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Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns

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Published on: August 30, 2013

Equivalent-neighbor Potts models in two dimensions.

Xiaofeng Qian¹, Youjin Deng^2,3, Yuhai Liu⁴

¹Lorentz Institute, Leiden University, P. O. Box 9506, 2300 RA Leiden, The Netherlands.

Physical Review. E

|December 15, 2016

Summary

The study reveals that longer interaction ranges in q-state Potts models can shift phase transitions from continuous to discontinuous. A tricritical point was found for the 3-state model, and the 4-state model

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Area of Science:

Statistical Mechanics
Condensed Matter Physics

Background:

The Potts model is a fundamental tool for studying phase transitions in magnetic systems and other statistical physics phenomena.
Understanding how interaction range affects critical behavior is crucial for developing accurate physical models.

Purpose of the Study:

To investigate the impact of variable interaction range on the phase transitions of two-dimensional q=3 and q=4 Potts models.
To identify universality classes and critical points for different interaction ranges (z).

Main Methods:

Monte Carlo simulations were employed to study the q=3 and q=4 Potts models.
Phase transitions were analyzed for varying numbers of equivalent neighbors (z).

Main Results:

For small z, transitions align with short-range Potts model universality classes.
As z increases, transitions become discontinuous.
A tricritical point was located for q=3 near z=80.
For q=4, transitions become discontinuous for z>16, with scaling behavior near a predicted fixed point at z=16.

Conclusions:

The interaction range significantly alters the nature of phase transitions in Potts models.
The findings provide insights into critical phenomena and universality in statistical physics.
The study highlights the existence of tricritical points and critical fixed points in extended Potts models.