Comparison of efficiency and bias of free energies computed by exponential averaging, the Bennett acceptance ratio, and thermodynamic integration

TLDR

Recent studies show the Bennett acceptance ratio (BAR) method is asymptotically unbiased for free‑energy estimation, yet its practical advantage over other atomistic simulation methods remains unclear. This work aims to compare the bias and efficiency of exponential averaging (EXP), thermodynamic integration (TI), and BAR by reviewing theoretical estimates and applying them to toy and realistic atomistic systems. The authors derive new bias‑variance formulas, link BAR to the weighted histogram analysis method, and evaluate the three estimators on analytically solvable toy problems and on insertion/deletion of Lennard‑Jones particles and larger molecules in TIP3P water. The results show no single method dominates across all scenarios; in toy models each method excels in different regimes, but in realistic atomistic tests BAR consistently yields lower bias, smaller variance, and greater efficiency, especially when phase‑space overlap is limited.

Abstract

Recent work has demonstrated the Bennett acceptance ratio method is the best asymptotically unbiased method for determining the equilibrium free energy between two end states given work distributions collected from either equilibrium and nonequilibrium data. However, it is still not clear what the practical advantage of this acceptance ratio method is over other common methods in atomistic simulations. In this study, we first review theoretical estimates of the bias and variance of exponential averaging (EXP), thermodynamic integration (TI), and the Bennett acceptance ratio (BAR). In the process, we present a new simple scheme for computing the variance and bias of many estimators, and demonstrate the connections between BAR and the weighted histogram analysis method. Next, a series of analytically solvable toy problems is examined to shed more light on the relative performance in terms of the bias and efficiency of these three methods. Interestingly, it is impossible to conclusively identify a “best” method for calculating the free energy, as each of the three methods performs more efficiently than the others in at least one situation examined in these toy problems. Finally, sample problems of the insertion/deletion of both a Lennard-Jones particle and a much larger molecule in TIP3P water are examined by these three methods. In all tests of atomistic systems, free energies obtained with BAR have significantly lower bias and smaller variance than when using EXP or TI, especially when the overlap in phase space between end states is small. For example, BAR can extract as much information from multiple fast, far-from-equilibrium simulations as from fewer simulations near equilibrium, which EXP cannot. Although TI and sometimes even EXP can be somewhat more efficient in idealized toy problems, in the realistic atomistic situations tested in this paper, BAR is significantly more efficient than all other methods.