Systematic errors in free energy perturbation calculations due to a finite sample of configuration space: sample-size hysteresis

Although it is well known that the free energy perturbation procedure is exact when an infinite sample of configuration space is used, for finite sample size there is a systematic error resulting in hysteresis for forward and backward simulations. The qualitative behavior of this systematic error is first explored for a Gaussian distribution, then a first-order estimate of the error for any distribution is derived. To first order the error depends only on the fluctuations in the sample of potential energies, ΔE, and the sample size, n, but not on the magnitude of ΔE. The first-order estimate of the systematic sample-size error is used to compare the effciencies of various computing strategies. It is found that slow-growth, free energy perturbation calculations will always have lower errors from this source than window-growth, free energy perturbation calculations for the same computing effort. The systematic sample-size errors can be entirely eliminated by going to thermodynamic integration rather than free energy perturbation calculations. When ΔE is a very smooth function of the coupling parameter, λ, thermodynamic integration with a relatively small number of windows is the recommended procedure because the time required for equilibration is reduced with a small number of windows. The present results give a method of estimating this sample-size hysteresis during the course of a slow-growth, free energyperturbation run. This is important because in these calculations time-lag and sample-size errors can cancel, so that separate methods of estimating and correcting for each are needed. When dynamically modified window procedures are used, it is recommended that the estimated sample-size error be kept constant, not that the magnitude of ΔE be kept constant. Tests on two systems showed arather small sample-size hysteresis in slow-growth calculations except in the first stages of creating a particle, where both fluctuations and sample-size hysteresis are large.