Connecting the Unstable Region of the Entropy to the Pattern of the Fisher's Zeros Map

Phase transitions are one of the most interesting natural phenomena. For finite systems, one of the concerns in the topic is how to classify a specific transition as being of first, second, or even of a higher order, according to the Ehrenfest classification. The partition function provides all the thermodynamic information about the physical systems, and a phase transition can be identified by the complex temperature where it is equal to zero. In addition, the pattern of the zeros on the complex temperature plane can provide evidence of the order of the transition. In this manuscript, we present an analytical and simulational study connecting the microcanonical analysis of the unstable region of the entropy to the canonical partition function zeros. We show that, for the first-order transition, the zeros accumulate uniformly in a vertical line on the complex inverse temperature plane as discussed in previous works. We illustrate our calculation using the $147$ particles Lennard-Jones cluster.