Monte Carlo methods a review

Monte Carlo methods can be classified as static, quasi-static or dynamic. Static methods are those that generate a sequence of statistically independent samples from the desired probability distribution tt. Quasi-static methods are those that generate a sequence of statistically independent batches of samples from the desired probability distribution ir the correlations within a batch are often difficult to describe. Dynamic methods are those that generate a sequence of correlated samples from some stochastic process (usually a Markov process) having the desired probability distribution TT as its unique equilibrium distribution. 

In this section we review briefly the principles of both static and dynamic Monte Carlo methods, with emphasis on the issues that determine the statistical efficiency of an algorithm. 

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Turner, C. H. et al. (2008). Simulation of Chemical Reaction Equilibria by die Reaction Ensemble Monte Carlo Method A Review. Molecular Simulation, 34(2), 119-146. 

Halton, j. H. 1970. A retrospective and prospective survey of the Monte Carlo method. SIAM Reviews, 12(1). 

The variational Monte Carlo method is reviewed here. It is in essence a... 

Bruce AD, Wilding NB, and Ackland GJ. 1997. Free energy of crystalline solids A lattice-switch Monte Carlo method. Physical Review Letters 79 3002-3005. 

Molecular dynamics and Monte Carlo simulations have been extensively applied to molten salts since 1968 to study structure, thermodynamic properties, and dynamic properties from a microscopic viewpoint. Several review papers have been published on computer simulation of molten salts. " Since the Monte Carlo method cannot yield dynamic properties, MD methods have been used to calculate dynamic properties. 

In tfiis chapter we address first the electrochemical application of the more familiar method of molecular (or atom) dynamics, and later turn to consider Monte Carlo methods, in each case giving a short introduction that should motivate the reader to pursue reading more specific works. Although the present research field is relatively new, the investigations are already too extensive to review in detail in a single chapter. For this reason, we discuss here the more extended research branches in the field and present a few representative examples. The application of simulations applied to nanostructuring problems is discussed in Chapter 36 liquid-liquid interfaces have been addressed by I. Benjamin (1997). 

The Boltzmann constant is represented by kB. It is more difficult to use Monte Carlo methods to investigate dynamic events as there is no intrinsic concept of time but an ensemble average over the generated states of the system should give the same equilibrium thermodynamic properties as the MD methods. A good review of both MD and the Monte Carlo methods can be found in the book by Frenkel and Smit [40]. 

This criterion requires a search through a nonconvex multidimensional conformation space that contains an immense number of minima. Optimization techniques that have been applied to the problem include Monte Carlo methods, simulated annealing, genetic methods, and stochastic search, among others. For reviews of the application of various optimization methods refer to Pardalos et al. (1996), Vasquez et al. (1994), or Schlick et al. (1999). 

We review Monte Carlo calculations of phase transitions and ordering behavior in lattice gas models of adsorbed layers on surfaces. The technical aspects of Monte Carlo methods are briefly summarized and results for a wide variety of models are described. Included are calculations of internal energies and order parameters for these models as a function of temperature and coverage along with adsorption isotherms and dynamic quantities such as self-diffusion constants. We also show results which are applicable to the interpretation of experimental data on physical systems such as H on Pd(lOO) and H on Fe(110). Other studies which are presented address fundamental theoretical questions about the nature of phase transitions in a two-dimensional geometry such as the existence of Kosterlitz-Thouless transitions or the nature of dynamic critical exponents. Lastly, we briefly mention multilayer adsorption and wetting phenomena and touch on the kinetics of domain growth at surfaces. 

Extensive reviews of the Monte Carlo method can be found in Refs. " here we restrict ourselves to brief comments. Suppose we wish to study the Ising Hamiltonia i, Eq. (9). Then a Monte Carlo simulation amounts to ... 

Simulations. In addition to analytical approaches to describe ion—solid interactions two different types of computer simulations are used Monte Carlo (MC) and molecular dynamics (MD). The Monte Carlo method relies on a binary collision model and molecular dynamics solves the many-body problem of Newtonian mechanics for many interacting particles. As the name Monte Carlo suggests, the results require averaging over many simulated particle trajectories. A review of the computer simulation of ion—solid interactions has been provided (43). 

Theoretical trends in the study of suspensions employ concepts and techniques originally developed in connection with theories of liquids, for example, equation hierarchies, closure problems, and Monte Carlo methods. In marked contrast with the definitive achievements reviewed in the previous section, the present section outlines a field currently under active development. 

Over the past ten years the numerical simulation of the behavior of complex reaction systems has become a fairly routine procedure, and has been widely used in many areas of chemistry, [l] The most intensive application has been in environmental, atmospheric, and combustion science, where mechanisms often consisting of several hundred reactions are involved. Both deterministic (numerical solution of mass-action differential equations) and stochastic (Monte-Carlo) methods have been used. The former approach is by far the most popular, having been made possible by the development of efficient algorithms for the solution of the "stiff" ODE problem. Edelson has briefly reviewed these developments in a symposium volume which includes several papers on the mathematical techniques and their application. [2]... 

The numerical results reviewed above were obtained for infinite lattices. How do the various quantities of interest behave near the percolation threshold in a large but finite lattice This problem has been studied by renormalization methods, which are essentially equivalent to finite-size scaling. For finite lattices the percolation transition is smeared out over a range of p, and one must expect a similar trend in other functions, including the conductivity. Computer simulations by the Monte Carlo method have been carried out for bond percolation on a three-dimensional simple cubic lattice by Kirkpatrick (1979). Five such experimental curves are shown in Fig. 40, each of which corresponds to a cube of size b, containing bonds. In Fig. 40 the vertical axis gives the fraction p of such samples that percolate (i.e., have opposite faces con-... 

The following five chapters deal with problems associated with solid phases, in some cases involving surface and interfacial problems. In Chapter 14, Steele presents a review of physical adsorption investigated by MD techniques. Jiang and Belak describe in Chapter 15 the simulated behavior of thin films confined between walls under the effect of shear. Chapter 16 contains a review by Benjamin of the MD equilibrium and non-equilibrium simulations applied to the study of chemical reactions at interfaces. Chapter 17 by Alper and Politzer presents simulations of solid copper, and methodological differences of these simulations compared to those in the liquid phase are presented. In Chapter 18 Gelten, van Santen, and Jansen discuss the application of a dynamic Monte Carlo method for the treatment of chemical reactions on surfaces with emphasis on catalysis problems. Khakhar in... 

We start with a short review of the Monte Carlo method for calculating integrals of the form... 

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