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Statistical Mechanics of Frustrated Assemblies and Incompatible Graphs.

José M Ortiz-Tavárez1, Zhen Yang1, Nicholas Kotov2,3,4

  • 1University of Michigan, Department of Physics, Ann Arbor, Michigan 48109-1040, USA.

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Summary
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We introduce a graph theory for geometrically frustrated assemblies, linking structure to frustration via incompatible flows. Numerical simulations reveal distinct transitions and length scales in these complex systems.

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Area of Science:

  • Physics
  • Materials Science
  • Graph Theory

Background:

  • Geometrically frustrated assemblies exhibit complex phenomena due to building block misfit.
  • Understanding the fundamental statistical mechanics of these systems is crucial.

Purpose of the Study:

  • To develop a graph theory formulation for geometrically frustrated assemblies.
  • To link structural connectivity to frustration using incompatible flows.
  • To connect this framework to tensor gauge theories of amorphous solids.

Main Methods:

  • Graph theory formulation.
  • Introduction of incompatible flows concept.
  • Numerical simulations.

Main Results:

  • Identified two distinct percolation transitions for structure and incompatible flows.
  • Observed a crossover between cumulative and noncumulative frustration controlled by disorder.
  • Revealed a divergent length scale in the response of frustrated assemblies.

Conclusions:

  • The graph theory provides a minimal yet comprehensive framework for frustrated assemblies.
  • The theory connects statistical mechanics to tensor gauge theories.
  • New characteristics of frustrated assemblies were uncovered through simulations.