Statistical simulations Monte Carlo framework

The goal of the present chapter is to describe some methods and approaches developed in the framework of the thermodynamic theory of adsorption. We confine ourselves to the thermodynamic approach, because this approach allows for direct engineering applications. The simulations in the framework of the thermodynamic approach are relatively simple and well repeatable, so that the algorithm for numerical solutions of the corresponding equilibrium problem may be generated on the basis of a relatively short and informal description. An alternative may be provided by ab initio calculations, direct appfication of the statistical mechanics (Monte Carlo) or other types of molecular simulations. These computations are much more complicated and they have not yet reached the stage where they may be directly used for modeling a wide variety of the practically important cases of mixed adsorption. 

In this review we put less emphasis on the physics and chemistry of surface processes, for which we refer the reader to recent reviews of adsorption-desorption kinetics which are contained in two books [2,3] with chapters by the present authors where further references to earher work can be found. These articles also discuss relevant experimental techniques employed in the study of surface kinetics and appropriate methods of data analysis. Here we give details of how to set up models under basically two different kinetic conditions, namely (/) when the adsorbate remains in quasi-equihbrium during the relevant processes, in which case nonequilibrium thermodynamics provides the needed framework, and (n) when surface nonequilibrium effects become important and nonequilibrium statistical mechanics becomes the appropriate vehicle. For both approaches we will restrict ourselves to systems for which appropriate lattice gas models can be set up. Further associated theoretical reviews are by Lombardo and Bell [4] with emphasis on Monte Carlo simulations, by Brivio and Grimley [5] on dynamics, and by Persson [6] on the lattice gas model. 

Statistical simulation methods can be basically separated into two approaches. The Monte Carlo (MC) framework [17,18,19] utilises random structural variations of single structural units (atoms, molecules, groups, etc.) followed by an evaluation of energies to decide whether the resulting new arrangement of atoms is accepted or should be discarded. Sampling of molecular dynamics (MD) employs equations... 

It should be noted that the hybrid quantum/classical schemes apply not only for determination of geometries, energies, and reaction mechanisms. The Monte Carlo [67, 68] and molecular dynamics (MD) [69-72] simulations are quite popular as frameworks for which various QM/MM procedures serve as subroutines . Before employing hybrid schemes the large-scale MD simulations were performed only with low-level approximations for force fields. The use of hybrid schemes extends significantly the scope of their application, improve precision of the results that allows to improve the understanding of statistical properties and dynamical processes in liquids and biopolymers. 

The situation of concern is when Monte Carlo simulation is used to quantify uncertainty in quantities in order to inform decisions. It is important to keep in mind that a Monte Carlo simulation should not be associated to a particular framework. Introductory text books and common computer programs for Monte Carlo simulation describe methods to quantify uncertainty in input parameters and provide tools to describe the uncertainty in output. Vose (2008) present different statistical principles and argue for uncertainty being subjective, at the same time as the impression is that principles from classical statistical statistics are seen as the first choice. The tutorial for RISK (Palisade Corporation, 2013) rightly present the quantification and interpretation of uncertainty in a neutral way e.g. by talking about Monte Carlo simulation as a technique to combine all the uncertainties you identify in your modeling situation leading to results presented in the form ofprobability distributions . However, statements like you can explicitly... 

Some issues arise. Risk assessment should consider all available information. Thus, a desirable characteristic of a conceptual framework would be to handle multiple sources of evidence. A nice feature offered by Monte Carlo simulation is that it is possible to account for correlations between input parameters. In its simplest form, these correlations are assigned assuming a pair-wise linear association between parameters. However, there are other ways to capture the correlation between parameters based on the same sources of evidence. Some sources of uncertainty are missed when parameters are predictions from other models. For example the link to the source of evidence is lost when not considering uncertainty in parameters that have been predicted by statistical regression models. 

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