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Stochastic fractal and Noether's theorem
Rakibur Rahman1,2, Fahima Nowrin2, M Shahnoor Rahman2
1Max-Planck-Institut für Gravitationsphysik, Am Mühlenberg 1, D-14476 Potsdam-Golm, Germany.
Physical Review. E
|March 19, 2021
Summary
This study reveals a conserved quantity in binary fragmentation, specifically the d_f-th moment (M_df), which remains constant regardless of survival probability (p) or fragmentation rate (α). The system exhibits fractal self-similarity, linked to a mathematical symmetry in its quantum mechanical interpretation.
Area of Science:
- Statistical Physics
- Fractal Geometry
- Stochastic Processes
Background:
- The binary fragmentation problem involves segments breaking, with daughter segments surviving with probability p or disappearing with probability 1-p.
- This process generates a stochastic dyadic Cantor set that evolves into a fractal over time.
Purpose of the Study:
- To investigate the fractal formation in binary fragmentation using analytical methods and Monte Carlo simulations.
- To identify conserved quantities and understand the role of survival probability (p) and fragmentation rate (α) in fractal dynamics.
Main Methods:
- Analytical investigation of a generic class of models for binary fragmentation.
- Monte Carlo simulations to observe the evolution of the stochastic dyadic Cantor set.
- Application of data collapse techniques to demonstrate self-similarity.
Main Results:
- Identified the d_f-th moment (M_df) as a conserved quantity, independent of p and α.
- Observed that scaling exponents are independent of p, while the self-similar distribution shows a weak p dependence.
- Demonstrated system self-similarity through data collapse, linked to dynamical scaling symmetry.
Conclusions:
- The binary fragmentation process leads to fractal structures with a conserved M_df moment.
- The system's self-similarity is robust, with scaling exponents independent of the survival probability.
- A connection was found between the conserved quantity and a quantum-mechanical phase rotation symmetry.
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