Approach to ergodicity in Monte Carlo simulations

Abstract

The approach to the ergodic limit in Monte Carlo simulations is studied using both analytic and numerical methods. With the help of a stochastic model, a metric is defined that enables the examination of a simulation in both the ergodic and nonergodic regimes. In the nonergodic regime, the model implies how the simulation is expected to approach ergodic behavior analytically, and the analytically inferred decay law of the metric allows the monitoring of the onset of ergodic behavior. The metric is related to previously defined measures developed for molecular dynamics simulations, and the metric enables the comparison of the relative efficiencies of different Monte Carlo schemes. Applications to Lennard-Jones 13-particle clusters are shown to match the model for Metropolis, J-walking, and parallel tempering based approaches. The relative efficiencies of these three Monte Carlo approaches are compared, and the decay law is shown to be useful in determining needed high temperature parameters in parallel tempering and J-walking studies of atomic clusters.

Heat capacity estimators for random series path-integral methods by finite-difference schemes

“…The optimal choice of the parameter R c for the constraining potential has been discussed in recent work. 32 If R c is taken to be too small, the properties of the system become sensitive to its choice, whereas large values of R c can result in problems attaining an ergodic simulation. To facilitate comparisons, in the current work, R c has been chosen to be identical to that used in Ref.…”

Phase changes in selected Lennard-Jones X13−nYn clusters

“…36 If R c is taken to be too small, the properties of the system become sensitive to its choice, whereas large values of R c can result in problems attaining an ergodic simulation. The classical and quantum Monte Carlo simulations presented here have been performed with R c ϭ2 XX and ⑀ϭ⑀ XX .…”

Energy estimators for random series path-integral methods

“…22 The optimal choice of the parameter R c for the constraining potential has been discussed in recent work. 23 If R c is taken to be too small, the properties of the system become sensitive to its choice, whereas large values of R c can result in problems attaining an ergodic simulation. To facilitate comparisons, in the current work, R c has been chosen to be identical to that used in Ref.…”