posted Oct 28, 2009 5:02 AM by Olaf Bochmann
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updated Oct 28, 2009 4:29 PM
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There are many views on what is emergence. At the same time it is one of the most seductive buzzwords in complex systems science. This summary is based on a talk from Robert MacKay, University of Warwick. It explains emergence as a property of a non-linear dynamical system, as non-unique statistical behavior without any topological reason. History- Wikipedia: is the way complex systems and patterns arise out of simple interactions (e.g. a termite mound produced by a termite colony)
- Aristotle, Metaphysics, Book 8: "... the totality is not, as it were, a mere heap, but the whole is something beside the parts ...", i.e., the whole is greater than the sum of the parts.
- Anderson (1972), More is Different: "... the whole becomes not merely more, but very different from the sum of its parts."
Weak vs. Strong emergence- "... a high-level phenomenon is strongly emergent with respect to a low-level domain when the high-level phenomenon arises from the low-level domain, but truths concerning that phenomenon are not deducible even in principle from truths in the low-level domain." (most common in philosophical discussions)
- "... a high-level phenomenon is weakly emergent with respect to a low-level domain when the high-level phenomenon arises from the low-level domain, but truths concerning that phenomenon are unexpected given the principles governing the low-level domain." (most common in recent scientific discussions on complex systems)
Q: Will something not deductible ever happen? It is unexpected to whom? Dynamic Systems ViewWhat emerges from a spatially extended dynamical system are probability distributions over space-time histories (space time phase) that arise from typical initial probabilities in the distant past. The amount of emergence is the "distance" of a space-time phase from the set of products for independent units. Strong emergence means non-unique space-time phase (but not due to decomposability).
Examples: - Climate is a probability distribution over space-time histories compatible with weather laws.
- For equilibrium statistical mechanical systems the allowed probability distributions are the Gibbs phases for the energy \beta H (in units of temerature).
- For Markov processes they are the Gibbs phases for -log p_{ij} (log of transitionprobabilities)
- for deterministic dynamical systems with symbolic dynamics they are the Gibbs phases for log det Df^u (=SRB measures)
- For spatially extended deterministic dynamical systems with symbolic dynamics they are the space-time Gibbs phases for tr (log Df^u)_{ss}.
Decomposability (we will not allow strong emergence to arrise from this): - Non-uniqueness can arise trivially for topological reasons, e.g. more than one attractor, or a 2-piece attractor - generally, because the system is decomposable.
- A system with a space-time symbolic description is "indecomposable" if any allowable configurations on two sufficiently separated space-time patches can be joined into an allowable configuration ("specification property").
Proved examples of strong emergence: - ferromagnetic phase of 2D lsing model
- ferromagnetic phase of Tom's NEC majority voter PCA
- period-2 phases of Toom's NEC voter PCA
- endemic infection v disease-free phases of contact processes
Crutchfield non-trivial collective behavior - make a zoo of possible phases
- study their correlation structure
- study robustness of phases/set of phases
- universality classes/aggregation
- bifurcations/scaling
- control/optimization of phases
- fit to data
- develop in contexts with dynamic network
- develop for game-theoretic contexts
- What about systems that never settle down?
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