Metropolis Monte Carlo randomness

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,... 

To overcome the inefficiency of Metropolis Monte Carlo, which searches all of the conformational space but does so very slowly, a procedure to move rapidly through the space of local minima has been devised.177 The energy of a random starting conformation is minimized. Then, a random change (selected in the range 0 to 2ir) is made in a randomly chosen dihedral angle, and the... 

The Metropolis Monte Carlo algorithm [47] simulates the evolution to thermal equilibrium of a solid for a fixed value of the temperature T. Given the current state of system, characterized by the parameters qt of the system, a move is applied by a shift of a randomly chosen parameter qi. If the energy after the move is less than the energy before, i.e. AE < 0, the move is accepted and the process continues from the new state. If, on the other hand, AE > 0, then the move may still be accepted with probability... 

Figure 5.7 Flow diagram of the Metropolis Monte Carlo scheme. RND represents a random number between 0 and 1.

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 following simple Metropolis Monte Carlo example demonstrates how correlations between successive pairs of random numbers can give incorrect results. Suppose we sample the movement of a particle along the x axis confined in a potential well that is symmetric about the origin 7(x) = V( — x). The classic Metropolis algorithm is outlined in Figure 1. At each step in our calculations, the particle is moved to a trial position with the first random number (sprng()) and then that step is accepted or rejected with the second random number. 

The well-known Metropolis Monte Carlo (MMC) procedure randomly samples conformational space according to the Boltzmann distribution of (distinguishable) conformations [34] ... 

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