Theory of a Systematic Computational Error in Free Energy Differences

Abstract

Systematic inaccuracy is inherent in any computational estimate of a nonlinear average, due to the availability of only a finite number of data values, $N$. Free energy differences $\Delta F$ between two states or systems are critically important examples of such averages. Previous work has demonstrated, empirically, that the “finite-sampling error” can be very large—many times $k_{B}T$—in $\Delta F$ estimates for simple molecular systems. Here we present a theoretical description of the inaccuracy, including the exact solution of a sample problem, the precise asymptotic behavior in terms of $1/N$ for large $N$, the identification of a universal law, and numerical illustrations. The theory relies on corrections to the central and other limit theorems.