Monte-Carlo simulation limits

Monte Carlo simulations, which include fluctuations, then yields Simulations of a coarse-grained polymer blend by Wemer et al find = 1 [49] in the strong segregation limit, in rather good... 

Monte Carlo simulations generate a large number of confonnations of tire microscopic model under study that confonn to tire probability distribution dictated by macroscopic constrains imposed on tire systems. For example, a Monte Carlo simulation of a melt at a given temperature T produces an ensemble of confonnations in which confonnation with energy E. occurs witli a probability proportional to exp (- Ej / kT). An advantage of tire Monte Carlo metliod is tliat, by judicious choice of tire elementary moves, one can circumvent tire limitations of molecular dynamics techniques and effect rapid equilibration of multiple chain systems [65]. Flowever, Monte Carlo... 

Monte Carlo simulations require less computer time to execute each iteration than a molecular dynamics simulation on the same system. However, Monte Carlo simulations are more limited in that they cannot yield time-dependent information, such as diffusion coefficients or viscosity. As with molecular dynamics, constant NVT simulations are most common, but constant NPT simulations are possible using a coordinate scaling step. Calculations that are not constant N can be constructed by including probabilities for particle creation and annihilation. These calculations present technical difficulties due to having very low probabilities for creation and annihilation, thus requiring very large collections of molecules and long simulation times. 

The idea of a finite simulation model subsequently transferred into bulk solvent can be applied to a macromolecule, as shown in Figure 5a. The alchemical transformation is introduced with a molecular dynamics or Monte Carlo simulation for the macromolecule, which is solvated by a limited number of explicit water molecules and otherwise surrounded by vacuum. Then the finite model is transferred into a bulk solvent continuum... 

Matthews, S.D., 1977, MOCARS A Monte Carlo Simulation Code for Determining Distribution and Simulation Limits, TREE-1138, July. 

The partition function Z is given in the large-P limit, Z = limp co Zp, and expectation values of an observable are given as averages of corresponding estimators with the canonical measure in Eq. (19). The variables and R ( ) can be used as classical variables and classical Monte Carlo simulation techniques can be applied for the computation of averages. Note that if we formally put P = 1 in Eq. (19) we recover classical statistical mechanics, of course. 

For the equihbrium properties and for the kinetics under quasi-equilibrium conditions for the adsorbate, the transfer matrix technique is a convenient and accurate method to obtain not only the chemical potentials, as a function of coverage and temperature, but all other thermodynamic information, e.g., multiparticle correlators. We emphasize the economy of the computational effort required for the application of the technique. In particular, because it is based on an analytic method it does not suffer from the limitations of time and accuracy inherent in statistical methods such as Monte Carlo simulations. The task of variation of Hamiltonian parameters in the process of fitting a set of experimental data (thermodynamic and... 

The relative fluctuations in Monte Carlo simulations are of the order of magnitude where N is the total number of molecules in the simulation. The observed error in kinetic simulations is about 1-2% when lO molecules are used. In the computer calculations described by Schaad, the grids of the technique shown here are replaced by computer memory, so the capacity of the memory is one limit on the maximum number of molecules. Other programs for stochastic simulation make use of different routes of calculation, and the number of molecules is not a limitation. Enzyme kinetics and very complex oscillatory reactions have been modeled. These simulations are valuable for establishing whether a postulated kinetic scheme is reasonable, for examining the appearance of extrema or induction periods, applicability of the steady-state approximation, and so on. Even the manual method is useful for such purposes. 

We recall that the MFT assumes that does not induce any correlation between separated sites if all of the sites are mutually uncorrelated at i = 0, MFT assumes that they remain uncorrelated at all later times t > 0. A virtue of this approach is that it permits an easy derivation of the limiting value density, pt->oo- Because the underlying assumption is generally not valid, however, we should hardly be surprised to learn that the limiting densities obtained for most of the interesting (i.e, nonlinear) rules differ significantly from those obtained by Monte Carlo simulations of those same rules. 

