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, DELTA-E, and the sample size, n, but not on the magnitude of DELTA-E. The first-order estimate of the systematic sample-size error is used to compare the efficiencies 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 DELTA-E is a very smooth function of the coupling parameter, lambda, 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 energy perturbation 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 DELTA-E be kept constant. Tests on two systems showed a rather 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.