Progress and Outlook in Monte Carlo Simulations

In 1953, Metropolis et al. introduced an ingenious stochastic method for sampling points in a multidimensional space, according to a prescribed probability distribution defined on that space. One of the great intellectual achievements of the 20th century, Metropolis Monte Carlo has found application in a wide range of scientific and engineering fields. Here, we are mainly concerned with the prediction of thermodynamic properties based on the principles of statistical mechanics. The multidimensional space sampled is the configuration space, spanned by the (generalized) coordinates of the molecules constituting the system plus possibly a few macroscopic extensive variables that are allowed to fluctuate, and the probability density Feq is set by an equilibrium ensemble. Sampled configuration-space points, or “states,” form a Markov chain, with each state being formed from the previous one in a Monte Carlo (MC) step. In the original MR2T2 algorithm, each MC step is executed in two stages. One first attempts an elementary move from the current state i into a new state j with probability R(ifj), where the stochastic matrix of attempt probabilities is symmetric, satisfying the condition