Drop an ice cube into a glass of water — within a minute it melts, and the temperature evens out. That's thermal equilibrium in action: energy spreads evenly, and the system "forgets" its past. In the quantum world, this is called the eigenstate thermalization hypothesis (ETH)—the idea that every particle within a large system eventually starts behaving as if it has a temperature.
But there are hermit particles that refuse to warm up. They retain memory of their initial state and don't mix with neighbors. How to tell one from the other? Physicists Nikolai Shakura and his colleagues came up with a clever method: they give the particle a slight "kick" with a random perturbation, like a gentle puff of air, and watch how fast it changes. They measure the speed for chunks of different sizes, from tiny to half the entire system.
They tested this on four types of quantum systems: chaotic (where everything mixes quickly), ordered (where particles move by strict rules), stuck in disorder (where energy is trapped in place), and systems with rare "forbidden" states (where some particles seem shielded from heat). In all cases, the method flawlessly separated thermal from non-thermal particles.
The practical value is that you no longer need to build cumbersome thermal models or wait for the system to reach equilibrium. One nudge and a second of observation, and you know what you're dealing with. It's like understanding a person's character not through long conversations, but by how they react to an unexpected knock at the door.
The S shape appears because in the middle of the system, the "friction" between parts is maximal, like stirring sugar into tea: in the center, the spoon meets the most resistance.