Sunday, January 6, 2008

Renormalization In Conway Life?

Just finnished skimming through: Robert B. Laughlin. "A Different Universe: reinventing physics from the bottom down".


Some comments.

first of all it was tantalizing but fustrating! hints, but no details! my guess is that to get a feel for what he's saying i'd have to get a degree in physics and then read the papers he quotes. oh well.

however it reminded me of this problem: For years I've had vague notions of finding cellular automata somewhere between John Horton Conway's game of life and chemistry. The inventiveness of Conway-life fascinates me. To be able to 'play' with a game like chemistry knowing PRECISELY all the microscopic rules would be fun. But there is one important element of Conway life that is disappointing in this regard. It is very brittle to random fluctuation. Change ONE cell, and you can create a disturbance that eventually destroys a whole 'life form'.

Chemistry is not ordinarily like this. Even in far from equilibrium conditions where random fluctuations can be amplified, they don't ordinarily send the whole ensemble crashing to a halt. So I've wanted to invent a cellular automata that is stable like chemistry or physics.

My intuition was that it has something to do with benzene etc... Atoms interact with each other locally but there are also long range interactions, which i do not understand microscopically. I have a vague notion that they have something to do with quantum mechanics. I also have a vague notion that you can't predict benzene from first principles in quantum mechanics. We certainly didn't predict Buckyballs after 60 years of quantum chemistry.

Then, in his book he mentions something about not being able to get the stable properties of macroscopic matter from microscopic quantum mechanics. That it comes from somewhere else. Something to do with Renormalization? I don't know ANY details about this! I have only a vague imagination about renormalization, so i make a guess: Do I have to introduce renormalization to Conway-life?

I imagine it means this: to calculate the next generation fate of each cell, do the following: calculate its fate according to its usual neighborhood of 8. Then look at each neighborhood of 4 surrounding that cell. Look at the whole Conway universe by 4's. There will be 4 such ways to partition the universe. Go through each one. For each 4-universe each new cell is the average (?) of the 4 original cells of the original universe that it contains. Now calculate the fate of the 4-cell containing your original cell for this new universe just as you do for the original Conway universe. Do it again: now partition the universe into 9-cells, there will be 9 such possible partitions. calculate the fate of the cell for each of those. go out to infinity. Now summation. produce a weighted sum of each of these fates like 1/16 (sum 1 to 4 of each 4-universe fates) + 1/81(sum 1 to 9 of 9 of each 9-universe fates) +... Can it be contrived to converge? This then must be done for each cell for each generation of recalculating Conway life.

Of course it seems impossibly impractical to me. I suppose you could do it in stages. Take a whole bunch of Conway life forms and work them out up to the 4-universe stage. Do it again up to the 9-universe stage. Again for the 25-universe stage... Does this procedure tend to some kind of behavior? Or is it just wildly different each time?

Is this how quantum mechanics is now done? I recall something about calculating all the possible feynman diagrams for each interaction at finer and finer scales each with more and more loops and summing them hoping it converges. is this where the infinities come in? how were they bannished?

Anyway for the Conway life scheme, if it COULD be done, what would happen? Would it so constrain the system that NOTHING interesting would happen?

This reminds me of another variation on Conway life I wanted to try, continuous Conway life: In R-2 you have a set of points. Say a disk of radius one centered at the origin. We will generalize conway life rules. The neighborhood of each point is a disk of radius... hmm... how did I choose the radius? Don't remember now. I'm not sure how to choose the neighborhood radius relative to the original shape i want to follow. That's bothersome. So hard to create a universe! Let's say r=1/2. So for each point look at the fraction f, of the disk r=1/2 that surrounds it that intersects with the original set. The continuous Conway life rules would be:

if f<2/8>3/8 (or 4/9) the original point 'dies'
if 2/8<=f<=3/8 (or /9etc..) the point stays
if for an empty point f=exactly 3/8? asking too much? 2.5/8<=f<=3.5/8? then 'turn that point on'

Well, you gotta decide the way you want to generalize the rules. Anyway It isn't hard to calculate the fate after the first generation or two for simple shapes like disk, square... but after a few generations the boundaries become complex and the formulas for them become complex and they are required to calculate the new boundaries. for instance a disk turns into an annulus which overlaps the edge of the disk. the annulus would turn into two concentric annuli? Square was harder.

Since it's difficult to calculate I thought up a scheme. Approach the system digitally. Break up the set into finer and finer digital approximations. At each stage see what the fate is under digital Conway life with larger and larger neighborhoods, with rules 'renormalized' (is that the appropriate use of the term?) appropriately. So the Conway life rules become more and more 'continous'. So to approach the fate of a square under continuous conway life, carry out the following procedure:

Approximate the square as 6x6 cells in original conway life and calculate its fate as many generations as needed, till it settles (or doesn't). Approximate the square as a square of 8x8 cells and calculate the fate (as many generations as needed...) under appropriately renormalized neighborhood-of-16 Conway life. (should the neighborhood be a 4x4 square or something closer to approximating a disk? So many choices!). Approximate the square as 10x10 cells and calculate its fate in neighborhood-of-25 Conway life. Continue. Will the behavior of each of these discrete Conway life universes CONVERGE to the behavior of the continuous one?

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