Model for small-sample bias of free-energy calculations applied to Gaussian-distributed nonequilibrium work measurements

We present a model for the bias of free-energy differences when determined using the nonequilibrium work (NEW) formalism due to Jarzynski. Input to the model is the distribution of work values underlying the NEW calculation, and the bias is estimated by assuming that all of the inaccuracy is incurred by failure to sample work values beyond a specific point in the tail of the distribution. The model is formulated considering both small- and large-sample NEW calculations. It is then applied to the study of bias for cases in which the work distribution is Gaussian. The model is shown to give an excellent description of the bias in situations where the bias is a more significant source of error than the sample variance. A scaling law is presented and it is shown that the bias as a function of sampling can be reduced to a single universal curve, approximately valid for all Gaussian work distributions. This result is used to formulate a simple criterion that can be applied to test if a NEW calculation is effectively free of bias. The criterion is shown to be effective even though it uses the measured (and perhaps biased) free energy as an input.