Analytical Derivation of Moment Equations in Stochastic Chemical Kinetics

The master probability equation captures the dynamic behavior of a variety of stochastic phenomena that can be modeled as Markov processes. Analytical solutions to the master equation are hard to come by though because they require the enumeration of all possible states and the determination of the transition probabilities between any two states. These two tasks quickly become intractable for all but the simplest of systems. Instead of determining how the probability distribution changes in time, we can express the master probability distribution as a function of its moments, and, we can then write transient equations for the probability distribution moments. In 1949, Moyal defined the derivative, or jump, moments of the master probability distribution. These are measures of the rate of change in the probability distribution moment values, i.e. what the impact is of any given transition between states on the moment values. In this paper we present a general scheme for deriving analytical moment equations for any N-dimensional Markov process as a function of the jump moments. Importantly, we propose a scheme to derive analytical expressions for the jump moments for any N-dimensional Markov process. To better illustrate the concepts, we focus on stochastic chemical kinetics models for which we derive analytical relations for jump moments of arbitrary order. Chemical kinetics models are widely used to capture the dynamic behavior of biological systems. The elements in the jump moment expressions are a function of the stoichiometric matrix and the reaction propensities, i.e. the probabilistic reaction rates. We use two toy examples, a linear and a non-linear set of reactions, to demonstrate the applicability and limitations of the scheme. Finally, we provide an estimate on the minimum number of moments necessary to obtain statistical significant data that would uniquely determine the dynamics of the underlying stochastic chemical kinetic system. The first two moments only provide limited information, especially when complex, non-linear dynamics are involved.     

Multi-zone TAP-reactors theory and application. III Multi-response theory and criteria of instantaneousness

A general theory of multi-response state-defining experiments for a multi-zone temporal analysis of products reactor, is developed using the Laplace transform and a generalization of the transfer matrix formalism previously introduced for single-response experiments. The theory provides a unified approach to multi-response modelling and an efficient means to compute the actual profiles of gases and surface species concentration in the reactor, as well as the values of the outlet fluxes numerically, using e.g. fast fourier transform. We investigate the kinetic/mathematical conditions under which explicit expressions for the moments of the outlet fluxes and series expansions for their transient values can be obtained. Several important examples are analyzed in detail, leading to simple criteria of instantaneousness for different reactants within a catalytic mechanism. In the case of thin-zone reactors, the criteria are significantly simplified, with equality (within experimental error) of “shifted residence times” of reactants being proposed as the criterion of instantaneousness. This is illustrated for the oxidation of propene on a vanadium oxide catalyst.     

The application of flow dispersion models to the FM01-LC laboratory filter-press reactor

The flow distribution in the rectangular channel of a laboratory filter-press electrochemical reactor was evaluated using three flow models namely: (a) axial dispersion, (b) sum of two phases and (c) fast and stagnant zones. In the case of the axial-dispersion model, several methods have been used to calculate the Peclet number; the moment method, the non-linear least-squares and the Laplace transform technique. Several boundary conditions, involving different physical and experimental assumptions of the flow were used to solve the partial differential equation that describes the flow behaviour. A total of nine expressions to examine flow dispersion has been used. The comparison of experimental and predicted response signals was made by evaluating the root mean squared error. A data fit in real time has been found to be a better choice as solutions based on the evaluation of moments are prone to error due the overweight of the signal at long times. Data fitting in the Laplace plane is very accurate but it does not guarantee a good fit in real time. Models based on the sum of a fast and a slow or stagnant phase resulted in solutions having very low values of the extension of the slow and stagnant phases, the assumption of a single phase with some degree of dispersion was considered more appropriate.     

A NOTE ON A MARKOV BILINEAR STOCHASTIC PROCESS IN DISCRETE TIME

Abstract. In this paper we study the ergodicity of a Markov bilinear stochastic process and prove that not all moments exist for a non-trivial bilinear stochastic process. We have derived expressions for the moments when they exist and diagrams are given which show the regions for which moments of various low orders exist. A modified bilinear process is briefly discussed.     

自缩聚开环超支化聚合反应的动力学分析

通过严格的动力学方法推导出阴离子型自缩聚开环超支化共聚合体系(SCROCP)的分布函数及各级统计矩数,进而给出了该共聚体系的二维重量分布函数、数均、重均分子量及多分散指数的解析公式。在SCROCP中的研究成果可推广到阴离子型自缩聚开环均聚体系(SCROP)中,此时活化剂的当量少于单体的当量,有一些单体没有被活化剂激活成引发剂单体,反应体系实际上是按共聚的机理进行的。计算结果表明,当初始单体含量较高时,移除残余单体可以使体系的多分散指数大大降低。     

Heat and mass transfer in packed beds—IV: Use of weighted moments to determine axial dispersion coefficients

The method of weighted moments is analysed on the basis of measured axial dispersion in a methanehydrogen mixture flowing through a bed of glass beads at various velocities. The optimum value of the Laplace parameter S is chosen as the one giving the minimum deviation between experiment and model in the integrand of the second weighted moment. The corresponding axial dispersion coefficients are lower than those determined with ordinary moments, and agree very closely with the best-fit values in the time domain. An equation proposed by Anderssen and White based on other considerations is shown to give a relatively good approximation of the optimum value of S. With the aid of this near-optimum S, calculating time for weighted moments can be shorter than for curve fitting in the time domain.     

