Monte-Carlo computer experiment

In Section 2 the results are presented for the energy band structure computations on periodic DNA related polymers. Furthermore the Monte Carlo computer experiments are described leading to the water and counter-ion structure of periodic single helix B-DNA models. 

For a cytosine stack model calculations have been performed to investigate the effect of water molecules on its electronic structure (16). In this first approach the water structure of a cytidine unit in aperiodic B-DNA obtained with the help of Monte Carlo computer experiments (17) has been used. This choice of the water structure, of course, does not give a realistic description for cytosine,but we get in a relatively simple model calculation an order of magnitude estimate of the effects. The bound water molecules act on the electronic system of the stacked nucleotide bases as an external field. By a proper representation of the electron distribution of the water molecules through point charges one can easily calculate the matrix elements of this perturbing field in terms of the Bloch functions of the finite polymer and include in this way its effect on the band structure. 

MONTE CARLO COMPUTER EXPERIMENTS ON THE WATER AND SODIUM ION... 

Whereas Bernal had to rely primarily on mechanically produced random sphere packings in his work on liquid structure, the molecular dynamics and Monte Carlo computer simulation techniques have, during the past three decades, provided researchers with new and powerful experimental tools enabling a much closer look into the structure of the liquid—one has available the trajectory of every atom. Despite this, computer simulation has been used principally to calculate liquid-state correlation functions. This situation, to quote Lumsden and Wilson [6], ... appears to stem in part from a peculiar and fundamental relation that has always existed between experiment and theory in science the importance of experimental data is judged by the theory to which it is applied. As the physicist Arthur Eddington said half seriously, no fact should be accepted as true until it has been confirmed by theory. Unless an attractive theory exists that decrees certain kinds of information to be important, few scientists will set out to acquire the information. Thus, it is only infrequently that computer simulations have been used to characterize liquid structure in ways other than those dictated by the prevailing liquid theory. 

On the basis of the model of a heterogeneous membrane, it is possible to create a simulation scheme based on dynamic Monte Carlo computer simulations of the adsorption and desorption process on heterogeneous surfaces to extract the involved rate constants as a function of the calcium ion concentration. A simple simulation based on a modified, partly reversible, random sequential adsorption (RSA) algorithm provides very good accordance between experiment and measurement. Figure 8 schematically depicts the assumed model. 

A computerized library ot references to integral neutron eiq )eriment8 has b n developed at tiie Lawrence Radiation Laboratory at Livermore. This library serves as a data base for the systematic retrieval of documents describing diverse critical and bulk nuclear experiments. The evaluation and reduction of the physical parameters of tee experiments to a standard numerical format, and their comparison with calculated results using Monte Carlo computer codes and established libraries of neutron cross sections, will make routine appraisal euler, and also make possible improvement of neutron trwsport cal culations. 

R.epresentations of fliese experiments were prepared for the KENO-iy Monte Carlo computer code. Cross sections were supplied from tte standard 27-energy-group subset of the Criticality Safety Reference Library (CSRL) and processed by the NITAWL module of fire AMPX-H cross-section proc ng system. The 27-group subset of the CSRL is a P library. However, flie version of KENO-IV used in these calculations computed K using only the Po and P portions of the scattering matrix. All axial and radial zones shown in Fig. 1 were included in die KENO-IV representation. 

Simulations on Fiiiers. Monte Carlo computer simulations have been carried out on a variety of filled elastomers (337-339) in an attempt to obtain a better molecular interpretation of how such dispersed phases reinforce elastomers. The approach taken enabled estimation of the effect of the excluded volume of the filler particles on the network chains and on the elastic properties of the networks. In the first step, distribution functions for the end-to-end vectors of the chains were obtained by applying Monte Carlo methods to rotational isomeric state representations of the chains (226). Conformations of chains that overlapped with any filler particle during the simulation were rejected. The resulting perturbed distributions were then used in the three-chain elasticity model (2) to obtain the desired stress—strain isotherms in elongation. These isotherms showed substantial increases in stress and modulus with the increase in filler content and elongation that are in at least quahtative agreement with the experiment. 

