Metric characterization of cluster dynamics on the Sierpinski gasket

TL;DR

This paper introduces an algorithm to analyze cluster dynamics on cellular automata, exemplified by the Ising model on a Sierpinski Gasket, revealing temperature-dependent regimes and critical behaviors through geometric metrics.

Contribution

It presents a novel information-based metric approach to characterize cluster dynamics on complex structures like the Sierpinski Gasket, linking geometric analysis with thermodynamic properties.

Findings

Identification of temperature regimes related to cluster size and change rate

Correlation between geometric cluster behavior and thermodynamic criticality

Detection of multiple time scales leading to chaos at high temperatures

Abstract

We develop and implement an algorithm for the quantitative characterization of cluster dynamics occurring on cellular automata defined on an arbitrary structure. As a prototype for such systems we focus on the Ising model on a finite Sierpsinski Gasket, which is known to possess a complex thermodynamic behavior. Our algorithm requires the projection of evolving configurations into an appropriate partition space, where an information-based metrics (Rohlin distance) can be naturally defined and worked out in order to detect the changing and the stable components of clusters. The analysis highlights the existence of different temperature regimes according to the size and the rate of change of clusters. Such regimes are, in turn, related to the correlation length and the emerging "critical" fluctuations, in agreement with previous thermodynamic analysis, hence providing a non-trivial geometric description of the peculiar critical-like behavior exhibited by the system. Moreover, at high temperatures, we highlight the existence of different time scales controlling the evolution towards chaos.