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Nonuniversality of invasion percolation in two-dimensional systems
Mark A Knackstedt1, Muhammad Sahimi, Adrian P Sheppard
1Department of Applied Mathematics, Research School of Physical Sciences and Engineering, Australian National University, Canberra ACT 0200, Australia.
Summary
This study shows invasion percolation (IP) fractal dimensions depend on lattice coordination number, not universal constants. Results suggest related problems like optimal paths also lack universal scaling properties.
Area of Science:
- Physics
- Statistical Mechanics
- Complex Systems
Background:
- Invasion percolation (IP) is a fundamental model for fluid flow in porous media and other disordered systems.
- Understanding the scaling properties, such as fractal dimensions, of clusters formed in IP is crucial for characterizing these systems.
- Previous studies often assumed universality, but the role of lattice structure, specifically coordination number, remained unclear.
Purpose of the Study:
- To precisely estimate fractal dimensions of the sample-spanning cluster, backbone, and minimal path in IP simulations.
- To investigate the dependence of these fractal dimensions on the coordination number (Z) of two-dimensional lattices.
- To determine if IP exhibits universal scaling properties or if they are nonuniversal and lattice-dependent.
Main Methods:
- Employed highly efficient algorithms for simulating invasion percolation (IP) with trapping.
- Conducted simulations on various two-dimensional lattices with different coordination numbers (Z).
- Calculated precise estimates for fractal dimensions of key IP structures: sample-spanning cluster, backbone, and minimal path.
Main Results:
- Fractal dimensions of the sample-spanning cluster, backbone, and minimal path were found to be nonuniversal.
- These quantities significantly vary with the coordination number (Z) of the underlying lattice.
- A crossover in the fractal dimension of the sample-spanning cluster was observed: ~1.82 for low Z, approaching ~1.896 (random percolation) for high Z.
Conclusions:
- The fractal dimensions in invasion percolation are not universal and are dependent on lattice coordination number.
- This nonuniversality implies that related problems, such as optimal paths in disordered media and minimum spanning trees, may also lack universal scaling properties.
- The findings necessitate a re-evaluation of universality assumptions in the study of disordered systems and percolation phenomena.