Locally implicit solution of a reaction-diffusion system with stiff kinetics

The Tyson-Fife reaction-diffusion equations are solved numerically using a locally implicit approach. Since the variables evolve at very different time scales, the resulting system of equations is stiff. The reaction term is responsible for the stiffness and the time step is increased by using an implicit method. The diffusion operator is evaluated explicitly and the system of implicit nonlinear equations is decoupled. The method is particularly useful for parameter values in which the equations are very stiff, such as the values obtained directly from the experimental reaction rate constants. Previous efforts modified the parameters on the equations to avoid stiffness. The equations then become a simplified model of excitable media and, for those cases, the locally implicit method gives a faster although less accurate solution. Nevertheless, since the modified equations no longer represent a particular chemical system an accurate solution is not as important. The algorithm is applied to observe the transition from simple motion to compound motion of a spiral tip.     

Sequences of ultrafast non-resonant multiphoton transitions in a three-electronic level molecule

Non-resonant multiphoton transitions between three electronic states of a molecular system are studied. Based on a projection operator formalism which is formulated in the framework of the so-called time-local as well as the time-nonlocal approach, time-dependent Schrödinger equations are obtained, which include effective couplings to the laser field. For both procedures a slowly varying amplitude approximation can be invoked. The resulting time-local equations are solved in a much more efficient way than the original effective Schrödinger equations. The validity of these approximations is verified numerically for a two-photon process. Furthermore, the effective Schrödinger equations are specified to sequences of two-photon and three-photon transitions. The derived equations are applied to a molecular system consisting of three electronic states with Morse-type potential energy curves. Using different laser pulse scenarios the conditions are discussed under which a sequence of two-photon and three-photon transitions can take place.     

Simulation of nonisothermal reactive liquid chromatography using two-dimensional lumped kinetic model

A nonisothermal two-dimensional lumped kinetic model of reactive liquid chromatography is formulated and applied to simulate the separation of multicomponent mixtures in a fixed-bed cylindrical column operating under nonisothermal condition. The axial and radial variations of concentration and temperature as well as reversibility of the chemical reactions are incorporated in the model equations. The model comprises a system of convection-diffusion-reaction partial differential equations coupled with algebraic and differential equations. Due to the nonlinearity of adsorption and reaction kinetics, it is required to apply an accurate numerical scheme for solving the model equations. In this study, an efficient and accurate high-resolution flux-limiting finite-volume scheme is proposed to solve the model equations. A number of stoichiometrical reactions are numerically simulated to determine the level of coupling between the temperature and concentration profiles. Moreover, the effects of various critical parameters on the process performance are examined. The results obtained are beneficial for understanding reaction and separation processes inside a liquid chromatographic reactor and to improve its performance.     

Non-Markovian stochastic Schrödinger equations in different temperature regimes: a study of the spin-boson model

Stochastic Schrodinger equations are used to describe the dynamics of a quantum open system in contact with a large environment, as an alternative to the commonly used master equations. We present a study of the two main types of non-Markovian stochastic Schrodinger equations, linear and nonlinear ones. We compare them both analytically and numerically, the latter for the case of a spin-boson model. We show in this paper that two linear stochastic Schrodinger equations, derived from different perspectives by Diosi, Gisin, and Strunz [Phys. Rev. A 58, 1699 (1998)], and Gaspard and Nagaoka [J. Chem. Phys. 13, 5676 (1999)], respectively, are equivalent in the relevant order of perturbation theory. Nonlinear stochastic Schrodinger equations are in principle more efficient than linear ones, as they determine solutions with a higher weight in the ensemble average which recovers the reduced density matrix of the quantum open system. However, it will be shown in this paper that for the case of a spin-boson system and weak coupling, this improvement does only occur in the case of a bath at high temperature. For low temperatures, the sampling of realizations of the nonlinear equation is practically equivalent to the sampling of the linear ones. We study further this result by analyzing, for both temperature regimes, the driving noise of the linear equations in comparison to that of the nonlinear equations.     

Mean field kinetic theory for a lattice gas model of fluids confined in porous materials

We consider the mean field kinetic equations describing the relaxation dynamics of a lattice model of a fluid confined in a porous material. The dynamical theory embodied in these equations can be viewed as a mean field approximation to a Kawasaki dynamics Monte Carlo simulation of the system, as a theory of diffusion, or as a dynamical density functional theory. The solutions of the kinetic equations for long times coincide with the solutions of the static mean field equations for the inhomogeneous lattice gas. The approach is applied to a lattice gas model of a fluid confined in a finite length slit pore open at both ends and is in contact with the bulk fluid at a temperature where capillary condensation and hysteresis occur. The states emerging dynamically during irreversible changes in the chemical potential are compared with those obtained from the static mean field equations for states associated with a quasistatic progression up and down the adsorption/desorption isotherm. In the capillary transition region, the dynamics involves the appearance of undulates (adsorption) and liquid bridges (adsorption and desorption) which are unstable in the static mean field theory in the grand ensemble for the open pore but which are stable in the static mean field theory in the canonical ensemble for an infinite pore.     

