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Coupling interactions are strongest between NMR-active nuclei bonded to each other, where spin information can be transmitted directly through the pair of bonding electrons. While nuclei polarize their electrons to the opposite spins, the bonding electron pair has opposite spins. Configurations with antiparallel nuclear spins are expected to be lower in energy. When coupling makes antiparallel states more favorable, J is considered to have a positive value. The one-bond coupling constant, 1J,...
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NMR-active nuclei have energy levels called 'spin states' that are associated with the orientations of their nuclear magnetic moments. In the absence of a magnetic field, the nuclear magnetic moments are randomly oriented, and the spin states are degenerate. When an external magnetic field is applied, the spin states have only 2 + 1 orientations available to them. A proton with = ½ has two available orientations. Similarly, for a quadrupolar nucleus with a nuclear spin value of one, the...
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Coordination compounds and complexes exhibit different colors, geometries, and magnetic behavior, depending on the metal atom/ion and ligands from which they are composed. In an attempt to explain the bonding and structure of coordination complexes, Linus Pauling proposed the valence bond theory, or VBT, using the concepts of hybridization and the overlapping of the atomic orbitals. According to VBT, the central metal atom or ion (Lewis acid) hybridizes to provide empty orbitals of suitable...
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The Stochastic Complexity of Spin Models: Are Pairwise Models Really Simple?

Alberto Beretta1, Claudia Battistin2, Clélia De Mulatier1,3

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Entropy (Basel, Switzerland)
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Model simplicity, measured by stochastic complexity, depends on interaction arrangements, not order. Localized dependencies in spin models create simple, falsifiable models, unlike complex, hard-to-falsify pairwise models.

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minimum description lengthmodel complexityspin modelsstatistical inference

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Area of Science:

  • Statistical physics
  • Information theory
  • Computational complexity

Background:

  • Model simplicity is crucial for interpretation and prediction.
  • Stochastic complexity quantifies model simplicity in information theory.
  • Spin models are used to study complex systems.

Purpose of the Study:

  • To investigate the factors determining model simplicity in spin models.
  • To understand how interaction structure influences stochastic complexity.
  • To identify characteristics of simple and complex models.

Main Methods:

  • Studied stochastic complexity of spin models with arbitrary order interactions.
  • Utilized bijections within interaction spaces to identify equivalence classes.
  • Analyzed the relationship between interaction arrangements and model complexity.

Main Results:

  • Model simplicity is determined by the arrangement, not the order, of interactions.
  • Localized, non-overlapping interactions lead to simple models.
  • Fully connected pairwise models exhibit high complexity.

Conclusions:

  • Simple models possess localized dependencies and yield falsifiable predictions.
  • Complex models, like fully connected pairwise models, are difficult to falsify.
  • Understanding interaction arrangements is key to designing simple, predictive models.