Shepherding Random Numbers

I enjoyed the link thanks. The patterns concerned there develop out of unstable states (random or patterned) by ~preferences of the interconnecting medium. I notice two dimensional mediums were only examined and 3d may provide scope for more complex phenomenon. The apparent patterning which Hoff mentions is entirely coincidental and uncoordinated by any medium/connection between random variates. Collections of random variates are able to present seemingly unlikely relationships or artefacts which can amount to meaningful or stable forms in other contexts - as monkeys at a keyboard chancing on a few pearls of wisdom.

I have spent a while scrutinising the appearance of patterns in test plots for my random number library: http://strainer.github.io/Fdrandom.js/ and learned that they can be expected to occur to some degree. There surely must be some theoretical and philosophical investigations of chaos capacity to chance on order.

There are a lot of very cool math properties hidden in each of these examples so I want to start a discussion on modeling some of them:

For the "memory" example, each particle is moving Uniform(-k, k) at each timestep (where k is some just some fixed distance). So the the distribution of the position of a particle at timestep t is the sum of t identically independent distributions, which will converge to a normal distribution, specifically with Uniform(a, b) having variance (b - a)^2 / 12, you converge to:

N(0, t * k^2 / 3)

So it turns out these particles end up behaving kind of like brownian motion! (Note I didn't take into account hitting walls. I am not sure how to model that?)

For the "velocity" example, empirically you can see that if you just let it run forever the points will just end up bunching at the walls. This makes sense since if there's a positive velocity in either direction it's hard to flip your velocity back into the other direction to get off the wall. By symmetry you expect half to be bunched on one wall and half on the other.

A fun question to ask then is if the velocities will accumulate so much in one direction such that it will always stay stuck on one wall after you run it a long enough time. I believe the answer is no! It will always switch directions again due to the fact that the probability that a random walk on a line returning to the origin has probability 1 as you run it forever (google recurrent random walk in 1 or 2 dimensions). This means the velocity (which is doing the random walk) will have to cross 0 and flip signs at some point. So although you will mostly find these particles hugging the walls, they will keep switching sides forever!

I think there are a lot of other really cool properties in these fancy visualizations waiting to be discovered.

Ah, that book very much aligns with my current interests. Thank you kindly!