Test of the Monte Carlo Method: Fast Simulation of a Small Ising Lattice

Abstract

A very fast stochastic procedure is used to generate samples of configurations of a 4 × 4 periodic Ising lattice in zero field. Running on the IBM 7094, we require only 15 μsec to process each site (cf. Yang who required 300 μsec on the 7090) and hence we generate 4000 completely new configurations each second. Our main results are based on samples of 10⁶ configurations at each of 10 temperatures; we also took samples of 10⁷ configurations at three temperatures. The 4 × 4 lattice can be solved exactly without much difficulty. Hence our data give information about the Monte Carlo method itself, especially its rate of convergence. We define the statistical inefficiency (SI) in a variable as the limiting ratio of the observed variance of its long‐term averages to their expected (Gaussian) variance. (Thus, if the SI is 3, then averages of the variable taken over Monte Carlo runs of three million configurations will be as accurate as averages over one million configurations drawn randomly from the true ensemble. The SI accounts for correlations between configurations closely following one another in the Monte Carlo run.) We find that for this lattice the SI in energy never exceeds $\frac{1}{2}$ , and that its maximum occurs slightly above the temperature which is critical for the infinite lattice. We confirm earlier statements that the influence of the initial configuration is lost very quickly, except when two phases coexist.