# Ping-pong progress through a quincunx

A quincunx or Galton board consists of staggered pegs from which ping-pong balls bounce and eventually display a binomial / normal distribution in catch-bins. I am wondering if the downward progress of each ball has been studied.

Assume just one ball falls, initially with some nonzero horizontal velocity and zero vertical velocity. Gravity increases its vertical velocity over time; assume no friction. Perhaps the ball radius and the peg radii are identical. The peg is immobile. Ball-peg collision is elastic. The ball's velocity vector is reflected from the common ball-peg tangent at a collision (like a lightray from a mirror).

(First bounce in animation below.)
Over time, the ball makes (sometimes slow) downward progress. E.g., in the example below, traversing the $10$ rows of pegs would take about $4.5$ time units in free-fall in the absence of collisions, but the $25$ collisions retard its progress considerably: it takes about $19$ time units to escape below.

(Repeats once; reload to replay animation. Browser dependent.)

Q1. Is there a simple argument to show that the ball actually does make downward progress, in that its expected vertical position at time $t_2 > t_1$ is below its expected position at $t_1$? Gravity is pulling the free paths downward, but collisions can shoot the ball upward.

Q2. Any insight into quantitative bounds, say, on the expected number of collisions per unit time, as a function of the ball/peg radii? Have analogous lattice reflections been studied?

## 1 Answer

A comprehensive study of the Galton board has been published by Arthur Lue and Howard Brenner, Phase flow and statistical structure of Galton-board systems (1993). Without any friction the influence of gravity vanishes in the long-time limit, because the kinetic energy overwhelms it. In that limit the Galton board reduces to a Lorentz gas, with a chaotic dynamics that has been studied extensively. With (even a little) friction the dynamics is richer, with attractors and repellers in phase space.

The dynamics in the Galton board without friction has been analyzed in The Galton Board: Limit theorems and recursion, by Chernov and Dolgopyat (2009). (See also this 2007 publication.) The displacement of the particle $x(t)$ grows as $t^{2/3}$, sublinearly because of the backscattering by the pegs. Conservation of energy then implies that the velocity grows as $t^{1/3}$.

As an answer to Q1, the motion in the Galton board without friction is recurrent: With probability one $$\lim_{t\rightarrow\infty}{\rm inf}\, x(t)\leq x(0).$$ Because of the diminishing effect of gravity in the long-time limit, the particle slows down and effectively returns to a vicinity of its initial state infinitely many times. As Chernov & Dolgopyat write: "We all know that a ball thrown into a real Galton board always rolls down and ends up on the floor. But on the idealized board, rather paradoxically, the ball will almost surely bounce all the way back up!"

• "the kinetic energy overwhelms": Ah, that makes sense! – Joseph O'Rourke Nov 26 '16 at 20:30
• "the motion in the Galton board without friction is recurrent": Wow! – Joseph O'Rourke Nov 26 '16 at 20:49