Branching of nucleation paths in a metastable lattice gas with Metropolis dynamics

Abstract

A cold, supersaturated gas on a square lattice with nearest-neighbour interaction is considered. The problem is equivalent to a metastable Ising ferromagnet with Metropolis dynamics in external magnetic field. Using the electric analogy established earlier for this problem by Shneidman (2003 J. Stat. Phys.112 293), the inverse of the nucleation rate is expressed as an equivalent resistance of a complex electric network. Explicit expressions for the equivalent resistance of the network can be obtained from an analysis of series–parallel connections, and the nucleation rate can be evaluated with accuracy which is sufficient not only for the exponential part, but also for the prefactor. At low temperatures and non-special values of the supersaturation S the path of the lowest energy dominates. On the other hand, for 1/2S = 1, 2, …, (with S ⩾ 1 corresponding to instability) paths with temporal excursions towards higher energies also contribute, leading to the renormalization of peaks in the prefactor of the inverse rate even in the limit of zero temperature. Results can be used to construct a Becker–Döring type picture which views nucleation as a random walk in a one-dimensional space of the cluster sizes. However, the ‘size’ cannot be characterized by any integer n, the number of particles in a cluster, but only by an n which is close to a perfect square. Other clusters are to be combined into physical ‘droplets’, which presumably will play the role of nuclei in the Becker–Döring description.