Freezing phase transition in a fractal potential

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 interval, which are influenced by a potential with a fractal structure. We prove that the system exhibits a phase transition at a finite temperature, which is characterized by the fact that the Gibbs–Boltzmann probability measure passes from being absolutely continuous with respect to Lebesgue (at high temperatures) 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.