Diffusion Part 1: the Drunken Walk

At its heart, diffusion is simply the random motion of particles. If you were to zoom in on the molecules in, for example, a glass of water, you’d observe a disordered and chaotic mess of particles that are constantly bumping into one another. While these collisions are deterministic in theory, they occur in such high numbers and with such frequency that we can effectively treat the motion of any individual molecule as random. A particle that moves in this way is aptly referred to as a random (or drunken) walker. Check out the particle below as an example; at each timestep, it moves either up, down, left, or right with equal probability. On average its displacement must be zero, but you can see that it still explores the space around it.

It turns out that we can calculate the probability that a drunken walker gets a certain distance from its starting point in some time interval. This is particularly useful when observing not a single walker, but large populations of walkers, because it means we can quantify how far a certain fraction of the population will walk in a given time interval. This is covered in part 2!

Here’s a fun fact to leave you with: random walks in 1D or 2D are recurrent, meaning that a drunken walker in a line or on a plane is guaranteed to return to its initial position after a while. Random walks in 3D or above, however, are transient, meaning that a drunken walker (or flyer?) moving unconstrained through space is not guaranteed to return to its initial position even if left for an infinite amount of time.

You now have full license to pretentiously utter the following quote at parties:

“A drunk man will find his way home, but a drunk bird may get lost forever.” -Shizuo Kakutani