Optimization of expanded ensemble methods.

Expanded ensemble methods, designed to sample a range of an order parameter lambda of interest, can be optimized to overcome the difficulties associated with traversing large free-energy barriers or rugged landscapes. The optimization strategy of Trebst et al. [Phys. Rev. E 70, 046701 (2004)] is based on finding suitable biasing weights for inter-lambda transitions that maximize the number of round trips that the system performs between the lower and upper lambda bounds. In this work, this optimized-ensemble methodology is extended by finding weights that minimize the mean round-trip time tau (between the lambda end states) for a Markovian walk. Applications are presented for an atomistically detailed model and for systems where one needs to sample a wide range of concentrations or compositions. A less rigorous method that implements a dual tau minimization (for both upward and downward trajectories) is found to be harder to converge but produce more round trips than a method based on a single tau minimization for all trajectories. While the proposed methods do not always minimize the true tau, they have performances that are either similar or better than those of the original optimized-ensemble method and provide useful information to characterize deviations from Markovian dynamics in the sampling of the lambda space.