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.

References

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.pdf
8.
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/15011053
10.
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.pdf
17.
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
.
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