Monte Carlo methods grid method

Fig. 5. The maximum figure of merit found with different protocols on systems with different number of composition (x) and noncomposition (z) variables (from Falcioni and Deem, 2000). The results are scaled to the maximum found by the grid searching method. Each value is averaged over scaled results on 10 instances of the random phase volume model with different random phases. The Monte Carlo methods are especially effective on systems with a larger number of variables, where the maximal figures of merit are more difficult to locate.

Cracial to the simulations presented here is the inclusion of surface reconstmction, together with correct time-dependence of the reactions. As such, the method provides an extension of earlier important computer simulations of CO oxidation on Pt surfaces " . A dynamic Monte Carlo method is used based on the solution of the master equation of the reaction system. The reaction system consists of a regular grid with periodic boundary conditions. The largest grid used in our simulations contained ca. eight million reaction sites. A short description of the model is presented in Fig. 3 and in Table I, that shows the parameters of the rate constants considered. 

In order to fully account for finite surface mobilities and the heterogeneous surface structure, we have to employ a stochastic description of the surface processes. The kinetic Monte Carlo method enables the incorporation of structural details at an atomistic level. This method has been apphed successfully in the field of heterogeneous (electro-) catalysis [59 1,65] and is further discussed in the chapter by Phil Ross in this book. In the model, hexagonal grids represent catalyst particles. The active sites are randomly distributed on the grid. Adsorbates are considered to bind to on-top sites. The first reaction method was used [66,67]. 

Reliability analysis of electric distribution grid by Monte Carlo method... 

Monte Carlo Methods and the Metropolis Algorithm If the diagnostic functional is discretised on a spatial grid the dimension of the integral is then of the order of N, the number of grid blocks. (It could be more as there may... 

In general, Monte Carlo methods refer to any procedures which involve sampling from random numbers. These methods are used in simulations of natural phenomena, simulation of experimental apparatus, and numerical analysis. An important feature is the simple structure of the computational algorithm. The method was developed by von Neuman, Ulam, and Metroplois during World War II to study the difiiision of neutrons in fissionable materials (ie, atomic bomb design)- Let us consider atom diffusion and demonstrate the principle of the Monte Carlo method. A two-dimensional square grid (Fig. 3.20A) represents interstitial sites in a sofid. 

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. 

V, ip, x, and t) in the PDF transport equation makes it intractable to solve using standard discretization methods. Instead, Lagrangian PDF methods (Pope 1994a) can be used to express the problem in terms of stochastic differential equations for so-called notional particles. In Chapter 7, we will discuss grid-based Eulerian PDF codes which also use notional particles. However, in the Eulerian context, a notional particle serves only as a discrete representation of the Eulerian PDF and not as a model for a Lagrangian fluid particle. The Lagrangian Monte-Carlo simulation methods discussed in Chapter 7 are based on Lagrangian PDF methods. 

Conformational isomers represent minima on an energy surface, and all structures and the corresponding strain energies can be obtained by a careful analysis. This can be performed manually (such as in Sections 17.3 and 17.4) or automatically. An automatic procedure may involve a systematic search (grid search methods), a stochastic search (e.g., torsional Monte Carlo or cartesian stochastic, i.e., the random kick method) or molecular dynamics (see Chapter 5 and Section 16.5). Implemented in MOMEC is a random kick stochastic search module, and this has been shown to lead to excellent results, not only for conformational equilibria, but also for distributions of configurational isomers[37]. 

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