Monte Carlo , generally simulations

Use parametrized interatomic potentials to sample the geometry, using classical Molecular Dynamics (MD) or Monte Carlo/Generalized Simulated Annealing (MC/GSA). 

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. 

Analytic combinatorial lattice statistics are used to calculate the packing and interactions of a molecule with the other molecules in the system. [The generalized lattice statistics used in this theory have been found to be very accurate (deviations less than 1%) compared (20,25) with Monte Carlo computer simulations in limiting cases presently amenable to such simulations.] Any continuum-space orientation of a molecule or molecular part or bond can be decomposed into its components parallel to the x, 2> And z axes of the system and then these components mapped onto the x, 2. and z axes of the SC lattice [see Figure 1(c)] in a manner analogous to normal coordinate analysis in, for example, molecular spectroscopy. 

Molecular Simulations Molecular simulations are useful for predicting properties of bulk fluids and solids. Molecular dynamics (MD) simulations solve Newton s equations of motion for a small number (on the order of 10 ) of molecules to obtain the time evolution of the system. MD methods can be used for equilibrium and transport properties. Monte Carlo (MC) simulations use a model for the potential energy between molecules to simulate configurations of the molecules in proportion to their probability of occurrence. Statistical averages of MC configurations are useful for equilibrium properties, particularly for saturated densities, vapor pressures, etc. Property estimations using molecular simulation techniques are not illustrated in the remainder of this section as commercial software implementations are not generally available at this time. 

Systematic transport studies of Li ions in oxide and chalcogenide glasses of the general formula, xLi2X (l-x)SiX2 (X = O, S or Se) by Pradel and Ribes (1994), revealed that conductivities in oxide glasses are significantly low. The d.c. conductivities were found to vary as yfx" and this behaviour found support in Monte Carlo computer simulation. Since... 

The properties of the generalized tiling model depend on two parameters, a dimensionless temperature t= kgTIE, and a dimensionless tiling fault energy r=EJE. We have obtained the statistical and thermodynamic properties of our model as a function of t and r from Metropolis Monte Carlo (MC) simulations. The basic variables used in the MC... 

The general solution of the model can be obtained using kinetic Monte Carlo (kMC) simulations. This stochastic method has been successfully applied in the field of heterogeneous catalysis on nanosized catalyst particles (Zhdanov and Kasemo, 2000,2003). It describes the temporal evolution of the system as a Markovian random walk through configuration space. This approach reflects the probabilistic nature of many-particle effects on the catalyst surface. Since these simulations permit atomistic... 

Here a is the standard error associated with the g r) constraint, and the summation is over the experimental and simulation (r] data points. It can be seen that in the absence of any experimental constraints, equivalent to A, = 0 in Eq. (5.3), the simulation is an energy-minimizing MMC simulation, while in the absence of a potential with only experimental constraints, the simulation becomes an RMC simulation. In the general case, it is a hybrid reverse Monte Carlo (HRMC) simulation [6]. In all cases, the simulations are left to run until the total cost function minimizes and oscillates around some average value. 

The main question is whether the surface-area measurement it provides is the true or correct value. Although the general consensus seems to be that the answer is in the negative, it has been impossible to evaluate by how much the BET area is off, because there has been no method that could independently provide that elusive true value. For example, Monte Carlo computer simulation has been performed in order to reproduce the BET model (i.e., without lateral interactions) and subsequently adding those lateral interactions [166]. The results indicate clearly than the BET equation leads to underestimated monolayer values. The error can be significantly reduced by applying the BET equation to a pressure range below the so-called B-point [2,13,37]. 

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