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.

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.

As you might have figured out, blocked edges are drawn in black, and the different clusters are colored different colors. The largest cluster is colored white - how does its size change at different probabilities? on different lattices?

When the probability is higher than 0.8, all of the lattices are almost completely colored white. Lower than 0.2, the white cluster doesn't tend to look much bigger than the others. At probabilities near the middle of the range, you probably found that some lattices look quite colorful, while others are almost all a single giant cluster. What do you notice about the transition between the two?

To make that question somewhat easier to answer, the results of your simulations are being recorded and plotted below. The plot has a point for each run of the simulation, plotting the fraction of cells that were in the largest (white) cluster against the open-edge probability. If you want to gather more data points quickly, just click.

The shape of the curves is very dramatic. There is an abrupt shift from the largest cluster being very small as a fraction of all cells, to it containing almost all cells. The shift happens at a different probability for each type of lattice, but the form is roughly the same. This looks very much like a phase transition!

If we used an infinite lattice instead of one small enough to fit comfortably on your screen, we would see similar results. There, however, we would be able to define a different quantity: the percolation probability. Percolation is said to occur when an infinite cluster appears, and so the percolation probability is the probability that a given cell is part of the infinite cluster. If we took larger and larger (finite) lattices, the phase transition would become more and more abrupt, and we could define a critical probability at which the percolation probability first becomes greater than zero. From the plot above, you can get a fairly good estimate of the critical probabilities \(p_T,p_S,p_H\) of the triangular, square, and hexagonal lattices, respectively, but simulation isn't the only option. Statistical mechanics has many tools to analyze phase transitions - continue on to read about transformations.