Percolation thresholds for discorectangles: Numerical estimation for a range of aspect ratios

Area of Science:

• Statistical physics
• Geometric probability
• Materials science

Background:

• Percolation theory studies the connectivity of random networks.
• Discorectangles, also known as stadiums or 2D spherocylinders, are shapes with unique geometric properties.
• Understanding the percolation of anisotropic shapes is crucial for various applications.

Purpose of the Study:

• To investigate the percolation thresholds of discorectangles using computational methods.
• To analyze how the aspect ratio of discorectangles influences their percolation behavior.
• To compare the percolation thresholds of discorectangles with those of simpler shapes like discs and sticks.

Main Methods:

• Monte Carlo simulation was employed to model the random placement and orientation of discorectangles.
• Scaling analysis was performed to determine the percolation thresholds in the thermodynamic limit.
• The study systematically varied the aspect ratio of the discorectangles.

Main Results:

• Percolation thresholds for discorectangles at aspect ratios of 1 (disc) and approaching infinity (stick) match established values.
• For intermediate aspect ratios, the percolation threshold of discorectangles falls between those of ellipses and rectangles.
• The percolation threshold of discorectangles increases with aspect ratio, asymptotically approaching the threshold for rectangles.

Conclusions:

• The study provides accurate percolation thresholds for discorectangles across a range of aspect ratios.
• The findings contribute to the understanding of percolation phenomena in systems with anisotropic particles.
• Discorectangles exhibit distinct percolation behavior influenced by their unique geometry.