Master equations chemical kinetics

Generalized first-order kinetics have been extensively reviewed in relation to teclmical chemical applications [59] and have been discussed in the context of copolymerization [53]. From a theoretical point of view, the general class of coupled kinetic equation (A3.4.138) and equation (A3.4.139) is important, because it allows for a general closed-fomi solution (in matrix fomi) [49]. Important applications include the Pauli master equation for statistical mechanical systems (in particular gas-phase statistical mechanical kinetics) [48] and the investigation of certain simple reaction systems [49, ]. It is the basis of the many-level treatment of... 

An interesting, but probably incorrect, application of the probabilistic master equation is the description of chemical kinetics in a dilute gas.5 Instead of using the classical deterministic theory, several investigators have introduced single time functions of the form P(n1,n2,t) where P(nu n2, t) is the probability that there are nl particles of type 1 and n2 particles of type 2 in the system at time t. They use the transition rate A(nt, n2 n2, n2, t) from the state with particles of type 1 and n2 particles of type 2 to the state with nt and n2 particles of types 1 and 2, respectively, at time t. The rates that are used are obtained by assuming that only uncorrelated binary collisions occur in the system. These rates, however, are only correct in the thermodynamic limit for a low density system. In this limit, the Boltzmann equation is valid from which the deterministic theory follows. Thus, there is no reason to attach any physical significance to the differences between the results of the stochastic theory and the deterministic theory.6... 

Since the formal chemical kinetics operates with large numbers of particles participating in reaction, they could be considered as continuous variables. However, taking into account the atomistic nature of defects, consider hereafter these numbers N as random integer variables. The chemical reaction can be treated now as the birth-death process with individual reaction events accompanied by creation and disappearance of several particles, in a line with the actual reaction scheme [16, 21, 27, 64, 65], Describing the state of a system by a vector N = TV),..., Ns, we can use the Chapmen-Kolmogorov master equation [27] for the distribution function P(N, t)... 

In a long series of papers on the master equation, Pritchard and his coworkers elucidated for the first time the effects of rotational and vibrational disequilibrium on the dissociation and recombination of a dilute diatomic gas. Ultrasonic dispersion in a diatomic gas was analyzed by similar computational experiments, and the first example of the breakdown of the linear mixture rule in chemical kinetics was demonstrated. A major difficulty in these calculations is that the eigenvalue of the reaction matrix (corresponding to the rate constant) differs from the zero eigenvalue (required by species conservation) by less than... 

Generally, the gas-phase concentrations Nk are described by a set of master equations of chemical kinetic (equations for active masses at the source/sink of the kth substance in the chemical reactions Wk)... 

A good review of the master equation approach to chemical kinetics has been given by McQuarrie [383]. Jacquez [335] presents the master equation for the general ra-compartment, the catenary, and the mammillary models. That author further develops the equation for the one- and two-compartment models to obtain the expectation and variance of the number of particles in the model. Many others consider the m-compartment case [342,345,384], and Matis [385] gives a complete methodological rule to solve the Kolmogorov equations. 

The chemical master equation (CME) for a given system invokes the same rate constants as the associated deterministic kinetic model. Yet the CME is more fundamental than the deterministic kinetic view. Just as Schrodinger s equation is the fundamental equation for modeling motions of atomic and subatomic particle systems, the CME is the fundamental equation for reaction systems. Remember that Schrodinger s equation is not a model for a specific mechanical system. Rather, it is a theoretical framework upon which models for particular systems can be developed. In order to write down a model for an atomic system based on Schrodinger s equation, one needs to know how to write down the Hamiltonian a priori. Similarly, the CME is not a model for a specific biochemical reaction system it is a theoretical framework. To determine the CME model for a reaction system, one must know what are the possible elementary reactions and the associated rate constants. 

Chemical master equation for Michaelis-Menten kinetics... 

Chemical kinetic rate methods including conventional transition state theory (TST), canonical variational transition state theory (CVTST) and Rice-Ramsper-ger-Kassel-Marcus in conjunction with master equation (RRKM/ME) and separate statistical ensemble (SSE) have been successfully applied to the hydrocarbon oxidation. Transition state theory has been developed and employed in many disciplines of chemistry [41 4]. In the atmospheric chemistry field, conventional transition state theory is employed to calculate the high-pressure-limit unimole-cular or bimolecular rate constants if a well-defined transition state (i.e., a tight... 

