Stochastic fractal and Noether's theorem

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

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