Variance minimization of free energy estimates from optimized expanded ensembles.

We explored the possibility of improving the accuracy and precision of free-energy differences estimated via expanded ensembles by manipulation of the biasing weights. Three different weighing approaches were compared: the flat histogram (FH) method, the optimized ensemble (OE) method, and a method introduced in this work, denoted MinVar, which aims to explicitly minimize the expected variance. The performance of these three methods was tested for the simulation of chemical potentials in systems of symmetric diblock copolymers with chain lengths of either 10 or 4 beads, and a system of one large hard sphere of diameter 10 d immersed in a fluid of hard spheres of diameter d. In addition, the effect of the weighing method on the observed accuracy was investigated for different choices of macrostate staging and for both optimized and nonoptimized acceptance ratio methods for calculating free-energy differences. In the diblock copolymer systems, we found that the maximum attainable accuracy can be limited by correlations between the samples, causing the "real" observed variances to be much larger than the expected "ideal" ones. Hence, if the formal minimization of the variance, as aimed by the MinVar method, occurs at the expense of increasing the correlations in the data, the accuracy may actually decrease. Although maximizing the number of round trips between initial and final macrostates (as aimed by the OE method) was found to be directly related to data decorrelation, this only translates into increased accuracy if the correlations are the major source of errors in the free energy estimates. Finally, for the hard sphere system, we found that the MinVar method performs better than both the OE and FH methods even though the MinVar method in this case never completes a round trip, illustrating that maximizing the number of round trips for fixed computational cost does not necessarily lead to increased precision.