A beautiful pattern that repeats in many places in nature is the process of diffusion, the more or less random movement of something through some medium, and diffusion-like equations can be used to model many diverse and seemingly unrelated phenomena.

Variations of it can be used to understand how chemicals spread in liquids, how aerosols or different gases mix and spread in the air, how heat spreads through materials(both solids, liquids, and gases), how different materials mix together(for example how different elements move through the Earth’s mantle), how mutations spread in populations of organisms over generations by drift or natural selection, how viruses or pheromones or other chemical signals move through populations of organisms, how changes in physical behavior spreads through a cluster of ants, how species and animals spread and migrate across continents, etc. etc.

This diffusion-type process is one of those amazing patterns that just seems to repeat everywhere in nature from the smallest to the largest scales, in everything that involves the movement of mass/chemicals, heat, and energy through time and space.

I don’t know the history of this in any detail, but I recall having been told that Ludwig Boltzmann actually was inspired by Darwin in some of his considerations of the randomness that applies to the movement of gas molecules through the air.

Now here is what becomes really interesting. Diffusion is just a random walk, and random walks produce fractals. These fractals are scale free and, yes, beautiful.

I love it. The concept of scale invariance blows my mind every time I think about it. There are many beautiful depictions of this phenomenon. Consider this famous comparison of dendritic neurons to the large scale structure of filaments of galaxy clusters in the universe:

On the left is a picture of a neuron in a network of dendrites. On the right is a picture of the a typical distribution of galaxies in the universe. The brighter the pixels, the more galaxies.

The diffusion equation was first published by Adolf Fick in 1855, 4 years before Darwin’s “On the Origin of Species”. Later on, Boltzmann proved that Fick’s equation can be derived from kinetic theory, but he did not come up with the diffusion equation. Boltzmann, however, did recognize the similarity of his understanding of entropy with Darwin’s evolution (though it is unclear how much his work is actually inspired by Darwin’s) The Hidden Connections Between Darwin and the Physicist Who Championed Entropy | Science| Smithsonian Magazine

Technically, mathematical entities called diffusion or Brownian Motion are not random walks, because there are not a finite number of steps, Between any two time points they take an infinite number of infinitely small steps. It is that which allows them to be fractal. Actual Brownian motion takes a finite number of steps, and as you get down to the scale of the particle being hit by a single molecule, the process is not fractal. Random walks that have discrete time or that have a finite number of steps in a finite amount of time are also not fractal. But I quibble.