Stochastic and Monte Carlo Methods

In stochastical methods the random kick is typically somewhat larger, and a standard minimization is carried out starting at the perturbed geometry. This may or may not produce a new minimum. A new perturbed geometry is then generated and minimized etc. There are several variations on how this is done. 

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Conformational Sampling with Poisson-Boltzmann Forces and a Stochastic Dynamics Monte Carlo Method Application to Alanine Dipeptide. 

Other methods which are applied to conformational analysis and to generating multiple conformations and which can be regarded as random or stochastic techniques, since they explore the conformational space in a non-deterministic fashion, arc genetic algorithms (GA) [137, 1381 simulation methods, such as molecular dynamics (MD) and Monte Carlo (MC) simulations 1139], as well as simulated annealing [140], All of those approaches and their application to generate ensembles of conformations arc discussed in Chapter II, Section 7.2 in the Handbook. 

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... 

Special considerations are required in estimating paraimeters from experimental measurements when the relationship between output responses, input variables and paraimeters is given by a Monte Carlo simulation. These considerations, discussed in our first paper 1), relate to the stochastic nature of the solution and to the fact that the Monte Carlo solution is numerical rather than functional. The motivation for using Monte Carlo methods to model polymer systems stems from the fact that often the solution... 

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... 

Although the collision and transition state theories represent two important methods of attacking the theoretical calculation of reaction rates, they are not the only approaches available. Alternative methods include theories based on nonequilibrium statistical mechanics, stochastic theories, and Monte Carlo simulations of chemical dynamics. Consult the texts by Johnson (62), Laidler (60), and Benson (59) and the review by Wayne (63) for a further introduction to the theoretical aspects of reaction kinetics. 

This criterion requires a search through a nonconvex multidimensional conformation space that contains an immense number of minima. Optimization techniques that have been applied to the problem include Monte Carlo methods, simulated annealing, genetic methods, and stochastic search, among others. For reviews of the application of various optimization methods refer to Pardalos et al. (1996), Vasquez et al. (1994), or Schlick et al. (1999). 

The Monte Carlo method permits simulation, in a mathematical model, of stochastic variation in a real system. Many industrial problems involve variables which are not fixed in value, but which tend to fluctuate according to a definite pattern. For example, the demand for a given product may be fairly stable over a long time period, but vary considerably about its mean value on a day-to-day basis. Sometimes this variation is an essential element of the problem and cannot be ignored. 

Thus, part of the energy transferred to a molecular medium by a charged particle is certainly delocalized. And though later this energy is localized on one of the molecules, this localization is stochastic, and thus the coordinates of the points of ionization and excitation cannot be determined more precisely than to within the magnitude of bpl or Ax we have presented previously. This circumstance is important, first of all, when one simulates tracks of charged particles using the Monte Carlo method, where the track is presented as a set of points where the interaction took place.302 303 Even if the plasmon states are not formed in the system, the... 

Studying the electron tracks with the Monte Carlo method, the authors of Refs. 302 and 303 have used the so-called stochastic approach, within which one fixes a simultaneous picture of the spatial distribution of excitation and ionization events. The tracks found this way are sets of spatial points where the inelastic scattering events took place. With this at hand it proves to be possible to calculate the energy absorption spectrum in sensitive volumes of the irradiated medium303 and to calculate the shape of the line and the slope of electronic spin echo signals.302 Such a... 

The most commonly used stochastic methods are the torsional Monte Carlo method11101 and the cartesian stochastic (or random kick) method11111. The two methods differ in the coordinate system in which they operate. The torsional Monte Carlo method uses internal coordinates, while the random kick method uses cartesian coordinates. The advantage of using internal coordinates is that the molecular degrees of freedom are reduced. The reason for choosing torsional angles as the vari-... 

Scanning of a potential energy surface (see potential energy surface). The methods currently used include random search (stochastic search, e. g., Monte Carlo methods) and molecular dynamics (see deterministic search, Monte Carlo search, stochastic search, molecular dynamics, scanning an energy surface). 

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