Collision statistics for a Sinai billiard with Poisson disc obstacles
Channel: Nils Berglund
Date: 01/10/2021

This fourth video with collision statistics in a Sinai billiard features circular obstacles generated by a Poisson disc algorithm. As in a Poisson point process, the position of each obstacle is chosen at random,  but with the additional constraint that there is a minimal distance between discs. The algorithm used to generate the discs is described here: https://bl.ocks.org/mbostock/dbb02448b0f93e4c82c3
The bottom half of the simulation shows a particle moving at constant speed, and bouncing off 366 circular pegs, as well as the rectangular box containing the pegs. To help tracking the motion of the particle, the pegs change color when they are hit: they become red when they are hit for the first time, and then their hue changes gradually, becoming red again after 32 collisions. The numbers at the bottom show the total number of collisions (pegs and outer rectangle), the maximal number of times a peg is hit, and the number of pegs hit as least once.
The two histograms on top show two different statistics. The left one shows how many pegs have been hit once, twice, three times, and so on. The colors in the histogram correspond to the colors of the hit pegs. The right plot shows the distribution of distances traveled between collisions, also called free paths. The vertical scale changes in the course of time in such a way that the maximum of the histogram remains constant. 
The mean free path is displayed at the top right. The scale is such that the rectangle has width 2, and length 7.6. The circles have radius 0.05.
The main difference with the simulation https://youtu.be/xTHOOWbPYQA of a Poisson point distribution is that the free path distribution increases for small distances, before starting to decrease at an exponential rate. This is due to the minimal distance requirement of the Poisson disc distribution, the smallest distances being due to collisions with the boundary. Periodic boundary conditions would eliminate the smallest distances altogether. 

Render time! 9 minutes

Music: Brandenburg Concerto No4-1 BWV1049, by Johann Sebastian Bach
Brandenburg Concerto No4-1 BWV1049 - Classical Whimsical by Kevin MacLeod is licensed under a Creative Commons Attribution 4.0 licence. https://creativecommons.org/licenses/by/4.0/
Source: http://incompetech.com/music/royalty-free/index.html?isrc=USUAN1100303
Artist: http://incompetech.com/

Current version of the C code used to make these animations: https://github.com/nilsberglund-orleans/YouTube-simulations
https://www.idpoisson.fr/berglund/software.html

Some outreach articles on mathematics:
https://images.math.cnrs.fr/_Berglund-Nils-1343_.html
(in French, some with a Spanish translation)