Stochastic Fractal and Noether's Theorem

• URL: http://arxiv.org/abs/2010.07953v1
• Date: Thu, 15 Oct 2020 18:00:21 GMT
• Title: Stochastic Fractal and Noether's Theorem
• Authors: Rakibur Rahman, Fahima Nowrin, M. Shahnoor Rahman, Jonathan A. D. Wattis and Md. Kamrul Hassan
• Abstract summary: We consider the binary fragmentation problem in which, at any breakup event, one of the daughter segments survives with probability $p$ or disappears with probability $1!!p$. 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 $alpha$. In an attempt to connect the symmetry with the conserved quantity, we reinterpret the continuity equation of a Euclidean quantum-mechanical system.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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