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Emergence in Quantitative Systems – towards a measurable definition

This article explores the concept of emergence in quantitative systems and proposes a measurable definition based on persistent mutual information. It discusses the limitations, challenges, and potential generalizations of this approach.

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Emergence in Quantitative Systems – towards a measurable definition

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  1. Emergence in Quantitative Systems – towards a measurable definition R C Ball, Physics Theory Group and Centre for Complexity Science University of Warwick R S MacKay, Maths M Diakonova, Physics&Complexity

  2. Emergence in Quantitative Systems – towards a measurable definition Input ideas: Shannon: Information -> Entropy transmission -> Mutual Information Crutchfield: Complexity <-> Information MacKay: Emergence = system evolves to non-unique state Emergence measure: Persistent Mutual Information across time. Work in progress …. still mostly ideas.

  3. Emergent Behaviour? • System + Dynamics • Many internal d.o.f. and/or observe over long times • Properties: averages, correlation functions • Multiple realisations (conceptually) time Statistical properties realisations Emergent properties - behaviour which is predictable (from prior observations) but not forseeable (from previous realisations).

  4. Strong emergence: different realisations (can) differ for ever MacKay: non-unique Gibbs phase (distribution over configurations for a dynamical system) Physics example: spontaneous symmetry breaking • system makes/inherits one of many equivalent choices of how to order • fine after you have achieved the insight that there is ordering (maybe heat capacity anomaly?) and what ordering to look for (no general technique).

  5. B B B A A A Entropy & Mutual Information Shannon 1948

  6. time Entropy density (rate) Shannon ? Excess Entropy Crutchfield & Packard 1982 space Statistical Complexity Shalizi et al PRL 2004 MI-based Measures of Complexity A B Persistent Mutual Information - candidate measure of Emergence

  7. Measurement of Persistent MI • Measurement of I itself requires converting the data to a string of discrete symbols (e.g. bits) • above seems the safer order of limits, and computationally practical • The outer limit may need more careful definition

  8. Examples with PMI • Oscillation (persistent phase) • Spontaneous ordering (magnets) • Ergodicity breaking (spin glasses) – pattern is random but aspects become frozen in over time Cases without with PMI • Reproducible steady state • Chaotic dynamics

  9. 0 PMI = 0 log 2 log 4 log 2 log 8 log 4 log 3 Logistic map

  10. Issue of time windows and limits PMI / log2 Long strings under- sampled Short time correl’n Length of past, future r=3.58, PMI / log2 = 2 Length of “present”

  11. First direct measurements PMI / ln2 r r

  12. Discrete vs continuous emergent order parameters This suggests some need to anticipate “information dimensionalities”

  13. A definition of Emergence • System self-organises into a non-trivial behaviour; • there are different possible instances of that behaviour; • the choice is unpredictable but • it persists over time (or other extensive coordinate). • Quantified by PMI = entropy of choice • Shortcomings • Assumes system/experiment conceptually repeatable • Measuring MI requires deep sampling • Appropriate mathematical limits need careful construction • Generalisations • Admit PMI as function of timescale probed • Other extensive coordinates could play the role of time

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