Full temporal density-matrix treatment of dual wavelength, arbitrary bandwidth, pulsed laser excitation of atoms with degenerate states in high collisional media

A full time-dependent theory, based on the Density-Matrix (D.M.) formalism, for two-step pulsed excitation of atoms in collision-dominated media by pulsed laser light of arbitrary bandwidth is presented. The atoms consist of three levels, of which each one, in turn, can consist of an arbitrary number of degenerate states. The atoms are exposed both to quenching collisions as well as elastic collisions. From a general set of density-matrix equations a more manageable, reduced set of fully time-dependent D.M. equations is formulated (the total number of equations is not more than 10, in contrast to the N² equations needed for a general set of D.M. equations, N being the total number of states within all three levels). The following approximations and assumptions have been made: the rotating-wave approximation; all individual transition probabilities between different states within a given pair of levels are the same (implying that the only input parameters for the transition probabilities are the spontaneous emission rates); all coherences between different states within each level are washed out by the high collisional rates; and the laser light is linearly polarized with an arbitrary bandwidth of Lorentzian shape. The full time-dependent equations are then solved in the steady-state limit of the non-diagonal elements, yielding time-dependent rate-equation-like population transfer equations. A few fully time-dependent simulations of some typical cases are given.     

On the Inversion of Quantum Mechanical Systems: Determining the Amount and Type of Data for a Unique Solution

The inverse problem of extracting a quantum mechanical potential from laboratory data is studied from the perspective of determining the amount and type of data capable of giving a unique answer. Bound state spectral information and expectation values of time-independent operators are used as data. The Schrödinger equation is treated as finite dimensional and for these types of data there are algebraic equations relating the unknowns in the system to the experimental data (e.g., the spectrum of a matrix is related algebraically to the elements of the matrix). As these equations are polynomials in the unknown parameters of the system, it is possible to determine the multiplicity of the solution set. With a fixed number of unknowns the effect of increasing the number of equations on the multiplicity of solutions is assessed. In general, if the number of the equations matches the number of the unknowns, the solution set is denumerable. A result on the solvability of polynomial equations is extended to the case where the number of equations exceeds the number of unknowns. We show that if one has more equations than the number of unknowns, generically a unique solution exists. Several examples illustrating these results are provided.     

Communication: limitations of the stochastic quasi-steady-state approximation in open biochemical reaction networks

It is commonly believed that, whenever timescale separation holds, the predictions of reduced chemical master equations obtained using the stochastic quasi-steady-state approximation are in very good agreement with the predictions of the full master equations. We use the linear noise approximation to obtain a simple formula for the relative error between the predictions of the two master equations for the Michaelis-Menten reaction with substrate input. The reduced approach is predicted to overestimate the variance of the substrate concentration fluctuations by as much as 30%. The theoretical results are validated by stochastic simulations using experimental parameter values for enzymes involved in proteolysis, gluconeogenesis, and fermentation.     

Generalized nematostatics

The generalized equations of bulk and interfacial nematostatics in terms of the tensor order parameter are derived using calculus of variations, taking into account long and short range nematic bulk free energies as well as anchoring and saddle-splay surface free energies. A general expression for the surface stress tensor order parameter for a nematic liquid crystal/isotropic fluid (NLC/I) interface has been derived, and found to represent normal, shear, and bending stresses. It is shown that the surface stress tensor is asymmetric. It is also found that anchoring energy contributes to bending and normal stresses, while saddle-splay energy contributes to normal and shear stresses. The rotational identifies governing the bulk and surface stress tensors are derived and used to show that the equations of nematostatics are fully consistent with the general balance equations of polar fluids. The equations presented provide a theoretical framework for solving interfacial problems involving NLCs that is applicable to cases where variations in liquid crystalline order and saddle-splay energy play significant roles.     

