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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.
Physical Review Letters
|April 25, 2025
Summary
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.
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.
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