Imagine you've just added a packet of dry flavored drink powder to a bottle of water without doing anything else to mix it together. How long will it take to fully dissolve so that you can no longer sense any of the texture of the powder on your tongue whenever you take a drink?

How long that might takes depends on the mechanics of diffusion, which is described as the net movement of particles from an area of higher concentration to an area of lower concentration. It doesn't matter whether those particles are atoms, molecules, aromas, viruses, drink mixes, foraging animals, et cetera, the process of diffusion ensures that eventually, the particles will go from being concentrated to being uniformly spread out in the medium to which they have been added.

The following video starts with an experiment the differences in how drops of colored dye become diffused in liquid water held at different temperatures, before introducing the math of random walks and how they lead to the equations that define the process of diffusion. The presenter also manages to throw in a biblical reference to explain what diffusion is, all in less than 13 minutes.

If you want to graduate to modeling random walks in diffusion, we'll point you to another video that provides both an introduction into two-dimensional random walks without any biblical references, but with its own special kind of goofy fun, before getting into how to program a random walk using Monte Carlo simulations in the Python programming language.

Solving diffusion problems represents a big challenge, because traditionally, their solution through modeling random walks requires lots of computing resources to approximate solutions using iterative numerical analysis. Or did, until this year, when Luca Giuggioli worked out how to more easily frame these kinds of problems for direct solution using mathematical tools originally developed to solve other problems, such as Chebyshev polynomials and the method of images (video).

The diffusion equation models random movement and is one of the fundamental equations of physics. To compare model predictions with empirical observations, the diffusion equation needs to be studied in finite space. When space and time are continuous, the analytic solution of the diffusion equation in finite domains has been known for a long time. But finding an exact solution when space and time are discrete has remained an outstanding problem for over a century—until now. I find the analytic solution of the discrete diffusion equation in confined domains and use it to predict how the probability for various reaction diffusion processes changes over time.

I make joint use of two techniques: special mathematical functions known as Chebyshev polynomials and a technique invented to tackle electrostatic problems, the so-called method of images. This approach allows me to construct hierarchically the solution to the discrete diffusion equation in higher dimensions from the one in lower dimensions.

The exact solution allows me to calculate transport quantities that, until now, could be derived only via time-consuming computer simulations or not at all because of prohibitive computational costs. In the context of random search processes, it is now possible to calculate accurately the probability for a “searcher” to reach a target for the first time, to return to its initial starting position, and to remain trapped at special defective locations.

These findings are directly relevant to a vast number of applications such as molecules moving inside a cell, animals foraging for resources in their home ranges, robots searching in a disaster area, and humans passing information or a disease.

That last bit explains the focus of the press release that accompanied Giuggioli's paper's publication, coming as it did during the global coronavirus pandemic of 2020, but which we think actually diminishes the accomplishment. In crafting a framework that allows direct solution of the equations describing diffusion processes, Giuggioli's approach has the potential to greatly reduce the amount of time and computing resources required to reach useful solutions across a wide range of applications. It's a leading contender to become one of the biggest math stories of the year.

References

Giuggioli, Luca. Exact Spatiotemporal Dynamics of Confined Lattice Random Walks in Arbitrary Dimensions: A Century after Smoluchowski and Pólya. Physical Review X. Vol. 10, Iss. 2 — April - June 2020 – Published 28 May 2020. DOI: 10.1103/PhysRevX.10.021045.

Johnston, Derek. An Introduction to Random Walks. [PDF Document]. 5 August 2011.

Bazant, Martin. 18.366 Random Walks and Diffusion. Fall 2006. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA. [Course Home | Study Materials]. Fall 2006.

Update 6 January 2021: Bonus video for background, featuring how to use the method of images with fluid flow problems: