Master equation modeling approach

For the examples described here, where the temperature dependence of was analyzed both by master equation modeling and by Equation (27), it is interesting to see how well the Tolman theorem approximation compares with the detailed modeling. This is shown in Table 7 for the cases of tetraethylsilane ion and (H20)jCr. It is seen that these two approaches give nearly identical results. 

If it cannot be guaranteed that the adsorbate remains in local equilibrium during its time evolution, then a set of macroscopic variables is not sufficient and an approach based on nonequihbrium statistical mechanics involving time-dependent distribution functions must be invoked. The kinetic lattice gas model is an example of such a theory [56]. It is derived from a Markovian master equation, but is not totally microscopic in that it is based on a phenomenological Hamiltonian. We demonstrate this approach... 

For adsorbates out of local equilibrium, an analytic approach to the kinetic lattice gas model is a powerful theoretical tool by which, in addition to numerical results, explicit formulas can be obtained to elucidate the underlying physics. This allows one to extract simplified pictures of and approximations to complicated processes, as shown above with precursor-mediated adsorption as an example. This task of theory is increasingly overlooked with the trend to using cheaper computer power for numerical simulations. Unfortunately, many of the simulations of adsorbate kinetics are based on unnecessarily oversimplified assumptions (for example, constant sticking coefficients, constant prefactors etc.) which rarely are spelled out because the physics has been introduced in terms of a set of computational instructions rather than formulating the theory rigorously, e.g., based on a master equation. 

Note here that the relation between mesoscopic and microscopic approaches is not trivial. In fact, the former is closer to the macroscopic treatment (Section 2.1.1) which neglects the structural characteristics of a system. Passing from the micro- to meso- and, finally, to macroscopic level we loose also the initial statement of a stochastic model of the Markov process. Indeed, the disadvantages of deterministic equations used for rather simplified treatment of bimolecular kinetics (Section 2.1) lead to the macro- and mesoscopic models (Section 2.2) where the stochasticity is kept either by adding the stochastic external forces (Section 2.2.1) or by postulating the master equation itself for the relevant Markov process (Section 2.2.2). In the former case the fluctuation source is assumed to be external, whereas in the latter kinetics of bimolecular reaction and fluctuations are coupled and mutually related. Section 2.3.1.2 is aimed to consider the relation between these three levels as well as to discuss problem of how determinicity and stochasticity can coexist. 

In such a representation of an infinite set of master equations for the distribution functions of the state of the surface and of pairs of surface sites (and so on) will arise. This set of equations cannot be solved analytically. To handle this problem practically, this hierarchy must be truncated at a certain level. In such an approach the numerical part needs only a small amount of computer time compared to direct computer simulations. In spite of very simple theoretical descriptions (for example, mean-field approach for certain aspects) structural aspects of the systems are explicitly taken here into account. This leads to results which are in good agreement with computer simulations. But the stochastic model successfully avoids the main difficulty of computer simulations the tremendous amount of computer time which is needed to obtain good statistics for the results. Therefore more complex systems can be studied in detail which may eventually lead to a better understanding of such systems. 

Real catalytic reactions upon solid surfaces are of great complexity and this is why they are inherently very difficult to deal with. The detailed understanding of such reactions is very important in applied research, but rarely has such a detailed understanding been achieved neither from experiment nor from theory. Theoretically there are three basic approaches kinetic equations of the mean-field type, computer simulations (Monte Carlo, MC) and cellular automata CA, or stochastic models (master equations). 

In Fig. 3, the simulation results for the same model problem are presented using the QCLE, the master equation, Tully s surface-hopping approach, the mean field approach, and adiabatic dynamics. The algorithmic details of each approach can be found elsewhere [2,40,79]. 

Chapter 8 provides a unified view of the different kinetic problems in condensed phases on the basis of the lattice-gas model. This approach extends the famous Eyring s theory of absolute reaction rates to a wide range of elementary stages including adsorption, desorption, catalytic reactions, diffusion, surface and bulk reconstruction, etc., taking into consideration the non-ideal behavior of the medium. The Master equation is used to generate the kinetic equations for local concentrations and pair correlation functions. The many-particle problem and closing procedure for kinetic equations are discussed. Application to various surface and gas-solid interface processes is also considered. 

Although a theoretical approach has been desecrated as to how one can apply the generalized coupled master equations to deal with ultrafast radiationless transitions taking place in molecular systems, there are several problems and limitations to the approach. For example, the number of the vibrational modes is limited to less than six for numerical calculations. This is simply just because of the limitation of the computational resources. If the efficient parallelization can be realized to the generalized coupled master equations, the limitation of the number of the modes can be relaxed. In the present approach, the Markov approximation to the interaction between the molecule and the heat bath mode has been employed. If the time scale of the ultrashort measurements becomes close to the characteristic time of the correlation time of the heat bath mode, the Markov approximation cannot be applicable. In this case, the so-called non-Markov treatment should be used. This, in turn, leads to a more computationally demanding task. Thus, it is desirable to develop a new theoretical approach that allows a more efficient algorithm for the computation of the non-Markov kernels. Another problem is related to the modeling of the interaction between the molecule and the heat bath mode. In our model, the heat bath mode is treated as... 

Apart from the heat bath mode, the harmonic potential surface model has been used for the molecular vibrations. It is possible to include the generalized harmonic potential surfaces, i.e., displaced-distorted-rotated surfaces. In this case, the mode coupling can be treated within this model. Beyond the generalized harmonic potential surface model, there is no systematic approach in constructing the generalized (multi-mode coupled) master equation that can be numerically solved. The first step to attack this problem would start with anharmonicity corrections to the harmonic potential surface model. Since anharmonicity has been recognized as an important mechanism in the vibrational dynamics in the electronically excited states, urgent realization of this work is needed. 

An important property of the stochastic version of compartmental models with linear rate laws is that the mean of the stochastic version follows the same time course as the solution of the corresponding deterministic model. That is not true for stochastic models with nonlinear rate laws, e.g., when the probability of transfer of a particle depends on the state of the system. However, under fairly general conditions the mean of the stochastic version approaches the solution of the deterministic model as the number of particles increases. It is important to emphasize for the nonlinear case that whereas the deterministic formulation leads to a finite set of nonlinear differential equations, the master equation... 

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 master equation approach has a long story tracing back to the pioneering work of Pauli [6] and van Hove [7], motivated by the need of reconciling quantum mechanical coherence with the evident randomness of statistical physics. In this section we illustrate this transition from coherent to incoherent behavior with a simple, but paradigmatic, model that will be repeatedly adopted in this review. Let us imagine a quantum system with two states, 11) and 2). The incoherent process described by the Pauli master equation is illustrated by the following set of two equations... 

In genetics, Delbriick s contribution to stochastic modeling, the Luria-Delbriick distribution, is well known [126]. That theory is also based on a master equation approach. 

Vibrational relaxation and excitation and usually the rate-limiting processes for molecular reaction in the gas phase, and their importance has led to many theoretical approaches. The use of an FPE such as described in Section II leads to a diffusion model in energy space, and only applies if the collision kernel P(E, E ) of the master equation is strongly peaked about the initial energy E. This is the weak collision limit in which the energy transfer is small, or comparable to kT. Other approaches, such as the model of Bhatnager, Gross and Krook propose that impulsive collisions randomize... 

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