Umbrella Sampling in the Long Range Ising Model

Umbrella sampling with the largest cluster size as the local order parameter has been used to study liquid to solid nucleation. In the absence of a rigorous definition of a solid-like cluster, an ad hoc definition of the clusters must be adopted. As a result it is not clear if the vanishing of the free energy barrier found by umbrella sampling should be interpreted in terms of a spinodal and how well the cluster found at the free energy maximum corresponds to the true nucleating droplet. To better understand umbrella sampling with a local order parameter we study nucleation in the long-range Ising model for which the cluster definition is given rigorously by a percolation mapping. We determine the free energy of the long-range Ising model as a function of the largest cluster size and the quench depth, and study the properties of the nucleating clusters as the spinodal is approached.