Model reduction of continuous and discrete multivariable systems by moments matching

Algorithms for the calculation of the moments of multiple-input-multiple-output continuous-data and discrete-data systems are derived. Moments matching is used to reduce the order of the Laplace transfer functions and the z-transfer functions in matrix form. Examples with satisfactory results are given to illustrate the power of the method.     

BIASES OF ESTIMATORS IN MULTIVARIATE NON-GAUSSIAN AUTOREGRESSIONS

Abstract. Expressions for the bias of the least-squares and modified Yule-Walker estimators in a correctly specified multivariate autoregression of arbitrary order are obtained without assuming that the innovations are Gaussian. Instead, the innovations are assumed to form a martingale difference sequence which is stationary up to sixth order and which has finite sixth moments. The errors in the expressions are shown to be O( n ^-3/2), as the sample size n under some moment conditions. The expressions obtained are the same in the Gaussian and non-Gaussian cases.     

Fast arbitrary order moments and arbitrary precision solution of the general rate model of column liquid chromatography with linear isotherm

Algorithms and software are presented for efficiently computing reference solutions of the general rate model with proven error bounds. Moreover, algorithms and software are presented for efficiently computing moments of arbitrary order. The methods are based on numerical inverse Laplace transform, and support both quasi-stationary and dynamic linear binding models. The inlet concentration profiles are treated in a most general way using piecewise cubic polynomials. Algorithmic differentiation obviates manual derivation of the required derivatives. Arbitrary precision arithmetics are applied for minimizing numerical roundoff errors, and several convergence acceleration techniques are evaluated. The implemented software package is freely available as open source on GitHub.     

Stationary Spatial Temperature Gradient in Gas Chromatography

Abstract

Imposing an increasing stationary temperature gradient along the length of a gas chromatographic column is proposed as a technique for improving separation. Side outlet ports with control valves would allow the process to have the same advantages of programmed temperature gas chromatography without the disadvantages of temperature transients. For a simple model of gas chromatography the analysis provides expressions for temporal moments at any point along the column as a function of the temperature gradient. Reduced retention times, sharpening of peaks, and higher symmetry are predicted due to increasing the temperature gradient. The relationship to chromathermography is discussed.     

MOMENTS OF CRYSTAL SIZE DISTRIBUTIONS IN SYSTEMS WITH SELECTIVE CRYSTAL REMOVAL OR SIZE DEPENDENT CRYSTAL GROWTH

Mathematical expressions that allow direct determination of moments of crystal size distributions and related quantities from kinetic parameters are derived. These expressions are developed for cases of size independent and size dependent crystal growth kinetics, or selective removal of fines and/or course product. Models used to describe size dependent growth and selective crystal removal are the Abegg, Stevens and Larson (1968) and the Randolph and Larson (1971) R-z crystallizer models, respectively. The derived equations can be used to evaluate moments and other characteristic properties of a distribution without lengthy and often inaccurate numerical integrations. This capability is particularly useful in on-line analysis of crystal size distributions and crystallizer performance.     

MOMENTS OF CRYSTAL SIZE DISTRIBUTIONS IN SYSTEMS WITH SELECTIVE CRYSTAL REMOVAL OR SIZE DEPENDENT CRYSTAL GROWTH

Mathematical expressions that allow direct determination of moments of crystal size distributions and related quantities from kinetic parameters are derived. These expressions are developed for cases of size independent and size dependent crystal growth kinetics, or selective removal of fines and/or course product. Models used to describe size dependent growth and selective crystal removal are the Abegg, Stevens and Larson (1968) and the Randolph and Larson (1971) R-z crystallizer models, respectively. The derived equations can be used to evaluate moments and other characteristic properties of a distribution without lengthy and often inaccurate numerical integrations. This capability is particularly useful in on-line analysis of crystal size distributions and crystallizer performance.     

