Exploring the Nexus between Thermodynamic Phase Transitions and Geometric Fractals through Systematic Lattice Point Classification

TL;DR

This paper introduces a probability-based approach to link fractal formation with phase transitions, demonstrating how specific lattice points in the Ising model undergo critical changes that lead to fractal structures.

Contribution

It presents a novel probability calculation method that directly connects fractal structures with phase transitions, especially identifying critical boundary points in the Ising model.

Findings

Boundary lattice points undergo phase transition at weight ~0.4

Fractal structures emerge at this critical weight

Probability manipulation supports the fractal-phase transition link

Abstract

Fractals are ubiquitous in the natural world, and their connection with phase transitions has been widely observed. This study investigates mechanisms of fractal formation from the perspective of phase transitions. A novel set of probability calculation methods is introduced to establish a direct link between fractals and phase transitions. Notably, in the Ising model, a specific category of boundary lattice points undergoes a phase transition when the associated weight reaches approximately 0.4. The identified correlation between phase transitions and fractals suggests the emergence of fractal structures at this critical weight. The paper offers supporting evidence for this conclusion through the deliberate manipulation of the proposed probability-based method. This research contributes to a deeper understanding of the interplay between fractals and phase transitions, providing valuable insights for further exploration in diverse scientific domains.