improvements to the exact stochastic simulation algorithm (SSA) and the approximate explicit tau-leaping procedure, as well as the development of two approximate strategies for simulating systems that are dynamically stiff: implicit tau-leaping and the slow-scale SSA.

Numerical Calculation of Multicoupled Fields in Large Salient Synchronous Generator

The physical fields of synchronous generator are very complicated because of the coupling of electromagnetic field, fluid field, and thermal field. In this paper, a large synchronous generator is taken for an example and the three types of fields are calculated. First, the mathematical model and physical model of the 2-D electromagnetic field are determined by which the losses in each part of the generator are calculated. Second, the fluid kinetic differential equations in stator radial ventilation duct are built. After determining the solved region and boundary conditions, the 2-D fluid field is computed and the different distributions of fluid field are compared and analyzed when the fluid has different entrance angles. Finally, the 3-D stator temperature fields are solved under different inlet angles respectively. The results of the coupled fields are analyzed, and some useful conclusions are presented

Dynamic reduction of a CH4/air chemical mechanism appropriate for investigating vortex–flame interactions

AbstractThis paper describes two methods, piecewise reusable implementation of solution mapping (PRISM) and dynamic steady‐state approximation (DYSSA), in which chemistry is reduced dynamically to reduce the computational burden in combustion simulations. Each method utilizes the large range in species timescales to reduce the dimensionality to the number of species with slow timescales. The methods are applied within a framework that uses hypercubes to partition multidimensional chemical composition space, where each chemical species concentration, plus temperature, is represented by an axis in space. The dimensionality of the problem is reduced uniquely in each hypercube, but the dimensionality of chemical composition space is not reduced. The dimensionality reduction is dynamic and is different for different hypercubes, thereby escaping the restrictions of global methods in which reductions must be valid for all chemical mixtures. PRISM constructs polynomial equations in each hypercube, replacing the chemical kinetic ordinary differential equation (ODE) system with a set of quadratic polynomials with terms related to the number of species with slow timescales. Earlier versions of PRISM were applied to smaller chemical mechanisms and used all chemical species concentrations as terms. DYSSA is a dynamic treatment of the steady‐state approximation and uses the fast–slow timescale separation to determine the set of steady‐state species in each hypercube. A reduced number of chemical kinetic ODEs are integrated rather than the original full set. PRISM and DYSSA are evaluated for simulations of a pair of counterrotating vortices interacting with a premixed CH4/air laminar flame. DYSSA is sufficiently accurate for use in combustion simulations, and when relative errors are less than 1.0%, speedups on the order of 3 are observed. PRISM does not perform as well as DYSSA with respect to accuracy and efficiency. Although the polynomial evaluation that replaces the ODE solver is sufficiently fast, polynomials are not reused sufficiently to enable their construction cost to be recovered. © 2007 Wiley Periodicals, Inc. 39: 204–220, 2007

Aims and methods in non-isothermal reaction kinetics

The majority of the works dealing with non-isothermal kinetics assumes only one kinetic differential equation, uses linearization techniques and do not check the fit between the simulated and the experimental data. It is not clear from the literature why we need kinetic evaluations at all in this field. Due to this controversial situation, the author has outlined his views on the aims and methods of the non-isothermal kinetics in this paper. Accordingly, the goal of the kinetic evaluation is to obtain better, more informative results from the experiments. If realistic models are used, numerous unknown parameters have to be determined during the evaluation. Since the most important errors of thermal analysis are not random, the laws of the mathematical statistics do not offer means to find the best set of model parameters. Nevertheless, the simultaneous evaluation of a series of thermoanalytical experiments by the method of least squares aims directly at the description of the sample behavior in a wide range of experimental conditions and helps the determination of a large number of unknown parameters. The outlined considerations are supported by examples from the work of the author and coworkers. As a comparison, a statistical survey is given on those papers that were published in journals specialized for thermal analysis, thermochemistry and pyrolysis in 2006 and contained the term “kinetic” or “kinetics” in their titles.

Simulação do equilíbrio: o método de Monte Carlo

We make several simulations using the Monte Carlo method in order to obtain the chemical equilibrium for several first-order reactions and one second-order reaction. We study several direct, reverse and consecutive reactions. These simulations show the fluctuations and relaxation time and help to understand the solution of the corresponding differential equations of chemical kinetics. This work was done in an undergraduate physical chemistry course at UNIFIEO.

Time scale and dimension analysis of a budding yeast cell cycle model.

