Stochastic simulation kinetic Monte Carlo

Makeev, A., D. Maroudas, and LG. Kevrekidis. 2002a. Coarse stability and bifurcation analysis using stochastic simulators kinetic Monte Carlo examples. Journal of Chemical Physics 116(23) 10083-10091. 

Although the collision and transition state theories represent two important methods of attacking the theoretical calculation of reaction rates, they are not the only approaches available. Alternative methods include theories based on nonequilibrium statistical mechanics, stochastic theories, and Monte Carlo simulations of chemical dynamics. Consult the texts by Johnson (62), Laidler (60), and Benson (59) and the review by Wayne (63) for a further introduction to the theoretical aspects of reaction kinetics. 

Coarse-grained Stochastic Processes and Kinetic Monte Carlo Simulation for the Diffusion of Interacting Particles. J. Chem. Phys., 119, 9412-9427. 

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... 

Katsoulakis, M.A. Vlachos, D.G. Coarse-grained stochastic processes and kinetic Monte Carlo simulators for the diffusion of interacting particles. J. Chem. Phys. 2003, 119, 9412-9428. 

More detailed studies of eleetroeatalytie processes, which incorporate heterogeneous surfaee geometries and finite surface mobilities of reactants, require kinetic Monte Carlo simulations. This stochastic method has been successfully applied in the field of heterogeneous catalysis on nanosized catalyst particles [59,60]. Since these simulations permit atomistic resolution, any level of structural detail may easily be incorporated. Moreover, kinetic Monte Carlo simulations proceed in real time. The simulation of current transients or cyclic voltammograms is, thus, straightforward [61]. 

An analytical solution to the master equation is only possible for very simple systems. The master equation, however, can readily be simulated by using stochastic kinetics or more specifically kinetic Monte Carlo simulation. Several Monte Carlo algorithms exist. More details on kinetic Monte Carlo simulation can be found in the Appendix. 

FIGURE 3.9 Approaches used to find the solution of the active site model of surface activity. The main distinction is made based on surface mobility. For the general case, the full interplay between on-site reactivity and extremely low COad surface diffusivity unfolds. All processes, including nucleation of active sites (rate constant kj ), forward and reverse rates of OH d formation (kf, kb), surface diffusion of COad (kdiff), and oxidative removal of COad (kox), are important for the overall kinetics. The solution for the general case, demands kinetic Monte Carlo simulations, where evolution of the system is described stochastically and positions of adsorbed COad and are relevant. Modeling is substantially simplified in the limit of... 

Gillespie s algorithm numerically reproduces the solution of the chemical master equation, simulating the individual occurrences of reactions. This type of description is called a jump Markov process, a type of stochastic process. A jump Markov process describes a system that has a probability of discontinuously transitioning from one state to another. This type of algorithm is also known as kinetic Monte Carlo. An ensemble of simulation trajectories in state space is required to accurately capture the probabilistic nature of the transient behavior of the system. 

In this section, we will present results of microldnetics simulations based on elementary reaction energy schemes deduced from quantum chemical studies. We use an adapted scheme to enable analysis of the results in terms of the values of elementary rate constants selected. For the same reason, we ignore surface concentration dependence of adsorption energies, whereas this can be readily implemented in the simulations. We are interested in general trends and especially the temperature dependence of overall reaction rates. The simulations will also provide us with information on surface concentrations. In the simulations to be presented here, we exclude product readsorption effects. Microldnetics simulations are attractive since they do not require an assumption of rate-controlling steps or equilibration. Solutions for overall rates are found by solving the complete set of PDFs with proper initial conditions. While in kinetic Monte Carlo simulations these expressions are solved using stochastic techniques, which enable formation... 

The relative fluctuations in Monte Carlo simulations are of the order of magnitude where N is the total number of molecules in the simulation. The observed error in kinetic simulations is about 1-2% when lO molecules are used. In the computer calculations described by Schaad, the grids of the technique shown here are replaced by computer memory, so the capacity of the memory is one limit on the maximum number of molecules. Other programs for stochastic simulation make use of different routes of calculation, and the number of molecules is not a limitation. Enzyme kinetics and very complex oscillatory reactions have been modeled. These simulations are valuable for establishing whether a postulated kinetic scheme is reasonable, for examining the appearance of extrema or induction periods, applicability of the steady-state approximation, and so on. Even the manual method is useful for such purposes. 

In Sect. 7.4.6, we discussed various stochastic simulation techniques that include the kinetics of recombination and free-ion yield in multiple ion-pair spurs. No further details will be presented here, but the results will be compared with available experiments. In so doing, we should remember that in the more comprehensive Monte Carlo simulations of Bartczak and Hummel (1986,1987, 1993,1997) Hummel and Bartczak, (1988) the recombination reaction is taken to be fully diffusion-controlled and that the diffusive free path distribution is frequently assumed to be rectangular, consistent with the diffusion coefficient, instead of a more realistic distribution. While the latter assumption can be justified on the basis of the central limit theorem, which guarantees a gaussian distribution for a large number of scatterings, the first assumption is only valid for low-mobility liquids. 

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). 

Monte Carlo simulations show that, during a simulation, due to stochastic fluctuations, synchronized kinetic oscillations spontaneously change into spatio-temporal patterns with steady state kinetic behavior, and vice versa. Also, rotating spirals can change into pulse generators and vice versa. 

Such stochastic modelling was advanced by Klein and Virk Q) as a probabilistic, model compound-based prediction of lignin pyrolysis. Lignin structure was not considered explicitly. Their approach was extended by Petrocelli (4) to include Kraft lignins and catalysis. Squire and coworkers ( ) introduced the Monte Carlo computational technique as a means of following and predicting coal pyrolysis routes. Recently, McDermott ( used model compound reaction pathways and kinetics to determine Markov Chain states and transition probabilities, respectively, in a rigorous, kinetics-oriented Monte Carlo simulation of the reactions of a linear polymer. Herein we extend the Monte Carlo... 

The simulation model development is divided into three sections. The first discusses the probabilistic modelling of lignin structure, and the use of probability distribution functions to generate representative lignin moieties. The second section details the depolymerization of lignin using stochastic kinetics. The final portion describes the combination of these elements into a Monte Carlo simulation and also presents representative predictions... 

When thermodynamic integration simulations and the thermodynamic cycle approach are used to evaluate free energy differences, the contribution of the kinetic energy usually cancels and therefore does not need to be calculated. Since Monte Carlo simulations generate ensembles of configurations stochastically, momenta are not available, and the contribution cannot be evaluated. 

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