Statistical simulations

Fan, J., Chen, C., Statistical simulation of low-speed unidirectional flaws in transitional... 

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). 

In this Section different analytical approximations for the static tunnelling recombination based on the recombination rate (3.1.2) are compared with the direct statistical simulations. For dimensions d = 1,2 considered here a new spatial scale ro = 5a was used (a is lattice spacing). All concentrations are given in dimensionless units r0d, where as time t is in units of 1. 

Analysis of the correlation functions demonstrates also impressive general agreement between the superposition approximation and computer simulations. Note, however certain overestimate of the similar particle correlations, X r,t), at small r, especially for d = 1. In its turn the correlation function of dissimilar particles, Y(r,t), demonstrates complete agreement with the statistical simulations. Since the time development of concentrations is defined entirely by Y(r, t), Figs 5.2 and 5.3 serve as an additional evidence for the reliability of the superposition approximation. An estimate of the small distances here at which the function Y (r, t) is no longer zero corresponds quite well to the earlier introduced correlation length o, equation (5.1.47) as one can see in fact that at moment t there are no AB pairs separated by r < o-... 

An increase of the standard deviation at r 3 due to small number of survived particles, demonstrates a limited possibility of the direct statistical simulations for a system with a variable number of particles. However, certain conclusions could be drawn even for such limited statistical information. Say, if for equal concentrations the analytical theory based on the superposition approximation seems to be quite adequate, for unequal concentrations its deviation from the computer simulations greatly increases in time. The superposition approximation gives the lower bound estimate of the actual kinetic curves tia( ) but if for d = 2 shown in Fig. 5.8 the deviation is considerable, for d, = 1 (Fig. 5.7) it is not observed, at least for the reaction depths considered. 

As it took place for the tunnelling recombination, divergence in results is not large. It will be shown in Chapter 6 that the reaction depths studied here are enough to establish appearance of the new asymptotic kinetic laws. The superposition approximation giving a lower bound estimate of the kinetics, reproduces correctly the kinetics at long times. Results of the linear approximation are not plotted since they diverge considerably from the statistical simulations. 

Quantitative information available from the correlation functions could be completed by the qualitative one, i.e., a visualized distribution of particles obtained by the direct statistical simulations [1-3]. Figures 1.20 and 1.21 demonstrate time development of spatial structures for d = 1 and 2. Note that computer simulations are restricted in time t since the scale should be considerably less that the system size. 

A comparision of the calculated correlation length for the infinite system with distinctive L used in computer simulations permits us to understand a nature of the instability of results observed in many statistical simulations. 

T. Naes and H. Martens, Comm, in statistics-simulation and computation, 1985,14, 545. 

Table 1 Comparison between theory and the results of the statistical simulation.

The atomic radii may be further refined to improve the agreement between experimental and theoretical solvation free energies. Work on this direction has been done by Luque and Orozco (see [66] and references cited therein) while Barone et al. [67] defined a set of rules to estimate atomic radii. Further discussion on this point can be found in the review by Tomasi and co-workers [15], It must be noted that the parameterization of atomic radii on the basis of a good experiment-theory agreement of solvation energies is problematic because of the difficulty to separate electrostatic and non-electrostatic terms. The comparison of continuum calculations with statistical simulations provides another way to check the validity of cavity definition. A comparison between continuum and classical Monte Carlo simulations was reported by Costa-Cabral et al. [68] in the early 1980s and more recently, molecular dynamics simulations using combined quantum mechanics and molecular mechanics (QM/MM) force-fields have been carried out to analyze the case of water molecule in liquid water [69],... 

Keywords Statistical simulations, Structure of liquids, Picosecond dynamics, Ab initio simulation... 

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... 

Rode BM, Schwenk CF et al (2005) The combination of quantum chemistry and statistical simulations a most powerful tool to access structure and dynamics of liquid systems. Coord Chem Rev 249 2993... 

Ferrer, A. J. and Romero, R. (1993). Small samples estimation of dispersion effects from unreplicated data. Communications in Statistics Simulation and Computation, 22, 975-995. 

Bettonvil, B. (1995). Factor screening by sequential bifurcation. Communications in Statistics—Simulation and Computation, 24, 165-185. 

Molecule-scale statistical simulations of chain molecular solutions are more specialized than simulations of small-molecule solutions. Here we discuss some of the special issues that come up. Dynamical simulations have also been pursued, and those results typically have been satisfactory, though fundamental questions have been raised (Madras and Sokal, 1987 Sokal, 1995). 

We suggest that a picture of this type can also include the dynamics of the alcohols if the features of the hydrogen bonding in these liquids are taken into account. A Monte Carlo statistical simulation of liquid methanol and ethanoP gives the following restilts ... 

Reiss et have recently carried out an extensive theoretical study of the linear and nonlinear properties of pNA in the gas phase, of three solvents, cyclohexane (CH), 1,4-dioxane (DI) and tetrahydrofuran (THF) and of solutions of pNA in these solvents. The aim of the work is to provide a treatment in which the liquid phase molecules are treated discretely via a statistical simulation of the liquid and solution structures. The general strategy for attaining this goal can be summarized as follows -... 

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. 

J. R. Macdonald and W. J. Thompson, "Strongly Heteroscedastic Nonlinear Regression," Communications in Statistics Simulation and Computation, 20 (1991) 843-885. 

---