Analyses of nucleation rates from molecular dynamics simulations. II. Weight functions, generation of Stochastic times, and realistic uncertainties

In paper I of this series it was shown how to take transient nucleation into account in the spontaneous freezing of large clusters when deriving nucleation rates and time lags from sets of nucleation times. This required an estimate of the "reduced moment" characterizing the period of transient nucleation. Also, a procedure was sketched for constructing sets of stochastic times simulating nucleation times, for purposes of determining statistical uncertainties in the derived kinetic parameters of nucleation rates. In the present paper, a considerably more precise method for generating stochastic nucleation times is presented and an optimum weighting scheme for least squares analyses of nucleation rates and time lags is formulated. In the prior literature no suitable means had been established for estimating the reduced moment. Alternative ways to estimate this moment from nucleation data are discussed. It is found that the true expectation values of uncertainties, sigma(epsilon), in rates and time lags are significantly larger than the uncertainties, sigma(ls), derived from residuals in least squares analyses of individual sets of nucleation times. Although the elements of the least squares error matrix are lower for the optimum weight function than for the unit weights and arctangent weights used in prior analyses, the actual uncertainties do not depend strongly upon which weighting scheme is employed. The derived kinetic parameters do, however, depend appreciably upon the weighting, and results of the optimum weighting are preferred. A virtue of the analysis of simulated stochastic nucleation times is that it provides a valid measure of the actual uncertainties in derived nucleation parameters as well as the smaller, and therefore misleading, uncertainties inferred from a conventional error matrix. The analysis presented leads to guidelines conveying how large a set of nucleation times must be in order to provide meaningful determinations of nucleation rates and time lags. The new procedure also provides the first estimates of the uncertainty in reduced moments derived from sets of nucleation times, including their dependence on sample size.