In a big city after rain, water collects in the same low spots no matter where the first drops fall. An hour later, the pattern of puddles hardly depends on where the downpour started. That is equilibrium — the outcome a system reaches regardless of its start, once the details are forgotten.

In mathematics, such situations are described by the thermodynamic formalism — a set of rules for systems that remember the past but gradually lose that memory. Take a deck of cards shuffled by a strict rule but with a tiny bit of randomness in the hand movements. After many shuffles, the order of cards becomes predictable on average, though the exact sequence cannot be guessed. Scientists call this the Gibbs measure: it shows how likely a given configuration is at rest.

The authors of the new work took five of the most common definitions of such a measure — via local rules for replacing one state with another, via large deviations from typical behavior, via eigenvectors of a special operator, via the principle of least energy, and via the good old “cylindrical” construction. And they proved that all of them yield exactly the same object. Moreover, they did it with explicit numbers: how fast everything converges, what the error is if you slightly change the system’s temperature.

The practical benefit lies in guarantees. When an engineer models turbulence in an engine or a biologist counts mutations in a population, they can pick any of the five definitions, whichever is most convenient for calculations, and be confident the result won’t fall apart. Moreover, the work provides estimates of how fast initial conditions are forgotten — a key to predicting how quickly a system settles after a disturbance. And this is only the first part of a large cycle of six articles on chaos and order. So we return to the puddles after rain: now it’s clear that mathematics can not only describe their inevitable appearance in low spots but also say how many minutes it takes for the picture to become stable.