Monte Carlo method simple sampling

In an unmodified Monte Carlo method, simple random sampling is used to select each member of the 777-tuple set. Each of the input parameters for a model is represented by a probability density function that defines both the range of values that the input parameters can have and the probability that the parameters are within any subinterval of that range. In order to carry out a Monte Carlo sampling analysis, each input is represented by a cumulative distribution function (CDF) in which there is a one-to-one correspondence between a probability and values. A random number generator is used to select probability in the range of 0-1. This probability is then used to select a corresponding parameter value. 

Various methods, such as influence sampling, can be used to reduce the number of calculations needed. See also Lapeyre, B., Introduction to Monte-Carlo Methods for Transport and Diffusion Equations, Oxford University Press (2003), and Liu, J. S., Monte Carlo Strategies in Scientific Computing, Springer (2001). Some computer programs are available that perform simple Monte Carlo calculations using Microsoft Excel. 

There are also hybrid methods which combine features from two or all three of the above. Opinions will freely be offered about which technique is best , but the reality is that different techniques will perform differently depending on the problem at hand. Except for very simple systems with only one or a few degrees of conformational freedom, systematic methods are not practical, and sampling techniques, which do not guarantee location of the lowest-energy structure (because they do not look everywhere), are the only viable alternative. By default, Spartan uses systematic searching for systems with only a few degrees of conformational freedom and Monte-Carlo methods for more complicated systems. 

This relatively simple model illustrates the viability of the straightforward analytical analysis. Most models, unfortunately, involve many more input variables and proportionally more complex formulae to propagate variance. Fortunately, the Latin hypercube sampling and Monte Carlo methods simplify complex model variance analysis. 

The numerical results reviewed above were obtained for infinite lattices. How do the various quantities of interest behave near the percolation threshold in a large but finite lattice This problem has been studied by renormalization methods, which are essentially equivalent to finite-size scaling. For finite lattices the percolation transition is smeared out over a range of p, and one must expect a similar trend in other functions, including the conductivity. Computer simulations by the Monte Carlo method have been carried out for bond percolation on a three-dimensional simple cubic lattice by Kirkpatrick (1979). Five such experimental curves are shown in Fig. 40, each of which corresponds to a cube of size b, containing bonds. In Fig. 40 the vertical axis gives the fraction p of such samples that percolate (i.e., have opposite faces con-... 

The molecular dynamics method is conceptually simpler than the Monte Carlo method. Here again, we can compute various averages of the form (2.107) and hence the RDF as well. The method consists in a direct solution of the equations of motion of a sample of N (j= 10 ) particles. In principle, the method amounts to computing time averages rather than ensemble averages, and was first employed for simple liquids by Alder and Wainwright (1957) [see review by Alder and Hoover (1968)]. The problem of surface effects is dealt with as in the Monte Carlo method. The sequence of events is now not random, but follows the trajectory which is dictated to the system by the equations of motion. In this respect, this method is of a more general scope, since it permits the computation of equilibrium as well as transport properties of the system. 

MATLAB programming tool allows a simple parallelization of loops with fixed number of recurrences using the parfor keyword. In case of the Monte Carlo method, the loop over all the sample size is parallelized and in case of the analytical algorithm it is the loop over the state space. Number of threads in which the calculation is executed equals at most the number of used processor cores. 

The Andersen thermostat is very simple. After each time step Si, each monomer experiences a random collision with a fictitious heat-bath particle with a collision probability / coll = vSt, where v is the collision frequency. If the collisions are assumed to be uncorrelated events, the collision probability at any time t is Poissonian,pcoll(v, f) = v exp(—vi). In the event of a collision, each component of the velocity of the hit particle is changed according to the Maxwell-Boltzmann distribution p(v,)= exp(—wv /2k T)/ /Inmk T (i = 1,2,3). The width of this Gaussian distribution is determined by the canonical temperature. Each monomer behaves like a Brownian particle under the influence of the forces exerted on it by other particles and external fields. In the limit i —> oo, the phase-space trajectory will have covered the complete accessible phase-space, which is sampled in accordance with Boltzmann statistics. Andersen dynamics resembles Markovian dynamics described in the context of Monte Carlo methods and, in fact, from a statistical mechanics point of view, it reminds us of the Metropolis Monte Carlo method. 

In general, Monte Carlo methods refer to any procedures which involve sampling from random numbers. These methods are used in simulations of natural phenomena, simulation of experimental apparatus, and numerical analysis. An important feature is the simple structure of the computational algorithm. The method was developed by von Neuman, Ulam, and Metroplois during World War II to study the difiiision of neutrons in fissionable materials (ie, atomic bomb design)- Let us consider atom diffusion and demonstrate the principle of the Monte Carlo method. A two-dimensional square grid (Fig. 3.20A) represents interstitial sites in a sofid. 

With this simple acceptance criterion, the Metropolis Monte Carlo method generates a Markov chain of states or conformations that asymptotically sample the XTT probability density function. It is a Markov chain because the acceptance of each new state depends only on the previous state. Importantly, with transition probabilities defined by Eqs. 15.23 and 15.24, the transition matrix has the limiting, equihbrium distribution as the eigenvector corresponding to the largest eigenvalue of 1. 

The correct treatment of boundaries and boundary effects is crucial to simulation methods because it enables macroscopic properties to be calculated from simulations using relatively small numbers of particles. The importance of boundary effects can be illustrated by considering the following simple example. Suppose we have a cube of volume 1 litre which is filled with water at room temperature. The cube contains approximately 3.3 X 10 molecules. Interactions with the walls can extend up to 10 molecular diameters into the fluid. The diameter of the water molecule is approximately 2.8 A and so the number of water molecules that are interacting with the boundary is about 2 x 10. So only about one in 1.5 million water molecules is influenced by interactions with the walls of the container. The number of particles in a Monte Carlo or molecular dynamics simulation is far fewer than 10 -10 and is frequently less than 1000. In a system of 1000 water molecules most, if not all of them, would be within the influence of the walls of the boundary. Clecirly, a simulation of 1000 water molecules in a vessel would not be an appropriate way to derive bulk properties. The alternative is to dispense with the container altogether. Now, approximately three-quarters of the molecules would be at the surface of the sample rather than being in the bulk. Such a situation would be relevcUit to studies of liquid drops, but not to studies of bulk phenomena. 

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