Rakibur Rahman
Stochastic fractal and Noether's theorem
Rahman, Rakibur; Nowrin, Fahima; Rahman, M Shahnoor; Wattis, Jonathan A D; Hassan, Md. Kamrul
Authors
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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