Monte Carlo method Metropolis sampling

Molecular model-building (conformational search) methods fall into two general classes systematic and random. - Systematic methods search all possible combinations of torsional angles, whereas random methods usually involve a Monte Carlo (with Metropolis sampling ) or molecular dynamics trajectory. Both approaches attempt to search large areas of conformational space and eventually converge on the desired conformation or structure. Dis-... 

A similar algorithm has been used to sample the equilibrium distribution [p,(r )] in the conformational optimization of a tetrapeptide[5] and atomic clusters at low temperature.[6] It was found that when g > 1 the search of conformational space was greatly enhanced over standard Metropolis Monte Carlo methods. In this form, the velocity distribution can be thought to be Maxwellian. 

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... 

With Monte Carlo methods, the adoption of the Metropolis sampling scheme intrinsically assumes equilibrium Boltzmann statistics, so special modifications are required to extend MC methods to non-equilibrium solvation as well. Fortunately, for a wide variety of processes, ignoring non-equilibrium solvation effects seems to introduce errors no larger than those already inherent from other approximations in the model, and thus both implicit and explicit models remain useful tools for studying chemical reactivity. 

Therefore, if graphs can be sampled with an unnormalised probability given by (using a Metropolis Monte Carlo method), then the energy estimator becomes ... 

The Monte Carlo method is easily carried out in any convenient ensemble since it simply requires the construction of a suitable Markov chain for the importance sampling. The simulations in the original paper by Metropolis et al. [1] were carried out in the canonical ensemble corresponding to a fixed number of molecules, volume and temperature, N, V, T). By contrast, molecular dynamics is naturally carried out in the microcanonical ensemble, fixed (N, V, E), since the energy is conserved by Newton s equations of motion. This implies that the temperature of an MD simulation is not known a priori but is obtained as an output of the calculation. This feature makes it difficult to locate phase transitions and, perhaps, gave the first motivation to generalize MD to other ensembles. 

In the Monte Carlo method to estimate a many-dimensional integral by sampling the integrand. Metropolis Monte Carlo or, more generally, Markov chain Monte Carlo (MCMC), to which this volume is mainly devoted, is a sophisticated version of this where one uses properties of random walks to solve problems in high-dimensional spaces, particularly those arising in statistical mechanics. 

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