Mathematical Properties of the Moments of Truncated Dynamic Magnetic Resonance Spectra and Their Potential Application

The moments of truncated dynamic magnetic resonance spectra, M_n(L), are expanded in terms of power series of the integration range −L to +L. The expansion consists of three contributions: 1) an L-independent term that, for the first three moments (n=1 to 3), is independent of the motion and equals the corresponding moment, M_n, of the rigid powder spectrum; 2) a limited number of (positive) terms, diverging as L^k (k<n, odd), reflecting the broadening effect due to motion; these terms vanish for the first three moments and become nonzero, with motion-dependent coefficients, only from the fourth moment on; and 3) an infinite series of converging (negative) terms, in powers of 1/L^k (k is odd), reflecting the reduction of the moments due to the truncation of the spectra; these terms are motion dependent for all moments. The convergence properties of this series are discussed and expressions for the lower (truncated) moments in the slow and fast motion limits of a secular Hamiltonian are derived. For the slow motion limit, it is shown how the L-dependence of the moments can be used to estimate the magnetic and dynamic parameters. The procedure is demonstrated using computer-simulated spectra. In the fast motion regime, closed expressions are obtained in a similar form to those of the relaxation equations. The effect of natural line width and strong-collision dynamics on the various moments as well as that of nonsecular terms in the Hamiltonian are also briefly discussed.     

Moment equation for the packed column with spherical composites

A mathematical model was proposed to describe the mass-transport phenomena in packed column with spherical composites which are consisted of the outer shells and the inner spheres. Any contact resistance between the former and the latter was assumed to be negligible. The first and the second moments were derived by means of the moment generating properties of Laplace transform.     

Matched asymptotic solutions of the coalescence-redispersion model for unmixed feed stream plug flow reactors

Matched asymptotic expansions are given for the first moments of the concentration distribution function of Curl's coalescence-redispersion population balance model when mixing is faster than reaction. The physical situation considered is an unmixed feed stream tubular reactor with a plug flow residence time distribution in which the single reaction A + nB → P occurs. The asymptotic expansions lead to analytical expressions for species conversion in such reactors and thus obviate the need for numerically tedious Monte Carlo simulations. It is also demonstrated that the perturbation technique applies in a straightforward manner to situations with competing reactions and that the analytical calculations compare favourably with Monte Carlo simulations.     

The Stationary Marginal Distribution of a Threshold AR(1) Process

An explicit analytic solution for the stationary marginal distribution of a simple threshold autoregressive process is given. Furthermore, closed form expressions for all moments of the process are presented. The derivation is based on the use of the Frobenius–Perron operator.     

Treatment of size-dependent aerosol transport processes using quadrature based moment methods

The use of moment methods for simulation of aerosol settling and diffusion phenomena in which the settling velocity and diffusion coefficient are functions of the size of the particles leads to difficult computational problems, especially if the moment equations need to be closed. In this study, a simple one dimensional problem of aerosol diffusion and gravitational settling is carried out using quadrature method of moments (QMOM) and the direct quadrature method of moments (DQMOM). Analytical solutions can be obtained for the number density function, and issues related to the integration of the solutions to get the moments are discussed. Comparison of the solutions of the moment equations to the moments obtained from the analytical solutions reveals that solutions depend on the initial choice of moments. Results also indicate that the proper choice of moments of the initial number density function may be a significant factor in obtaining more accurate solutions from QMOM or DQMOM.     

Exact Geometry of Autoregressive Models

Exact expressions for the statistical curvature and related geometric quantities in first-order autoregressive models are derived. We present a method for calculating moments that is applicable in general autoregressive models. It combines the algebra of differential and difference operators to simplify the problem, and to obtain results valid for all sample sizes. The exact covariance matrix for the minimal sufficient statistic is also derived.     

On the limiting form of the residence-time distribution for a constant-volume recycle system

The residence-time density function for a recycle system usually tends to exponential form, with mean equal to the ratio of the volume to the volumetric flow-rate, when the recycle rate is increased at constant throughput. Two conditions each sufficient to guarantee this behaviour are: (i) that the normalized once-through residence-time density functions become independent of the recycle rate for recycle rates in excess of some finite value; or (ii) that the moments of all orders of the normalized density functions are bounded. In the former case the existence of only the zeroth and first moments for the once-through density functions is required which conditions are themselves consequences of the principle of conservation of mass. In the latter case the form of the once-through normalized density functions may change with the recycle rate. When the first n moments of the normalized once-through densities are bounded, the limits of the first n moments of system residence-time density exist and are the same as those of the exponential density function.     

Moment analysis of nonlinear systems: A kinetic study

The moment method has been used extensively to evaluate parameters of chemical reactions, adsorption or reactor design, but only for linear systems. Moment analysis of nonlinear systems is studied in the present work. A general complicated reaction system involving nonlinear terms in a batch slurry reactor is chosen to demonstrate the method. The first and second improved solutions in Laplace domain and moment expressions are obtained. The improved solution can converge to the exact solution if the iterative procedures are repeated several times.