Wolfram Science Group
Phoenix, AZ USA
Registered: Aug 2003
First of all let me say welcome, and thanks for an outstanding post. This is good stuff. I have several somewhat disconnected comments to offer.
You say -
all state spaces which are not members of such loops must eventually evolve to states which are members of such loops
For a finite system run for a long enough time this is true. All eventually must repeat, though the repetition period can get abritrarily long rather quickly. It seems to me the more fundamental fact is that some state space configurations which are not members of such loops can evolve to states which are members of such loops, and can moreover do so quickly. It is not necessary that this be exhaustive, just that it can happen. You will hit some stable cycles. The NKS book talks about this in the section on class 2 behavior and systems of limited size.
One thing I've noticed repeatedly in NKS experiments is how number theoretic effects can crop up, related to things like system size. Not just large scale effects, 1000 wide being different from 10 wide because the possibility space is enourmously larger. But tiny ones, 359 wide being different from 360 wide, because 360 has many divisors and therefore many possible sub-cycles, while 359 is prime. You are more likely to hit short repetition periods in a 360 wide system - or any other local maximum of divisor sigma - than in the 359 system - or any other prime number width.
More basically, you are right that local reversibility can arise in pockets of class 1 or 2 behavior, that it can be an apparent property of subsystems. What I find fascinating is the possibility we may have sifted for just such subsystems with previous formal methods, based on continuous mathematics and geometrical thinking. Symmetry is a particularly useful analytic tool in those formal methods. Are essentially all our basic physical laws - except of course the Second - symmetric because only such laws somehow arise, or merely because such symmetries are particularly noticable to traditional mathematical methods?
It happens I've been reading the pragmatist CS Pierce. In a piece entitled "Causation and Force" (an interesting piece, incidentally, particularly for its day) he noticed the following about common sense ideas on causality compared to the physics of his day (classical mechanics - this was around 1900).
The grand principle of causation...involves three propositions to which I beg (your) particular attention. The first is, that the state of things at any one instant of time is completely and exactly determined by the state of things at one other instant. The second is that the cause, or determining state of things, precedes the effect or determined state of things in time. The third is that no fact determines a fact preceding it in time in the same sense in which it determines a fact following it in time. These propositions are generally held to be self-evident truths; but it is further urged that whether they be so or not, they are indubitably proved by modern science. In truth however all three of them are in flat contradiction to the principles of mechanics.
He goes on to explain what he means, and surprisingly enough he is right. Even with classical mechanics. Mechanics was reversible. It moreover involves second derivatives, and as such more than one preceding instant in time (that is, it requires not only positions but also velocities to be specified, and velocity involves a time spannedness). The mathematical relations involved do not favor the time relations our intuitions about casuality lay down. In field-like theories, causes do not precede effects but are instead instantaneous relations. Locality is made to do all the work, not precedence.
Pierce noticed that our common sense ideas of causality do correspond well to our psychological experience, and also to our experience of large scale aggregates, which of course involve the action of the Second Law. But do not correspond to reversible classical mechanics.
One can bend over backwards to find apparent irreversibility in underlying rules that are reversible, provided their behavior is complicated enough. The NKS book shows some ways to do that in the physics chapter. But you are correct that it is in many respects much easier to find reversibility as an emergent phenomenon of limited sub-systems or cases, within rules that are in principle irreversible.
And if that were really the case, it would mean (pace Pierce above) that much of our intuition about casuality might be saved. The simple programs NKS considers are typically set up to obey the three criteria of causality he lays down.
If it were so, then we would suspect the existence of laws relating to unsymmetric behaviors that previous mathematical formalisms have failed to spot. The second law is the leading known example, and it may be useful to consider why it is known. It deals with a different form of symmetry - a statistical one over initial conditions rather than a time related one within a single case.
A stable attractor is found as the evolutionary end point of a whole ensemble of possible previous states. It is "many to one"-ness of the paths (to an average) that produces irreversibility. Our mathematical methods spot it because there is a symmetry between time changes within the system and a different possible initial condition - both characterised by the same mathematical "object", a continuous distribution.
So some questions would be - can traditional mathematical methods spot laws that aren't symmetric, one way or another? And, can simple program models do so, more readily? If so to the last, what are some examples?
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