In recent decades, self-organizing phenomena have attracted the interest of workers in many fields of physics (pattern formation, non-linear dynamics, statistical mechanics, non-equilibrium thermodynamics, plasma physics, etc.), as well as mathematicians, biologists, engineers, complex systems theorists, and even social scientists. Surprisingly little work, however, has been done on quantifying self-organization, or even providing rigorous definitions and criteria. The aim of this research is to formulate a quantitative, theoretically well-justified indicator or measure of self-organization, which could be feasibly applied to actual experimental data or theoretical models; or, failing that, to show that apparently promising paths towards such an indicator are really dead ends. Cellular automata will be used as a ``test-bed''. In addition to their intrinsic interest, CAs are at once simple enough for their ``mechanisms'' to be understood in detail, readily computable, and sufficiently general that they can capture the essentials of many physical situations, including important instances of self-organization.Well. If that hasn't put you off completely (oi, those scare-quotes!), I'll try to unpack it.
At first sight self-organization is a really ancient idea --- after all, Democritus and Lucretius didn't admit any Organizer. But they also didn't admit any real tendency towards organization. They were saying, quite correctly, that an infinity of monkeys at typewriters, typing for eternity, will produce every possible book. The modern idea seems to originate with Descartes, and is that the normal operation of natural laws will tend to organize the universe, at better-than-chance rates; no Great Architects need apply. (Of course Descartes' didn't call it ``self-organization''; in fact that word isn't even in the OED, though ``self-assembly'' is.) Nothing really solid came of this until the last century, with Darwin and Wallace (Diderot was brilliant, but D'Almbert's Dream is pure hand-waving); then statistical mechanics; then D'Arcy Thompson; Needham &c.; and any number of other developments which in retrospect were precursors. The real boom, and I believe the word ``self-organization'' itself, came after the second world war --- the founders of cybernetics were very interested --- and most especially since the '70s.
Without exception, everyone has relied on a ``I know it when I see it'' test for self-organization, even those (such as Ilya Prigogine, NL and philosopher-manqué) who think it is the Key to the Mysteries of the Universe. I have discovered one attempt at a rigorous definition; said definition entails that self-organization is a contradiction in terms, and the author --- with admirable courage for someone publishing in a proceedings volume entitled Principles of Self-Organizing Systems --- recommends that the name be abandoned. (That paper is also remarkable for being the only one in that volume to discuss general principles.) [Addendum, 27 February 1996: Klimonotvich's book on The Structure of Turbulence does attempt to frame a rather general measure of self-organization. I'm still reading it, so I'm not quite confident of the details. Watch this space.]
This sort of reliance on intuition is unsatisfactory for two reasons. First, we're mathematical scientists, dammit, and we want numbers; at the very least we want rigorous tests. Second, intuitions about old subjects are more reliable than intuitions about new ones, for obvious evolutionary reasons. We recognize colors almost infallibly, emotions pretty well (but not well enough to defeat actors, con-men or poker faces), and are even decent when it comes to art, smut and visual symmetry. To expect us to have strong, reliable intuitions about an idea which wasn't even explicitly formulated sixty years ago, is absurd. (There have been big arguments about whether turbulent fluids, or ecological succession, are self-organizing.) Obviously self-organization does happen, and is important in pattern formation and biology (termite mounds are a really classic instance) and even, it appears, economics. (I shall write more about uses of self-organization anon.)
Feynman has this beautiful passage about how you get the same sort of mathematics in the most different parts of physics, and isn't it wonderful and mysterious for nature to do this? That's fucking bullshit; the motherfucking physicists just think of the same goddam thing over and over again.(You understand why I don't give his name.) In this spirit, in my talk on this subject last spring, I mentioned focused on some of the ideas which occur to physicists over and over again --- entropy, correlations, and scaling. (In fact I bored my audience to tears talking about scaling and correlation in phase transitions.) Ultimately I'll write something elegant and lucid about them here, and explain why decreasing entropy does not (always) violate the second law of thermodynamics, but for the moment see the notes to my talk.
Since giving the talk, I've read Wolfram's interesting paper on ``Statistical Mechanics of Cellular Automata'' (in his Cellular Automata and Complexity); he uses entropy and correlation functions as evidence that some CAs self-organize, but others do not. He doesn't see any need --- at least not there --- to justify these measures, and complains they're crude. I suspect their crudity is a veritable feature; see below.
Ashby's ideas about dynamical systems selecting against unstable states and for stable ones are (I think) most naturally formalized by saying that the entropy of the distribution of states decreases as that distribution accumulates on the attractors.
One thing I need to work out is whether an increase in a length-scale or stronger correlations imply a lowered entropy.
A problem with pattern-recognition is that there are no recognizers-of-pattern-in-general, merely recognizers-of-some-more-or-less-general-pattern, so they're really only useful if we already know the kinds of patterns are likely to emerge. The most vague and general patterns reduce to things like ``not every possibility is equally likely'' or ``there are connections between one part of the pattern and another'', which is to say, to things like the entropy and correlations. This is why I said that I suspect the crudeness of entropies and length scales as pattern-detectors is an advantage: they can't tell you much about what the pattern is, but they do tell you it's there.
The biologists seem happy with this story, so let's assume it works for living things. We don't know how to recognize a really new level of organization, and it's not at all clear whether the concept can be usefully extended beyond biology. We may agree that a snowflake is at a higher level than its constituent water molecules (one? two? more?) but what about a blizzard, or a snow-drift, or a glacier?
See further the notebook on this.