A Galton board is an instrument invented in 1873 by Francis Galton (1822–1911). It is a box with a glass front and many horizontal nails or pins embedded in the back and a funnel. Galton and many modern statisticians claimed that a lead ball descending to the bottom of the Galton board would display random walk. In this study, a new mathematical model of Galton board is developed, to further improve three very recently proposed models. The novel contribution of this paper is the introduction of the velocity-dependent coefficient of restitution. The developed model is then analyzed using symbolic dynamics. The results of the symbolic dynamics analysis prove that the developed Galton board model does not behave the way Galton envisaged.
Issue Section:
Technical Brief
References
1.
Hoover
, W. G.
, and Moran
, B.
, 1992
, “Viscous Attractor for the Galton Board
,” Chaos
, 2
(4
), pp. 599
–602
.2.
Hoover
, W. G.
, 1991
, Computational Statistical Mechanics
(Studies in Modern Thermodynamics), Elsevier Science
, Amsterdam, The Netherlands
.3.
Hoover
, W. G.
, and Hoover
, C. G.
, 2012
, Time Reversibility, Computer Simulation, Algorithms, Chaos
, 2nd ed., World Scientific
, Singapore.4.
Mat Daud
, A. A.
, 2014
, “Mathematical Modelling and Symbolic Dynamics Analysis of Three New Galton Board Models
,” Commun. Nonlinear Sci. Numer. Simul.
, 19
(10
), pp. 3476
–3491
.5.
Barnes
, G.
, 1958
, “Study of Collision—Part I: A Survey of the Periodical Literature
,” Am. J. Phys.
, 26
(5
), pp. 5
–8
.6.
Barnes
, G.
, 1958
, “Study of Collisions—Part II: Survey of the Textbooks
,” Am. J. Phys.
, 26
(1
), pp. 9
–12
.7.
Kozlov
, V. V.
, and Mitrofanova
, M. Y.
, 2003
, “Galton Board
,” Regular Chaotic Dyn.
, 8
(4
), pp. 431
–439
.https://arxiv.org/pdf/nlin/0503024.pdf8.
Lue
, A.
, and Brenner
, H.
, 1993
, “Phase Flow and Statistical Structure of Galton-Board Systems
,” Phys. Rev. E
, 47
(5
), pp. 3128
–3144
.9.
Bruno
, L.
, Calvo
, A.
, and Ippolito
, I.
, 2003
, “Dispersive Flow of Disks Through a Two-Dimensional Galton Board
,” Eur. Phys. J. E
, 11
(2
), pp. 131
–140
.https://www.ncbi.nlm.nih.gov/pubmed/1501105310.
Judd
, K.
, 2007
, “Galton's Quincunx: Random Walk or Chaos?
,” Int. J. Bifurcation Chaos
, 17
(12
), pp. 4463
–4467
.11.
Rosato
, A. D.
, Blackmore
, D.
, Buckley
, L.
, Oshman
, C.
, and Johnson
, M.
, 2004
, “Experimental, Simulation and Nonlinear Dynamics Analysis of Galton's Board
,” Int. J. Nonlinear Sci. Numer. Simul.
, 5
(4
), pp. 289
–312
.12.
Goldsmith
, W.
, 1960
, Impact the Theory and Physical Behavior of Colliding Solids
, Edward Arnold Publishers, Ltd.
, London.13.
Hodgkinson
, E.
, 1834
, “On the Collision of Imperfectly Elastic Bodies
,” Br. Assoc. Rep.
, 4
, pp. 534
–543
.14.
Vincent
, J. H.
, 1900
, “Experiments on Impact
,” Proc. Cambridge Philos. Soc.
, 10
, pp. 332
–357
.15.
Judd
, K.
, 2003
, “Chaotic-Time-Series Reconstruction by the Bayesian Paradigm: Right Results by Wrong Methods
,” Phys. Rev. E
, 67
(2
), p. 026212
.16.
Judd
, K.
, Reynolds
, C.
, and Rosmond
, T.
, 2004
, “Towards Shadowing in Operational Weather Prediction
,” Naval Research Laboratory, Monterey, CA, Technical Report No. NRL/MR/7530-04-18
.http://www.lse.ac.uk/CATS/Talks%20and%20Presentations/EGUAbstracts/TowardsShadowingInOperationalWeatherModels.pdf17.
Judd
, K.
, and Smith
, L. A.
, 2001
, “Indistinguishable States I: Perfect Model Scenario
,” Physica D
, 151
(2–4
), pp. 125
–141
.18.
Judd
, K.
, and Smith
, L. A.
, 2004
, “Indistinguishable States II: The Imperfect Model Scenario
,” Physica D
, 196
(3–4
), pp. 224
–242
.19.
Judd
, K.
, and Stemler
, T.
, 2009
, “Failures of Sequential Bayesian Filters and the Success of Shadowing Filters in Tracking Nonlinear Deterministic and Stochastic Systems
,” Phys. Rev. E
, 79
(6
), p. 066206
.Copyright © 2017 by ASME
You do not currently have access to this content.