# Freezing phase transition in a fractal potential

## TL;DR

This paper presents a one-dimensional thermodynamic model with a fractal potential that undergoes a phase transition, where the probability measure shifts from continuous to singular, indicating a freezing phenomenon at a finite temperature.

## Contribution

It introduces a simple fractal potential model demonstrating a phase transition and freezing behavior at a positive temperature, with rigorous proofs of measure properties.

## Key findings

- System exhibits a phase transition at finite temperature.
- Below critical temperature, measure supported on the Cantor set.
- System 'freezes' before reaching zero temperature.

## Abstract

In this work we propose a simple example of a one-dimensional thermodynamic system where non-interacting particles are allowed to move over the $[0,1]$ interval, which are influenced by a potential with a fractal structure. We prove that the system exhibits a phase transition at a finte temperature, which is characterized by the fact that the Gibbs-Boltzmann probability measure passes from being absolutely continuous with respect to Lebesgue (at high temperature) to being singular continuous (at low temperatures). We prove that below the critical temperature (when the Gibbs-Boltzmann probability measure is singular continuous) the probability measure is supported on the middle-third Cantor set and that further lowering the temperature, the probability measure does not change anymore. This means that, in some sense, the system reaches the ground-state before the zero temperature, indicating that the system "freezes" at a positive temperature.