Method of Monte-Carlo

There have been some significant advances in the methods of Monte Carlo and 193 0022-4596/87 3.00... 

The method of Monte Carlo integrations over configuration space seems to be a feasible approach to statistical mechanical problems as yet not analytically soluble. For the computing time of a few hours with presently available electronic computers, it seems possible to obtain pressure for a given volume and temperature to an accuracy of a few percent." ... 

A new algorithm which enables one to search for the global minima of Gibbs functions with many independent variables, the method of Monte Carlo annealing, has been introduced... 

The method of Monte Carlo simulation is often called the Metropolis method, since it was introduced by Metropolis and coworkers (64). Monte Carlo techniques in general provide data on equilibrium propaties only, wha-eas MD gives nonequilibrium properties, such as transport properties, as well as equilibrium properties. 

Binder has written an introduction to the theory and methods of Monte Carlo simulation techniques in classical statistical mechanics that are capable of providing measurements of equilibrium properties and of simulating transport and relaxation phenomena. The standard Metropolis algorithm of system sampling has latterly been supplemented by the force bias, Brownian dynamics, and molecular dynamics techniques, and, as noted in the first report, with the aid of these the study has commenced of the behaviour of polymeric systems. 

By sampling phase space directly, the methods of Monte Carlo and molecular dynamics can avoid the restrictive approximations of lattice dynamics. In principle, at least, polymorphism, anharmonicity, static and dynamic disorder may all be rigorously captured by direct simulation. However, other limitations arise which differ from those encountered in a lattice dynamical approach. Comparison of both approaches can provide insight into the significance of these limitations [46]. 

Understanding the hydrogen bond properties of water has been a grand challenge for liquid state theories and molecular simulations. The first simulation of liquid water was carried out more than four decades ago by using the method of Monte Carlo, where the energies of different configurations were calculated from empirical interaction potentials. However, the many-body interactions and their participation in... 

In addition to the statistical response, the effectiveness of the active control system is further demonstrated using the method of Monte Carlo simulation. Sample functions of the components of the buffeting loads in the normal coordinates are simulated using the Fast Fourier transform (FFT) technique [25]. Then, a system of simultaneous coupled differential equations is solved using a 4 order Runge-Kutta numerical integration method to obtain the sample function of bridge response quantities [11]. 

Figure 6.11 Schematic of Monte Carlo simulation 6.2.5 The parametric method...

Gil-Villegas A, McGrother S C and Jackson G 1997 Reaction-field and Ewald summation methods in Monte Carlo simulations of dipolar liquid crystals Mol. Phys. 92 723-34... 

Lattice models have been studied in mean field approximation, by transfer matrix methods and Monte Carlo simulations. Much interest has focused on the occurrence of a microemulsion. Its location in the phase diagram between the oil-rich and the water-rich phases, its structure and its wetting properties have been explored [76]. Lattice models reproduce the reduction of the surface tension upon adsorption of the amphiphiles and the progression of phase equilibria upon increasmg the amphiphile concentration. Spatially periodic (lamellar) phases are also describable by lattice models. Flowever, the structure of the lattice can interfere with the properties of the periodic structures. 

Demidov A A 1999 Use of Monte-Carlo method in the problem of energy migration in molecular complexes Resonance Energy Transfer e6 D L Andrews and A A Demidov (New York Wiley) pp 435-65... 

Monte Carlo Method The Monte Carlo method makes use of random numbers. A digital computer can be used to generate pseudorandom numbers in the range from 0 to 1. To describe the use of random numbers, let us consider the frequency distribution cui ve of a particular factor, e.g., sales volume. Each value of the sales volume has a certain probabihty of occurrence. The cumulative probabihty of that value (or less) being realized is a number in the range from 0 to 1. Thus, a random number in the same range can be used to select a random value of the sales volume. 

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... 

K. Binder, ed. Applications of Monte Carlo Methods in Statistical Physics. Berlin Springer, 1984. 

In addition to the MD method, a wealth of Monte Carlo methods is used also at the atomistic level [6]. They use essentially the same models, force fields, for polymers. Their main advantage, however, is that by introduction of clever moves one can beat the slow physical dynamics of the systems and can run through phase space much faster than by MD. These methods are still in their infancy, but will certainly become more important. 

The bond fluctuation model (BFM) [51] has proved to be a very efficient computational method for Monte Carlo simulations of linear polymers during the last decade. This is a coarse-grained model of polymer chains, in which an effective monomer consists of an elementary cube whose eight sites on a hypothetical cubic lattice are blocked for further occupation (see... 

Applications of Monte Carlo Methods to Sequence Distributions in Polymers... 

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... 

The electrophoretic mobilities of flexible macromolecnles (e.g., DNA, oligonucleotides, and other polymers) in gel media have also been extensively stndied by a nnm-ber of methods, including Monte Carlo simnlations [159,165,208,357,358,361,362,447]. In general, the mobility is expected to vary with the length of the polymer to the -1 power (p N y, however, there are complicating effects of the applied electric field as well as the... 

In structure matching methods, potentials between the CG sites are determined by fitting structural properties, typically radial distribution functions (RDF), obtained from MD employing the CG potential (CG-MD), to those of the original atomistic system. This is often achieved by either of two closely related methods, Inverse Monte Carlo [12-15] and Boltzmann Inversion [5, 16-22], Both of these methods refine the CG potentials iteratively such that the RDF obtained from the CG-MD approaches the corresponding RDF from an atomistic MD simulation. 

A new idea has recently been presented that makes use of Monte Carlo simulations [60,61], By defining a range of parameter values, the parameter space can be examined in a random fashion to obtain the best model and associated parameter set to characterize the experimental data. This method avoids difficulties in achieving convergence through an optimization algorithm, which could be a formidable problem for a complex model. Each set of simulated concentration-time data can be evaluated by a goodness-of-fit criterion to determine the models that predict most accurately. 

The quantity GN is an estimation of G, and the fundamental theorem of Monte Carlo guarantees that the expected value of GN is G, if G exists [Kalos, M. H., and P. A. Whitlock, Monte Carlo Methods, vol. 1, Wiley, New York (1986)]. The error in the calculation is given by... 

Binder, K., Applications of Monte Carlo methods to statistical physics, Rep. Prog. Phys. 1997, 60, 487-559... 

Up to this point and in the following sections and as long as the contrary is not specified, all the discussion will refer to the study of closed, isothermal systems (N,V,T). Though in the applications of Monte Carlo method to the study of solutions... 

The main drawback with the application of Monte Carlo method in this ensemble lies in the fact that, due to the perturbation [34] that must be applied to the volume, it takes approximately 15% more of computing time than in the canonical (N,V,T) ensemble. Another possible problem is that some interaction potentials may lead to unreasonable densities in the calculation. 

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