On Stochastic Error and Computational Efficiency of the Markov Chain Monte Carlo Method

In Monte Carlo simulations, the thermal equilibria quantities are estimated by ensemble average over a sample set containing a large number of correlated samples. As the stochastic error of the simulation results is significant, it is desirable to understand the variance of the estimation by ensemble average, which depends on the sample size (i.e., the total number of samples in the set) and the sampling interval (i.e., cycle number between two consecutive samples). Although large sample sizes reduce the variance, they increase the computational cost of the simulation. For a given CPU time, the sample size can be reduced greatly by increasing the sampling interval while the corresponding increase of the variance is negligible. In this work, we report a few general rules that relate the variance with the sample size and the sampling interval. These relations were observed in our numerical results. The main contribution of this work is the theoretical proof of these numerical observations and the set of assumptions that lead to them