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Stochastic fractal and Noether's theorem

Rahman, Rakibur; Nowrin, Fahima; Rahman, M Shahnoor; Wattis, Jonathan A D; Hassan, Md. Kamrul

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Authors

Rakibur Rahman

Fahima Nowrin

M Shahnoor Rahman

JONATHAN WATTIS jonathan.wattis@nottingham.ac.uk
Professor of Applied Mathematics

Md. Kamrul Hassan



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 α, 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 α. While the scaling exponents do not depend on p, the self-similar distribution shows a weak p-dependence. 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.

Citation

Rahman, R., Nowrin, F., Rahman, M. S., Wattis, J. A. D., & Hassan, M. K. (2021). Stochastic fractal and Noether's theorem. Physical Review E, 103(2), Article 022106. https://doi.org/10.1103/physreve.103.022106

Journal Article Type Article
Acceptance Date Jan 13, 2021
Online Publication Date Feb 4, 2021
Publication Date Feb 4, 2021
Deposit Date Jan 13, 2021
Publicly Available Date Feb 4, 2021
Journal Physical Review E
Print ISSN 2470-0045
Electronic ISSN 2470-0053
Publisher American Physical Society
Peer Reviewed Peer Reviewed
Volume 103
Issue 2
Article Number 022106
DOI https://doi.org/10.1103/physreve.103.022106
Public URL https://nottingham-repository.worktribe.com/output/5225295
Publisher URL https://journals.aps.org/pre/abstract/10.1103/PhysRevE.103.022106

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