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Phase transitions are one of the most interesting natural phenomena. Forfinite systems, one of the concerns in the topic is how to classify a specifictransition as being of first, second, or even of a higher order, according tothe Ehrenfest classification. The partition function provides all thethermodynamic information about the physical systems, and a phase transitioncan be identified by the complex temperature where it is equal to zero. Inaddition, the pattern of the zeros on the complex temperature plane can provideevidence of the order of the transition. In this manuscript, we present ananalytical and simulational study connecting the microcanonical analysis of theunstable region of the entropy to the canonical partition function zeros. Weshow that, for the first-order transition, the zeros accumulate uniformly in avertical line on the complex inverse temperature plane as discussed in previousworks. We illustrate our calculation using the $147$ particles Lennard-Jonescluster.

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