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

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 $x=e^{-\beta \epsilon }$ 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.

13 pages, 10 figures

References in corpus (28)

• Machine learning phases of matter
• Statistical Mechanics of Equilibrium and Nonequilibrium Phase Transitions: The Yang-Lee Formalism
• Classification of phase transitions in small systems
• A generic, hierarchical framework for massively parallel Wang-Landau sampling
• Microcanonical entropy inflection points: Key to systematic understanding of transitions in finite systems
• Scaling Relations for Logarithmic Corrections
• Self-Consistent Scaling Theory for Logarithmic Correction Exponents
• Phase Transition Strength through Densities of General Distributions of Zeroes
• Classification of Phase Transitions by Microcanonical Inflection-Point Analysis
• Critical and tricritical singularities from small-scale Monte Carlo simulations: The Blume-Capel model in two dimensions
• Energy Probability Distribution Zeros: A Route to Study Phase Transitions
• A New Thermodynamics from Nuclei to Stars III
• Numerical comparison of two approaches for the study of phase transitions in small systems
• Finite-size behaviour of the microcanonical specific heat
• A new look at the 2D Ising model from exact partition function zeros for large lattice sizes
• Evidence for additional third-order transitions in the two-dimensional Ising model
• A Look Back at the Ehrenfest Classification. Translation and Commentary of Ehrenfest's 1933 paper introducing the notion of phase transitions of different order
• Exact microcanonical statistical analysis of transition behavior in Ising chains and strips
• Using zeros of the canonical partition function map to detect signatures of a Berezinskii-Kosterlitz-Thouless transition
• Critical and tricritical behavior of the $d=3$ Blume-Capel model: Results from small-scale Monte Carlo simulations
• Exploring Replica-Exchange Wang-Landau sampling in higher-dimensional parameter space
• Microcanonical finite-size scaling in specific heat diverging 2nd order phase transitions
• Phase Transitions in the Two-Dimensional Ising Model from the Microcanonical Perspective
• Locating Critical Points Using Ratios of Lee-Yang Zeros
• Phase Diagram of Flexible Polymers with Quenched Disordered Charged Monomers
• Connecting the Unstable Region of the Entropy to the Pattern of the Fisher's Zeros Map
• Finding the Dominant Zero of the Energy Probability Distribution
• Estimating the Number of States via the Rodeo Algorithm for Quantum Computation