Tom Mazanec wrote:dvgrn:
What are the formulae for calculating those "magic constants"?
There's no formula, for sure. Those numbers are the result of a big statistics-collection project: for gradually increasing values of N, run random N%-fill soups for one tick each, and take the average of the T=1 densities for a lot of these random soups.
When you do that, you find that
-- for small percentages and large percentages, the T=1 average density is going to be smaller than the T=0 density.
-- for percentages in some middle range, the T=1 average density is bigger than the T=0 density.
(if I'm understanding Achim's notes correctly -- I haven't tried this myself, so I could be misinterpreting somehow.)
Achim's observation was that the upper end of that middle range seemed to produce the largest final ash density. I don't know if that's actually true or not. I'm not sure why it would necessarily be
precisely true -- maybe it's just approximately true. At least, it would be hard to show statistically that the point of maximum ash density is precisely the same as the point of maximum T=1 density, to the six decimal places that Achim quoted!
The numbers might come out a little different for different initial soup sizes and/or different universe geometries.
E.g., the number might shift a bit depending on if the universe is a bounded torus, or if it has empty space around the edges. Presumably you wouldn't use the empty space in final density calculations, and maybe to be safe you'd just do the density measurement on a smaller central area.
I should also be clear that I don't know any more than I've said already, and maybe I actually know less than I've said! Quite possibly someone has done a statistical survey like Achim's more recently than 2003 -- I haven't necessarily been paying attention.