Exact Partition Functions for the Primitive Droplet Nucleation Model in 2 and 3 Dimensions

TL;DR

This paper derives exact closed-form expressions for the grand canonical partition functions of primitive droplet nucleation models in two and three dimensions, applicable for all complex chemical potentials, by leveraging their PDE properties.

Contribution

It provides the first exact solutions for these partition functions in 2D and 3D, advancing understanding of droplet nucleation energetics.

Findings

Exact partition functions obtained for d=2 and 3.

Partition functions satisfy simple PDEs.

Results applicable for all complex chemical potentials.

Abstract

The grand canonical partition functions for primitive droplet nucleation models with an excess energy epsilon_n = - mu n + sigma n^{1-eta}, eta = 1/d, for droplets of n constituents in d dimensions are calculated exacly in closed form in the cases d=2 and 3 for all (complex) mu by exploiting the fact that the partition functions obey simple PDE.