Toss a handful of sand on the floor: there are few large grains, many medium ones, and even more tiny ones. This skew, where small events are common and large ones are rare, is described by a power law. It shows up in city sizes, earthquake magnitudes, and citation counts of scientific papers.
To understand where this law comes from, physicists took the simplest model: hard spheres that collide and exchange energy. At equilibrium, their speeds follow a different law — the bell-shaped Maxwell–Boltzmann distribution. But if the spheres are just starting to speed up, or if new fast particles are constantly being added, the picture changes: energies arrange themselves into a power-law distribution.
The key condition is the absence of a characteristic scale. At equilibrium, the system has a typical energy around which everything clusters. Far from equilibrium, no such point exists: energies can be anything, and this gives rise to self-similarity — a small piece of the graph looks like the whole graph. This is the same self-similarity we see in a jagged coastline or a branching tree.
Openness also matters. If you keep adding particles with different energies, the system never reaches equilibrium and stays in an intermediate state where the power law holds. The exponent depends only on how quickly particles are added, not on the details of their collisions. That's why so many different systems, from dust to galaxies, obey the same law.
A power law describes word frequencies in language: the second most common word appears about half as often as the first, the third about a third as often, and so on. This is Zipf's law, discovered using Joyce's Ulysses.