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Published on: November 9, 2012
Rigidity percolation in a field
1Departmento de Física Aplicada, CINVESTAV del IPN, Avenida Tecnológico Km 6, 97310 Mérida, Yucatán, Mexico. cristian@mda.cinvestav.mx
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 20, 2003
Summary
Rigidity percolation on random graphs with varying connectivity and an applied field exhibits a phase transition line. This line separates rigid and flexible states, ending at a critical point with predictable behaviors.
Area of Science:
- Statistical Mechanics
- Network Science
- Materials Science
Background:
- Rigidity percolation is a critical phenomenon in network science.
- Understanding phase transitions in disordered systems is crucial.
Purpose of the Study:
- To analyze rigidity percolation on Erdös-Rényi graphs with varying connectivity and an applied field.
- To determine the phase transition line and critical exponents.
Main Methods:
- Analysis of rigidity percolation on randomly diluted Erdös-Rényi graphs.
- Exact determination of the first-order transition line in the (gamma,h) plane.
- Calculation of densities of uncanceled degrees of freedom and redundant bonds.
Main Results:
- The study precisely maps the phase transition line separating rigid and flexible phases.
- Analytic expressions for densities of uncanceled degrees of freedom and redundant bonds are derived.
- The density of uncanceled degrees of freedom is shown to behave as a free energy for rigidity percolation.
Conclusions:
- The research provides an exact solution for rigidity percolation on diluted random graphs with a field.
- Analogies to liquid-vapor transitions are discussed, offering new insights.
- The findings are relevant for understanding the mechanical properties of complex networks.
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