Percolation theory is a branch of statistical physics and mathematics that studies how fluids, particles, or information move through porous or randomly connected systems. It examines the conditions under which a material allows a substance to percolate—that is, to move or filter from one side to another through randomly occupied paths or pores.
The system is typically modeled as a grid or network, where each site or bond is either open (allowing flow) or closed (blocking flow) with a certain probability. Percolation theory seeks to understand when a large-scale connected pathway forms, enabling flow across the system.
Key Concepts:
- Percolation threshold (pc): The critical probability at which a spanning cluster first forms, allowing global connectivity.
- Describes phase transitions, where a system shifts from non-percolating (isolated clusters) to percolating (a continuous path exists).
- Applies to 2D, 3D, and higher-dimensional lattices.
Examples of Applications:
- Fluid flow in soils and rocks for groundwater movement or oil recovery.
- Spread of diseases in populations (epidemic percolation).
- Electrical conductivity in composite materials.
- Forest fires, where ignition spreads based on percolating clusters of trees.
Percolation theory provides powerful tools for understanding connectivity, transport, and critical phenomena in disordered systems across physics, biology, geology, and network science.