Then numerical methods of matrix diagonalization are used to find the eigenvalues of the matrix operator 0)(P —I) — K, which are the time constants that determine both the chemical kinetics and the energy relaxation. Part three of this work deals in detail with the formulation of the Master Equation for a number of different systems, for example termolecular association reactions and reversible reactions. It then deals with methods for finding the time constants and simulating the kinetics. The Master Equation is the method of choice at present for modelling the competition between energy transfer and reaction. 

The kinetics of chemical reactions on surfaces is described using a microscopic approach based on a master equation. This approach is essential to correctly include the effects of surface reconstruction and island formation on the overall rate of surface reactions. The solution of the master equation using Monte Carlo methods is discussed. The methods are applied to the oxidation of CO on a platinum single crystal surface. This system shows oscillatory behavior and spatio-temporal pattern formation in various forms. 

The kinetics of chemical reactions on surfaces is normally described using macroscopic rate equations. The master equation can be used to derive such macroscopic rate equations. Sometimes this derivation is exact, but we often will have to make approximations, which may or may not be appropriate. This will depend on the system. If the approximation to derive the macroscopic rate equations are too crude, the master equation shows, however, how to add corrections to rate equations. It is in general necessary to make approximations even with these corrections, but one has the choice what approximations to make. Of course, in practice one may... 

It is often stated that MC methods lack real time and results are usually reported in MC events or steps. While this is immaterial as far as equilibrium is concerned, following real dynamics is essential for comparison to solutions of partial differential equations and/or experimental data. It turns out that MC simulations follow the stochastic dynamics of a master equation, and with appropriate parameterization of the transition probabilities per unit time, they provide continuous time information as well. For example, Gillespie has laid down the time foundations of MC for chemical reactions in a spatially homogeneous system.f His approach is easily extendable to arbitrarily complex computational systems when individual events have a prescribed transition probability per unit time, and is often referred to as the kinetic Monte Carlo or dynamic Monte Carlo (DMC) method. The microscopic processes along with their corresponding transition probabilities per unit time can be obtained via either experiments such as field emission or fast scanning tunneling microscopy or shorter time scale DFT/MD simulations discussed earlier. The creation of a database/lookup table of transition... 

High Pressure limit kinetic parameters are obtained from canonical Transition State Theory calculations. Multifrequency Quantum Rice-Ramsperger-Kassel (QRRK) analysis is used to calculate k(E) data and master equation analysis is applied to evaluate fall-off in this chemically activated reaction system. 

The present study calculates thermochemical properties of intermediates, transition states and products important to the degradation of the aromatic ring in the phenyl radical + O2 reaction system. Kinetic parameters are developed for the important elementary reaction paths through each channel as a function of temperature and pressure. The calculation is done via a bimolecular chemical activation and master equation analysis for fall-ofif. 

The kinetic parameters of each path are determined as a function of temperature and pressure using the bimolecular chemical activation analysis. High Pressure limit kinetic parameters from the calculation results are obtained with the canonical Transition State Theory. The multifrequency Quantum Rice-Ramsperger-Kassel analysis is utilized to obtain k(E) and Master Equation analysis is used for the evaluation of pressure fall-off in this complex bimolecular chemical activation reaction. Results are applicable to elementary experiments at low pressures, ambient combustion studies at one atmosphere, as well as higher-pressure turbine systems. 

In Chap. 2 we analyzed single variable linear and non-linear systems with single and multiple stable stationary states by use of the deterministic equations of chemical kinetics. We introduced species-specific affinities and the concept of an excess work with these we showed the existence of a thermodynamic state function 4> and compiled its many interesting properties, see (2.15 2.19), including its relation to fluctuations as given by the stationary solution of the master equation, (2.34). We continue this approach here by turning to systems with more than one intermediate, [1]. 

In this chapter we formulate the thermodynamic and stochastic theory of the simple transport phenomena diffusion, thermal conduction and viscous ffow (1) to present results parallel to those listed in points 1-7, Sect. 8.1, for chemical kinetics. We still assume local equilibrium with respect to translational and internal degrees of freedom. We do not assume conditions close to chemical or hydrodynamic equilibrium. For chemical reactions and diffusion the macroscopic equations for a given reaction mechanism provide sufficient detail, the fluxes in the forward and reverse direction, to write a birth-death master equation with a stationary solution given in terms of For thermal conduction and viscous flow we derive the excess work and then find Fokker-Planck equations with stationary solutions given in terms of that excess work. 

Miller JA, Klippenstein SJ (2006) Master equation methods in gas phase chemical kinetics. J Phys Chem A 110 10528-10544... 

---