On the surface of a tiny sphere live two invisible fields, like two kinds of ripples on a soap bubble. These fields don't just oscillate on their own: they constantly influence each other, like musicians playing in unison. In a new study, physicists derived exact equations that describe how this ripple changes over time, heats up, and descends into chaos.

Using these equations, the scientists measured the "divergence rate" of nearby trajectories — the so-called entropy (the largest Lyapunov exponent). They found that at a certain finite temperature, it tends to zero, as if the system calms down. This means that quantum chaos never violates the known Maldacena-Shenker-Stanford bound — a kind of "ceiling" for the rate of disorder growth. At high temperatures, the behavior approaches ordinary classical chaos, as if quantum effects gradually fade away.

Finally, the researchers traced how different parts of the system become entangled. It turned out that information mixes extremely quickly — almost instantly by the standards of such systems. This is reminiscent of how a drop of ink spreads in a glass of water, but here the process is much faster, affecting the very fabric of the quantum world.