A general relation between the largest nucleus and all nuclei distributions for free energy calculations

Prediction of nucleation rates in first order phase transitions requires the knowledge of the barrier associated with the free energy profile W. Molecular simulations offer a direct route through W = -kT ln p_a, where k is Boltzmann's constant, T is temperature, and p_a is the probability distribution of the size of any nucleus. However, in practice, the extremely scarce spontaneous occurrence of large nuclei impedes the full determination of p_a, and a numerical bias must be introduced, which is generally done on the size of the largest nucleus in the system, leading to the probability size distribution of the largest nucleus p_l. Although p_l is known to be system size dependent, unlike p_a, it has extensively been used as an approximation for p_a. This communication demonstrates an improved relation between p_a and p_l, which cures this approximation and allows an accurate calculation of free energy barriers from biased simulations.