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A perfect crystal, in theory, has a uniform structure with the same unit cell and lattice points throughout. However, any deviation from this periodic arrangement is known as an imperfection or defect. These defects can be categorized into three types: point, line, and plane defects.Point defects occur when there is a deviation from the ideal due to missing atoms, displaced atoms, or additional atoms. These imperfections might occur due to imperfect packing during crystallization or because of...
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Using Microwave and Macroscopic Samples of Dielectric Solids to Study the Photonic Properties of Disordered Photonic Bandgap Materials
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Information pathways in a disordered lattice.

Lawrence R Frank1, Vitaly L Galinsky2

  • 1Center for Scientific Computation in Imaging, University of California at San Diego, La Jolla, California 92037-0854, USA and Center for Functional MRI, University of California at San Diego, La Jolla, California 92037-0677, USA.

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|April 16, 2014
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Summary
This summary is machine-generated.

Maximum entropy random walks on disordered lattices reveal that transition probabilities characterize lattice information, not necessarily physical processes. Localization arises from Laplacian solutions, not prior interpretations, and generalizes to multiple modes.

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Area of Science:

  • Statistical Mechanics
  • Information Theory
  • Condensed Matter Physics

Background:

  • Random walks are fundamental models for diffusion and transport in disordered systems.
  • The principle of maximum entropy is a powerful tool for inferring probability distributions from limited information.
  • Previous interpretations of localization phenomena in disordered lattices have been debated.

Purpose of the Study:

  • To derive the maximum entropy random walk on a disordered lattice without step restrictions.
  • To re-interpret the phenomenon of localization in defective lattices.
  • To establish a theoretical framework for information flow in disordered systems.

Main Methods:

  • Application of the principle of maximum entropy with a specific prior.
  • Analysis of the Laplacian operator on the lattice structure.
  • Utilizing a Fokker-Planck formalism to connect microscale dynamics to macroscale structure.

Main Results:

  • Transition probabilities characterize information on defective lattices, independent of physical processes.
  • Localization is a consequence of Laplacian solutions, generalizing to multiple modes.
  • Information flow dynamics are linked to lattice structure via Fokker-Planck equations.

Conclusions:

  • A novel theoretical framework for information flow in disordered systems has been developed.
  • This framework offers potential solutions for inferring connectivity from discrete imaging data, such as neural data.
  • The findings challenge previous interpretations of localization phenomena in disordered lattices.