Lattice Percolation

On the following pages you will explore one specific problem with a statistical mechanics flavor: percolation. Percolation generally refers to when liquids filter through solid materials, such as water through coffee grounds. Percolation can also describe systems where nearest-neighbor interactions are important, such as the magnetic interaction between atoms in iron, or where bonds can be formed spontaneously, such as cross linking between polymers, leading to gels. We will consider a more abstract version of the problem, which has the advantage of being simple to understand while still exhibiting interesting behavior, and, most importantly, is fun to play with!

To start off, we need a lattice. For our purposes, a lattice is a regular arrangement of points (we will call them vertices), some of which are connected by lines, or edges. Lattices can be constructed in any number of dimensions. In two dimensions, a lattice resembles an arrangement of tiles. The lines divide the plane into many small shapes which we will call cells. For percolation, each edge will be either open or closed. An open edge allows the cells on other side to connect, while a blocked edge keeps them separate. We will call a collection of cells that are connected through open edges a cluster. All that's left is to say how we determine the open and closed edges. We will decide this for each edge randomly, so that each is open with some probability \(p\).

If that was a bit too much, take a look at the interactive percolation system below. You can choose the type of lattice (triangular, square, or hexagonal) by clicking the corresponding shape, change the probability that edges are open by moving the slider on the right, and randomly assign the bonds with the same lattice and probability by clicking in the center.

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Play around with the simulation a bit more before reading on. This will be a lot more fun if you get a sense for how percolation works and try to find interesting things on your own.