Statistical simulations methods

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... 

The DSMC method is a molecule-based statistical simulation method for rarefied-gas flows introduced by Bird [3]. The method solves the dynamical equations for the gas flow numerically, using thousands of simulated molecules. Each simulated molecule represents a large number of real molecules. Assuming molecular chaos and a rarefied gas, only binary collisions need be considered, and so the molecular motion and the collisions are uncoupled if the computational time step is smaller than the physical collision time. Interactions with botmdaries and with other... 

Monte Carlo method is formally defined by the following quote as Numerical methods that are known as Monte Carlo methods ean be loosely described as statistical simulation methods, where statistieal simulation is defined in quite general terms to be any method that uses sequenees of random numbers to perform the simulation [122],... 

Monte Carlo methods are statistical simulation methods that use sequences of random numbers to perform the simulation. With these methods, a large model of a system is sampled in a number of random parametric configurations, and the resulting data is used to describe the emerging properties of a system as a whole. 

The Monte-Carlo method is a static statistical simulation method. 

This chapter concentrates on describing molecular simulation methods which have a counectiou with the statistical mechanical description of condensed matter, and hence relate to theoretical approaches to understanding phenomena such as phase equilibria, rare events, and quantum mechanical effects. 

Monte Carlo search methods are stochastic techniques based on the use of random numbers and probability statistics to sample conformational space. The name Monte Carlo was originally coined by Metropolis and Ulam [4] during the Manhattan Project of World War II because of the similarity of this simulation technique to games of chance. Today a variety of Monte Carlo (MC) simulation methods are routinely used in diverse fields such as atmospheric studies, nuclear physics, traffic flow, and, of course, biochemistry and biophysics. In this section we focus on the application of the Monte Carlo method for... 

The computation of quantum many-body effects requires additional effort compared to classical cases. This holds in particular if strong collective phenomena such as phase transitions are considered. The path integral approach to critical phenomena allows the computation of collective phenomena at constant temperature — a condition which is preferred experimentally. Due to the link of path integrals to the partition function in statistical physics, methods from the latter — such as Monte Carlo simulation techniques — can be used for efficient computation of quantum effects. 

The density of states is the central function in statistical thermodynamics, and provides the key link between the microscopic states of a system and its macroscopic, observable properties. In systems with continuous degrees of freedom, the correct treatment of this function is not as straightforward as in lattice systems - we, therefore, present a brief discussion of its subtleties later. The section closes with a short description of the microcanonical MC simulation method, which demonstrates the properties of continuum density of states functions. 

Thermodynamic perturbation theory represents a powerful tool for evaluating free energy differences in complex molecular assemblies. Like any method, however, FEP has limitations of its own, and particular care should be taken not only when carrying out this type of statistical simulations, but also when interpreting their results. We summarize in a number of guidelines the important concepts and features of FEP calculations developed in this chapter ... 

A rigourous way to evaluate the total interaction potential energy, U(q(N- ), would be the formulation and resolution of the Schrodinger equation for the whole system at each configuration. However, given the size of the samples where the statistical simulations are performed, this method is impracticable. 

Revenue management is not a phrase-based management concept but a discipline based on quantitative methods such as statistics, simulation and optimization as well as systems including steps for data collection, estimation and forecasting, optimization and sales control (Cross 2001, pp. 17-18). 

Voigt-Martin, I.G., Zhang, Z.H., Kolb, U. and Gihnore, C.J. (1997) The use of maximum entropy statistics combined with simulation methods to determine the structure of 4-dimethylamino-3-cyanobiphenyl Ultramicroscopy, 68,43-59. 

Contemporary computer-assisted molecular simulation methods and modern computer technology has contributed to the actual numerical calculation of solvent effects on chemical reactions and molecular equilibria. Classical statistical mechanics and quantum mechanics are basic pillars on which practical approaches are based. On top of these, numerical methods borrowed from different fields of physics and engineering and computer graphics techniques have been integrated into computer programs running in graphics workstations and modem supercomputers (Zhao et al., 2000). 

Chapter 5 details the modified statistical thermodynamic prediction method of van der Waals and Platteeuw (1959). The application of molecular simulation methods to hydrates is outlined in Section 5.3. 

The lowest-lying potential energy surfaces for the 0(3P) + CH2=C=CH2 reaction were theoretically characterized using CBS-QB3, RRKM statistical rate theory, and weak-collision master equation analysis using the exact stochastic simulation method. The results predicted that the electrophilic O-addition pathways on the central and terminal carbon atom are dominant up to combustion temperatures. Major predicted end-products are in agreement with experimental evidence. New H-abstraction pathways, resulting in OH and propargyl radicals, have been identified.254... 

Ray Kapral came to Toronto from the United States in 1969. His research interests center on theories of rate processes both in systems close to equilibrium, where the goal is the development of a microscopic theory of condensed phase reaction rates,89 and in systems far from chemical equilibrium, where descriptions of the complex spatial and temporal reactive dynamics that these systems exhibit have been developed.90 He and his collaborators have carried out research on the dynamics of phase transitions and critical phenomena, the dynamics of colloidal suspensions, the kinetic theory of chemical reactions in liquids, nonequilibrium statistical mechanics of liquids and mode coupling theory, mechanisms for the onset of chaos in nonlinear dynamical systems, the stochastic theory of chemical rate processes, studies of pattern formation in chemically reacting systems, and the development of molecular dynamics simulation methods for activated chemical rate processes. His recent research activities center on the theory of quantum and classical rate processes in the condensed phase91 and in clusters, and studies of chemical waves and patterns in reacting systems at both the macroscopic and mesoscopic levels. 

Sensitivity analysis methods can be used in combination with methods for variance propagation. For example, Cullen Frey (1999) describe how variance in the sum of random numbers can be apportioned among the inputs to the sum. All of the statistical sensitivity methods mentioned above can be applied to the results of Monte Carlo simulation, in which... 

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