Robust-first Computing Wiki

What would be the short and long term behavior if you started a simulation with a filled circle of the Box atoms (occupying perhaps 4% of the grid sites) in the center of an otherwise empty grid? What would you expect see to over 1K, 10K, 100K AEPS?

Answer: I start with two boxes side by side. There are four possible formations. In some cases, their interactive swapping will let them jump out of the box. I guess in the short term, like 1K AEPS, they can stay in the circle. But after 10K AEPS, they may spread out. Maybe still in small clusters or pairs. After a very long time, 100K AEPS, they can find some place and do their box move on their own.

Left 30kAEPS Right:100kAEPS

Comments There are a few more key insights. For example, under what conditions could two interacting Boxes split up and both start doing clean non-interacting boxes on their own? It is as another kind of neutral dynamics, like this one.

Under what condition will some Boxes dance alone and some other dance in pair? We guess if we put a solid circle of Boxes in the middle of the canvas, after a very long time, maybe months, years or decades, all the Boxes dance alone except there is a last pair of them.

Is this pair of Boxes stay together forever? No, When the pair randomly walk around and pass by a single Box, one of the pair maybe dropped out and the single one will be picked up. Isn't it interesting? A Box is trapped in a 2 by 2 area, however when two of them are together, they are not bound to a fixed position any more and they begin to walk.

A Box repeats the west, north, east, east pattern. After FOUR events, it returns to the origin location. When we are talking about move west, the Atom actually swap with its west neighbor Atom. This is same for the other three directions.

We can break down the pattern into x-direction moves and y-direction moves. The y direction-moves are actually irrelevant to x-direction moves. A Box returns to its origin x position after TWO 'Big events'–each big event is worth TWO normal events. So it is possible to focus just on one direction now and combine the transition of two directions later. Here we choose x-direction for analysis. Assume we can use a x-Half-Box Element to represent only the x-direction moves of Box. Actually we already have that Flip Element which moves one step back and forth along the x-axis.

Let's first make some short hand to record the move that Flip Atoms will take. For Atom a. We denote a moving west as a and a moving east as A. Similarly for Atom b, b means b moves west and B means b moves east. If we observe the moves of a standalone Atom a, the moves we record is aAaAaA….. When a and b are side by side, we can observe the moves of this pair. The record will look like aAaAbBbBabABbBbBaAbaAB….. due to the random and asynchronous feature of the MFM simulator.

Fortunately, we can simplify this string into one of the six basic patterns: aAbB, abAB, abBA, baAB, baBA, bBaA. Then we can see how this pair behaves under these six patterns.

Four out of the six patterns actually preserve the origin location of the pair of a and b. They are aAbB, abBA, baAB, bBaA. After four events, the pair returns to its start position. The abAB pattern will take the pair ONE step to the west and the baBA pattern will take the pair ONE step to the east. Now we see why a pair is no longer bound to the origin place and begins to move. This is a 1-D random walk with the probability of 1/6 to go west, chance of 1/6 to go east and chance of 2/3 to stay. The y-direction moves are just transposes of the x-direction moves. The combination of x-direction and y-direction moves will let the pair walk in the 2-D canvas and the pair will never stop.

The diagonal formations of a and b in 2-D space has no effect in moving the pair. Only the horizontal or vertical position of a and b can move the pair.

Will we have 3-D and other high dimensional MFM in the future? The spatial concept is so essential for the MFM, and if we have high dimensional MFM, this HD MFM seems to me maybe like the structure of human brain.

To collect data about how Boxes move we can create a new Pbox Element. This Pbox means pair of Boxes. A single Atom of Pbox behaves very similar to a Box. This new element is just for statistic usage. We observed that a single Box is trapped in a box range. It only swaps with Empty sites. On the other hand, if a pair is formed, one Box in this pair swaps with not only Empty but also another Box. According to this, we can use a counter Int(8) countAlone to record this difference. If a Box swaps with Empty, we reduce the value of countAlone. If a Box swaps with another Box, we increase this value.

We let a Box transform to a Pbox if its countAlone value is positive. We also let a Pbox transform to a Box if this value is less than -15. These threshold values comes from some experiments and can be tuned in the future.

At the start of out experiment, we put 1% Boxes. They began to interact immediately and most of them transformed to Pboxes in a short time. After some time, some Pboxes were dropped out of a pair and they became Boxes again. The transformation kept going. At some time, there was only one pair left.