In analytical chemistry, a number of identical measurements are taken and then an error is estimated by computing the standard deviation. With computational experiments, repeating the same step should always give exactly the same result, with the exception of Monte Carlo techniques. An error is estimated by comparing a number of similar computations to the experimental answers or much more rigorous computations. 

Monte Carlo simulation is a numerical experimentation technique to obtain the statistics of the output variables of a function, given the statistics of the input variables. In each experiment or trial, the values of the input random variables are sampled based on their distributions, and the output variables are calculated using the computational model. The generation of a set of random numbers is central to the technique, which can then be used to generate a random variable from a given distribution. The simulation can only be performed using computers due to the large number of trials required. 

In his early survey of computer experiments in materials science , Beeler (1970), in the book chapter already cited, divides such experiments into four categories. One is the Monte Carlo approach. The second is the dynamic approach (today usually named molecular dynamics), in which a finite system of N particles (usually atoms) is treated by setting up 3A equations of motion which are coupled through an assumed two-body potential, and the set of 3A differential equations is then solved numerically on a computer to give the space trajectories and velocities of all particles as function of successive time steps. The third is what Beeler called the variational approach, used to establish equilibrium configurations of atoms in (for instance) a crystal dislocation and also to establish what happens to the atoms when the defect moves each atom is moved in turn, one at a time, in a self-consistent iterative process, until the total energy of the system is minimised. The fourth category of computer experiment is what Beeler called a pattern development... 

Fig. 7 gives an example of such a comparison between a number of different polymer simulations and an experiment. The data contain a variety of Monte Carlo simulations employing different models, molecular dynamics simulations, as well as experimental results for polyethylene. Within the error bars this universal analysis of the diffusion constant is independent of the chemical species, be they simple computer models or real chemical materials. Thus, on this level, the simplified models are the most suitable models for investigating polymer materials. (For polymers with side branches or more complicated monomers, the situation is not that clear cut.) It also shows that the so-called entanglement length or entanglement molecular mass Mg is the universal scaling variable which allows one to compare different polymeric melts in order to interpret their viscoelastic behavior. 

In this situation computer simulation is useful, since the conditions of the simulation can be chosen such that full equihbrium is established, and one can test the theoretical concepts more stringently than by experiment. Also, it is possible to deal with ideal and perfectly flat surfaces, very suitable for testing the general mechanisms alluded to above, and to disregard in a first step all the complications that real substrate surfaces have (corrugation on the atomistic scale, roughness on the mesoscopic scale, surface steps, adsorbed impurities, etc.). Of course, it may be desirable to add such complications at a later stage, but this will not be considered here. In fact, computer simulations, i.e., molecular dynamics (MD) and Monte Carlo (MC) calculations, have been extensively used to study both static and dynamic properties [11] in particular, structural properties at interfaces have been considered in detail [12]. 

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... 

Another important insight obtained from this example is related to the number of Monte Carlo trials which must be averaged to obtain a comparison value to the experimentally observed quantities. In order to produce a reasonable estimate of the distribution a suitable ratio of shimmer to measurement error must be achieved. A reasonable value based on experience only was found to be 0.2. In this example 100 Monte Carlo trials were required. With such a large number of trials computer logistics are an important concern. The details of the computer run and of the mapping procedure are discussed by Duever (7 ). ... 

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. 

Sometimes the theoretical or computational approach to description of molecular structure, properties, and reactivity cannot be based on deterministic equations that can be solved by analytical or computational methods. The properties of a molecule or assembly of molecules may be known or describable only in a statistical sense. Molecules and assemblies of molecules exist in distributions of configuration, composition, momentum, and energy. Sometimes, this statistical character is best captured and studied by computer experiments molecular dynamics, Brownian dynamics, Stokesian dynamics, and Monte Carlo methods. Interaction potentials based on quantum mechanics, classical particle mechanics, continuum mechanics, or empiricism are specified and the evolution of the system is then followed in time by simulation of motions resulting from these direct... 

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