Generalized nematostatics

The generalized equations of bulk and interfacial nematostatics in terms of the tensor order parameter are derived using calculus of variations, taking into account long and short range nematic bulk free energies as well as anchoring and saddle-splay surface free energies. A general expression for the surface stress tensor order parameter for a nematic liquid crystal/isotropic fluid (NLC/I) interface has been derived, and found to represent normal, shear, and bending stresses. It is shown that the surface stress tensor is asymmetric. It is also found that anchoring energy contributes to bending and normal stresses, while saddle-splay energy contributes to normal and shear stresses. The rotational identifies governing the bulk and surface stress tensors are derived and used to show that the equations of nematostatics are fully consistent with the general balance equations of polar fluids. The equations presented provide a theoretical framework for solving interfacial problems involving NLCs that is applicable to cases where variations in liquid crystalline order and saddle-splay energy play significant roles.     

Numerical calculation of the Onsager transport coefficients for isothermal binary electrolytes

The Ebeling-Falkenhagen diffusion equations are applied to calculate the Onsager transport coefficients as well as the electrical conductances, transference numbers, and mutual diffusion coefficients for isothermal binary electrolytes. For this purpose the hierarchy of diffusion equations is closed on the level of the binary distribution functions by the superposition approximation. The resulting system of common differential equations is solved by numerical methods. Hydrodynamic interactions are taken into account up to first order. Some results are given for symmetrically charged binary electrolytes with hard-core ions (restricted primitive interaction model). The model parameters (Bjerrum parameter and Debye screening length) are chosen to represent strong electrolytes up to the molar region.     

三组分共聚酯数均聚合度和数均分子量的计算方程

为便于生产控制以制得预定分子量的产物，根据直接酯化法合成聚对苯二甲酯乙二酯（ＰＥＴ）共聚酯的特点，综合考虑各种主、副反应，从统计观点出发，对三组分ＰＥＴ共聚酯体系的齐聚物和缩聚物，分别建立了其数均聚合度Ｘｎ和数均分子量Ｍｎ的简化和非简化计算方程．将简化Ｘｎ方程与Ｃａｒｏｔｈｅｒｓ方程进行计算比较，其结果完全相同；将简化Ｘｎ方程应用于二组分聚酯体系，其形式与传统的二组分线型缩聚物Ｘｎ方程形式相同，各Ｘｎ和Ｍｎ方程可推广应用于二组分聚酯体系和三组分以上的多组分共聚酯体系．     

A complete decoupling of Faddeev-like equations for three arrangement systems; applications to HH2 collision

A complete decoupling of time-dependent Faddeev-like equations for three identical particles is presented in terms of an operator formalism. No restriction to pairwise additive interaction needs to be made. Our decoupled equations can be looked at as being a generalization of the special decoupled versions of Faddeev and Lovelace in so far as they include all irreducible representations of the permutation group S₃. Thus, the new equations also apply to three composite particles of any spin. An application is made for the system composed of three H-atoms. In particular we show how the cross sections of ortho—para transitions are directly related to the transition operators obtained from the decoupled equations.     

The application of the concept of reaction layer to the study of electrode processes coupled with the second-order chemical reactions at microelectrodes under steady-state conditions

Under steady-state conditions, the current equations of the second-order EC, ECE and DISPI reactions at microdisk, microspherical and microring electrodes are derived with the aid of the concept of the reaction layer. The conditions under which these equations would be valid are also discussed. Using these equations, methods to determine the kinetic parameters for the second-order EC, ECE and DISPI reactions are presented. The reduction of 2,6-diphenyl-pyrylium cation and oxidation of triphenylamine were investigated as examples of the second-order EC and ECE reactions.     

Charge/discharge of an electrochemical supercapacitor electrode pore; non-uniqueness of mathematical models

A thermodynamic analysis has been done to enhance understanding of the relation between various mathematical models for electrochemical supercapacitor pores. For the same capacitive charge/discharge experiment a variety of one-dimensional mathematical model equations concerning the transport of ions and double layer charge/discharge along the pore are shown to be indistinguishable. Some of those indistinguishable equations could be interpreted as derived from diffusional mechanisms while others appear as derived from migrational mechanisms. Ohmic resistivities and diffusivities obtained in such case are not contradicting results but characterize identical physical processes. The results are valid as long as the assumptions of irreversible thermodynamics of local equilibrium along the pore and of linearization of the flux equations hold.     

A coupled hartree-fock calculation of the paramagnetic contribution to the cotton-mouton constant of helium

The perturbation equations necessary for the evaluation of the paramagnetic part of the Cotton-Mouton constant of helium are derived as an example of coupled Hartree-Fock theory for a double perturbation and the ensuing integro-differential equations solved numerically. A comment is made on the errors in the numerical solution of such equations.