Galton Board

Dropping a set of beads on a board with evenly distributed pegs results in a binomial distribution. Is it possible to generate other kinds of distributions by varying some parameters (Pegs size, pegs distribution, bead format, etc.)? Is it possible to achieve a distribution that does not obey the central limit theorem in an i.i.d. scenario? What happens to the distribution when one makes the board vibrate?

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Objectives:.

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• Describe the emergence of the binomial distribution in a typical Galton Board.

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• Characterize the limit distribution of the sum of independent random variables.

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• Create a simulation that allows us to vary the parameters of interest in the physical system in order to obtain deviations from the normal law.

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• Analyze the effects of vibration of the board.

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Research

Numerical Developments

Experimental Findings

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Random walk and Binomial distribution

All graphics and illustrations presented above were generated by the research group IPT Univalle.

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Random walk and Binomial distribution

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Random walk and Binomial distribution

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Experimental Findings

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Random walk and Binomial distribution

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Experimental Findings

Numerical Developments

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Random walk and Binomial distribution

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Experimental Findings

Numerical Developments

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Random walk and Binomial distribution

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Experimental Findings

Numerical Developments

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Random walk and Binomial distribution

Random variable

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Random walk and Binomial distribution

Random variable

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Random walk and Binomial distribution

Random variable

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Random walk and Binomial distribution

Random variable

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Random walk and Binomial distribution

Random variable

Binomial distribution

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How one must rescale the variable X, in order to obtain a limit distribution?

What is the limit distribution?

General characterization of the attraction basin

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General characterization of the attraction basin

For to be a possible limiting distribution for the reduced variable, there must exist, for all , , and , two quantities

and , such that:

[1] Bouchaud, J., & Georges, A. (1990). Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications. Physics Reports, 195, 127-293.

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THEOREM. Stable law (Levy, 1925):

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THEOREM. Stable law (Levy, 1925):

For to be a possible limiting distribution for the reduced variable, there must exist, for all , , and , two quantities

and , such that:

DEFINITION. Characteristic function:

We define the characteristic function of a probability distribution as the Fourier transform of the probability density function.

[1] Bouchaud, J., & Georges, A. (1990). Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications. Physics Reports, 195, 127-293.

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General characterization of the attraction basin

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Experimental Findings

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[1] Bouchaud, J., & Georges, A. (1990). Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications. Physics Reports, 195, 127-293.

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THEOREM. Canonical representation of stable laws

(Levy, Khintchine):

General characterization of the attraction basin

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is stable if and only if its characteristic function reads:

[1] Bouchaud, J., & Georges, A. (1990). Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications. Physics Reports, 195, 127-293.

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THEOREM. Canonical representation of stable laws

(Levy, Khintchine):

General characterization of the attraction basin

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Experimental Findings

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is stable if and only if its characteristic function reads:

Where , , and are real numbers such that

and

,

,

for

for

[1] Bouchaud, J., & Georges, A. (1990). Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications. Physics Reports, 195, 127-293.

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THEOREM. Canonical representation of stable laws

(Levy, Khintchine):

General characterization of the attraction basin

Research

Experimental Findings

Numerical Developments

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THEOREM (Gnedenko)

belongs to the attraction basin , with

If and only if

and

[1] Bouchaud, J., & Georges, A. (1990). Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications. Physics Reports, 195, 127-293.

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General characterization of the attraction basin

Research

Experimental Findings

Numerical Developments

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[1] Bouchaud, J., & Georges, A. (1990). Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications. Physics Reports, 195, 127-293.

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General characterization of the attraction basin

Furthermore

Non-Gaussian large-x behaviour (Heavy Tail)

Gaussian large-x behaviour

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Physical Motivation

[2] Bouchaud, JP., Le Doussal, P. (1985). Numerical study of a D-dimensional periodic Lorentz gas with universal properties. J Stat Phys 41, 225–248.

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Levy walks [1] .

Sinai billiard on a square lattice [2].

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Physical Motivation

[2] Bouchaud, JP., Le Doussal, P. (1985). Numerical study of a D-dimensional periodic Lorentz gas with universal properties. J Stat Phys 41, 225–248.

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Levy walks [1] .

Sinai billiard on a square lattice [2].

