From: Harold V. McIntosh (MCINTOSH@unamvm1.dgsca.unam.mx)
Date: Sun May 16 1993 - 02:23:27 UTC
According to Howard Andrew GUTOWITZ <gutowitz(at)AMOCO.SACLAY.CEA.FR>: > > or maybe mean field theory has to be improved :-) > Quite so, but how shall we go about it? In one dimension, there are probability measures, and Bayes Theorem approximations to them. That was an interesting adventure. No doubt n-dimensions has them too, but without Kolmogorff's conditions to help derive some equations. Seems to me that the subject hasn't been exhausted; whoever wants to look into this should consult Gutowitz' articles in Physica D (and elsewhere) from years past. - There may be another direction, which is to look at those mean field curves for advanced generations. Up to a point (tens, but hardly thousands, of generations) the mean field curve can be deduced by monte carlo calculations - taking statistics on sample evolutions (They could just be calculated, since they are deterministic, but the amount of computation required makes that approach impractical). When different generations are compared, it is sometimes found that the curves group themselves into families, although the relation between families is not necessary iteration. - Iteration would be expected if probabilities were truly independent, but Nature being obligingly Perverse, the more interesting the rule, the more it deviates from iteration. Yet, although there are unlimited ways to pack functions into the unit interval, maybe the same reasoning that convinces us that iterates have limits could be turned to justifying the expectation that evolutes also have limits which could be discerned by numerical experiments. - All in all, there is as much to be learned from generating statistics for the evolution of automata as there is in cataloging their stable and periodic artifacts. - This afternoon's additions to the 3D-Life program include displaying (and printing, so as to include them in Messages) cross sections of a field as it evolves. In some ways this is more satisfactory than trying to create a credible three dimensional display, and in any event is surely preferable to proceding blindly. - Shown below are planar cross sections of three successive generations of the evolution of a 20x20x20 cube using the Chat\'e-Mannevill rule 16/19/0/0. The not evolution of a 20x20x20 cube using the Chat\'e-Mannevill rule 16/19/0/0. The notorious period 3 collective behavior is clearly in evidence; as further generations evolve, the pattern is repeated, but not exactly. Thus, the density has period 3 but the configuration does not. - - ...**..........*....- - ....................- - .***********........ - *............*******- - .............*...**.- - **...*******........ - *....*..*...********- - ..............*****.- - **...*******........ - .....*.....*........- - ....................- - .....****........... - ..**......**........- - ..........*..*...**.- - ....*****........... - ..**......**........- - ..*........*......*.- - ....**.............. - ..**.....***...**...- - ..*........*......*.- - ....**.........**... - **...*....**........- - .*.........*........- - ....**.............. - **..*.....**.......*- - *.........*..*...*..- - ...**............... - *..*.......*.......*- - ....................- - ..**................ - ..*.....*...*******.- - .....**.......***...- - .**.....*........... - .*...........*****..- - ....***...*..*...*..- - **..*..............* - .*...........*****..- - ....***...*..*...*..- - **..*..............* - ...........*.......*- - ..********..........- - ..*......*.*.......* - ....................- - .*********..........- - .*.......*.......... - ....................- - ************.......*- - *...***...*......... - ..............*.*...- - *************.....**- - .*.......*....***..* - ....................- - ..........**.......*- - .................... - ....................- - ....................- - ...**............... - ..*........*........- - ....................- - .****...****........ - In point of fact, there is much more to a Ch-M automaton than this: it has been observed that the phase can slowly drift, that the amplitude can vary over long periods of times, that the injection of slight impurities can either stabilize or unsettle the phase, and so on. - Harold V. McIntosh |Depto. de Aplicaci'on de Microcomputadoras MCINTOSH@UNAMVM1.BITNET |Instituto de Ciencias/UAP mcintosh@unamvm1.dgsca.unam.mx |Apdo. Postal 461 (+52+22)43-6330 |72000 Puebla, Pue., MEXICO
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