Figure 1.21. Monte Carlo simulation of six groups of eight normally distributed measurements each raw data are depicted as x,- vs. i (top) the mean (gaps) and its upper and lower confidence limits (full lines, middle) the confidence limits CL(s ) of the standard deviation converge toward a = 1 (bottom, Eq. 1.42). The vertical divisions are in units of 1 a. The CL are clipped to +5a resp. 0. .. 5ct for better overview. Case A shows the expected behavior, that is for every increase in n the CL(x,nean) bracket /r = 0 and the CL(i t) bracket a - 1. Cases B, C, and D illustrate the rather frequent occurrence of the CL not bracketing either ii and/or ff, cf. Case B n = 5. In Case C the low initial value (arrow ) makes Xmean low and Sx high from the beginning. In Case D the 7 measurement makes both Cl n = 7 widen relative to the n 6 situation. Case F depicts what happens when the same measurements as in Case E are clipped by the DVM.

The method for estimating parameters from Monte Carlo simulation, described in mathematical detail by Reilly and Duever (in preparation), uses a Bayesian approach to establish the posterior distribution for the parameters based on a Monte Carlo model. The numerical nature of the solution requires that the posterior distribution be handled in discretised form as an array in computer storage using the method of Reilly 2). The stochastic nature of Monte Carlo methods implies that output responses are predicted by the model with some amount of uncertainty for which the term "shimmer" as suggested by Andres (D.B. Chambers, SENES Consultants Limited, personal communication, 1985) has been adopted. The model for the uth of n experiments can be expressed by... 

Enzyme reactions, like all chemical events, are dynamic. Information coming to us from experiments is not dynamic even though the intervals of time separating observations may be quite small. In addition, much information is denied to us because of technological limitations in the detection of chemical changes. Our models would be improved if we could observe and record all concentrations at very small intervals of time. One approach to this information lies in the creation of a model in which we know all of the concentrations at any time and know something of the structural attributes of each ingredient. A class of models based on computer simulations, such as molecular dynamics, Monte Carlo simulations, and cellular automata, offer such a possibility. 

The lattice gas has been used as a model for a variety of physical and chemical systems. Its application to simple mixtures is routinely treated in textbooks on statistical mechanics, so it is natural to use it as a starting point for the modeling of liquid-liquid interfaces. In the simplest case the system contains two kinds of solvent particles that occupy positions on a lattice, and with an appropriate choice of the interaction parameters it separates into two phases. This simple version is mainly of didactical value [1], since molecular dynamics allows the study of much more realistic models of the interface between two pure liquids [2,3]. However, even with the fastest computers available today, molecular dynamics is limited to comparatively small ensembles, too small to contain more than a few ions, so that the space-charge regions cannot be included. In contrast, Monte Carlo simulations for the lattice gas can be performed with 10 to 10 particles, so that modeling of the space charge poses no problem. In addition, analytical methods such as the quasichemical approximation allow the treatment of infinite ensembles. 

The first analysis is one with AS-level precision, the second with TIMS-level precision. The first order 2a error for the resulting 331 ka age is 96 ka, but examination of the distribution of a Monte Carlo simulation (Fig. 2) shows that the actual age distribution is strongly asymmetric, with 95% confidence limits of 158/-79 ka. For either younger ages or more-precise analyses, however, the first-order age errors are more than adequate, as shown by the Monte Carlo results for the same data, but with TIMS-level precision (Fig. 2B). 

Figure 2. Histograms of Monte Carlo simulations for two synthetic analyses (Table 1) of a 330 ka sample. The lower precision analysis (A) has a distinctly asymmetric, non-Gaussian distribution of age errors and a misleading first-order error calculation. The higher precision analysis (B) yields a nearly symmetric, Gaussian age distribution with confidence limits almost identical those of the first-order error expansion.

A Monte Carlo simulation (Fig. 3) can be made as usual (that is, without constraints on the output age), in which case only about 24% of the trials will yield ratios corresponding to a finite age, and a younger limit of >821 ka (95% confidence) or >531 ka (68% conf) is indicated. If, however, the a priori assumption of a closed system with no initial °Th is made, the failed trials can be ignored (since they violate the a priori constraints), and solution of both age and age-error (630 +370/-210 ka at 95% conf., or +150/-140 ka at 68% conf) can be obtained from the Monte Carlo simulations. 

The MEM is a powerful new method which is especially useful in cases with limited data sets (powder diffraction). Monte Carlo simulations have shown that the MEM introduces systematic features into the reconstructed density and caution should be exercised when interpreting fine details of an MEM density. It must be emphasized that because the present MEM algorithms do not contain any models, they cannot filter out inconsistencies in the data stemming from systematic errors. The MEM densities may therefore contain non-physical features not only because of systematic bias in the calculation but also because of systematic errors in the data. 

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