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Stochastic Fractal and Noether's Theorem

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

Authors

Rakibur Rahman

Fahima Nowrin

M Shahnoor Rahman

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. (in press). Stochastic Fractal and Noether's Theorem. Physical Review E,

Journal Article Type Article
Acceptance Date Jan 13, 2021
Deposit Date Jan 13, 2021
Journal Physical Review E
Print ISSN 2470-0045
Publisher American Physical Society
Peer Reviewed Peer Reviewed
Public URL https://nottingham-repository.worktribe.com/output/5225295