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 to 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 $p_a$ the probability distribution of the size of any nucleus. But in practice, the extremely scarce spontaneous occurrence of large nuclei impedes the full determination of $p_a$, and a numerical bias must be introduced, e.g. 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 been extensively used as an approximation for $p_a$. This paper proposes an exact relation between $p_a$ and $p_l$, which cures this approximation and allows an exact calculation of free energy barriers from biased simulations.