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Hi Ionel: Right... but I hope I've adequately stipulated my aversion to the notion of� "initial and final states" in my earlier reply to John M. I'm not dealing with "chaotic processes" anyway. They are complex, but not chaotic. I've got some fairly well-defined coherent systems, whose interactions are characterized by sufficiently coherent processes that they can be quantified, their interactive processes can be replicated, their behavior can be predicted, and those predictions can be corroborated as to their accuracy. The butterfly effect is a hypothesis; I don't know of any methodology by which it can be quantified, replicated, predicted, or corroborated. As I said in my "Wrap-Up" post to John M., that's a very different context than the one that motivated my original question. As an aside, the constraint that a system "come back to where it started from" is not relevant to any of the systems I'm studying. In principle, I'm not even sure that's possible in complex systems. Probably not. Irreversibility is ubiquitous in real-world systems. Whether the systems are "deterministic" may or may not be applicable. In chaotic processes, one can assume determinism until the cows come home. For my purposes, the question is one of utility, by which I mean "usefulness". In any case, your reference to such processes is intriguing. I would be fascinated to read some of the literature. Can you point me in the direction of some sources you'd be willing to recommend? Thanks! Pete Ionel wrote: RE: Process definition There is a growing literature on "chaotic processes" also, that don't seem to fit the proposed definition, it's a deterministic type of process that may end up a long way from where it started and will 'never' come back to where it satrted from. Lorentz's story of the butterfly's wings air flow in Brazil causing a Tornado in North America! Everyone on vacation... beware of tornadoes! Ionel |