Monte Carlo method boundary conditions

The alternative simulation approaches are based on molecular dynamics calculations. This is conceptually simpler that the Monte Carlo method the equations of motion are solved for a system of A molecules, and periodic boundary conditions are again imposed. This method pennits both the equilibrium and transport properties of the system to be evaluated, essentially by numerically solvmg the equations of motion... 

With the Monte Carlo method, the sample is taken to be a cubic lattice consisting of 70 x 70 x 70 sites with intersite distance of 0.6 nm. By applying a periodic boundary condition, an effective sample size up to 8000 sites (equivalent to 4.8-p.m long) can be generated in the field direction (37,39). Carrier transport is simulated by a random walk in the test system under the action of a bias field. The simulation results successfully explain many of the experimental findings, notably the field and temperature dependence of hole mobilities (37,39). 

Barker, J.A., A quantum-statistical Monte Carlo method Path integrals with boundary conditions, J. Chem. Phys. 1979, 70, 2914-2918... 

There are two basic approaches to the computer simulation of liquid crystals, the Monte Carlo method and the method known as molecular dynamics. We will first discuss the basis of the Monte Carlo method. As is the case with both these methods, a small number (of the order hundreds) of molecules is considered and the difficulties introduced by this restriction are, at least in part, removed by the use of artful boundary conditions which will be discussed below. This relatively small assembly of molecules is treated by a method based on the canonical partition function approach. That is to say, the energy which appears in the Boltzman factor is the total energy of the assembly and such factors are assumed summed over an ensemble of assemblies. The summation ranges over all the coordinates and momenta which describe the assemblies. As a classical approach is taken to the problem, the summation is replaced by an integration over all these coordinates though, in the final computation, a return to a summation has to be made. If one wishes to find the probable value of some particular physical quantity, A, which is a function of the coordinates just referred to, then statistical mechanics teaches that this quantity is given by... 

Recent years have seen the extensive application of computer simulation techniques to the study of condensed phases of matter. The two techniques of major importance are the Monte Carlo method and the method of molecular dynamics. Monte Carlo methods are ways of evaluating the partition function of a many-particle system through sampling the multidimensional integral that defines it, and can be used only for the study of equilibrium quantities such as thermodynamic properties and average local structure. Molecular dynamics methods solve Newton s classical equations of motion for a system of particles placed in a box with periodic boundary conditions, and can be used to study both equilibrium and nonequilibrium properties such as time correlation functions. 

We examine the orbital compass model by utilizing the quantum Monte-Carlo method is a finite-size cluster [4], The simulations have been performed on a square lattice of Lx L sites with periodic-boundary conditions. 

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 a staged multi-scale approach, the energetics and reaction rates obtained from these calculations can be used to develop coarse-grained models for simulating kinetics and thermodynamics of complex multi-step reactions on electrodes (for example see [25, 26, 27, 28, 29, 30]). Varying levels of complexity can be simulated on electrodes to introduce defects on electrode surfaces, composition of alloy electrodes, distribution of alloy electrode surfaces, particulate electrodes, etc. Monte Carlo methods can also be coupled with continuum transport/reaction models to correctly describe surfaces effects and provide accurate boundary conditions (for e.g. see Ref. [31]). In what follows, we briefly describe density functional theory calculations and kinetic Monte Carlo simulations to understand CO electro oxidation on Pt-based electrodes. 

In ordinary applications of the Monte Carlo method the boundary conditions are carefully contrived to make the environment of the sample mimic that in the interior of a macroscopic system (cf. Chapter 4, Section 4). At the same time, one of the attractive features of such calculations is that one can extract from them, along with thermodynamic averages, microscopic information on the structure of the system. The most common such data concern the details of the pair correlation function, although more complicated information is also available. Along this line, for instance. Hoover have proposed a study... 

Figure 18 The irradiance field G within the photobioreactor shown in Fig. 6, with the same parameters as in Fig. 16, but the Lambertian emission is replaced by collimated emission at 6, — 0. Comparison between the PI approximation (Eq. (88)) and the reference solution (Monte Carlo method, MCM). For collimated incidence, only the boundary condition atz = 0 is modified, in comparison with the solution used in Fig. 16. We still have g(°)(z = 0) =qn/ but the ballistic irradiance becomes G ° z 0) —qn/ni- Therefore, the same solution as in Fig. 16 can be used, but with replacement of 4gn with 2+ /fl )qn in Eq. (88).

In the case of polar liquids computer experiments whether by molecular dynamics or Monte Carlo methods present difficulties arising from the long range character of the dipole interaction energy and resolution of the problems for the modest sizes of systems in simulations is a task for statistical mechanics rather than ingenious computation methods. Vertheim (35) has summarized the results of the two main kinds of effort to improve on truncation, whether by cutoff of the potential or by periodic boundary conditions namely use of mean field approximations for longer distances and use of Ewald or other summation methods. There is much of interest and instruction value in comparison of results so far... 

The second simulation technique is molecular dynamics. In this technique, which was pioneered by Alder, initial positions of theparticles of a system of several hundred particles are assigned in some way. Displacements of the particles are determined by numerically simulating the classical equations of motion. Periodic boundary conditions are applied as in the Monte Carlo method. The first molecular dynamics calculations were done on systems of hard spheres, but the method has been applied to monatomic systems having intermolecular forces represented by the square-well and Lennard-Jones potential energy functions, as well as on model systems representing molecular substances. Commercial software is now available to carry out molecular dynamics simulations on desktop computers. ... 

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