Stochastic Fractal and Noether's Theorem

TL;DR

This paper studies a stochastic binary fragmentation process leading to fractal structures, revealing a conserved moment related to fractal dimension and exploring the connection between symmetry and conservation laws through a quantum-mechanical analogy.

Contribution

It introduces a stochastic fragmentation model with a conserved moment at the fractal dimension and links dynamical scaling symmetry to a mathematical conserved quantity via quantum mechanics.

Findings

The $d_f$-th moment $M_{d_f}$ is conserved regardless of parameters.

The system exhibits self-similarity through data collapse.

The Noether charge for scaling symmetry is trivial, but $M_{d_f}$ relates to phase rotation symmetry.

Abstract

We consider the binary fragmentation problem in which, at any breakup event, one of the daughter segments either survives with probability $p$ or disappears with probability $1\!-\!p$. It describes a stochastic dyadic Cantor set that evolves in time, and eventually becomes a fractal. We investigate this phenomenon, through analytical methods and Monte Carlo simulation, for a generic class of models, where segment breakup points follow a symmetric beta distribution with shape parameter $\alpha$, which also determines the fragmentation rate. For a fractal dimension $d_f$, we find that the $d_f$-th moment $M_{d_f}$ is a conserved quantity, independent of $p$ and $\alpha$. We use the idea of data collapse -- a consequence of dynamical scaling symmetry -- to demonstrate that the system exhibits self-similarity. In an attempt to connect the symmetry with the conserved quantity, we reinterpret the fragmentation equation as the continuity equation of a Euclidean quantum-mechanical system. Surprisingly, the Noether charge corresponding to dynamical scaling is trivial, while $M_{d_f}$ relates to a purely mathematical symmetry: quantum-mechanical phase rotation in Euclidean time.