Monte Carlo-type simulations

The Monte Carlo method therefore simulates by means of a system model an individual sampling value. For evaluation of the results, the known procedures of mathematical statistics can be used. A very important instrument in Monte-Carlo-type simulation is the randomizer. It generates random values within the numerical interval (0,1). The allocation of the random number to a specific value of the random variables is effected via a given distribution function in accordance with Figure 3.20. 

MONTE CARLO-TYPE SIMULATIONS OF CORROSION FATIGUE LIFETIMES FROM THE EVOLUTION OF SHORT SURFACE CRACKS... 

Monte Carlo-Type Simulations of Corrosion Fatigue Lifetimes from the... 

Much of this theory has been incorporated into computer simulations of sputtering and ion damage processes. Thus, it is rare to actually employ these expressions directly given the avadabdity of programs such as TRIM, a Monte Carlo type simulation of the collision cascade in sohds struck with ions. [15]... 

These apparent restrictions in size and length of simulation time of the fully quantum-mechanical methods or molecular-dynamics methods with continuous degrees of freedom in real space are the basic reason why the direct simulation of lattice models of the Ising type or of solid-on-solid type is still the most popular technique to simulate crystal growth processes. Consequently, a substantial part of this article will deal with scientific problems on those time and length scales which are simultaneously accessible by the experimental STM methods on one hand and by Monte Carlo lattice simulations on the other hand. Even these methods, however, are too microscopic to incorporate the boundary conditions from the laboratory set-up into the models in a reahstic way. Therefore one uses phenomenological models of the phase-field or sharp-interface type, and finally even finite-element methods, to treat the diffusion transport and hydrodynamic convections which control a reahstic crystal growth process from the melt on an industrial scale. 

The Monte Carlo method as described so far is useful to evaluate equilibrium properties but says nothing about the time evolution of the system. However, it is in some cases possible to construct a Monte Carlo algorithm that allows the simulated system to evolve like a physical system. This is the case when the dynamics can be described as thermally activated processes, such as adsorption, desorption, and diffusion. Since these processes are particularly well defined in the case of lattice models, these are particularly well suited for this approach. The foundations of dynamical Monte Carlo (DMC) or kinetic Monte Carlo (KMC) simulations have been discussed by Eichthom and Weinberg (1991) in terms of the theory of Poisson processes. The main idea is that the rate of each process that may eventually occur on the surface can be described by an equation of the Arrhenius type ... 

We suggest that a picture of this type can also include the dynamics of the alcohols if the features of the hydrogen bonding in these liquids are taken into account. A Monte Carlo statistical simulation of liquid methanol and ethanoP gives the following restilts ... 

Of the statistical simulations, two major types are distinguished cellular automata (CA) and Monte Carlo (MC) simulations. The basic ideas concerning CA go back to Wiener and Rosenblueth [1] and Von Neumann [2]. CA exist in many variants, which meikes the distinction between MC and CA not always clear. In general, in both techniques, the catalyst surface is represented by a matrix of m x n elements (cells, sites) with appropriate boundary conditions. Each element can represent an active site or a collection of active sites. The cells evolve in time according to a set of rules. The rules for the evolution of cells include only information about the state of the cells and their local neighborhoods. Time often proceeds in discrete time steps. After each time step, the values of the cells are updated according to the specified rules. In cellular automata, all cells are updated in each time step. In MC simulations, both cells and rules are chosen randomly, and sometimes the time step is randomly chosen as well. Of course, all choices have to be made with the correct probabilities. 

Most molecular simulation techniques can be categorized as being among three main types (1) quantum mechanics, (2) molecular dynamics (MD) and (3) kinetic Monte Carlo (KMC) simulation. Quantum mechanics methods, which include ah initio, semi-empirical and density functional techniques, are useful for understanding chemical mechanisms and estimating chemical kinetic parameters for gas-phase... 

In the models classifiable into the first group, the system analyzed is represented by means of a group of interacting particles and the statistical distribution of any property is calculated as the the average over the different configurations generated in the simulation. Especially notable among these models are those of Molecular Dynamics and those of the Monte Carlo type. 

DSMC method is one form of Monte Carlo type of method that has been applied to study gas flows in microdevices (Liou and Fang [1]). Bird [2] first applied DSMC to simulate homogeneous gas relaxation problem. The fundamental idea is to track thousands or millions of randomly selected, statistically representative particles and to use their motions and interactions to modify their positions and states appropriately in time. Each simulated particle represents a number of real molecules. Collision pairs of molecule in a small computational cell in physical space are randomly selected based on a probability distribution after each computation time step. In essence, particle motions are modeled deterministically, while collisions are treated statistically. A significant advantage of DSMC is that the total computation required is proportional to the number of molecules simulated N, in contrast to for the molecular dynamics simulations. 

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