Monte Carlo Approaches

The Monte Carlo approach, although much slower than the Hybrid method, makes it possible to address very large systems quite efficiently. It should be noted that the Monte Carlo approach gives a correct estimation of thermodynamic properties even though the number of production steps is a tiny fraction of the total number of possible ionization states. 

Dynamical simulations monitor time-dependent processes in molecular systems in order to smdy their structural, dynamic, and thennodynamic properties by numerically solving an equation of motion, which is the formulation of the rules that govern the motion executed by the molecule. That is, molecular dynamics (MD) provides information about the time dependence and magnitude of fluctuations in both positions and velocities, whereas the Monte Carlo approach provides mainly positional information and gives only little information on time dependence. 

A simple, time-honoured illustration of the operation of the Monte Carlo approach is one curious way of estimating the constant n. Imagine a circle inscribed inside a square of side a, and use a table of random numbers to determine the cartesian coordinates of many points constrained to lie anywhere at random within the square. The ratio of the number of points that lies inside the circle to the total number of points within the square na l4a = nl4. The more random points have been put in place, the more accurate will be the value thus obtained. Of course, such a procedure would make no sense, since n can be obtained to any desired accuracy by the summation of a mathematical series... i.e., analytically. But once the simulator is faced with a eomplex series of particle movements, analytical methods quickly become impracticable and simulation, with time steps included, is literally the only possible approach. That is how computer simulation began. 

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... 

Both the molecular dynamics and the Monte Carlo approaches have great strengths and often lead to quite similar results for the properties of the systems investigated [10]. However, these methods depend on rather elaborate models for the molecular interactions. As a result, as noted above, both methods are... 

As stated above, the most important missing piece in protein folding theory is an accurate all-atom potential. Recently there has been much effort in this direction, and much more is needed [48,55,72-77]. The existence of a potential satisfying minimal criteria such as folding and stability for a single protein was demonstrated in [73]. It is not a realistic potential by any means, but its existence validates the all-atom, implicit solvent, Monte Carlo approach as a serious candidate for theory. The method used to derive this potential was ad hoc, and has recently been compared with other standard methods in a rigorous and illuminating study [77]. 

Figure 33. The stability of yeast glycolysis A Monte Carlo approach. A Shown in the distribution of the largest positive real part within the spectrum of eigenvalues, depicted from above (contour plot). Darker colors correspond to an increased density of eigenvalues. Instances with > 0 are unstable. B The probability that a random instance of the Jacobian corresponds to an unstable metabolic state as a function of the feedback strength 0, . The loss of stability occurs either via in a saddle node (SN) or via a Hopf (HO) bifurcation.

The Monte Carlo approach can now be used for the same problem. The algorithm for the z velocity component and position in dimensionless form can be written as... 

Keep it simple. Generally, uncertainty analyses need not be complex, although many investigators tend to make them so. While available software enables rather complicated Monte Carlo approaches, simple approaches usually provide defendable results that can be easily communicated to others. 

Fedra K. 1983. A Monte Carlo approach to estimating and prediction. In Beck MB, van Straten G, editors. Uncertainty and forecasting of water quality. Berlin Springer-Verlag. 

FIGURE 7.1 Use of 2nd-order Monte Carlo approach to distinguish between variability and uncertainty for mathematical expressions involving constants and random variables. Five hypothetical values or distributions from the outer loop simulation are shown for the inputs and output. For the well-characterized input constants and random variables, the values and distributions, respectively, do not change from one outer loop simulation to the next. 

Kinetic theory of gases (collision model or Monte Carlo approach)... 

An alternative to solving the Boltzmann equation direcdy is the use of particle simulation techniques, sometimes referred to as Monte Carlo methods (58, 59). Major difficulties with the Monte Carlo approach include self-consistency, inclusion of ions, and extension to two spatial dimensions. However, these difficulties are probably not insurmountable, and the Monte Carlo approach may well turn out to be a very powerful tool for discharge analysis. 

Conformational expansion of molecules (also called conformation hunting in Cressets XedeX software module) applies a Monte Carlo approach combined with fast molecular dynamics for ring conformations. The minimization of the conformations is done using the XED force field, in order to assign correct charges. Based on the results obtained by Bostrom [99], this method performs comparably to other available methods when considering the RMS difference... 

In the Monte Carlo approach, there are no inherent limitations on the complexity of the exposure equation, the number of component variables, the probability distributions for the variable components, or the number of iterations. This freedom from limitations is especially useful in simulating the distributions of a LADD for the different exposure scenarios considered here. As its name suggests, a LADD is the average over all the days in an individual s lifetime of the dose of a chemical (e.g., atrazine, simazine, or both) received as a result of his or her exposure from one or more exposure pathways (e.g., water, diet, or herbicide handling). Because the exposure equation can explicitly consider each day individually, the values of the equation s variable components can vary from day to day and have different distributions for different ages and different lifespan projections. 

Below we concentrate on the Monte Carlo approach which allows to investigate the movement of each electron. Figs. 3(a) and 3(b) present the results of a simulation for a selected particle and the total spin of the system for a symmetrically and asymmetrically doped well, respectively. As demonstrated in this Figure, the relaxation times are similar in both cases. The randomness of the spin movement is clearly seen in the figure. The relaxation of the in-plane components of the spin is determined by the same random mechanism, and, therefore, occurs at the same time scale. 

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