Percolation
Hi everyone!
Welcome to my blog. I think I’ve got something very cool for you today. It’s a model from theoretical physics known as percolation, and it involves structures that look like this:
Let’s get into it!
The Question: Can you predict whether fluid will flow through some porous substance?
The Model: Consider an infinite grid of pairs of integers. With probability p, connect a point in the grid with one of its nearest neighbors (excluding diagonals), and otherwise do not connect them. Repeat this process between all adjacent pairs of points in the grid. If two points are connected by an edge, we say that fluid can flow between them. Otherwise, fluid cannot flow. Having done this, you will have a random graph whose appearance will depend on p. We will say that fluid can flow macroscopically through this substance if, somewhere in our infinite grid, there is an infinite cluster of connected points.
To make things concrete, first let’s take a look at the grids.
As you can see, the value of p (know as the porosity parameter) controls the number/density of connections on the graph. Recall that we are imagining that fluid can flow along the connections between the “pores.”
Just by eye, we can see that fluid could certainly flow through the grid when p = 1, and it certainly could not flow when p = 0.25. It seems likely as well that fluid could flow when p = 0.75, although perhaps not certain. How about p = 0.5? This case is significantly less clear.
The requirement for fluid to flow on “macroscopic scales” is that there needs to be some cluster (i.e. a connected region) that has an infinite number of nodes when our grid extends off to infinity. Let’s take a look at the behavior of clusters.
These are the five largest clusters for a random grid with p = 0.4. For this value of p, none of our clusters extends more than half way across the grid. It therefore seems unlikely that we will find an infinite cluster as we extend our grid farther out. Let’s see what happens at p = 0.5.
Here, we have a slightly different story. Now the cluster has extended all the way across our grid. To make sure this isn’t a fluke, let’s see what happens on some larger grids.
So what’s the verdict? Will fluid flow at p = 0.5? Somehow, it still feels uncertain. It seems to cross these 40x40 grids most of the time, but an infinite grid is a different animal. Could one of those clusters really extend to infinity at this level of connectivity?
Well it turns out that it is possible to prove that there cannot be an infinite cluster for p = 0.5. However, it is the case that p = 0.5 is a very special value for this model because the system actually experiences a phase transition at p = 0.5. What this means is that for all p < 0.5, no fluid can flow macroscopically on any such random grid (i.e. there can be no infinite clusters). Moreover, for all p > 0.5, fluid can flow macroscopically on all random grids. In fact, it is possible to prove that there will be exactly one infinite cluster for all random grids with p > 0.5.
Such a discontinuous transition between “impermeable” and “permeable” is surprisingly rich behavior for such an apparently simple model!
However, this is not the end of the surprises to be found in the percolation model. People also study variants of the model in higher dimensions, and variants of the model where the “sites”/nodes are randomly populated instead of the “bonds”/edges. Each of these variants presents its own interesting behavior and subtleties.
It is also known that this system exhibits universality at the phase transition. I am not very familiar with universality, but what little I understand suggests that it is a truly fascinating phenomenon. Systems that exhibit universality have macroscopic properties which do not depend sensitively on the details of the underlying physics and instead only depend on the dimension of space. One consequence of universality is that vastly different systems may behave in similar ways during phase transitions.
However, I will say no more, because I am now out of my depth yet again!
Unfortunately, as you probably have noticed, all I can do is quote results and run small simulations here. Percolation theory is apparently quite technical to explore analytically, and I have been getting my information for this blog post from The Princeton Companion to Mathematics (which is an excellent and approachable book, by the way).
I hope that in the future I can learn more about the physics of percolation models. If I learn anything cool, I promise you guys will be the first to know :)
Thank you for visiting my blog!
Until next time, be safe and be well.
-Matt