BackgroundThe progress through the eukaryotic cell division cycle is driven by an underlying molecular regulatory network. Cell cycle progression can be considered as a series of irreversible transitions from one steady state to another in the correct order. Although this view has been put forward some time ago, it has not been quantitatively proven yet. Bifurcation analysis of a model for the budding yeast cell cycle has identified only two different steady states (one for G1 and one for mitosis) using cell mass as a bifurcation parameter. By analyzing the same model, using different methods of dynamical systems theory, we provide evidence for transitions among several different steady states during the budding yeast cell cycle.ResultsBy calculating the eigenvalues of the Jacobian of kinetic differential equations we have determined the stability of the cell cycle trajectories of the Chen model. Based on the sign of the real part of the eigenvalues, the cell cycle can be divided into excitation and relaxation periods. During an excitation period, the cell cycle control system leaves a formerly stable steady state and, accordingly, excitation periods can be associated with irreversible cell cycle transitions like START, entry into mitosis and exit from mitosis. During relaxation periods, the control system asymptotically approaches the new steady state. We also show that the dynamical dimension of the Chen's model fluctuates by increasing during excitation periods followed by decrease during relaxation periods. In each relaxation period the dynamical dimension of the model drops to one, indicating a period where kinetic processes are in steady state and all concentration changes are driven by the increase of cytoplasmic growth.ConclusionWe apply two numerical methods, which have not been used to analyze biological control systems. These methods are more sensitive than the bifurcation analysis used before because they identify those transitions between steady states that are not controlled by a bifurcation parameter (e.g. cell mass). Therefore by applying these tools for a cell cycle control model, we provide a deeper understanding of the dynamical transitions in the underlying molecular network.

EVIDENCE OF TRAP DISTRIBUTION IN BORATE GLASS

In order to elucidate the type of trap distribution in borate glass we have analyzed a set of glow peaks around 110°C recorded with different doses of β-ray irradiation. The glow curves are subjected to the rigorous Computerized Glow Curve Deconvolution (CGCD) both in the framework of kinetic formalism and differential equation formalism to judge which type of trap distribution favors in borate glass. It is observed that the distribution of traps is exponential, which is also well supported by standard error analyses i.e. Figure Of Merit (FOM) and Chi-square (χ2) test.

A fourth-order accurate, three-point compact approximation of the boundary gradient, for electrochemical kinetic simulations by the extended Numerov method

The fourth-order accurate, three-point compact (extended Numerov) finite-difference scheme of Chawla [J. Inst. Math. Appl. 22 (1978) 89] has been recently found superior (in terms of accuracy and efficiency) to the conventional second-order accurate spatial discretisation commonly used in electrochemical kinetic simulations. However, the two-point compact boundary gradient approximation, accompanying the scheme, is difficult to apply in the case of time-dependent kinetic partial differential equations, because it introduces unwanted second temporal derivatives into calculations. The conventional five-point gradient formula is free from this drawback, but it is also not very convenient, owing to the locally increased bandwidth of the matrix of linear equations arising from the spatio-temporal discretisation. A new three-point compact boundary gradient approximation derived in this work, avoids the above inconveniences and economically re-uses expressions utilised by the extended Numerov discretisation. The fourth-order accuracy of the new approximation is proven theoretically and verified in computational experiments performed for examples of kinetic models.

Treatment of polymer surfaces in plasma Part I. Kinetic model

The surface tension of the polymer materials depends on functional groups over its surface. As a result from the plasma treatment the kind and concentration of the functional groups can be changed. In the present work, the possible kinetic reactions are defined. They describe the interaction between the plasma and the polymer surface of polyethylene terephthalate (PET). Basing on these reactions, the systems of differential kinetic equations are suggested. The solutions are obtained analytically for the system kinetic equations at defined circumstances.

Constitutive modeling and flow simulation of polytetrafluoroethylene (PTFE) paste extrusion

A constitutive rheological equation is proposed for the paste extrusion of polytetrafluoroethylene (PTFE) that takes into account the continuous change of the microstructure during flow, through fibril formation. The mechanism of fibrillation is captured through a microscopic model for a structural parameter, ξ. This model essentially represents a balance of fibrillated and unfibrillated domains in the PTFE paste through a first-order kinetic differential equation. The rate of fibril formation is assumed to be a function of the strain rate and a flow type parameter, which describes the relative strength of straining and rotation in mixed type flows. The proposed constitutive equation consists of a shear-thinning and a shear-thickening terms, the relative contribution of the two being a function of ξ. Finite element simulations using the proposed constitutive relation predict accurately the variations of the extrusion pressure with the apparent shear rate and die geometrical parameters.