### Abstract

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.

Original language | English |
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Article number | 056216 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 86 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2012 Nov 27 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Condensed Matter Physics
- Statistical and Nonlinear Physics
- Statistics and Probability

### Cite this

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*,

*86*(5), [056216]. https://doi.org/10.1103/PhysRevE.86.056216