Research

Experimental Findings

Numerical Developments

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Physical Motivation

[2] Bouchaud, JP., Le Doussal, P. (1985). Numerical study of a D-dimensional periodic Lorentz gas with universal properties. J Stat Phys 41, 225–248.

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Sinai billiard on a square lattice [2].

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Experimental Findings

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Physical Motivation

Anomalous diffusion regime,

[2] Bouchaud, JP., Le Doussal, P. (1985). Numerical study of a D-dimensional periodic Lorentz gas with universal properties. J Stat Phys 41, 225–248.

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Sinai billiard on a square lattice [2].

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Experimental Findings

Numerical Developments

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Galton Board Simulation

• Interactions among the beads are always neglected.

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• Elastic collisions are implemented.

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• The beads are under the action of the gravitational force.

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• Particles can be released with a given initial velocity

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• The bin at which it is collected is defined by the last horizontal position of the bead.

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Galton Board Simulation

Fixed parameters:

• number of Beads → 100,000
• number of Rows → 30
• relative sizes of pegs and beads → 1 and 0.5
• Time interval → 0.1s with 5 substeps.
• Damping coefficient → 0.2
• Velocity → 30 as standard

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Experimental Findings

Numerical Developments

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Galton Board Simulation

Fixed parameters:

• number of Beads → 100,000
• number of Rows → 30
• relative sizes of pegs and beads → 1 and 0.5
• Time interval → 0.1s with 5 substeps.
• Damping coefficient → 0.2
• Velocity → 30 as standard

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Experimental Findings

Numerical Developments

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Deviations from the Normal Law

Angle of incidence of the beads.

Relevant parameters:

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Deviations from the Normal Law

Pegs distribution.

Relevant parameters:

Hexagonal lattice.

Rectangular lattice.

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Deviations from the Normal Law

Results

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Effects of Velocity of the particles

With taking random values between 0 and 180

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Effects of Velocity of the particles

With taking random values between 0 and 180.

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Effects of board vibration

Angular Frequency

Allows to simulate sinusoidal behavior on the pegs

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Effects of board vibration

It is found that for values greater than 16 Rad/s the vibration prevents the uninterrupted passage of particles.

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Visual Representation of Concepts.

Interactive Learning.

Social Commitment

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Effects of board vibration

Results

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Visual Representation of Concepts.

Interactive Learning.

Social Commitment

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Visual Representation of Concepts.

Interactive Learning.

Social Commitment

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Accessible Learning Resources.

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Github: https://github.com/niaggar/GaltonBoard

Social Commitment

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• We explained why the Binomial distribution is the limiting function for the Galton Board with evenly distributed pegs, by treating the system as a random walk.

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• We characterized the limit distribution of the sum of independent random variables as being stable functions completely defined by the parameters and .

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• We created a simulation that allows us to vary the parameters of interest, in particular; the angle of incidence and the distribution of pegs.

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Summary

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• It is possible to produce other kinds of distributions by varying the physical parameters of the Board. This asymptotic distributions belong to the family of Stable functions.

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• It is possible to obtain a limit distribution with parameter , deviating from the normal law and therefore violating the Central Limit Theorem in its conventional form (i.i.d scenario).

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• Vibration of the board at a fixed frequency induced a change in the angles of incidence of the beads, equivalent to the random variation performed in the simulation. The vibration thus increased the deviation from the normal law.

Conclusions and Perspectives

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Bouchaud, J., & Georges, A. (1990). Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications. Physics Reports, 195, 127-293.

Bouchaud, JP., Le Doussal, P. (1985). Numerical study of a D-dimensional periodic Lorentz gas with universal properties. J Stat Phys 41, 225–248.

Zolotarev, V. M. (1986). One-dimensional stable distributions (Vol. 65). American Mathematical Society, Providence, RI. ISBN: 0-8218-4519-5

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Arfken, G. B. 1., & Weber, H. (2005). Mathematical methods for physicists. 6th ed. / Boston, Elsevier.

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Amir Bar. (2013). Brownian motion and the Central Limit Theorem. Weizmann Institute of Science.

Bibliography

Thank you!

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Galton Board Colombian Team

• Nicolás Aguilera García
• Andrés Felipe Valencia
• William Salazar
• Prof. Diego Luis Gonzales
• Kevin Giraldo

❤️🫂

Acknowledgments