{
  "id": "2010.07953",
  "version": "v1",
  "published": "2020-10-15T18:00:21.000Z",
  "updated": "2020-10-15T18:00:21.000Z",
  "title": "Stochastic Fractal and Noether's Theorem",
  "authors": [
    "Rakibur Rahman",
    "Fahima Nowrin",
    "M. Shahnoor Rahman",
    "Jonathan A. D. Wattis",
    "Md. Kamrul Hassan"
  ],
  "comment": "11 pages, 6 captioned figures each containing 2 subfigures",
  "categories": [
    "cond-mat.stat-mech",
    "hep-th",
    "quant-ph"
  ],
  "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 $\\alpha$, 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 $\\alpha$. 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-15T18:00:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "noethers theorem",
      "stochastic fractal",
      "stochastic dyadic cantor set",
      "conserved quantity",
      "monte carlo simulation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}