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Козлов, Валерий Васильевич; Mitrofanova, M.Yu.

doi:10.1070/rd2003v008n04abeh000255

Cited by 13 publications

(6 citation statements)

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“…Galton board has been extensively studied in various asymptotic regimes, see [2,3,4] and references therein. In this letter we discuss an idealized infinite Galton board; our ball is a point particle of unit mass moving according to equationsq = v andv = g = const and colliding elastically with immobile convex obstacles of infinite mass (scatterers), which are positioned periodically on the board.…”

Section: Introductionmentioning

confidence: 99%

Diffusive Motion and Recurrence on an Idealized Galton Board

Chernov

Dolgopyat

2007

Phys. Rev. Lett.

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We study a mechanical model known as a Galton board--a particle rolling on a tilted plane under gravitation and bouncing off a periodic array of rigid pegs. Incidentally, this model is identical to a periodic Lorentz gas where an electron is driven by a uniform electric field. Previous heuristic and experimental studies have suggested that the particle's speed v(t) should grow as t(1/3) and its coordinate x(t) as t(2/3). We find exact limit distributions for the rescaled velocity t(-1/3)v(t) and position t(-2/3)x(t). In addition, we determine that the particle's motion is recurrent; i.e., the particle comes back to the top of the board with a probability of one.

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Section: Introductionmentioning

confidence: 99%

Diffusive Motion and Recurrence on an Idealized Galton Board

Chernov

Dolgopyat

2007

Phys. Rev. Lett.

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“…But on the idealized board, rather paradoxically, the ball will almost surely bounce all the way back up! (Naturally, this spectacular phenomenon is never observed in practice because physical collisions are non-elastic and the ball is subject to friction [4,23,27]. )…”

Section: Velocity T −1/3 V(t) and Rescaled Position T −2/3 X(t)mentioning

confidence: 99%

“…Such a non-stationary behavior makes mathematical studies very difficult and may explain the lack of rigorous results (until now), despite persistent interest in the physics community [4,23,24,27,28,30,31]. To make things more tractable, one can remove the excess energy in various ways (deterministically or stochastically).…”

Section: Introductionmentioning

confidence: 99%

The Galton board: Limit theorems and recurrence

Chernov

Dolgopyat

2008

J. Amer. Math. Soc.

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“…In a real Galton board balls bounce off pegs as they fall. For ease of computation, we consider an equivalent dynamical model of a Galton board where point particles bounce off circular disks fixed on a plane, as in previous works [17][18][19][20][21][22]. Particles fall under the influence of a constant gravitational field and elastically bounce off circular disks arranged in a hexagonal array, as shown in Fig.…”

Section: Galton Board Modelmentioning

confidence: 99%

“…In this paper, we discuss the randomness in the ideal Galton board, which has not yet been studied from the point of view of random output generation, although some features of the Galton board have been studied from the view point of a model of physical materials [17][18][19][20][21][22]. We deal with an ideal dynamical model which is inspired by the Galton board.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2012

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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.

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Abstract: In this paper, we present results of simulations of a model of the Galton board for various degrees of elasticity of the ball-to-nail collision.

Cited by 13 publications

References 0 publications

Diffusive Motion and Recurrence on an Idealized Galton Board

Diffusive Motion and Recurrence on an Idealized Galton Board

The Galton board: Limit theorems and recurrence

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

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