A Microcanonical Inflection Point Analysis via Parametric Curves and its Relation to the Zeros of the Partition Function

TL;DR

This paper introduces a new method for analyzing phase transitions in statistical physics by parametrizing the entropy function and relating Fisher's zeros to transition order, demonstrated on multiple models.

Contribution

It presents an alternative protocol for phase transition analysis using entropy parametrization and clarifies the relation between Fisher's zeros and transition order.

Findings

Fisher's zeros form a circle in the complex inverse temperature map.

Latent heat is inversely related to the zeros' distance.

Key features like loops indicate phase transition presence.

Abstract

In statistical physics, phase transitions are arguably among the most extensively studied phenomena. In the computational approach to this field, the development of algorithms capable of estimating entropy across the entire energy spectrum in a single execution has highlighted the efficacy of microcanonical inflection point analysis, while Fisher's zeros technique has re-emerged as a powerful methodology for investigating these phenomena. This paper presents an alternative protocol for analyzing phase transitions using a parametrization of the entropy function in the microcanonical ensemble. We also provide a clear demonstration of the relation of the linear pattern of the Fisher's zeros on the complex inverse temperature map (a circle in the complex $x=e^{-\beta \varepsilon}$ map) with the order of the transition, showing that the latent heat is inversely related to the distance between the zeros. We study various model systems, including the Lennard-Jones cluster, the Ising, the XY, and the Zeeman models. By examining the behavior of thermodynamic quantities such as entropy and its derivatives in the microcanonical ensemble, we identify key features-such as loops and discontinuities in parametric curves-which signal phase transitions' presence and nature. This approach can facilitate the classification of phase transitions across various physical systems.