  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 non-ergodic regimes. In the non-ergodic 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.
