Metropolis Monte Carlo, generalized

Because generalized Metropolis Monte Carlo methods are based on random sampling from probability distribution functions, it is necessary to use a high-quality random-number generator algorithm to obtain reliable results. A review of such methods is beyond the scope of this chapter, " but a few general considerations merit discussion. 

In general, Monte Carlo simulations are such calculations in which the values of some parameters are determined by the average of some randomly generated individuals.45-54 In chemistry applications, the most prevalent methods are the so called Metropolis Monte Carlo (MMC)55 and Reverse Monte Carlo (RMC) ones. The most important quantities in these methods are some kinds of U energy-type potentials (e.g. internal energy, enthalpy,... 

Molecular Dynamics simulation is one of many methods to study the macroscopic behavior of systems by following the evolution at the molecular scale. One way of categorizing these methods is by the degree of determinism used in generating molecular positions [134], On the scale from the completely stochastic method of Metropolis Monte Carlo to the pure deterministic method of Molecular Dynamics, we find a multitude and increasingly diverse number of methods to name just a few examples Force-Biased Monte Carlo, Brownian Dynamics, General Langevin Dynamics [135], Dissipative Particle Dynamics [136,137], Colli-sional Dynamics [138] and Reduced Variable Molecular Dynamics [139]. 

The properties of the generalized tiling model depend on two parameters, a dimensionless temperature t= kgTIE, and a dimensionless tiling fault energy r=EJE. We have obtained the statistical and thermodynamic properties of our model as a function of t and r from Metropolis Monte Carlo (MC) simulations. The basic variables used in the MC... 

Kikuchi K, Yoshida M, Maekawa T, Watanabe H. Metropolis Monte Carlo method for Brownian dynamics simulation generalized to include hydrodynamic interactions. Chem Phys Lett 1992 196 57-61. 

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. 

The Metropolis Monte Carlo method attempts to sample a representative set of equilibrium states in a manner that facilitates the calculation of meaningful averages for properties of the system. This method is discussed in the following subsections Basic Aspects of the Metropolis Monte Carlo Method, Monte Carlo Moves, and General Pointers for Conducting Monte Carlo Simulations. 

The Metropolis Monte Carlo Method could also be referred to as a metaheuristic, that is, a heuristic that is general enough to apply to a broad range of problems. Similar to heuristics, these are not guaranteed to produce an optimal solution, so are often used in situations either where this is not crucial, or a suboptimal solution can be modified. 

The strategy in Variational Monte Carlo (VMC) is therefore to pick a proper form for a trial wave function based on physical insight for the particular system under study. In general, a number of parameters (oi,..., a ) will appear in the wave function to be treated as variational parameters. For any given set of a the Metropolis algorithm is used to sample the distribution... 

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-... 

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. 

---