A pot of soup forgotten on the stove cools to room temperature. It seems everything has come to rest. But if you look inside with a quantum magnifying glass, you'll find that water molecules haven't frozen: they continue their dance. It's just that now the dance is steady, not turbulent like when boiling. Scientists from a new study built a model showing that on the path to such equilibrium, classical leveling and quantum jitters take different routes.

The authors represented the system's energy as a ladder, where each rung is a specific energy level. Particles hop from rung to rung, like water droplets that merge and then splash apart. In the classical version, such jumps quickly lead to a uniform filling of all rungs — the very thermal equilibrium described by the Gibbs distribution (a universal law for temperature and energy).

But when quantum effects come into play, the picture changes. Quantum particles can occupy several rungs simultaneously: this is called coherence (consistency of states). It turns out that coherence doesn't prevent particles from filling the rungs according to the classical law. However, the quantum system itself doesn't freeze: it keeps jittering, and this jitter doesn't vanish even in full equilibrium. The authors call this effect 'equilibrium without convergence.'

To measure this deviation from classical rest, the scientists introduced the concept of thermal distance — how far a quantum state is from the ideal Gibbs distribution. It never reaches zero, but it can be precisely estimated. This is like a clock's pendulum: it swings around the equilibrium position but never stops at the lowest point. The quantum world remains alive even in its calmest state.