Randomness in a Galton board from the viewpoint of predictability: Sensitivity and statistical bias of output states

Kenichi Arai*, Takahisa Harayama, Satoshi Sunada, Peter Davis

*この研究の対応する著者

研究成果査読

4 被引用数 (Scopus)

抄録

The Galton board is a classic example of the appearance of randomness and stochasticity. In the dynamical model of the Galton board, the macroscopic motion is governed by deterministic equations of motion, and predictability depends on uncertainty in the initial conditions and its evolution by the dynamics. In this sense the Galton board is similar to coin tossing. In this paper, we analyze a simple dynamical model which is inspired by the Galton board. Especially, we focus on the predictability, considering the relation between the uncertainty of initial states and the structure of basins of initial states that result in the same exit state. The model has basins with fractal basin structure, unlike the basins in coin tossing models which have only finite structure. Arbitrarily small uncertainty of initial conditions can cause unpredictability of final states if the initial conditions are chosen in fractal regions. In this sense, our model is in a different category from the coin tossing model. We examine the predictability of a small Galton board model from the viewpoint of the sensitivity and the statistical bias of final states. We show that it is possible to determine the radii of scatterers corresponding to a given predictability criterion, specified as a statistical bias, and a given uncertainty of initial conditions.

本文言語English
論文番号056216
ジャーナルPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
86
5
DOI
出版ステータスPublished - 2012 11 27
外部発表はい

ASJC Scopus subject areas

  • 統計物理学および非線形物理学
  • 統計学および確率
  • 凝縮系物理学

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