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Published on: May 6, 2010
Fractal geometry of critical systems
Antoniou1, Contoyiannis, Diakonos
1Department of Physics, University of Athens, GR-15771 Athens, Greece.
Summary
We reveal fractal clusters in critical systems undergoing phase transitions. Their size, determined by fractal dimension and system properties, equals the correlation length at criticality.
Area of Science:
- Statistical Mechanics
- Condensed Matter Physics
- Phase Transitions
Background:
- Second-order thermal phase transitions involve critical phenomena.
- Understanding the geometry of systems at critical points is crucial.
- Fractal geometry often describes complex systems near criticality.
Purpose of the Study:
- To investigate the fractal geometry of clusters in critical systems.
- To analyze the statistical mechanics of these clusters using instanton-like configurations.
- To relate cluster size to correlation length at criticality.
Main Methods:
- Local description of system dynamics at the critical point.
- Analysis of instanton-like configurations within clusters.
- Calculation of fractal dimension and scaling properties.
- Inclusion of finite-size effects for cluster size determination.
Main Results:
- Formation of clusters with fractal geometry at the critical point.
- Instanton-like configurations dominate cluster statistical mechanics.
- Fractal dimension depends on embedding dimension and scaling properties.
- Critical cluster size is determined by system size, critical temperature, and coupling.
- Identified critical cluster size with the correlation length.
Conclusions:
- The study provides a local and global description of critical systems.
- Fractal geometry and instanton-like configurations are key to understanding critical phenomena.
- The correlation length at criticality is directly related to the critical cluster size.
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